Exhaustive source inventory
Declaration Catalog
Every named declaration scanned from ASTIS production roots and tests appears here. Registry leaves and 12 reviewed interfaces have richer cards; internal helpers stay concise and link to their exact module anchor.
| Declaration | Kind | Local status | Route status | Module | Source |
|---|---|---|---|---|---|
AutoSamplingTheory.AutomationStage
|
inductive | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:5 |
AutoSamplingTheory.TaskKind
|
inductive | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:14 |
AutoSamplingTheory.TaskStatus
|
inductive | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:23 |
AutoSamplingTheory.AgentRole
|
inductive | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:32 |
AutoSamplingTheory.AcceptanceGate
|
structure | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:39 |
AutoSamplingTheory.ArtifactSpec
|
structure | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:46 |
AutoSamplingTheory.AutomationTask
|
structure | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:53 |
AutoSamplingTheory.AgentContract
|
structure | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:66 |
AutoSamplingTheory.PostCycleArtifactKind
|
inductive | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:73 |
AutoSamplingTheory.PostCycleArtifactSpec
|
structure | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:81 |
AutoSamplingTheory.WorkflowCheckSpec
|
structure | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:88 |
AutoSamplingTheory.leanBuildGate
|
def | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:95 |
AutoSamplingTheory.forbiddenPatternGate
|
def | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:101 |
AutoSamplingTheory.defaultGates
|
def | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:107 |
AutoSamplingTheory.postCycleArtifactSpecs
|
def | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:109 |
AutoSamplingTheory.workflowCheckSpecs
|
def | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:137 |
AutoSamplingTheory.threeLayerAgentContracts
|
def | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:153 |
AutoSamplingTheory.conversionArtifacts
|
def | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:181 |
AutoSamplingTheory.seedAutomationTasks
|
def | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:203 |
AutoSamplingTheory.automationTaskCount
|
def | Partial | Not mapped | AutoSamplingTheory.Automation |
AutoSamplingTheory/Automation.lean:231 |
AutoSamplingTheory.ArtifactLanguage
|
inductive | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:14 |
AutoSamplingTheory.ProofStatus
|
inductive | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:22 |
AutoSamplingTheory.SourceKind
|
inductive | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:31 |
AutoSamplingTheory.SourceAnchor
- Stable pointer to the source of a mathematical claim. |
structure | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:41 |
AutoSamplingTheory.ProofObligation
- An honest record for content that is not yet proved in Lean. |
structure | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:50 |
AutoSamplingTheory.TheoremContract
- Paper theorem or lemma translated into a Lean-facing contract. |
structure | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:60 |
AutoSamplingTheory.ProofDagBlock
- A reusable proof-DAG block, usually one node in a paper proof. |
structure | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:72 |
AutoSamplingTheory.forbiddenProofPatterns
- Patterns that are not allowed to close mathematical content. |
def | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:83 |
AutoSamplingTheory.sourceAnchor
|
def | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:86 |
AutoSamplingTheory.localTexAnchor
|
def | Partial | Not mapped | AutoSamplingTheory.Core |
AutoSamplingTheory/Core.lean:100 |
AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalBochner.conditional_bochner_energy
- Actual conditional normalized Bochner identity and curvature-energy bound.
The displayed Hessian square term uses genuine directional derivatives in
an orthonormal basis, not an assumed operator or a spectral-gap premi |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalBochner |
AutoSamplingTheory/ExampleCases/ProximalBPS/ConditionalBochner.lean:25 |
AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalGradient.conditional_gradient_closable
- The actual conditional Gibbs gradient has a dense smooth compact domain
and a single-valued closed graph extension. The displayed graph equivalence
fixes the genuine gradient, not an arbitrary abstract closable operato |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalGradient |
AutoSamplingTheory/ExampleCases/ProximalBPS/ConditionalGradient.lean:26 |
AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalResolvent.conditional_weak_resolvent
- On each actual fiber, one fixed genuine gradient closure solves every
positive-epsilon weak equation; the operator precedes epsilon and input f. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalResolvent |
AutoSamplingTheory/ExampleCases/ProximalBPS/ConditionalResolvent.lean:21 |
AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScore.reflected_conditional_covariance
- The reflected actual backward Gaussian conditional kernel has the stated
normalized density, and expectations of smooth compactly supported tests have
the centered score derivative. Normalization and domination are con |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScore |
AutoSamplingTheory/ExampleCases/ProximalBPS/ConditionalScore.lean:36 |
AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScoreDomain.conditional_curvature_and_score_domain
- The actual reflected conditional law has the source curvature and sharp
score derivative bound, with directional scores in L2 and the stated finite
variance/gradient-energy domain. No Poincare inequality is assumed or |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScoreDomain |
AutoSamplingTheory/ExampleCases/ProximalBPS/ConditionalScoreDomain.lean:30 |
AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianAugmentation.augmentation_eq_withDensity
- The law of independent `X ~ μ`, `Z ~ stdGaussian E` and
`Y = X + sqrt η • Z` has the displayed joint density relative to
`μ.prod volume`. The input law may be singular with respect to volume. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianAugmentation |
AutoSamplingTheory/ExampleCases/ProximalBPS/GaussianAugmentation.lean:25 |
AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianReflection.reflection_preserves_augmentation
- Reflecting the auxiliary point through the position is involutive and
preserves the generative Gaussian augmentation. All maps used in the
pushforward calculation are proved measurable. The positive-scale hypothesis
re |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianReflection |
AutoSamplingTheory/ExampleCases/ProximalBPS/GaussianReflection.lean:33 |
AutoSamplingTheory.ExampleCases.ProximalBPS.GibbsAugmentation.normalized_augmentation_density
- The Gibbs normalizer is strictly positive, its generative Gaussian
augmentation is a probability measure, and this measure has the exact normalized
source density. Integrability, positive normalization and measurable t |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.GibbsAugmentation |
AutoSamplingTheory/ExampleCases/ProximalBPS/GibbsAugmentation.lean:38 |
AutoSamplingTheory.ExampleCases.ProximalBPS.MacroscopicRepresentative.macroscopic_reflection_smooth_representative
- The actual reflected pair has conditional kernel S, and the actual
compressed reflection PUP of each smooth compactly supported macroscopic test
has the S-expectation as an L2 representative with the score covariance
d |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.MacroscopicRepresentative |
AutoSamplingTheory/ExampleCases/ProximalBPS/MacroscopicRepresentative.lean:27 |
AutoSamplingTheory.ExampleCases.ProximalBPS.ReflectionL2.actual_reflection_block_identities
- Actual reflection is an L2 self-adjoint isometric involution. The actual
conditional projection has the normalized quadratic-tilt kernel representation
almost everywhere, and its reflection blocks satisfy the source al |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.ProximalBPS.ReflectionL2 |
AutoSamplingTheory/ExampleCases/ProximalBPS/ReflectionL2.lean:26 |
AutoSamplingTheory.ExampleCases.SampleWiki.Cases.IdealProximalChain.kl_rate_from_reciprocal_step
- Exact algebraic tail of the inverse-time proximal-sampler argument.
If every proximal step increases reciprocal KL by at least `h / R2`, then after
any positive number `n` of steps the KL value is at most `R2 / (n h)` |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SampleWiki.Cases.IdealProximalChain |
AutoSamplingTheory/ExampleCases/SampleWiki/Cases/IdealProximalChain.lean:39 |
AutoSamplingTheory.ExampleCases.SampleWiki.SourceIdentity
- Stable source identity attached to a SampleWiki case before mathematical
formalization begins. The hashes are supplied by the source watcher rather
than trusted as mathematical evidence by themselves. |
structure | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SampleWiki |
AutoSamplingTheory/ExampleCases/SampleWiki.lean:30 |
AutoSamplingTheory.ExampleCases.SampleWiki.VerificationStage
- Verification stages are intentionally finer than a Boolean `verified`
flag. In particular, successful Lean elaboration precedes semantic source
review and graph assimilation. |
inductive | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SampleWiki |
AutoSamplingTheory/ExampleCases/SampleWiki.lean:40 |
AutoSamplingTheory.ExampleCases.SampleWiki.admissibleForScientificGraph
- Only a source-reviewed or already assimilated case is eligible to feed the
scientific theorem graph. A merely compiled theorem-shaped declaration is not
enough. |
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SampleWiki |
AutoSamplingTheory/ExampleCases/SampleWiki.lean:53 |
AutoSamplingTheory.ExampleCases.SampleWiki.assimilated_admissible
- An assimilated SampleWiki case satisfies the graph-admission contract. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SampleWiki |
AutoSamplingTheory/ExampleCases/SampleWiki.lean:59 |
AutoSamplingTheory.ExampleCases.SampleWiki.compiled_not_admissible
- Compilation alone does not discharge the source-review boundary. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SampleWiki |
AutoSamplingTheory/ExampleCases/SampleWiki.lean:64 |
AutoSamplingTheory.ExampleCases.SampleWiki.discovered_not_admissible
- Discovery alone is never treated as a formal mathematical certificate. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SampleWiki |
AutoSamplingTheory/ExampleCases/SampleWiki.lean:70 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.AdaptiveCenterRGO.adaptive_center_recovery
- Construct globally measurable target, Gaussian forward and center-retaining
backward kernels before every probability center law, with exact joint recovery.
The general probability base explicitly abstracts the source |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.AdaptiveCenterRGO |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/AdaptiveCenterRGO.lean:26 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.AdaptiveKLError.adaptive_center_kl_error
- Construct the ideal center-dependent kernels and bound the actual composed
output KL by the input KL plus the conditional KL integrated under that actual
input law. No finite-divergence or separately assumed measurabil |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.AdaptiveKLError |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/AdaptiveKLError.lean:28 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment.gaussian_square
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ApproximateInitialGradientMoment.lean:38 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment.gaussian_perturbation_square
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ApproximateInitialGradientMoment.lean:70 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment.lipschitz_square_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ApproximateInitialGradientMoment.lean:115 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment.smoothing_square
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ApproximateInitialGradientMoment.lean:126 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment.coupling_square
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ApproximateInitialGradientMoment.lean:179 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment.wasserstein_square
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ApproximateInitialGradientMoment.lean:225 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment.regularized_gradient
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ApproximateInitialGradientMoment.lean:254 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment.approximate_initial_gradient_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ApproximateInitialGradientMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ApproximateInitialGradientMoment.lean:274 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.clip
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:27 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.clip_bounds
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:29 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.clip_error
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:35 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.clipped_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:52 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.clipped_normalizer
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:84 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.estimator
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:107 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.actual_clipped_input
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:113 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.Attempt
|
abbrev | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:160 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.attempts
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:162 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.accepted
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:167 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.output
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:170 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.queryCount
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:175 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.actual_retry
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:180 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram.clipped_gradient_program
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedGradientProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedGradientProgram.lean:235 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.product_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:30 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.exponential_mean_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:52 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.integrated_exponential_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:68 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.clipped_density_transfer
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:94 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.estimator
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:139 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.actual_time_product
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:145 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.swap_time_preserving
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:185 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.swap_time_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:194 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.actual_proposal_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:208 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential.clipped_mean_exponential
- Actual proposal and clipped-output exponential mean-error bounds. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedMeanExponential |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedMeanExponential.lean:249 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.exponential_domination
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:33 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.power_mean_exponential
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:46 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.integrable_of_exponential_abs
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:58 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.exponential_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:68 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.inverse_normalizer_power
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:75 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.integral_exp_le_abs
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:90 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.exponential_mean_square
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:102 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.normalized_moments
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:108 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.tilt_density_pair
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:167 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.forward_density_power
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:183 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.reverse_density_power
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:199 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.rn_power_lintegral
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:237 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.tilt_power_bounds
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:248 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.step_implies_ideal
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:299 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.estimator
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:320 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison.clipped_renyi_comparison
- Actual clipped and ideal laws have bounded bidirectional normalized RN powers. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ClippedRenyiComparison |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ClippedRenyiComparison.lean:327 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.klFun_le_square
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:48 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.kl_le_second_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:56 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.actual_terminal_target
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:101 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.kernel_error_sum
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:133 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.composed_error_sum
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:177 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.terminal_precision_large
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:214 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.firstIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:247 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.scaledCached
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:251 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.Attempt
|
abbrev | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:262 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.attemptLaw
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:264 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.accepted
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:269 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.output
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:272 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL.finite_output_kl
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedFiniteOutputKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedFiniteOutputKL.lean:280 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep.measurable_fiber_kl
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedKLOneStep.lean:52 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep.conditional_kl_integral
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedKLOneStep.lean:71 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep.composed_kl_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedKLOneStep.lean:111 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep.mixture_kl_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedKLOneStep.lean:120 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep.conditional_step
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedKLOneStep.lean:148 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep.actual_gibbs_posterior
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedKLOneStep.lean:200 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep.firstIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedKLOneStep.lean:260 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep.one_step_kl_error
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedKLOneStep |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedKLOneStep.lean:266 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.Precision.compatible
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:46 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.Paths.termination
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:111 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.Actual.firstIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:299 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.Actual.execution
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:303 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.firstIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:405 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.scaledCached
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:409 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.Attempt
|
abbrev | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:420 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.attemptLaw
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:422 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.accepted
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:427 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.output
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:430 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.queryCount
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:435 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.terminal_extension
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:440 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution.enhanced_terminal_execution
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.EnhancedTerminalExecution |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/EnhancedTerminalExecution.lean:494 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.FiniteRGOKLError.finite_rgo_kl_error
- The actual finite RGO output KL is bounded by accumulated observation error
under its own state laws plus the terminal residual, including infinite values. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.FiniteRGOKLError |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/FiniteRGOKLError.lean:40 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.FiniteRGOProgram.finite_rgo_program
- Construct finite threshold-absorbed RGO execution, identify its terminal-output
marginal with recursive evaluation, and derive ideal recovery for every finite cap. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.FiniteRGOProgram |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/FiniteRGOProgram.lean:40 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw.centered_arc_rotation
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianArcLaw.lean:16 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw.affine_arc_law
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianArcLaw.lean:32 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw.arc_derivative
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianArcLaw.lean:56 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw.arc
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianArcLaw.lean:66 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw.velocity
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianArcLaw.lean:69 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw.arc_joint_measurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianArcLaw.lean:74 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw.arc_endpoints
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianArcLaw.lean:81 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw.gaussian_arc_law
- Actual isotropic Gaussian arc law and calculus, at the source variance scale.
The joint product equality proves independence of position and velocity. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianArcLaw |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianArcLaw.lean:89 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianKL.gaussian_kl_reverse_transport
- Actual ENNReal KL smoothing bound from the genuine quadratic transport budget. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianKL |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianKL.lean:28 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianMixture.bounded_displacement_reverse_transport
- An actual bounded-displacement coupling controls the actual Gaussian
smoothed RN power moment, its integrability and its normalized logarithm. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianMixture |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianMixture.lean:26 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianPowerMoment.gaussian_power_moment
- The actual Gaussian likelihood has finite qth moment with exact coefficient
q(q-1)/(2 tau), and its normalized log-moment is q norm(x-y)^2/(2 tau). |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianPowerMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianPowerMoment.lean:25 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianRGOErrorBudget.gaussian_rgo_error_budget
- Source-scaled Wasserstein accuracy implies an actual Gaussian observation KL
budget and its finite recursive accumulation, with terminal error retained. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GaussianRGOErrorBudget |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GaussianRGOErrorBudget.lean:36 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GibbsPositionMoment.stationary_position_integrability
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GibbsPositionMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GibbsPositionMoment.lean:36 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GibbsPositionMoment.coordinate_position_ibp
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GibbsPositionMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GibbsPositionMoment.lean:101 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GibbsPositionMoment.gibbs_position_moment
- Actual Gibbs normalization, position/pairing integrability, exact
position-gradient moment, and sharp dimension/curvature position bound. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GibbsPositionMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GibbsPositionMoment.lean:177 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.arc
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:23 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.velocity
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:25 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.estimator
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:28 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.path_integral
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:31 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.potential_growth
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:63 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.potential_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:88 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.estimator_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:106 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.estimator_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:145 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.auxiliary_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:172 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean.gradient_arc_mean
- The actual unclipped path estimator has a common log-weight mean under
the independent uniform-time and auxiliary Gaussian input. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.GradientArcMean |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/GradientArcMean.lean:224 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.smooth_lower
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:30 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.regularized_lower
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:76 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.ideal_volume_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:104 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.translated_density
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:127 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.ideal_gaussian_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:147 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.tilt_change_density
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:173 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.affine_gaussian_tilt
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:211 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.ideal_volume_law
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:245 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.estimator
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:269 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.actual_mean_tilt
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:275 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification.ideal_rgo_identification
- Positive finite normalization and equality of the actual ideal RGO laws. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.IdealRGOIdentification |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/IdealRGOIdentification.lean:302 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent.joint_iterates
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/JointReferenceGradientDescent.lean:36 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent.jointIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/JointReferenceGradientDescent.lean:46 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent.joint_stop
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/JointReferenceGradientDescent.lean:52 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent.actual_joint_program
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/JointReferenceGradientDescent.lean:90 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent.expected_count
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/JointReferenceGradientDescent.lean:148 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent.joint_reference_gradient_descent
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.JointReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/JointReferenceGradientDescent.lean:207 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.LogarithmicDepth.terminal_depth
- The actual schedule reaches the source terminal variance threshold at the
prescribed logarithmic stage. The upper-depth coefficient is explicit and is
distinct from the coefficient defining that stage. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.LogarithmicDepth |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/LogarithmicDepth.lean:19 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall.differential_scaling
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/NormalizedReferenceCall.lean:38 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall.scaled_gibbs
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/NormalizedReferenceCall.lean:76 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall.scaled_smoothing
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/NormalizedReferenceCall.lean:108 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall.scaled_wasserstein
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/NormalizedReferenceCall.lean:138 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall.regularized_hessian
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/NormalizedReferenceCall.lean:179 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall.normalized_laws
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/NormalizedReferenceCall.lean:212 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall.actualIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/NormalizedReferenceCall.lean:292 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall.normalized_reference_call
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.NormalizedReferenceCall |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/NormalizedReferenceCall.lean:297 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel.joint
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ObservationConditionalKernel.lean:47 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel.firstIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ObservationConditionalKernel.lean:87 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel.actual_joint
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ObservationConditionalKernel.lean:94 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel.actual_conditional
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ObservationConditionalKernel.lean:131 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel.observation_conditional_kernel
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ObservationConditionalKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ObservationConditionalKernel.lean:195 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.poisson_power_integral
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:16 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.iid_prefix_law
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:33 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.unitUniform
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:40 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.unitUniform_probability
|
instance | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:42 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.totalCost
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:46 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.failure_prefix_probability
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:51 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.totalCost_le_prefix
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:63 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.cost_tail_subset
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:80 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.poisson_exponential_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:92 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.poisson_exponential_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:105 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.fixed_prefix_exponential_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:116 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.fixed_prefix_chernoff
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:148 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.fixed_prefix_chernoff_shift
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:172 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.full_batch_cost_tail
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:185 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.failure_power_exponential_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:212 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.actual_attempt_count_law
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:229 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.full_batch_cost_tail_of_budget
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:239 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.poisson_tail_budget
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:264 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.delta_log_budget
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:275 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.full_batch_cost_delta_tail
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:284 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.ceil_threshold_le_single_log
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:307 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.Attempt
|
abbrev | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:325 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.attemptLaw
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:327 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.attemptLaw_probability
|
instance | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:333 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.accepted
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:339 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.measurable_accepted
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:342 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail.poisson_query_tail
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonQueryTail |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonQueryTail.lean:356 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.poisson_power_integral
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:31 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.finite_auxiliary_product_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:48 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.poisson_auxiliary_product_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:60 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.bounded_estimator_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:78 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.auxiliary_product_probability
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:90 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.unitUniform
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:105 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.unitUniform_Iic
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:111 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.uniform_acceptance_submeasure
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:124 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.uniform_acceptance_mass
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:148 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.finite_uniform_acceptance
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:157 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.poisson_uniform_acceptance
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:178 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.iid_prefix_law
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:221 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.infinite_uniform_acceptance
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:228 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.actual_poisson_acceptance
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:249 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.actual_accepted_proposal
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:277 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.failure_prefix_probability
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:313 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.first_hit_submeasure
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:325 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.never_hit_null
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:356 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.firstOutput
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:372 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.measurable_firstOutput
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:377 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.firstOutput_on_hit
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:395 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.firstOutput_law
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:405 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.surviving_current_law
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:457 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.totalCost
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:473 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.expected_totalCost
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:478 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.bounded_acceptance_mass
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:506 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.normalized_density_kernel
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:523 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.bounded_estimator_output_kernel
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:546 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.poisson_nat_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:569 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.normalized_exp_tilt
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:593 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.Attempt
|
abbrev | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:616 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.attemptLaw
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:618 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.accepted
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:630 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.measurable_accepted
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:633 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.attempt_count_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:647 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection.poisson_rejection_output
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.PoissonRejection |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/PoissonRejection.lean:656 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalEstimatorLipschitz.nonexpansive_of_monotone_optimality
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalEstimatorLipschitz |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ProximalEstimatorLipschitz.lean:22 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalEstimatorLipschitz.proximal_estimator_lipschitz
- The actual exact proximal estimator is nonexpansive in its center and
`sqrt eta`-Lipschitz in its noise input, on the existing construction range.
The proximal map is constructed, not assumed Lipschitz; its equation an |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalEstimatorLipschitz |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ProximalEstimatorLipschitz.lean:52 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator.parameterized_contraction_point
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ProximalGaussianEstimator.lean:47 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator.actual_proximal_minimizer
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ProximalGaussianEstimator.lean:85 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator.gaussian_square
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ProximalGaussianEstimator.lean:150 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator.actual_gaussian_gradient_moments
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ProximalGaussianEstimator.lean:182 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator.proximal_gaussian_estimator
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProximalGaussianEstimator |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ProximalGaussianEstimator.lean:239 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProxyReverseTransport.proxy_reverse_transport
- One actual proxy and coupling simultaneously satisfy eventwise TV,
bounded displacement and all positive-time Gaussian RN moment/log bounds. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ProxyReverseTransport |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ProxyReverseTransport.lean:29 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RGOBackward.rgo_backward_recovery
- One actual backward kernel with every-point precision update, target recovery
and all quadratic-budget KL guarantees for its actual smoothed-input outputs. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RGOBackward |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/RGOBackward.lean:29 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RGOCalculus.rgo_calculus
- The source's regularized potential has the stated curvature and smoothness,
positive finite Gibbs mass, and an actual probability law. A further quadratic
regularization gives exactly the updated normalized source pote |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RGOCalculus |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/RGOCalculus.lean:34 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RGOClosure.quadratic_tilt_tilt
- Applying two normalized quadratic exponential reweightings gives a single
reweighting with summed precision and precision-weighted centre. All
normalizing integrals are positive and finite by boundedness of the weights |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RGOClosure |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/RGOClosure.lean:40 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RecursiveCondition.contraction_bounds
- The ill-conditioned recursive update lies between half and four fifths
of its previous condition number. The denominator is proved positive from
the hypotheses, so no totalized-division exceptional case is used. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RecursiveCondition |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/RecursiveCondition.lean:23 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RecursiveDepth.parameter_control
- Actual scheduled precision recursion has persistent well-conditioning and
an explicit geometric terminal-parameter bound. This is not an error or cost
bound for a random sampler. All finite variance readings occur at p |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RecursiveDepth |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/RecursiveDepth.lean:20 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RecursiveVariance.variance_update_bounds
- One well-conditioned precision update yields a positive finite variance
bounded by `2*c`. If the previous variance is finite, it contracts by
`2*c/(1+2*c)`. Positivity of every denominator follows from the hypotheses. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.RecursiveVariance |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/RecursiveVariance.lean:28 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost.regularized_hessian
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingCost.lean:55 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost.firstIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingCost.lean:88 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost.statewise_cost
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingCost.lean:92 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost.integrate_actual_cost
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingCost.lean:158 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost.reference_carrying_cost
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingCost.lean:195 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel.state_iterates
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingKernel.lean:44 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel.firstIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingKernel.lean:55 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel.stopped_family
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingKernel.lean:60 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel.variable_reference
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingKernel.lean:82 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel.quadratic_source_alignment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingKernel.lean:137 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel.scaled_reference_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingKernel.lean:168 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel.reference_carrying_kernel
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.ReferenceCarryingKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/ReferenceCarryingKernel.lean:201 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping.clipping_exp_domination
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcClipping.lean:15 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping.clipping_integral_domination
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcClipping.lean:38 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping.clipping_parameter_lower
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcClipping.lean:57 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping.clipping_parameter_bounds
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcClipping.lean:81 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping.clipArc
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcClipping.lean:118 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping.clipVelocity
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcClipping.lean:121 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping.smooth_gradient_arc_clipping
- Measurability, integrability and the explicit two-scale exponential bound
for the actual gradient estimator clipping excess under the source step range. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcClipping |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcClipping.lean:126 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.gaussian_weighted_exp
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:19 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.quadratic_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:26 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.quadratic_integral_raw
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:35 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.quadratic_integral
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:45 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.basis_exp_product
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:61 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.quadratic_stdGaussian
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:69 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.sqrt_inv_le_exp
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:87 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.quadratic_stdGaussian_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:98 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.linear_exp
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:110 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.linear_abs_exp
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:127 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.scaled_quadratic_bound
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:149 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.gaussian_product_abs_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:170 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.positional_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:197 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.source_constants
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:242 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.field_product_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:273 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.true_gradient_product_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:308 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.gradient_output_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:334 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.momentArc
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:388 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.momentVelocity
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:391 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.actual_gradient_arc_moment
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:394 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment.smooth_gradient_arc_moment
- Actual Gaussian-input estimator: measurability, exponential integrability,
the source-proof-supported factor-2 moment bound and its logarithmic form. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothGradientArcMoment |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/SmoothGradientArcMoment.lean:431 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.StateDependentRGO.state_dependent_recovery
- Construct measurable state-dependent target, Gaussian observation and retained-state
posterior kernels, with exact recovery of every ideal joint state-target law.
This supplies kernel semantics for an RGO stage, not an |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.StateDependentRGO |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/StateDependentRGO.lean:31 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.StoppedGaussianRGOError.stopped_gaussian_rgo_error
- Align the actual Gaussian error program with the stopped program, retain its
terminal-set residual, and derive stopped output accuracy from terminal-only KL. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.StoppedGaussianRGOError |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/StoppedGaussianRGOError.lean:38 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.StoppedRGODepth.stopped_rgo_depth
- With the source variance schedule and fixed initial-state threshold, the actual
absorbed RGO program reaches the terminal set by the explicit logarithmic stage bound.
Its output law at that initial state is unchanged b |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.StoppedRGODepth |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/StoppedRGODepth.lean:37 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.state_iterates
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:55 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.firstIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:66 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.stopped_family
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:71 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.variable_reference
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:93 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.gaussian_proposal
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:159 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.scaledCached
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:174 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.measurable_scaledCached
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:185 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Attempt
|
abbrev | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:191 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.attemptLaw
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:193 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.accepted
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:198 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.output
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:201 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.queryCount
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:206 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.actual_kernel_program
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:211 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.cachedEstimator
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:264 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.auxiliary_scale
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:270 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.scaled_mean
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:281 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Stream.Attempt
|
abbrev | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:297 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Stream.attemptLaw
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:298 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Stream.attemptMap
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:302 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Stream.accepted
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:304 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Stream.output
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:306 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Stream.queryCount
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:310 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Stream.stream_law
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:315 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Stream.stream_program
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:337 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.Stream.stream_output_ae
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:355 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.output_eq_stream
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:373 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.source_correspondence
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:383 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel.terminal_fors_kernel
- Construct the actual terminal reference and Markov sampler, identify its
retry program and source reparameterization, and prove two-way RN accuracy
and sampling-stage expected cached query cost on the terminal state do |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalFORSKernel |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalFORSKernel.lean:438 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent.gradient_decay
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalReferenceGradientDescent.lean:44 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent.stopping_certificate
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalReferenceGradientDescent.lean:93 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent.firstIndex
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalReferenceGradientDescent.lean:112 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent.stopped_program
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalReferenceGradientDescent.lean:116 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent.expected_count
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalReferenceGradientDescent.lean:172 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent.quadratic_upper
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalReferenceGradientDescent.lean:231 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent.regularized_data
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalReferenceGradientDescent.lean:259 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent.terminal_reference_gradient_descent
- Construct the actual first gradient-descent reference point. Prove the
stopping/output measurability, exact terminal center residual, first-hit and
query-count bounds, and conditional initial-law expected cost. The exp |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalReferenceGradientDescent |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalReferenceGradientDescent.lean:303 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.scaled_parameters
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:38 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.parameter_bounds
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:62 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.accuracy_error
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:84 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.rn_moment_positive
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:101 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.logarithmic_accuracy
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:121 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.cached_expected_cost
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:135 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.cached_cost_tail
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:143 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.cachedEstimator
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:176 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.Attempt
|
abbrev | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:182 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.attemptLaw
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:184 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.accepted
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:189 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.output
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:192 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.queryCount
|
def | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:197 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.actual_program
|
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:202 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost.terminal_sampler_accuracy_cost
- Actual terminal output accuracy and full-batch cached gradient query bounds. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TerminalSamplerAccuracyCost |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TerminalSamplerAccuracyCost.lean:275 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.Truncation.truncated_proxy
- A finite infimum p-cost budget produces an actual optimal coupling and
an explicit truncated proxy within delta in eventwise total variation, coupled
to Q at displacement at most r * delta^(-1/p). |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.Truncation |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/Truncation.lean:29 |
AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TwoNoiseRGO.two_noise_rgo
- Actual two-noise target recovery and added-time KL control, using one
Markov kernel chosen before every proposal law and radius. |
theorem | Partial | Not mapped | AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.TwoNoiseRGO |
AutoSamplingTheory/ExampleCases/SmoothedPicardHMC/TwoNoiseRGO.lean:28 |
AutoSamplingTheory.ImplementationStatus
|
inductive | Partial | Not mapped | AutoSamplingTheory.Literature |
AutoSamplingTheory/Literature.lean:5 |
AutoSamplingTheory.PaperMode
|
inductive | Partial | Not mapped | AutoSamplingTheory.Literature |
AutoSamplingTheory/Literature.lean:13 |
AutoSamplingTheory.PaperEntry
|
structure | Partial | Not mapped | AutoSamplingTheory.Literature |
AutoSamplingTheory/Literature.lean:19 |
AutoSamplingTheory.literature
|
def | Partial | Not mapped | AutoSamplingTheory.Literature |
AutoSamplingTheory/Literature.lean:31 |
AutoSamplingTheory.literatureCount
|
def | Partial | Not mapped | AutoSamplingTheory.Literature |
AutoSamplingTheory/Literature.lean:79 |
AutoSamplingTheory.OpenProblem
|
structure | Partial | Not mapped | AutoSamplingTheory.OpenProblems |
AutoSamplingTheory/OpenProblems.lean:5 |
AutoSamplingTheory.openProblems
|
def | Partial | Not mapped | AutoSamplingTheory.OpenProblems |
AutoSamplingTheory/OpenProblems.lean:14 |
AutoSamplingTheory.openProblemCount
|
def | Partial | Not mapped | AutoSamplingTheory.OpenProblems |
AutoSamplingTheory/OpenProblems.lean:32 |
AutoSamplingTheory.lawMapEqOfAEEq
- Pushforward-law equality from almost-everywhere equality of random variables.
This is the measure-level version of the endpoint-law bookkeeping used by the
SALD Euler--Maruyama interpolation blocks: once two process r |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:35 |
AutoSamplingTheory.lawMapIntegral
- Integrating a test against a pushforward law is the same as integrating
the composed test on the original probability space.
This is the weak-test bookkeeping used before differentiating EM
interpolation laws. It doe |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:48 |
AutoSamplingTheory.lawMapIntegralHasDerivAtOfSample
- Transport a supplied sample-space derivative to the corresponding
pushforward-law weak-test integral.
The analytic derivative is still an explicit hypothesis. This lemma only
packages the `Measure.map` integral rewri |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:63 |
AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndSample
- Transport a sample-space derivative to a named law path equal to a
`Measure.map` path.
This is the named-law variant used when a paper first writes
`hat rho_s = Law(hat X_s)` and the Lean target keeps `hatRhoS` as a s |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:86 |
AutoSamplingTheory.lawMapIntegralHasDerivAtOfDominated
- Transport a dominated pointwise derivative to a pushforward-law weak-test
derivative.
This is the first parametric-integral step below the cycle-79 law-map handoff:
Mathlib's dominated derivative-under-integral theore |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:120 |
AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated
- Named-law version of `lawMapIntegralHasDerivAtOfDominated`.
If a paper keeps a named law path `ρ s` with `ρ s = Measure.map (X s) P`,
this combines the dominated sample-space derivative-under-integral step with
the na |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:161 |
AutoSamplingTheory.lawMapProdEqOfAEEq
- Pushforward-law equality for paired random variables from componentwise
almost-everywhere equality.
This is a narrow endpoint-law helper for stitched EM paths: once two endpoint
representatives agree almost everywhere |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:210 |
AutoSamplingTheory.lawMapProdFst
- First marginal of a paired pushforward law.
This is endpoint-law bookkeeping for common-space EM arguments: after a joint
endpoint law has been represented as a paired pushforward, projecting the first
coordinate reco |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:227 |
AutoSamplingTheory.lawMapProdSnd
- Second marginal of a paired pushforward law.
This is the right-endpoint analogue of `lawMapProdFst`; it keeps marginal-law
extraction separate from the conditional-drift and Fokker--Planck obligations. |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:241 |
AutoSamplingTheory.lawMapProdSwap
- Swap the coordinate order of a paired pushforward law.
Mathlib conditional-distribution APIs usually represent the joint law for
`Y | X` in the order `(X,Y)`. Some paper proofs first name the joint law in the
opposit |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:257 |
AutoSamplingTheory.condDistribAeEqCondExpKernelMap
- Mathlib orientation bridge between `condDistrib` and `condExpKernel`.
For the SALD conditional drift, instantiate `Y` with `X_k^eta` and `X` with
`hat X_s`: the conditional distribution of `X_k^eta` given `hat X_s` ag |
theorem | Partial | Partial | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:274 |
AutoSamplingTheory.condDistribIntegralSampleAeEqOfCondExpKernelMap
- Sample-space component-version bridge from `condExpKernel.map` to
`condDistrib`.
For the SALD `condC` field, this isolates the remaining Mathlib-facing
boundary after `condDistrib` and `condExpKernel.map` have been al |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:298 |
AutoSamplingTheory.condDistribIntegralAEStronglyMeasurable
- Strong measurability of a vector-valued conditional integral against
`condDistrib`.
This packages the Mathlib theorem in the orientation used by the SALD
component fields: conditioning variable `X`, sampled variable ` |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:328 |
AutoSamplingTheory.condDistribIntegralIntegrable
- Integrability of a vector-valued conditional integral against
`condDistrib`.
For SALD this is the Mathlib-local handoff needed to turn integrable frozen
drift summands into integrable component conditional fields befo |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:348 |
AutoSamplingTheory.condDistribIntegralMapAEStronglyMeasurable
- Strong measurability of the state-space conditional integral under the
conditioning law `μ.map X`.
This is the law-space version needed for the SALD named `hat rho_s` field:
Mathlib's `condDistrib` backend already giv |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:370 |
AutoSamplingTheory.condDistribIntegralMapIntegrable
- Integrability of the state-space conditional integral under the
conditioning law `μ.map X`.
For SALD this is the Mathlib-local input that turns an integrable frozen
drift or score summand on the joint law of `(hat X_s |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:390 |
AutoSamplingTheory.condDistribIntegralMapIntegral
- Disintegrate an integral through the `condDistrib` kernel.
For the SALD conditional drift in `appendix.tex:1368-1377`, instantiate `X`
with `hat X_s`, `Y` with `X_k^eta`, and `f` with the weak test-gradient
pairing ag |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:412 |
AutoSamplingTheory.condDistribIntegralNamedLawIntegral
- Named-law variant of `condDistribIntegralMapIntegral`.
This is the paper-oriented form for `\hat\rho_s = Law(\hat X_s)`. |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:445 |
AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable
- Named-law variant of `condDistribIntegralMapAEStronglyMeasurable`.
Instantiate `hatRho` with `Law(hat X_s)`, `X` with `hat X_s`, and `Y` with
`X_k^eta`. The hypothesis `hatRho = μ.map X` is the paper's
`\hat\rho_s = |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:466 |
AutoSamplingTheory.condDistribIntegralNamedLawIntegrable
- Named-law variant of `condDistribIntegralMapIntegrable`. |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:482 |
AutoSamplingTheory.condDistribIntegralNamedFieldRegularity
- Versioning theorem for a named conditional-integral component field.
If a SALD component field such as `condC_{k,s}` or `condScore_{k,s}` is chosen
as a `hatRho`-a.e. version of the canonical `condDistrib` integral, t |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:505 |
AutoSamplingTheory.MeasureContract
- A named probability measure or time-indexed law in a paper proof. |
structure | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:532 |
AutoSamplingTheory.KLContract
- Forward KL divergence contract `KL(rho || pi)`. |
structure | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:541 |
AutoSamplingTheory.FIContract
- Fisher information contract `FI(rho || pi)`. |
structure | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:550 |
AutoSamplingTheory.LSIContract
- Log-Sobolev inequality contract. |
structure | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:559 |
AutoSamplingTheory.PIContract
- Poincare inequality contract. |
structure | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:568 |
AutoSamplingTheory.TransportVelocityContract
- Transport velocity field satisfying a continuity equation. |
structure | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:577 |
AutoSamplingTheory.GuidedTiltContract
- Guide tilt `pi_t proportional to p_t exp(-F_t)`. |
structure | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:586 |
AutoSamplingTheory.DvVariationalFormulaInterface
- Source-cited interface for the Donsker--Varadhan entropy duality formula.
This is data, not a proof. It records the exact analytic shape needed by the
SALD paper before theorem-specific finite-log-mgf witnesses insta |
structure | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:600 |
AutoSamplingTheory.dvVariationalObligation
- Donsker--Varadhan variational formula as a cited-result contract. |
def | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:616 |
AutoSamplingTheory.dvVariationalFormulaInterface
- Precise source-cited DV interface matching `appendix.tex:73-79`.
Downstream proof obligations may depend on this interface only as a cited
analytic result until an actual Lean proof or imported theorem replaces it. |
def | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:628 |
AutoSamplingTheory.lsiKlFiSqrtDensitySquareScalar
- Pointwise square identity for the LSI density test `phi = sqrt(r)`.
In the SALD source step `main_body.tex:208-215`, this is the local scalar
part of replacing `phi^2` by the Radon-Nikodym density ratio `r = rho/pi`.
|
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:650 |
AutoSamplingTheory.lsiKlFiSqrtDensityEntropyIntegrandScalar
- Pointwise entropy-integrand rewrite for `phi = sqrt(r)`.
This proves only the scalar rewrite
`phi^2 log(phi^2) = r log r` after nonnegativity of the density ratio is
available. Integrability, zero-density conventions |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:661 |
AutoSamplingTheory.lsiKlFiSqrtDensityNormalizationScalar
- Scalar normalization handoff for the LSI test `phi = sqrt(r)`.
After an integral backend has shown that the mass of `phi^2` equals the mass of
the density ratio `r`, probability normalization of `r` gives the LSI test |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:671 |
AutoSamplingTheory.lsiKlFiRnDerivLIntegralMassOne
- Radon-Nikodym mass normalization for the LSI density ratio.
For probability measures `rho << pi`, the density ratio `d rho / d pi` has
unit `pi`-mass. This is the measure-level backend behind the source line
`int phi |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:683 |
AutoSamplingTheory.lsiKlFiRnDerivDensityMassOne
- Real-integral normalization of the Radon-Nikodym density ratio.
This supplies the real mass input used by the scalar normalization bridge for
the LSI test `phi=sqrt(d rho/d pi)`. |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:695 |
AutoSamplingTheory.lsiKlFiSqrtRnDerivTestMassOne
- Normalization of the source LSI test `phi=sqrt(d rho/d pi)`.
This combines the pointwise square identity for the square-root density test
with the Radon-Nikodym mass theorem. Smooth/admissible-test and approximation
|
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:708 |
AutoSamplingTheory.lsiKlFiRnDerivEntropyIntegral
- Entropy transport from the density-ratio integral to the KL log-likelihood integral.
For `rho << pi`, Mathlib's log-likelihood-ratio backend identifies
`int (d rho/d pi) log(d rho/d pi) d pi` with the paper's KL integ |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:724 |
AutoSamplingTheory.lsiKlFiSqrtRnDerivEntropyIntegral
- Entropy transport for the square-root density test used by LSI.
This rewrites the LSI entropy integrand for
`phi=sqrt(d rho/d pi)` and then uses the Radon-Nikodym entropy transport
identity. It still does not prove a |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:738 |
AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainScalar
- One-dimensional pointwise Fisher-chain coefficient for the LSI test.
For a positive density ratio `r`, the source substitution
`phi=sqrt(r)` has differential coefficient
`d phi = (2*sqrt(r))^{-1} d r`, while
`d log r |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:758 |
AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainOfDerivativesScalar
- Pointwise Fisher-chain handoff with named derivative identities.
This packages the scalar part of
`nabla sqrt(r) = (2*sqrt(r))^{-1} nabla r` and
`nabla log r = r^{-1} nabla r`. It does not prove differentiability,
gr |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:774 |
AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumScalar
- Finite-coordinate Fisher-chain handoff for the LSI density test.
This lifts the pointwise scalar identity for `phi=sqrt(r)` to a finite sum of
coordinate-square terms. It is still not the vector Sobolev chain rule or |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:790 |
AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainFiniteSumHandoffScalar
- Finite-coordinate handoff to the Dirichlet/Fisher identity.
Once a density backend identifies the Dirichlet term with the finite coordinate
sum of `d sqrt(r)` squares and the Fisher term with
`r * sum_i (d log r_i)^2` |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:820 |
AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralFiniteSum
- Integral handoff for the finite-coordinate Fisher chain rule.
After a Sobolev backend supplies coordinate derivative identities almost
everywhere for `sqrt(r)` and `log r`, this pushes the cycle-38 finite-sum
coeffici |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:841 |
AutoSamplingTheory.lsiKlFiSqrtDensityFisherChainIntegralHandoffScalar
- Scalar Dirichlet/Fisher handoff after the integral finite-sum identity.
This packages the exact `dirichlet=(1/4)*FI` input consumed by the existing
LSI/KL/FI scalar bridges when the analytic backend represents the Dir |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:861 |
AutoSamplingTheory.dvVariationalOneSidedConsequenceScalar
- Scalar rearrangement behind the one-sided use of the cited DV formula.
This is not a proof of Donsker--Varadhan. It starts after a cited or
eventually formalized entropy-duality theorem has supplied the variational
u |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:881 |
AutoSamplingTheory.dvVariationalOneSidedFromSupremumScalar
- Scalar supremum step behind the one-sided use of the cited DV formula.
This does not prove Donsker--Varadhan. It starts after a cited or eventually
formalized theorem has identified `kl` with the supremum of the admi |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:892 |
AutoSamplingTheory.dvFiniteLogMgfOfLeAlpha
- Finite-log-mgf monotonicity for the scaled tests used before DV.
If the exponential moment for `alpha0 * q` is integrable under a finite
measure, then the exponential moment for `alpha * q` is integrable for
`0 <= alp |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:913 |
AutoSamplingTheory.dvVariationalOneSidedOfTiltedRight
- Mathlib-backed one-sided Donsker--Varadhan inequality via exponential tilting.
This proves only the admissible-test upper bound
`E_nu[Z] - log E_mu[exp Z] <= KL(nu || mu)` under explicit Mathlib
measure-theoretic hypo |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:929 |
AutoSamplingTheory.dvVariationalOneSidedOfScaledTest
- One-sided DV inequality for a SALD-style scaled selected test.
This packages the theorem-instance side conditions for tests of the form
`Z = alpha * q`. The `alpha0` exponential-moment assumption supplies the
finite- |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:973 |
AutoSamplingTheory.dvVariationalScaledTestEnergyBound
- Energy form of the one-sided DV bound for a scaled selected test.
For SALD use sites, `q` is a squared velocity or residual norm. This theorem
starts after the selected-test hypotheses have been supplied, applies the |
theorem | Partial | Partial | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:999 |
AutoSamplingTheory.dvVariationalScaledTestEnergyBoundWithCoeff
- Coefficient-preserving energy form of the selected scaled-test DV bound.
This is the local algebraic shape used before Gronwall in SALD proofs after a
nonnegative prefactor, such as `(1/2)*dot{s}(t)^(-1)`, multiplies |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:1047 |
AutoSamplingTheory.dvVariationalTiltedRightOneSidedConsequence
- One-sided Donsker--Varadhan consequence from the tilted backend.
This is the form consumed by SALD after a selected test has supplied the
explicit Mathlib hypotheses. It remains a one-sided theorem only; the
Bouchero |
theorem | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:1079 |
AutoSamplingTheory.lsiToKlFiObligation
- Log-Sobolev implies KL-FI comparison as a reusable proof target. |
def | Partial | Not mapped | AutoSamplingTheory.Probability |
AutoSamplingTheory/Probability.lean:1094 |
AutoSamplingTheory.RMFLD.rmfldPaperRoot
|
def | Partial | Not mapped | AutoSamplingTheory.RMFLD |
AutoSamplingTheory/RMFLD.lean:10 |
AutoSamplingTheory.RMFLD.rmfldSource
|
def | Partial | Not mapped | AutoSamplingTheory.RMFLD |
AutoSamplingTheory/RMFLD.lean:12 |
AutoSamplingTheory.RMFLD.exploratorySeedLabels
|
def | Partial | Not mapped | AutoSamplingTheory.RMFLD |
AutoSamplingTheory/RMFLD.lean:16 |
AutoSamplingTheory.RMFLD.rmfldExploratoryContract
|
def | Partial | Not mapped | AutoSamplingTheory.RMFLD |
AutoSamplingTheory/RMFLD.lean:25 |
AutoSamplingTheory.RMFLD.rmfldProofDag
|
def | Partial | Not mapped | AutoSamplingTheory.RMFLD |
AutoSamplingTheory/RMFLD.lean:34 |
AutoSamplingTheory.SALD.saldPaperRoot
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27 |
AutoSamplingTheory.SALD.saldMainSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29 |
AutoSamplingTheory.SALD.saldAppendixSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33 |
AutoSamplingTheory.SALD.saldIterationSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37 |
AutoSamplingTheory.SALD.saldGronwallSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41 |
AutoSamplingTheory.SALD.saldGronwallExponentRewriteSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45 |
AutoSamplingTheory.SALD.saldDvVariationSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49 |
AutoSamplingTheory.SALD.saldPiSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53 |
AutoSamplingTheory.SALD.saldPiVelocityNormSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57 |
AutoSamplingTheory.SALD.saldKlFiLsiSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61 |
AutoSamplingTheory.SALD.saldContinuousSdeSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65 |
AutoSamplingTheory.SALD.saldFokkerPlanckSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69 |
AutoSamplingTheory.SALD.saldAlphaComplexitySource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:73 |
AutoSamplingTheory.SALD.saldForwardKlSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:77 |
AutoSamplingTheory.SALD.saldForwardKlProofSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:81 |
AutoSamplingTheory.SALD.saldForwardKlDerivativeSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:85 |
AutoSamplingTheory.SALD.saldForwardKlDvEnergySource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:89 |
AutoSamplingTheory.SALD.saldForwardKlGronwallSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:93 |
AutoSamplingTheory.SALD.saldForwardKlEndpointScheduleSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:97 |
AutoSamplingTheory.SALD.saldForwardKlDependencyChainSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:101 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:105 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteLipSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:109 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteInterpolationSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:113 |
AutoSamplingTheory.SALD.saldFrozenDeltaCrossLipSaldSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:117 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteProofSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:121 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteDerivativeSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:125 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteConditionalFpSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:129 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteDvVelocitySource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:133 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteGronwallSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:137 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteAccumulatedErrorSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:141 |
AutoSamplingTheory.SALD.saldForwardKlDiscreteCoefficientChainSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:145 |
AutoSamplingTheory.SALD.saldGuidedResidualSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:149 |
AutoSamplingTheory.SALD.saldGuidedResidualProofSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:153 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:157 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDerivativeSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:161 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDvGronwallSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:165 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetResidualDvSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:169 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetPureContractionSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:173 |
AutoSamplingTheory.SALD.saldUnifiedForwardKlSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:177 |
AutoSamplingTheory.SALD.saldUnifiedForwardKlProofSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:181 |
AutoSamplingTheory.SALD.saldVaSaldItoSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:185 |
AutoSamplingTheory.SALD.saldGuidedResidualMainSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:189 |
AutoSamplingTheory.SALD.saldCorrectionFieldSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:193 |
AutoSamplingTheory.SALD.saldUnifiedTransportBridgeSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:197 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:201 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:205 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteDeltaSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:209 |
AutoSamplingTheory.SALD.saldFrozenDeltaCrossLipGeneralSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:213 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteDerivativeSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:217 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalDriftSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:221 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:225 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:229 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:233 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteCondDistribIntegralMathlibSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:241 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:249 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:257 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteKlLogRatioMathlibSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:265 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteYoungSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:273 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteResidualDvSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:277 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteGronwallSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:281 |
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteGronwallSideConditionSource
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:285 |
AutoSamplingTheory.SALD.firstFaithfulLabels
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:289 |
AutoSamplingTheory.SALD.GronwallCandidateContract
- Lean-facing calculus interface for the appendix Gronwall lemma.
This is contract data, not a theorem. The eventual proof should instantiate
these fields using Mathlib's interval-integral and derivative APIs while
pre |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:309 |
AutoSamplingTheory.SALD.GronwallEndpointCalculusContract
- Endpoint-safe calculus ledger for the appendix Gronwall proof.
The source proof differentiates an integrating factor on a closed interval,
integrates a pointwise derivative inequality, and rewrites the exponential
fac |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:330 |
AutoSamplingTheory.SALD.GronwallExponentRewriteContract
- Lower-level ledger for the final exponent rewrite in `lem:gronwall`.
This keeps the interval-additivity and exponential-product algebra separate
from the derivative/FTC part of the Gronwall proof, because later SALD t |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:351 |
AutoSamplingTheory.SALD.gronwallNegIntegralRewriteScalar
- Formal scalar algebra for the final exponent rewrite in `lem:gronwall`.
The interval-integral equality `i0 = it + it1` is still a separate analytic
obligation; this lemma only closes the real additive negation part. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:369 |
AutoSamplingTheory.SALD.gronwallExpProductRewriteScalar
- Formal scalar `Real.exp` product algebra for the Gronwall rewrite.
Once interval additivity has produced `i0 = it + it1`, this proves the
pointwise exponential factor used in `appendix.tex:65-69`. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:380 |
AutoSamplingTheory.SALD.gronwallIntervalIntegralAdditivityScalar
- Interval-integral additivity bridge for the final Gronwall exponent rewrite.
The source uses this with `0 <= t <= t1`; Mathlib's oriented interval integral
version only needs interval-integrability on the adjacent pie |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:393 |
AutoSamplingTheory.SALD.gronwallExpProductRewriteIntervalIntegral
- Compiled bridge from interval-integral additivity to the Gronwall
`Real.exp` product rewrite.
This closes only the pointwise exponential factor. Rewriting the whole
`b_t` integral remains the separate congruence obli |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:406 |
AutoSamplingTheory.SALD.gronwallExpProductRewriteIntegralCongr
- Push the Gronwall exponent rewrite through the outer source integral.
This formalizes only the congruence step from `appendix.tex:67` to
`appendix.tex:69`, assuming the adjacent interval-integrability needed by the
po |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:425 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorProductDerivative
- Product derivative for the Gronwall integrating factor.
This is the Lean form of `appendix.tex:58-60`, after the derivative of
`A(t)=int_0^t a` has been supplied by the interval-integral FTC backend. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:452 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorDerivativeInequalityScalar
- Scalar order core for `appendix.tex:60-61`.
Once the source differential inequality `K' <= -a*K+b` is available and the
integrating factor is known to be nonnegative, this closes the real algebra
turning the product d |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:469 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorDerivativeLe
- Pointwise derivative inequality for the Gronwall integrating factor.
This packages the source line 58-61 step after an antiderivative derivative
`d/dt int_0^t a = a(t)` is supplied. The subsequent integration from `0 |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:488 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorDerivativeLeOfIntegral
- FTC-backed version of `gronwallIntegratingFactorDerivativeLe`.
This discharges the local derivative of `int_0^t a` using Mathlib's
interval-integral fundamental theorem at the point `t`. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:510 |
AutoSamplingTheory.SALD.gronwallOrderIntegrationOfHasDerivAt
- Order-integration backend for the Gronwall integrating-factor proof.
This is the source step from `appendix.tex:62-63`: once the derivative of
`F(t)=exp(int_0^t a)*K(t)` is represented by `f'` on the source interval a |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:532 |
AutoSamplingTheory.SALD.gronwallOrderIntegrationOfHasDerivRight
- Endpoint-safe order-integration backend for the Gronwall proof.
This version matches the closed-interval issue in `appendix.tex:62-63` more
closely than `gronwallOrderIntegrationOfHasDerivAt`: it only differentiates o |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:553 |
AutoSamplingTheory.SALD.gronwallEndpointEvaluationScalar
- Endpoint evaluation for the integrated Gronwall factor.
After integrating the derivative inequality, the source uses
`exp(int_0^0 a)=1` to turn the left endpoint into `K_0`. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:574 |
AutoSamplingTheory.SALD.gronwallEndpointMultiplyByExpNegScalar
- Scalar multiplication by the inverse integrating factor.
This is the endpoint algebra immediately before the final source exponent
rewrite in `appendix.tex:65-69`. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:591 |
AutoSamplingTheory.SALD.gronwallEndpointIntegralRewrite
- Move the endpoint inverse integrating factor through the source `b_t`
integral and apply the final Gronwall exponent rewrite.
This is the compiled version of the passage from `appendix.tex:65` to
`appendix.tex:69` aft |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:612 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorBoundOfDerivatives
- Global integrating-factor assembly for the appendix Gronwall proof.
This theorem threads the proof-producing local Gronwall helpers into the
paper's displayed bound under explicit global calculus and interval-integral |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:645 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorBoundOfInteriorDerivatives
- Endpoint-safe global Gronwall assembly with interior derivatives.
This is the same displayed bound as
`gronwallIntegratingFactorBoundOfDerivatives`, but the FTC/order-integration
step only requires continuity of the i |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:725 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorBoundOfIntegral
- FTC-backed form of the global Gronwall assembly.
This uses Mathlib's right-endpoint derivative theorem for the integral
`t ↦ ∫_0^t a`, then delegates the rest of the proof to
`gronwallIntegratingFactorBoundOfDerivativ |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:807 |
AutoSamplingTheory.SALD.gronwallCoefficientSideConditionsOfContinuous
- Continuous coefficients supply the interval-integrability and local FTC
side conditions needed by the appendix Gronwall assembly.
This is still below the source lemma: it assumes global continuity of the
coefficient ` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:844 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorBoundOfContinuousData
- Continuous-data wrapper for the appendix Gronwall display.
Compared with `gronwallIntegratingFactorBoundOfIntegral`, this theorem proves
the integrability of the derivative-side and right-hand-side integrands from
glo |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:866 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorBoundOfDifferentiable
- Source-facing derivative wrapper for the appendix Gronwall display.
This version writes the paper's derivative term as `deriv K`. It keeps one
explicit interval-integrability hypothesis for the product-derivative
int |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:904 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorBoundOfC1
- C1-style source-facing Gronwall wrapper.
When the selected derivative witness is continuous, the interval-integrability
left explicit in `gronwallIntegratingFactorBoundOfDifferentiable` is produced
from continuity. T |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:941 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorBoundOfInteriorContinuousData
- Continuous-data Gronwall assembly with only interior derivatives for `K`.
This discharges the interval-integrability and integral-FTC side conditions as
in `gronwallIntegratingFactorBoundOfContinuousData`, but it uses |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:975 |
AutoSamplingTheory.SALD.gronwallIntegratingFactorBoundOfInteriorC1
- C1-compatible source wrapper with no endpoint derivative hypothesis on `K`.
This uses `deriv K` for the paper's `dK_t/dt` term, but only assumes
differentiability of `K` on the open source interval. Continuity of `K` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1024 |
AutoSamplingTheory.SALD.forwardKlGronwallCoeffIntervalIntegrable
- Assemble interval-integrability of the continuous forward-KL Gronwall
coefficient from its source LSI and alpha pieces.
For `thm:forward-KL`, `lsiPart` is `dot{s}(t)*C_LSI(t)` and `alphaPart` is
`(1/2)*dot{s}(t)^(-1)* |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1050 |
AutoSamplingTheory.SALD.forwardKlGronwallCoeffAdjacentIntervalIntegrable
- Adjacent-interval version of
`forwardKlGronwallCoeffIntervalIntegrable` for the continuous forward-KL
Gronwall exponent bridge.
The hypotheses are exactly the theorem-specific interval-integrability data
still owed fo |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1064 |
AutoSamplingTheory.SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces
- Continuous forward-KL use site for the compiled Gronwall exponent
congruence.
Once the LSI and alpha pieces of
`a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1)` are interval
integrable on the adjacent interval |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1096 |
AutoSamplingTheory.SALD.forwardKlGronwallCoeffIntegralSub
- Integral subtraction for the continuous forward-KL Gronwall coefficient.
This is the local interval-integral algebra behind the source split
`a(t)=dot{s}(t)*C_LSI(t)-(1/2)*dot{s}(t)^(-1)*alpha^(-1)`. The theorem-spec |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1122 |
AutoSamplingTheory.SALD.forwardKlGronwallInitialExponentSplitScalar
- Scalar split of the initial Gronwall exponent in `thm:forward-KL`.
This proves only the Real exponential algebra in `appendix.tex:249-250`; the
integral identities producing the two pieces are supplied by
`forwardKlGr |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1136 |
AutoSamplingTheory.SALD.forwardKlGronwallInitialExponentSplitOfPieces
- Initial-term exponent split for the continuous forward-KL theorem display.
Given interval-integrability of the LSI and alpha pieces on `[0,T]`, this
matches the source's two exponential factors multiplying the initial |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1149 |
AutoSamplingTheory.SALD.forwardKlGronwallResidualExponentDropScalar
- Pointwise residual-exponent drop for the final forward-KL display.
The source drops the nonpositive LSI contribution inside
`exp(-int_t^T a)`. This lemma starts after the interval integral of the LSI
piece has been s |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1170 |
AutoSamplingTheory.SALD.forwardKlGronwallResidualExponentDropIntegral
- Integral residual-exponent drop for `thm:forward-KL`.
This packages the last display-matching inequality in `appendix.tex:248-251`
under explicit side conditions: adjacent interval-integrability for the LSI and
alpha |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1187 |
AutoSamplingTheory.SALD.forwardKlDvPositiveAlphaScalingScalar
- Scalar positive-alpha division for the continuous forward-KL DV step.
After the cited DV formula gives
`alpha * energy <= kl + logMgf`, the paper divides by `alpha > 0` and rewrites
`alpha^(-1) * logMgf` as the alpha- |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1224 |
AutoSamplingTheory.SALD.forwardKlDvPositiveAlphaCoefficientScalar
- Coefficient-preserving form of
`forwardKlDvPositiveAlphaScalingScalar`.
This is the scalar handoff to the Gronwall coefficient audit: once the
nonnegative prefactor, later `(1/2) * dot{s}(t)^(-1)`, has been supplied, |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1247 |
AutoSamplingTheory.SALD.forwardKlPostDvGronwallCoefficientScalar
- Post-DV scalar handoff to the continuous forward-KL Gronwall coefficient.
This is the source step in `appendix.tex:230-244`: after the pre-DV
derivative inequality has the form
`dK/dt <= -(dot{s}*C_LSI)*K + coeff*ener |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1272 |
AutoSamplingTheory.SALD.forwardKlPostDvGronwallCoefficientOfScheduleScalar
- Source-shaped post-DV handoff for `thm:forward-KL`.
This specializes `forwardKlPostDvGronwallCoefficientScalar` to the coefficient
`coeff=(1/2)*dot{s}(t)^(-1)` that appears immediately before the Gronwall
application |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1296 |
AutoSamplingTheory.SALD.generalMovingTargetGronwallCoeffAdjacentIntervalIntegrable
- Adjacent-interval coefficient package for the continuous general VA-SALD
Gronwall side conditions.
For `thm:general-moving-target-SALD`, `lsiPart` is
`(sigma_t^2/2)*dot{s}(t)*C_LSI(t)`, `alphaPart` is
`sigma_t^(-2)*do |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1322 |
AutoSamplingTheory.SALD.generalMovingTargetGronwallExpProductRewriteIntegralCongrOfPieces
- Continuous general VA-SALD use site for the compiled Gronwall exponent
congruence.
The theorem-specific hypotheses expose the adjacent interval-integrability of
the sigma/LSI coefficient, the alpha coefficient, and th |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1359 |
AutoSamplingTheory.SALD.discreteForwardKlGronwallCoeffIntervalIntegrable
- Scalar order core for the discrete forward-KL residual exponent bound.
In the source application, `lsiTerm` is the nonnegative LSI contribution,
`alphaTerm` is the interval alpha contribution, and `gammaTerm` is the i |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1392 |
AutoSamplingTheory.SALD.discreteForwardKlGronwallCoeffIntegralSubSub
- Integral subtraction for the three-piece discrete Gronwall coefficient.
For `thm:forward-KL-discrete`, the coefficient is
`a(t)=dot{s}(t)*C_LSI(t)-dot{s}(t)^(-1)*alpha^(-1)
-2*dot{s}(t)*eta^2*alpha'^(-1)*Gamma(t)`. T |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1410 |
AutoSamplingTheory.SALD.discreteForwardKlGronwallInitialExponentSplitScalar
- Scalar split of the initial Gronwall exponent for discrete forward-KL.
This is the real exponential algebra behind `main_body.tex:309-315`: the
initial term keeps the LSI contraction as one factor and collects the pos |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1428 |
AutoSamplingTheory.SALD.discreteForwardKlGronwallInitialExponentSplitOfPieces
- Initial-term exponent split for the discrete forward-KL theorem display.
Given interval-integrability of the LSI, alpha, and Gamma coefficient pieces on
`[0,T]`, this matches the source's two exponential factors multi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1444 |
AutoSamplingTheory.SALD.discreteForwardKlResidualExponentBoundScalar
|
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1462 |
AutoSamplingTheory.SALD.discreteForwardKlResidualExpBoundScalar
- Exponential form of `discreteForwardKlResidualExponentBoundScalar`.
This compiles only the monotone `Real.exp` wrapper around the scalar residual
exponent inequality used in the accumulated-error bridge. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1475 |
AutoSamplingTheory.SALD.discreteForwardKlAlphaComplexityCollectionScalar
- Constant-factor integral core for the `A_alpha` term in the discrete
forward-KL accumulated-error bridge.
After the linear slowdown supplies `dot{s}(t)⁻¹ = r⁻¹`, this formalizes only the
interval-integral algebra turn |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1494 |
AutoSamplingTheory.SALD.discreteForwardKlDeltaAccumulationScalar
- Constant-factor integral core for the `barDelta` term in the discrete
forward-KL accumulated-error bridge.
After the linear slowdown supplies `dot{s}(t)=r`, this formalizes only the
source algebra collecting `2*r*eta* |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1508 |
AutoSamplingTheory.SALD.discreteForwardKlAccumulatedErrorCollectionScalar
- Combined scalar/integral collection for the two additive residual terms in
the discrete forward-KL accumulated-error bridge.
This packages the compiled lower slice for cycle 27: once the linear-slowdown
coefficient id |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1524 |
AutoSamplingTheory.SALD.discreteForwardKlResidualIntegralDisplayBoundScalar
- Residual-integral display bridge for discrete forward-KL.
This is the cycle-61 lower scalar wrapper for the last additive term in
`main_body.tex:309-323`. Once the residual Gronwall kernel has already been
bounded by |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1543 |
AutoSamplingTheory.SALD.discreteForwardKlMainDisplayBoundScalar
- Main-display scalar wrapper for the discrete forward-KL accumulated-error
bridge.
This is the cycle-66 lower proof-producing step for `main_body.tex:309-323`
after the appendix Gronwall display in `appendix.tex:557-59 |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1567 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConstantScheduleSquareScalar
- Scalar inverse-schedule square identity for the discrete general VA-SALD
time-change coefficient.
The analytic fact that `dotT` is the derivative of the inverse schedule remains
part of `sald.general_moving_target_dis |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1611 |
AutoSamplingTheory.SALD.discreteForwardKlTimeChangeSquareCoefficientRewriteScalar
- Scalar time-change coefficient rewrite for the discrete forward-KL proof.
In `appendix.tex:526-553`, the source multiplies the `s`-time DV coefficient
by `dot{s}(t)` and rewrites
`dot{s}(t) * dot t(s(t))^2 * coeff` as |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1628 |
AutoSamplingTheory.SALD.discreteForwardKlPostDvTimeChangedDerivativeScalar
- Scalar post-DV time-change handoff for discrete forward-KL.
This is the proof-producing lower core for the cycle-56
`sald.discrete_forward_kl.gronwall_accumulation` packet. It starts after the
EM/KL derivative backen |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1652 |
AutoSamplingTheory.SALD.discreteForwardKlPointwiseGronwallInputOfPostDvTimeChanged
- Pointwise Gronwall-input wrapper for the discrete forward-KL time change.
The scalar theorem above handles one fixed time. This wrapper is the exact
lower-facing shape needed by the Gronwall accumulation obligation: |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1697 |
AutoSamplingTheory.SALD.discreteForwardKlEmInterpolationLeftEndpointVector
- Left-endpoint algebra for the frozen EM interpolation in
`appendix.tex:260-266`.
This proves only the pointwise vector identity behind
`\hat X_{s_k}=X_k^\eta`: at the left endpoint the time increment and Brownian
incr |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1728 |
AutoSamplingTheory.SALD.discreteForwardKlEmInterpolationRightEndpointVector
- Right-endpoint algebra for the frozen EM interpolation in
`appendix.tex:260-266`.
Once the mesh identity `s_{k+1}-s_k=eta` and the EM update definition for
`X_{k+1}^\eta` are supplied, this identifies the interpolatio |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1742 |
AutoSamplingTheory.SALD.discreteForwardKlLawEqOfPointwise
- Law-level handoff from pointwise equality of random variables.
This is the abstract step needed to use the endpoint-vector identities in the
EM interpolation proof: once two random variables are pointwise equal, any
c |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1757 |
AutoSamplingTheory.SALD.discreteForwardKlEmInterpolationLeftEndpointLawHandoff
- Left-endpoint law handoff for the frozen EM interpolation.
Combines the pointwise identity `\hat X_{s_k}=X_k^\eta` with an abstract law
operator. It does not construct `Law`, Brownian motion, or densities. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1769 |
AutoSamplingTheory.SALD.discreteForwardKlEmInterpolationRightEndpointLawHandoff
- Right-endpoint law handoff for the frozen EM interpolation.
After the mesh identity and pointwise EM update definition are supplied, this
turns the right-endpoint vector identity into the law equality used at
`appendi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1784 |
AutoSamplingTheory.SALD.discreteForwardKlEmEndpointLawPairHandoff
- Endpoint-law pair handoff for the frozen EM interpolation.
This is the lower cycle-40 instantiation layer for
`sald.discrete_forward_kl.em_endpoint_laws`: once the repository supplies
named law representations for `ha |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1804 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmInterpolationLeftEndpointLawHandoff
- General VA-SALD left-endpoint law handoff for
`eq:general_moving_target_SALD_frozen_interp`.
This is the same abstract law transport used by the discrete forward-KL EM
block, specialized to the general moving-target n |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1837 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmInterpolationRightEndpointLawHandoff
- General VA-SALD right-endpoint law handoff for the frozen interpolation.
After the mesh identity and the pointwise general EM update are supplied, this
turns the source endpoint identity into the law equality
`\hat\rh |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1854 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmEndpointLawPairHandoff
- Endpoint-law pair handoff for the discrete general moving-target VA-SALD
EM interpolation.
Once named law representations for `hat rho_s`, `rho_k^eta`, and
`rho_{k+1}^eta` are supplied, this proves the two endpoint la |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1876 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmEndpointLawPairOfNamedInterpolation
- Named-interpolation endpoint-law handoff for the discrete general
moving-target VA-SALD EM path.
This is the lower cycle-49 endpoint slice for `appendix.tex:1354-1357`.
It starts from the repository's eventual named p |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1913 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmEndpointMeasureMapPairOfNamedInterpolation
- Measure-level endpoint-law handoff for the discrete general
moving-target VA-SALD EM path.
This is the first concrete measure-theory backfill below the abstract law
operator handoffs. It uses `Measure.map` and almost |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1952 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmJointEndpointMeasureMapOfNamedInterpolation
- Joint endpoint-law handoff for the discrete general moving-target VA-SALD
EM path.
This packages the two endpoint a.e. identities into a paired pushforward law.
It is still only endpoint bookkeeping below `appendix.te |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:1993 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmJointEndpointMarginalLawsOfNamedInterpolation
- Marginal endpoint-law extraction from the joint endpoint law for the
discrete general moving-target VA-SALD EM path.
This composes the paired endpoint-law equality with the first/second
projection lemmas. It is still |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2035 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteHatRhoMarginalOfJointMap
- Marginal compatibility for the joint law used in the conditional drift.
For a fixed EM interpolation time `s`, the source defines
`\hat\rho_s=Law(\hat X_s)` and then conditions `X_k^eta` on `\hat X_s=x`.
This lemma pr |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2079 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap
- First-marginal compatibility for Mathlib's conditional-distribution
orientation.
For `condDistrib X_k^eta hatX_s P`, Mathlib names the joint law in the order
`(hatX_s, X_k^eta)`, so the conditioning law is the first m |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2103 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalKernelCompatibilityOfJointMapMarginal
- Transport a supplied conditional-kernel compatibility predicate to the
named `\hat\rho_s` marginal.
The analytic backend must still supply the regular conditional kernel and prove
that it disintegrates the joint law. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2128 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityOfJointMap
- Bundle the second-marginal equality with conditional-kernel compatibility.
This is the lower endpoint-to-conditional wrapper used before the weak
Fokker--Planck statement: if a supplied kernel compatibility predicate |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2168 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalDriftLinearCombination
- Conditional-expectation linearity wrapper for the frozen general
VA-SALD drift.
The analytic backend must still provide the regular conditional law and the
linearity hypotheses for the selected conditional expectation |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2210 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalDriftFieldOfLinearCombination
- Named-field version of
`generalMovingTargetDiscreteConditionalDriftLinearCombination`.
If a later analytic backend supplies `barB` as the selected conditional
expectation in the source definition, this theorem rewrite |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2234 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents
- Named conditional-drift component handoff for `bar b_{k,s}`.
After the analytic backend supplies named conditional fields for the frozen
guide drift and frozen score summands, this wrapper rewrites the selected
source |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2261 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityHandoff
- Abstract regularity handoff for the named frozen drift field.
The analytic backend chooses the concrete meanings of `FieldMeasurable` and
`FieldIntegrable` (for example, measurability and local integrability under
`\h |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2296 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents
- Component-regularity handoff for the named frozen drift field.
This is the next local bookkeeping step after naming
`condC_{k,s}` and `condScore_{k,s}` in `appendix.tex:1368-1377`.
If the concrete backend supplies mea |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2327 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity
- Mathlib `condDistrib` version of the named frozen-drift regularity handoff.
This is the source-specific backfill for `appendix.tex:1368-1377` after the
cycle-85 law-space conditional-integral lemmas. If `condC` and ` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2375 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity
- Canonical `condDistrib` version of the frozen conditional drift field.
This cycle-106 theorem removes the old supplied component-field regularity
premise for the canonical representative in `appendix.tex:1368-1377`. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2464 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq
- Named-field bridge from the canonical `condDistrib` drift representative.
This is the strict versioning handoff left by
`generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity`: if the paper's
selected `barB` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2552 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedBarBEqMeasOfStronglyMeasurable
- Equality-set measurability for the named `barB` source representative.
Cycle 110 removes one supplied side condition from the cycle-109
`ae_map_iff` bridge: once the canonical conditional-drift representative and
the |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2622 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq
- Conditional-expectation uniqueness bridge for the named `barB` representative.
Cycle 112 narrows the remaining `hbarBCondExp` side condition from
`generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2660 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpKernelSourceDef
- Source-definition bridge for the named `barB` representative.
This narrows the remaining `appendix.tex:1368-1377` version-selection
boundary. If the source-level conditional-expectation definition of `barB`
is given |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2742 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef
- Source-definition bridge for the named `barB` representative using
Mathlib's product conditional-expectation theorem.
This is a stricter `appendix.tex:1368-1377` backend than assuming a direct
`hatRhoS`-a.e. equality |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2804 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef
- Source-definition bridge with the `hbarBCondExp` premise replaced by a
conditional-expectation uniqueness boundary.
This is the cycle-112 downstream handoff for `appendix.tex:1368-1377`.
Instead of assuming directly t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:2932 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents
- Convert the source-facing state-event set-integral criterion into the
`comap hatXAtS` criterion used by Mathlib conditional-expectation uniqueness.
The paper defines `bar b_{k,s}` by conditioning on `hat X_s = x`, so |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3008 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBCondExpOfCondDistrib
- Canonical `condDistrib` drift satisfies the sample-space conditional
expectation identity.
This factors the `hbarBCondExp` part of the cycle-115 boundary for the
canonical Mathlib representative used in `appendix.tex: |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3046 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBStateSetIntegralOfCondDistrib
- Canonical `condDistrib` drift satisfies the state-event set-integral
criterion.
This is the cycle-114 narrowing of the remaining `hbarBStateSetIntegral`
boundary for `appendix.tex:1368-1377`: if `barB` is chosen to be |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3150 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalAeEq
- Selected named `barB` inherits the canonical state-event set-integral
criterion.
Cycle 115 narrows the post-canonical blocker for `appendix.tex:1368-1377`.
Once the paper-selected `barB` is identified `hatRhoS`-a.e. w |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3254 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCondExpSourceDef
- Source conditional-expectation version of the selected `barB` bridge.
This is the cycle-115 lower packet for `appendix.tex:1368-1377`. It removes
the supplied selected-to-canonical `hbarBAe` input from
`generalMoving |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3377 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBStateEventIntegralAndIntegrableOfCondDistrib
- Canonical `condDistrib` drift inherits the selected-bridge integrability
and state-event set-integral conclusion.
This specializes the cycle-115 selected `barB` bridge to the canonical Mathlib
conditional-distribution |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3455 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteNamedBarBStateEventIntegralAndIntegrableOfCanonicalPointwiseEq
- Pointwise source selection of the canonical `condDistrib` `barB` representative.
This is the cycle-117 lower handoff for the selected-version boundary at
`appendix.tex:1368-1377`. If the paper's named field is chosen |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3546 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface
- Direct canonical `barB` state-event interface for the EM backend.
Cycle 118 consumes the source-supported canonical representative choice instead
of keeping a separate named `barB` version open. The theorem existenti |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3614 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef
- Source-definition bridge with the conditional-expectation set-integral
boundary restricted to state events.
This is a narrower form of
`generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef`:
inst |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3681 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField
- Candidate regularity for the named `barB` representative pulled back to
the sample space.
Cycle 113 discharges the `hbarBMeas`/`hbarBInt` inputs of
`generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSet |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3758 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef
- Source-definition bridge with candidate regularity pulled back from the
named state marginal.
This is the cycle-113 lower handoff for `appendix.tex:1368-1377`. It removes
the older sample-space candidate-regularity h |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3795 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample
- Source-specific map-a.e. transfer for named conditional component fields.
The Mathlib conditional-expectation facts often first produce a sample-space
a.e. statement after composing with `hatXAtS`. Since the paper na |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3864 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfCondExpKernelMap
- Source-specific component-version bridge from `condExpKernel.map`.
This cycle-103 lower theorem targets `appendix.tex:1368-1377` for one
component such as `condC_{k,s}`. It no longer takes the old sample-space
`hguid |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3895 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions
- Cycle-91 lower theorem using sample-space version equalities.
This removes the direct `hatRhoS`-a.e. component-version hypotheses from
`generalMovingTargetDiscreteCondDistribNamedDriftRegularity`: it is enough to
prov |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:3943 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents
- Cycle-74 lower handoff from a supplied conditional-kernel backend.
Once the cited Mathlib/disintegration layer supplies kernel compatibility for
the joint law, and supplies the component integral fields with their
mea |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4019 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents
- Cycle-75 lower handoff for the `condDistrib` orientation.
Mathlib's conditional distribution for `X_k^eta | hatX_s` supplies the joint
law in the order `(hatX_s, X_k^eta)`. Existing SALD endpoint compatibility
uses ` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4098 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointMeasureMapToSwappedConditionalCompatibility
- Cycle-76 endpoint-law to swapped conditional-kernel compatibility.
This is the endpoint-to-conditional bridge for the active EM backend. It
packages the already compiled endpoint `Measure.map` handoff for
`\hat X_{s_ |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4193 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointMeasureMapToConditionalCompatibility
- Cycle-76 lower handoff with the original joint-law marginal exposed.
The previous endpoint-to-swapped wrapper returns the first marginal in the
Mathlib-style `(hat X_s, X_k^eta)` orientation. The weak Fokker--Planck
|
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4288 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointMeasureMapWeakFpPrereqHandoff
- Cycle-81 lower handoff from endpoint `Measure.map` compatibility to the
weak-FP prerequisite layer.
This is the endpoint-only part of the cycle-81 backend: once the named
endpoint laws, the original/swapped `hatRhoS` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4366 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff
- Cycle-80 endpoint/conditional drift-regularity handoff.
This composes the cycle-76 endpoint-to-conditional compatibility wrapper with
the named component-field regularity wrapper from the conditional-drift layer.
It i |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4465 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpReadinessHandoff
- Cycle-81 middle handoff from endpoint/conditional bookkeeping to weak-FP
readiness.
This wrapper does not prove the weak conditional Fokker--Planck theorem. It
only packages the endpoint `Measure.map` law equalities, |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4571 |
AutoSamplingTheory.SALD.discreteForwardKlConditionalFpDivergenceDriftSplit
- Divergence-linearity algebra for the conditional-drift FP regrouping.
In `appendix.tex:377-385`, after the analytic Laplacian split and linearity of
the divergence operator have been supplied, the source regroups
`-di |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4695 |
AutoSamplingTheory.SALD.discreteForwardKlConditionalFpLaplacianSplitHandoff
- Lower handoff algebra for the EM conditional Fokker--Planck split.
This composes the two analytic inputs used in `appendix.tex:357-385`: the
conditional-drift Fokker--Planck equation and the Laplacian split relative t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4713 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsHandoff
- Weak-test source-sign handoff for the discrete general EM
Fokker--Planck equation.
For each admissible weak test, the analytic backend should supply the
conditional Fokker--Planck identity with the drift contribution |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4740 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsAdmissibleHandoff
- Admissible-test version of the weak conditional Fokker--Planck source-sign
handoff.
The paper's weak form is only meant for an admissible test class. This local
wrapper keeps that predicate explicit while doing the s |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4759 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorHandoff
- Generator-level handoff for the weak conditional Fokker--Planck source
signs.
This is the cycle-77 refinement of the cycle-72 wrapper. It separates the
analytic input into two supplied facts: first, the EM interpolat |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4783 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfGeneratorPiecesHandoff
- Component-split generator handoff for the weak conditional Fokker--Planck
source signs.
This lower cycle-77 wrapper is one step closer to the source invocation than
`generalMovingTargetDiscreteWeakConditionalFpSourceS |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4828 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianAction
- Source-facing diffusion action handoff for the weak conditional
Fokker--Planck identity.
The direct downstream hypothesis `diffusionAction phi = sigmaCoeff •
laplacian phi` is narrowed into two source steps: the EM/Br |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4889 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianActionOfIntegrationByParts
- Weak Laplacian integration-by-parts interface for the diffusion action.
This narrows the cycle-129 `hlaplacianAction` boundary. The source
Fokker--Planck display in `appendix.tex:1379-1387` contributes the positive
` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4930 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfLaplacianIntegrationByParts
- Diffusion-source handoff with the weak Laplacian integration-by-parts
boundary exposed.
This keeps the EM/Brownian generator action `hdiffusionAction` separate and
replaces the broader cycle-129 `hlaplacianAction` pre |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:4971 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity
- Green-identity scout route for the weak Laplacian integration-by-parts
boundary.
This narrows the direct `hweakLaplacianIbP` input exposed in cycle 130 to the
two no-boundary Green steps used by the source proof: move |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5024 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfGreenLaplacianIbP
- Diffusion-source handoff with the weak Laplacian IBP route exposed as two
Green identities.
This removes the direct `hweakLaplacianIbP` premise from the cycle-130
diffusion-source helper. The remaining analytic leave |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5070 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfFirstGreenNoBoundaryFlux
- Diffusion-source handoff with the first Green identity factored through
no-boundary flux algebra.
This lower helper narrows the direct `hfirstGreen` premise from the lower_1
Green route. The first Green identity is r |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5133 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenNoBoundaryFlux
- Diffusion-source handoff with both Green identities factored through
no-boundary flux algebra.
This narrows the remaining direct `hsecondGreen` premise left by
`generalMovingTargetDiscreteWeakConditionalFpDiffusionSou |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5220 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenTraceBoundary
- Second-Green diffusion-source handoff with zero boundary flux narrowed to
a boundary trace-product condition.
This keeps the cycle-131 residual and divergence facts explicit, but replaces
the direct `hsecondGreenZeroB |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5319 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleGeneratorPiecesHandoff
- Sample-space generator derivative handoff for the weak conditional
Fokker--Planck source signs.
This cycle-86 refinement removes the coarse supplied generator/time-derivative
equality from the cycle-77 source-sign wra |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5431 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorHandoff
- Sample-space generator handoff with a definitionally split generator
action.
This cycle-92 refinement removes the explicit `hgeneratorSplit` input from the
cycle-86 theorem. The generator action consumed by the law-t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5524 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorHandoff
- Mapped-law weak derivative from a sample-space split generator.
This cycle-92 companion removes the separate supplied `hlawDerivative` input
when the immediate goal is the weak-test Fokker--Planck derivative itself. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5594 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfSampleSplitGeneratorHandoff
- Named-law weak derivative from a sample-space split generator.
Cycle 104 removes the remaining bookkeeping gap between the paper notation
`\hat\rho_s = Law(\hat X_s)` and the cycle-92 `Measure.map` derivative route.
I |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5672 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpNamedLawDerivativeOfDominatedSplitGeneratorHandoff
- Named-law weak derivative from dominated pointwise sample-path derivatives.
This cycle-110 generator-to-law refinement removes the integral-level
`hsampleGenerator` premise from
`generalMovingTargetDiscreteWeakConditi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5752 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction
- Drift source action from the named conditional drift weak pairing.
This cycle-94 helper replaces the primitive `hdriftSource` shape used by the
weak conditional Fokker--Planck handoffs. The new inputs expose the two |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5882 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound
- Integrability of the `barB` weak-test contraction from integrability of
the conditional drift field.
This cycle-98 lower theorem removes the primitive paired-integrability input
from the divergence/no-boundary route w |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5925 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral
- `barB` weak divergence from the integral no-boundary identity.
This cycle-98 handoff narrows the remaining divergence half of
`ASTIS.SALD.cycle94.remaining_barB_divergence_boundary`. Instead of assuming
directly that |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:5969 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing
- `barB` weak divergence with paired integrability discharged by a norm
bound against the integrable conditional drift field.
This composes the local integrability theorem
`generalMovingTargetDiscreteBarBPairIntegrableO |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6021 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral
- Drift source action with the `barB` divergence fact reduced to an integral
no-boundary theorem.
This composes the cycle-94 `barB` weak-action handoff with the cycle-98
integral no-boundary handoff. The downstream wea |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6085 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral
- Drift source action with the `barB` divergence fact reduced to a bounded
law-integral no-boundary theorem.
Compared with
`generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral`,
this version |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6145 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef
- `barB` weak divergence when the weak pairing is the law-integral
definition.
This cycle-100 helper removes the separate `hweakGradIntegral` supplied
hypothesis from the bounded no-boundary route. It specializes
`weak |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6214 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef
- Drift source action with the weak-gradient pairing definition aligned to
the law integral.
This is the cycle-100 lower-ready version of
`generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIn |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6267 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteBarBPairNormBoundOfInnerGradientBound
- Inner-product weak-test contraction from a gradient norm bound.
This cycle-100 lower theorem removes the remaining `hpairNormBound` supplied
hypothesis when the paper-facing weak pairing is the real inner product of t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6330 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryInnerGradientBound
- `barB` weak divergence with inner-product contraction supplied by
Cauchy--Schwarz.
Compared with
`generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef`,
this version no longer takes `hpai |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6369 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound
- Drift source action with both weak-pairing definition alignment and the
inner-product contraction bound discharged locally.
The remaining analytic boundary is now the weak-test gradient norm estimate
plus the no-bound |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6426 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDriftDivNoBoundaryOfProductRule
- Product-rule/no-boundary algebra for the `barB` drift-divergence term.
This lower helper narrows the monolithic `hdivNoBoundary` premise to the
source-facing pieces expected from the no-boundary divergence theorem:
th |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6488 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientProductRuleBoundary
- Drift source action with the no-boundary premise factored through the
product-rule and boundary-flux identities.
Compared with
`generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound`,
this ve |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6534 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated
- Dominated named-law weak derivative with canonical `barB` integrability
supplied by the conditional-drift state-event interface.
This cycle-119 lower theorem consumes
`generalMovingTargetDiscreteCanonicalBarBEmStateEv |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6615 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated
- Canonical `barB` dominated weak derivative on the source EM interval.
This cycle-120 lower theorem removes the supplied `hsampleNeighborhood`
premise from the canonical `barB` weak-FP consumer by specializing the loca |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6795 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasDominated
- Canonical `barB` dominated weak derivative on the source EM interval,
with sample measurability derived from law-space test measurability.
This cycle-121 lower theorem removes the supplied `hsampleMeas` premise from
t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:6944 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDominated
- Canonical `barB` dominated weak derivative on the source EM interval,
with sample measurability and sample integrability transported from the named
law.
This cycle-122 lower theorem removes the supplied `hsampleInt` p |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:7092 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasDominated
- Canonical `barB` dominated weak derivative on the source EM interval,
with sample derivative measurability derived from a concrete EM derivative
representative.
This cycle-123 lower theorem removes the supplied `hsamp |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:7238 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundDominated
- Canonical `barB` dominated weak derivative on the source EM interval,
with the local sample-derivative bound transported from the concrete EM
derivative representative.
This cycle-124 lower theorem removes the supplie |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:7387 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntDominated
- Canonical `barB` dominated weak derivative on the source EM interval,
with the dominating bound integrability transported from the joint EM law.
This lower theorem removes the supplied `hboundInt` premise from the
cyc |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:7547 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathDominated
- Canonical `barB` dominated weak derivative on the source EM interval,
with the path derivative obtained from differentiability of the concrete EM
weak-test path.
This cycle-125 lower theorem removes the supplied `hpat |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:7715 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDominated
- Canonical `barB` dominated weak derivative on the source EM interval,
with the derivative-value identity transported from the concrete EM derivative.
This cycle-126 lower theorem removes the supplied `hderivValue` pre |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:7876 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionDominated
- Canonical `barB` dominated weak derivative on the source EM interval,
with the drift weak-action identity derived from guide/score component
pairings.
This cycle-127 dynamic-leaf theorem removes the first post-`canoni |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:8047 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasDominated
- Canonical `barB` dominated weak derivative on the source EM interval,
with the drift weak-action identity and pairing measurability derived from
separate field measurability facts.
This lower refinement removes the ra |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:8319 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceDominated
- Canonical `barB` dominated weak derivative with no-boundary narrowed to
trace/product-rule facts.
This cycle-128 refinement removes the direct `hdivNoBoundary` continuation
from the post-cycle-127 pair-measurability t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:8534 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDominated
- Canonical `barB` dominated weak derivative with no-boundary narrowed and
canonical-field measurability derived from the condDistrib regularity theorem.
This follow-on removes the separate `hcanonicalBarBMeas` premise |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:8796 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalMeasIntDerivMeasBoundIntPathValueDriftActionPairMeasNoBoundaryTraceCanonicalMeasDiffusionSourceDominated
- Canonical `barB` dominated weak derivative with the diffusion source action
split into the EM/Brownian weak diffusion action and weak Laplacian action.
This cycle-129 lower continuation removes the direct `hdiffusionS |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:9028 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero
- Zero boundary flux from a boundary-integral trace product that vanishes
almost everywhere.
This is the source-facing no-boundary specialization used for the
`hatRhoS * barB` drift term: after the divergence theorem ha |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:9239 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero
- Boundary trace-product vanishing from zero admissible-test trace.
This is the compact-support/zero-trace lower handoff for the cycle-102
boundary packet. It does not prove the analytic trace theorem; it only
removes |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:9280 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox
- Boundary-flux integral representation from Mathlib's box divergence
theorem.
This is the lower-ready Mathlib specialization for the
`appendix.tex:1379-1387` no-boundary drift packet. It does not prove the
weighted-fi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:9315 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFlux
- Second-Green diffusion-source handoff with the boundary-flux integral
represented by the local box divergence theorem interface.
This lower helper removes the direct `hsecondGreenBoundaryFluxIntegral`
premise from
`ge |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:9426 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTestTraceZero
- Second-Green diffusion-source handoff with the trace-product condition
narrowed to zero admissible-test trace.
This is the cycle-132 source-facing continuation of the cycle-131
box-boundary-flux packet. It keeps the |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:9584 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfTraceEqTestTraceZero
- Second-Green diffusion-source handoff with the zero trace narrowed to a
trace-identification theorem plus the standard admissible-test zero trace.
This lower-scout continuation does not prove the analytic trace theore |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:9737 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqTestTraceZero
- Second-Green diffusion-source handoff with trace identification narrowed
from an a.e. boundary statement to pointwise equality of the selected traces.
This lower_2 continuation keeps the analytic source task as the po |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:9900 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero
- Second-Green diffusion-source handoff with admissible-test zero trace
narrowed from an a.e. boundary statement to pointwise zero trace.
This cycle-133 continuation keeps the selected second-Green trace
identification |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10058 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfTestLaplacianNormalization
- Second-Green diffusion-source handoff with test-Laplacian normalization
narrowed to a test-local premise.
This lower-scout continuation removes the broad `htestLaplacian` premise from
the cycle-133 pointwise-trace con |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10217 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization
- Test-local Laplacian normalization from an operator-level source identity.
This lower_2 theorem narrows the remaining analytic
`htestLaplacianLocal` leaf from the cycle-133 second-Green route. Instead of
asking for a |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10370 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfSourcePullback
- Operator-level test-Laplacian normalization from shared source definitions.
This cycle-134 helper is the next source-facing boundary below
`generalMovingTargetDiscreteTestLaplacianLocalOfOperatorNormalization`.
It doe |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10390 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfSourcePullback
- Pointwise test-Laplacian normalization from source-pullback definitions.
This lower_1 scout bridge adapts the cycle-134 operator-level source-pullback
normalization to the pointwise `htestLaplacianPointwise` leaf expo |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10421 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteSourceTestLaplacianEqStdIteratedFDeriv
- Mathlib source formula for the selected weak-test Laplacian.
This cycle-134 scout theorem narrows the remaining source-definition leaves
`htestLaplacianActionDef` and `hweakFpLaplacianDef`: once the selected source
te |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10453 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula
- Weak-FP Laplacian definition from the standard-basis source formula.
This lower_2 helper narrows the `hweakFpLaplacianDef` leaf left by the
cycle-134 source-pullback packet. Once the selected weak test is represented |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10471 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula
- Density-Laplacian source formula from a pointwise standard-basis field.
This cycle-136 lower_1 scout theorem narrows the remaining
`hdensityLaplacianStdBasisDef` source boundary. It separates the weak-action
definiti |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10520 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField
- Pointwise density-Laplacian source formula from the Mathlib Laplacian.
This lower_2 helper narrows the remaining `hsourceDensityLaplacianStdBasis`
source boundary from lower_1. It is enough to identify the named sour |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10572 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula
- Weak-FP standard-basis source formula from the density-Laplacian action.
This cycle-136 helper targets the remaining weak-FP side of the standard-basis
source pair. It derives the `hweakFpStdBasisDef` shape consumed |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10612 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP
- Weak-FP density-Laplacian action from pointwise weak Laplacian IBP.
This cycle-137 helper narrows the remaining `hweakFpDensityLaplacianAction`
boundary from cycle 136. The source-facing analytic leaf is now the poin |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10651 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP
- Weak-FP standard-basis source formula from pointwise weak Laplacian IBP.
This cycle-137 downstream bridge removes the direct
`hweakFpDensityLaplacianAction` premise from the cycle-136 standard-basis
consumer. The rem |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10669 |
AutoSamplingTheory.SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity
- Pointwise weak Laplacian IBP from the Green identity chain.
This lower_1 proof-scout helper narrows the pointwise weak Laplacian
integration-by-parts leaf exposed in cycle 137. The source-facing theorem to
prove next |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10712 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity
- Weak-FP standard-basis formula from pointwise Green identities.
This downstream bridge feeds the pointwise Green/test-calculus split directly
into the cycle-137 standard-basis consumer. The remaining analytic leaves |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10741 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero
- Pointwise second-Green identity from box-divergence and zero test trace.
This lower_2 helper narrows the direct pointwise `hsecondGreenPointwise` leaf
exposed by the cycle-137 Green-identity scout. It reconstructs th |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10797 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux
- Weak-FP standard-basis formula with the second-Green pointwise leaf
factored through box-divergence and pointwise zero trace.
This downstream bridge instantiates
`generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwis |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:10940 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteFirstGreenPointwiseOfBoundaryFluxZero
- Pointwise first-Green identity from boundary-flux cancellation.
This cycle-138 helper narrows the direct `hfirstGreenPointwise` leaf exposed by
the cycle-137 Green-identity scout. The pointwise first Green equality i |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11054 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfFirstGreenBoundaryFluxAndSecondGreenBoxBoundaryFlux
- Weak-FP standard-basis formula with both Green pointwise leaves narrowed.
This downstream bridge removes the direct `hfirstGreenPointwise` premise from
`generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSeco |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11092 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTestLaplacianActionDefOfStdBasisSourceFormula
- Test-calculus Laplacian definition from the standard-basis source formula.
This cycle-135 helper is the test-action sibling of
`generalMovingTargetDiscreteWeakFpLaplacianDefOfStdBasisSourceFormula`. It
narrows the re |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11216 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula
- Pointwise test-Laplacian normalization from the test standard-basis formula.
This lower_2 bridge continues the cycle-138 pointwise source-pullback scout
without using the weak-FP standard-basis conclusion as an input. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11265 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfStateIntegral
- Weak-FP source-action definition from the EM state-law integral.
This lower_1 scout bridge narrows `hweakFpSourceActionDef` to the concrete
state-law integral interface behind `appendix.tex:1379-1387`. Once
`\hat\rho |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11312 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegral
- Weak-FP source-action definition from the selected source Laplacian field.
This lower_2 continuation narrows the lower_1 state-integral inputs. Once the
named weak-FP source field is identified with Mathlib's Laplaci |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11364 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteSourceLaplacianFieldMeasOfSelectedTestLaplacianMeasurable
- Source-Laplacian field measurability from ordinary measurability.
Cycle 199 narrows the direct weak-Fokker--Planck side condition
`hsourceLaplacianFieldMeas` to a source-facing test-class regularity premise
that does |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11432 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteSelectedTestLaplacianMeasurableOfContinuous
- Selected-test Laplacian measurability from continuity.
Lower_3's cycle 199 API bridge keeps the remaining source-facing regularity
honest: if the original test class supplies continuity of the selected-test
Laplacian, |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11454 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteLaplacianSourceStateIntegralOfEmGeneratorStateIntegral
- Source-Laplacian state integral from the frozen EM generator component.
This cycle-140 bridge narrows the remaining
`hlaplacianSourceStateIntegral` input exposed by the cycle-139 lower_2 theorem.
It separates the sour |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11476 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfLawIntegral
- Frozen EM generator Laplacian state integral from its law-integral form.
This lower_1 scout bridge narrows the `hemGeneratorStateIntegral` premise to a
law-space source fact. Once the paper-selected marginal is repre |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11513 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfSourceFunctional
- Frozen EM generator Laplacian state integral from its source-functional form.
This cycle-153 bridge narrows the direct `hemGeneratorLaplacianStateIntegral`
leaf selected by the EM conditional-law/state-event illness a |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11562 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStateIntegral
- Weak-FP source-action definition from the frozen EM generator state integral.
This consumer feeds the cycle-140 source-integral bridge into
`generalMovingTargetDiscreteWeakFpSourceActionDefOfSourceLaplacianStateIntegr |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11618 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorLawIntegral
- Weak-FP source-action definition from the frozen EM generator law integral.
This lower_1 consumer removes the sample-space `hemGeneratorStateIntegral`
premise from the cycle-140 bridge. The remaining source-cited ana |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11677 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional
- Weak-FP source-action definition from the frozen EM generator source functional.
This lower_2 continuation narrows lower_1's remaining law-space
`hemGeneratorLawIntegral` fact. If the frozen EM generator's Laplacian
|
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11740 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorSourceActionDefOfStdBasisSourceFormula
- Frozen EM generator source action from the standard-basis source formula.
This cycle-141 illness-area bridge narrows the remaining
`hemGeneratorSourceActionDef` leaf from the cycle-140 source-functional route.
For the |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11808 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfStdBasisSourceFunctional
- Frozen EM generator Laplacian state integral from the standard-basis source
formula.
This lower_2 continuation narrows the cycle-153
`hemGeneratorSourceActionDef` input to the source-cited standard-basis formula
for t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11858 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorStdBasisSourceFormula
- Weak-FP source-action route from the EM generator standard-basis formula.
This consumer feeds the cycle-141 split into
`generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorSourceFunctional`.
It replaces the d |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11917 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorStdBasisDefOfTraceField
- Frozen EM generator standard-basis formula from a named trace field.
This lower_1 scout bridge narrows the remaining `hemGeneratorStdBasisDef`
source theorem. The analytic Brownian-generator work is now the smaller p |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:11982 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceFieldSourceFunctional
- Frozen EM generator Laplacian state integral from the trace-field split.
This cycle-154 bridge discharges the `hemGeneratorStdBasisDef` premise left by
the cycle-153 state-integral packet. The remaining source-facing |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12035 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceFieldSourceFormula
- Weak-FP source-action route from the EM generator trace-field split.
This feeds the lower_1 trace-field narrowing into the existing cycle-141
standard-basis consumer. It replaces `hemGeneratorStdBasisDef` by the smal |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12097 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLawIntegralSourceFormula
- Weak-FP source-action route from a law-space EM generator trace integral.
This lower_2 continuation narrows the trace-action source boundary exposed by
the lower_1 trace-field split. Instead of assuming directly that |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12169 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorTraceLawIntegralOfStateIntegral
- Frozen EM generator trace law integral from the EM state integral.
This cycle-142 middle bridge narrows the remaining
`hemGeneratorTraceLawIntegral` boundary to the sample-space trace integral
along the frozen EM inte |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12241 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceStateIntegralSourceFormula
- Weak-FP source-action route from the EM trace state integral.
This cycle-142 consumer feeds the state-integral trace narrowing into the
cycle-141 trace-law route. It replaces the law-space
`hemGeneratorTraceLawIntegr |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12285 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldMeasOfSourceLaplacianFieldMeas
- Trace-field measurability from the selected-test Laplacian field.
This lower_1 scout bridge narrows the trace-field measurability side condition
left by the state-integral EM generator route. Once the named trace fie |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12357 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegral
- Trace state integral from the selected-test Laplacian state integral.
This lower_1 scout bridge narrows the remaining `hemGeneratorTraceStateIntegral`
source theorem. It does not prove the EM generator state integral |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12401 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorTraceStateIntegralOfLaplacianStateIntegralLaplacianField
- Trace state integral from a direct trace-field/Laplacian identity.
This cycle-155 dynamic-leaf bridge narrows the remaining
`hemGeneratorTraceStateIntegral` boundary exposed by the cycle-154 state
integral route. It |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12456 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateIntegralSourceFormula
- Weak-FP source-action route from the EM Laplacian state integral.
This lower_1 consumer removes the trace-specific state-integral and
measurability premises exposed by the cycle-142 middle packet. The remaining
EM an |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12498 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianLawIntegralSourceFormula
- Weak-FP source-action route from the EM Laplacian law integral.
This lower_2 continuation narrows the cycle-142
`hemGeneratorLaplacianStateIntegral` leaf exposed by the trace-state route.
The remaining source-cited EM |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12575 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianLawIntegralOfStateEventFormula
- Frozen EM generator Laplacian law integral from a state-event formula.
This cycle-143 bridge narrows the remaining `hemGeneratorLaplacianLawIntegral`
boundary. Instead of assuming the law-space selected-test Laplacia |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12648 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateEventEqOfPointwise
- State-event Laplacian equality from a pointwise event-field definition.
This lower-1 cycle-143 bridge narrows the remaining measurable-state-event
boundary. If the named frozen-generator event field is pointwise the
|
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12690 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfActionDef
- Total-event generator formula from the source action definition.
This lower-2 cycle-143 helper narrows the remaining
`hemGeneratorLaplacianTotalEventIntegral` premise. The source-facing boundary
is now the function-l |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12717 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfSourceFunctional
- Total-event generator formula from the source-functional Laplacian action.
This lower-2 cycle-148 helper targets the genuine source leaf left by the
state-event route. The total-event action formula follows from the |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12744 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceFunctional
- Total-event generator formula from the standard-basis source action.
This cycle-149 dynamic-leaf bridge narrows the direct
`hemGeneratorSourceActionDef` input left by the cycle-148 source-functional
route. The remain |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12795 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqOfStdBasisSourceFormula
- Frozen EM generator Laplacian event-field identity from the source
standard-basis formula.
This cycle-144 bridge narrows the remaining pointwise event-field identity.
Instead of requiring the named frozen-generator La |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12853 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfStdBasisSourceAndEventFormula
- Total-event generator formula from standard-basis source and event fields.
This cycle-149 lower_1 bridge removes the direct
`hemGeneratorLaplacianEventFieldEqLaplacian` premise left by the cycle-149
standard-basis sou |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12891 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStateEventFormula
- Weak-FP source-action route from an EM generator state-event formula.
This cycle-143 consumer feeds the state-event narrowing into the cycle-142
trace-Laplacian law-integral route. The old
`hemGeneratorLaplacianLawIn |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:12949 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventFormula
- Weak-FP source-action route from a pointwise EM event-field formula.
This lower-1 cycle-143 consumer removes the all-state-events integral equality
as a primitive premise. It reconstructs that equality from the point |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13024 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefFormula
- Weak-FP source-action route from a pointwise EM event-field definition and
the source action definition.
This lower-2 cycle-143 consumer removes the total-event formula as a primitive
premise. It derives that total-e |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13100 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventActionDefFormula
- Weak-FP source-action route from a standard-basis EM event-field
definition and the source action definition.
This cycle-144 consumer removes the pointwise
`hemGeneratorLaplacianEventFieldEqLaplacian` premise left by |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13178 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionFormula
- Frozen EM generator Laplacian action definition from its standard-basis
event-action formula.
This lower-2 cycle-144 helper narrows the remaining
`hemGeneratorLaplacianActionDef` premise. If the paper source gives th |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13258 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventStdBasisActionFormula
- Weak-FP source-action route from standard-basis formulas for both the
frozen EM event field and action.
This lower-2 cycle-144 consumer removes `hemGeneratorLaplacianActionDef` as a
primitive premise under the standar |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13311 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDefOfLawIntegralFormula
- Frozen EM generator standard-basis action definition from the law-space
Laplacian integral.
This cycle-145 bridge narrows the remaining
`hemGeneratorLaplacianStdBasisActionDef` boundary from the cycle-144 lower_2
pack |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13394 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianStdBasisEventLawIntegralFormula
- Weak-FP source-action route from the EM law integral and standard-basis
event-field formula.
This cycle-145 consumer removes `hemGeneratorLaplacianStdBasisActionDef` as a
primitive premise under the cycle-144 standard |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13440 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfTraceField
- Frozen EM generator Laplacian event-field standard-basis definition from
the named trace field.
This lower_2 cycle-145 bridge narrows the remaining
`hemGeneratorLaplacianEventFieldStdBasisDef` premise. If the paper i |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13521 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldStdBasisOfLaplacianField
- Trace-field standard-basis formula from the Mathlib Laplacian field.
This cycle-146 lower_1 scout helper narrows the remaining
`htraceFieldStdBasis` boundary. It is enough to identify the named frozen EM
trace field |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13555 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldEqLaplacianOfPointwise
- Trace-field equality from the pointwise trace-field identity.
This cycle-156 illness-area refiner narrows the direct
`htraceFieldEqLaplacian` boundary. The field-level equality for the named
frozen EM trace field is |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13595 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseStdBasisOfEventFieldStdBasis
- Pointwise trace-field standard-basis display from the event-field display.
This lower_2 illness-area refiner narrows the remaining
`htraceFieldPointwiseStdBasis` leaf. It is enough to prove that the named
frozen EM L |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13622 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorTraceFieldPointwiseEqLaplacianOfStdBasis
- Pointwise trace-field Laplacian identity from the standard-basis display.
This lower_1 scout bridge narrows the remaining cycle-156 pointwise leaf. It
is enough to prove the paper's explicit Hessian-trace formula for |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13663 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldStdBasisDefOfPointwise
- Event-field standard-basis definition from a pointwise event-field display.
This cycle-157 illness-area refiner narrows the direct
`hemGeneratorLaplacianEventFieldStdBasisDef` boundary. The field-level
definition of |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13704 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDefOfPointwiseLaplacian
- Pointwise event-field standard-basis display from a pointwise Laplacian identity.
This lower_1 scout split keeps the remaining analytic source fact at the
paper's `Delta` notation. Once the named frozen EM Laplacian |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13739 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfPointwiseScalar
- Event-field Laplacian identity from a statewise pointwise source display.
This lower_2 cycle-157 bridge narrows the remaining
`hemGeneratorLaplacianEventFieldEqLaplacian` boundary without using the older
weak-FP sourc |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13780 |
AutoSamplingTheory.SALD.emFrozenBrownianLaplacianEventField
- Source-facing frozen Brownian generator Laplacian event field.
For the EM interpolation in `appendix.tex:984-995`, the Brownian diffusion
generator contributes the selected-test Laplacian field appearing in the
Fokker |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13807 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseEqLaplacianOfBrownianDef
- Pointwise event-field Delta identity from the Brownian event-field definition.
This lower_2 cycle-158 theorem implements the lower_1 scout route. It narrows
`hEmGeneratorLaplacianEventFieldPointwiseEqLaplacian` to th |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13821 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefOfPointwise
- Brownian event-field definition from a pointwise source display.
This cycle-159 middle bridge narrows the remaining
`hEmGeneratorLaplacianEventFieldBrownianDef` boundary to the pointwise
Brownian-generator event-field |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13851 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDefOfStdBasis
- Pointwise Brownian event-field definition from the coordinate trace display.
This lower_2 cycle-159 bridge narrows the remaining pointwise
Brownian-generator event-field boundary to the paper's coordinate Hessian-trac |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13878 |
AutoSamplingTheory.SALD.emFrozenScalarBrownianItoGeneratorEventField
- Source-facing frozen scalar Brownian Ito generator event field.
For the Brownian increment in `appendix.tex:984-995`, the scalar diffusion
coefficient is handled by the surrounding weak-FP action. This named event
fi |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13916 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDefOfFrozenScalarBrownianItoGenerator
- Brownian pointwise standard-basis display from the named scalar Ito generator.
This cycle-160 middle bridge narrows the remaining
`hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisDef` boundary to one
named sou |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13936 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoGeneratorDefOfPointwise
- Function-level frozen scalar Ito generator definition from its pointwise form.
This lower-scout bridge keeps the active boundary on the Brownian/Ito generator
source theorem. It narrows the remaining function equalit |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13971 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseDefOfCoordinateGenerator
- Pointwise frozen scalar Ito generator display from coordinate-generator data.
This lower_2 cycle-160 bridge narrows the remaining pointwise Brownian/Ito
generator boundary to the paper's two smaller stochastic-generat |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:13999 |
AutoSamplingTheory.SALD.emFrozenScalarBrownianItoOneDimTaylorGenerator
- One-dimensional scalar Brownian Ito/Taylor second-order term.
For a fixed coordinate direction `e`, this is the diagonal second derivative
term produced by the scalar Brownian second moment in the frozen interpolation |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14042 |
AutoSamplingTheory.SALD.gaussianRealZeroSecondMoment
- Centered real Gaussian second moment for the scalar Brownian coordinate.
This is the Mathlib-backed moment fact needed below the one-dimensional
Brownian/Ito Taylor boundary for `eq:general_moving_target_SALD_frozen_i |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14055 |
AutoSamplingTheory.SALD.gaussianRealZeroOneDimTaylorMomentContribution
- Centered scalar Gaussian Taylor moment contribution.
This lower_2 bridge combines the zero first moment of
`ProbabilityTheory.gaussianReal 0 v` with the compiled centered second moment.
It is the local moment-algebra |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14075 |
AutoSamplingTheory.SALD.gaussianRealLinearQuadraticTaylorSummandsIntegrable
- Gaussian integrability of the linear and quadratic scalar Taylor summands.
This removes the polynomial-summand integrability bookkeeping from the
Taylor-integral source boundary. The normalized-remainder integrabilit |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14090 |
AutoSamplingTheory.SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefs
- Taylor integral split for the frozen scalar Brownian coordinate.
This bridge narrows the source-facing
`hFrozenScalarBrownianItoTaylorMomentDecomposition` input to a direct Taylor
integral definition of the coordinate |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14138 |
AutoSamplingTheory.SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndGaussianPolynomialIntegrability
- Taylor integral split with Gaussian polynomial summand integrability.
This bridge removes the two polynomial summand integrability hypotheses from
`selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfInt |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14237 |
AutoSamplingTheory.SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder
- Taylor integral split with dominated remainder integrability.
This lower_2 bridge removes the remaining `hRemainderInt` bookkeeping field
from the Taylor moment split. The stochastic source equality and the
remainder |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14301 |
AutoSamplingTheory.SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE
- Source-integrand/a.e. equality bridge for the Brownian coordinate integral.
This middle packet narrows the remaining coordinate-generator Taylor integral
definition to two source-facing fields: a source integral defin |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14375 |
AutoSamplingTheory.SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise
- Pointwise source Taylor identity supplies the Brownian-coordinate a.e.
integrand equality.
This lower packet is the narrow pointwise-to-a.e. adapter for the source
Taylor integrand. The analytic content remains the s |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14429 |
AutoSamplingTheory.SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementAndLineDef
- Raw source Taylor integrand from selected-line increment naming.
This lower_2 bridge narrows the source-facing field
`hSourceTaylorIntegrandRawDef`. The remaining paper content is split into two
smaller source-cited |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14464 |
AutoSamplingTheory.SALD.selectedWeakTestSelectedIncrementCoordinateLineDefOfEndpointAndLineDef
- Selected increment coordinate-line identity from endpoint naming.
This cycle-190 bridge narrows the source-facing
`hSelectedIncrementCoordinateLineDef` field. The remaining paper content is
split into the selected-en |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14509 |
AutoSamplingTheory.SALD.selectedWeakTestSourceTaylorIntegrandRawDefOfSelectedIncrementEndpointAndLineDef
- Raw source Taylor integrand from selected-increment endpoint fields.
This cycle-190 bridge removes the older supplied field
`hSelectedIncrementCoordinateLineDef` from the raw source-integrand route by
deriving it from |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14554 |
AutoSamplingTheory.SALD.selectedWeakTestSourceTaylorIntegrandDefOfRawAndLineTaylorSplit
- Source Taylor integrand definition from the raw selected-line increment.
This cycle-188 bridge narrows the source-facing field
`hSourceTaylorIntegrandDef`. The source correspondence below it now has two
smaller piece |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14611 |
AutoSamplingTheory.SALD.selectedWeakTestSelectedLineTaylorSplitDefOfRawTaylorAndTermDefs
- Selected-line Taylor split from raw Taylor terms and source-term naming.
This lower_2 bridge narrows the source-facing `hSelectedLineTaylorSplitDef`
leaf. The remaining analytic Taylor content is the raw scalar expan |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14657 |
AutoSamplingTheory.SALD.selectedWeakTestSourceTaylorIntegrandPointwiseOfLineTermDefs
- Source-term split for the Brownian-coordinate pointwise Taylor integrand.
This cycle-186 bridge narrows the source-facing identity
`hSourceTaylorIntegrandPointwise`. It keeps the actual scalar Taylor
correspondence e |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14739 |
AutoSamplingTheory.SALD.selectedWeakTestSourceLinearTermDefOfScalarLineFirstCoeffDef
- Source linear term from the scalar line Taylor term and coefficient convention.
This lower_2 bridge narrows the source-facing field `hSourceLinearTermDef`.
The analytic paper content remains in two smaller source-cite |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14785 |
AutoSamplingTheory.SALD.selectedWeakTestSourceQuadraticTermDefOfScalarLineTaylorCoeffDef
- Source quadratic term from the scalar Taylor quadratic term and coefficient
convention.
This cycle-187 bridge is the quadratic analogue of
`selectedWeakTestSourceLinearTermDefOfScalarLineFirstCoeffDef`. It narrows th |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14835 |
AutoSamplingTheory.SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralRawTaylorAndTermDefs
- Brownian coordinate Taylor integral from the raw selected-line Taylor data.
This cycle-189 bridge composes the already compiled source-integral, a.e.,
source-term, and raw selected-line Taylor bridges. It narrows the |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:14887 |
AutoSamplingTheory.SALD.selectedWeakTestQuadraticCoeffDefOfSecondTaylorCoeffDef
- Quadratic-coefficient definition from the diagonal second Taylor coefficient.
This cycle-175 lower_2 bridge only unfolds the local name
`emFrozenScalarBrownianItoOneDimTaylorGenerator`. The source-facing analytic
ide |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15043 |
AutoSamplingTheory.SALD.selectedWeakTestVarianceOneOfNormalizedBrownianVarianceDef
- Real-valued variance-one field from the normalized Brownian variance definition.
This bridge discharges the downstream `hVarianceOne` shape once the source
correspondence has defined the normalized scalar Brownian coo |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15073 |
AutoSamplingTheory.SALD.selectedWeakTestNormalizedVarianceDefOfGaussianRealUnitLaw
- Normalized Brownian coordinate law supplies the `NNReal` variance definition.
This cycle-178 middle bridge narrows the source-facing `hNormalizedVarianceDef`
field. Once the source correspondence identifies the norma |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15097 |
AutoSamplingTheory.SALD.selectedWeakTestNormalizedCoordinateLawOfStdGaussianVectorLaw
- Normalized scalar coordinate law from a vector standard Gaussian law.
This cycle-178 lower_2 bridge narrows the scalar coordinate-law source field:
once the paper correspondence supplies the normalized vector incremen |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15138 |
AutoSamplingTheory.SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfStdGaussianVectorLaw
- Source coordinate-generator integral from the normalized Brownian law.
This cycle-184 bridge narrows `hBrownianCoordinateGeneratorSourceIntegralDef`.
The source work below it is the stochastic definition of the scalar |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15187 |
AutoSamplingTheory.SALD.selectedWeakTestRemainderGeneratorLimitDefOfStdGaussianVectorLaw
- Remainder generator integral from the normalized Brownian coordinate law.
This cycle-185 lower_2 bridge narrows `hRemainderGeneratorLimitDef`: once the
source correspondence defines the normalized remainder contributi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15262 |
AutoSamplingTheory.SALD.selectedWeakTestRemainderMeasOfStdGaussianVectorLaw
- Remainder measurability transported from the normalized Brownian coordinate law.
This cycle-194 lower_2 bridge discharges the downstream `hRemainderMeas`
shape once the source correspondence has supplied measurability |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15336 |
AutoSamplingTheory.SALD.selectedWeakTestRemainderBoundOfStdGaussianVectorLaw
- Remainder domination transported from the normalized Brownian coordinate law.
This cycle-194 bridge narrows the downstream `hRemainderBound` leaf to the
same source-side normalized scalar-coordinate law used for the r |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15399 |
AutoSamplingTheory.SALD.selectedWeakTestRemainderBoundIntegrableOfStdGaussianVectorLaw
- Remainder-bound integrability transported from the normalized Brownian coordinate law.
This cycle-195 bridge narrows the downstream `hRemainderBoundInt` leaf to the
same source-side normalized scalar-coordinate law us |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15460 |
AutoSamplingTheory.SALD.selectedWeakTestBrownianCoordinateGeneratorNormalizedLawDefOfScalarPushforward
- Normalized-law coordinate generator from a scalar pushforward law.
This cycle-184 lower_2 bridge narrows the source-facing
`hBrownianCoordinateGeneratorNormalizedLawDef`: once the frozen-interpolation
source correspon |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15526 |
AutoSamplingTheory.SALD.selectedWeakTestBrownianCoordinateGeneratorSourceIntegralDefOfScalarPushforwardAndStdGaussianVectorLaw
- Source coordinate-generator integral from scalar pushforward and Gaussian coordinate law.
This cycle-191 bridge discharges the supplied
`hBrownianCoordinateGeneratorNormalizedLawDef` field from the source-facing
`hBro |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15597 |
AutoSamplingTheory.SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfScalarPushforwardRawTaylorAndTermDefs
- Brownian coordinate Taylor integral from scalar pushforward and raw Taylor fields.
This cycle-192 bridge discharges the supplied
`hBrownianCoordinateGeneratorSourceIntegralDef` field from the
`hBrownianCoordinateGener |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15678 |
AutoSamplingTheory.SALD.selectedWeakTestRemainderGeneratorNormalizedLawDefOfScalarPushforward
- Normalized-law remainder generator from a scalar pushforward law.
This cycle-185 lower_2 bridge narrows the source-facing
`hRemainderGeneratorNormalizedLawDef`: once the frozen-interpolation source
correspondence supp |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15834 |
AutoSamplingTheory.SALD.selectedWeakTestRemainderGeneratorLimitDefOfScalarPushforwardAndStdGaussianVectorLaw
- Remainder generator integral from scalar pushforward and Gaussian coordinate law.
This cycle-190 lower_2 bridge discharges the supplied
`hRemainderGeneratorNormalizedLawDef` field from the source-facing
`hRemainderGen |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15904 |
AutoSamplingTheory.SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsDominatedRemainderAndRemainderLimitScalarPushforward
- Taylor moment split from an explicit Taylor integral and scalar-pushforward remainder law.
This cycle-193 lower_2 bridge removes only the primitive
`hRemainderGeneratorLimitDef` supplied hypothesis from the dominated- |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:15983 |
AutoSamplingTheory.SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfScalarPushforwardRawTaylorAndDominatedRemainder
- Taylor moment decomposition from scalar pushforward, raw Taylor fields, and dominated remainder.
This cycle-193 bridge removes the primitive
`hBrownianCoordinateGeneratorTaylorIntegralDef` and
`hRemainderGeneratorLimi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16105 |
AutoSamplingTheory.SALD.selectedWeakTestQuadraticVariationNormalizationOfCoeffDefAndVarianceOne
- Quadratic-variation normalization from source coefficient and variance fields.
This cycle-174 lower_2 bridge is only the algebraic assembly below
`hFrozenScalarBrownianItoQuadraticVariationNormalization`: once the sou |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16302 |
AutoSamplingTheory.SALD.selectedWeakTestQuadraticVariationNormalizationOfSecondTaylorCoeffAndNormalizedVarianceDef
- Quadratic-variation normalization from second Taylor and normalized variance fields.
This cycle-176 lower_2 bridge composes the cycle-175 coefficient bridge with
the normalized Brownian variance bridge. It removes th |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16335 |
AutoSamplingTheory.SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT
- Dominated-convergence handoff for the normalized scalar Taylor remainder.
This lower_2 cycle-162 theorem is the integral-limit block below
`hFrozenScalarBrownianItoNormalizedTaylorRemainderVanishes`. It isolates the
|
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16383 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondOrderTaylorRemainderPointwiseAE
- Source-shaped scalar Taylor pointwise limit for the normalized remainder.
Cycle 163 narrows the remaining `hPoint` input of
`gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfDCT`. For each
fixed scalar Brown |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16436 |
AutoSamplingTheory.SALD.gaussianRealNormalizedTaylorRemainderIntegralTendstoZeroOfSelectedTestLineEq
- DCT integral limit from identifying the paper remainder with the source line.
This lower_1 scout bridge keeps the remaining analytic content at the source
boundary. Once the paper's `normalizedRemainder` is eventuall |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16478 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainder
- Source-shaped selected-test scalar normalized Taylor remainder.
This is the concrete normalized remainder used by the paper's scalar
Brownian/Ito Taylor line after writing the Brownian coordinate increment as
`r = h * |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16525 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderSourceEq
- The source-shaped selected-test remainder supplies the `hSourceEq` input.
Cycle 163 lower_2 removes the source-equality placeholder for the concrete
selected scalar line: once `normalizedRemainder` is the source-shape |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16541 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero
- DCT limit for the concrete selected-test normalized remainder.
This lower_2 bridge discharges the `hSourceEq` placeholder from the lower_1
source-identification theorem for the actual selected scalar Taylor remainder. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16567 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderEventuallyAEStronglyMeasurable
- Measurability of the concrete selected-test normalized Taylor remainder.
Cycle 164 discharges the `hMeas` input for the source-shaped scalar remainder
below `gaussianRealSelectedTestLineSecondOrderNormalizedRemainderI |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16612 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
- Split the selected scalar second-order Taylor quotient bound.
Cycle 165 lower_2 narrows the deterministic `hTaylorQuotientBound` input for
the concrete normalized-remainder domination theorem. It is enough to supply
|
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16676 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound
- Quadratic domination of the concrete selected-test normalized remainder.
This cycle-165 bridge narrows the remaining DCT `hBound` input to the
deterministic scalar Taylor quotient estimate for the selected one-dimensi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16736 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder
- First-order selected-test quotient bound from a quadratic remainder bound.
Cycle 166 narrows the `hFirst` input of
`gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff`.
The source-fac |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16782 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor
- Nonnegative-side first-order selected-line Taylor remainder from Mathlib Taylor.
This is the source-facing positive half of the remaining cycle-166 `hFirst`
boundary. Mathlib's `taylor_mean_remainder_bound` is interv |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16830 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor
- Signed first-order selected-line Taylor remainder from interval Taylor data.
This lower_2 bridge combines the compiled nonnegative interval Taylor lemma
with the reflected line `q ↦ sourceTest (x + q • (-e))` for nega |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16867 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderTaylorReflectSetUniv
- Reflection compatibility for the first-order selected-line Taylor polynomial on `Set.univ`.
The negative-side interval Taylor argument uses the line
`q ↦ sourceTest (x + q • (-e))` at `-r`. For the first-order Taylor |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16946 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt
- First-order interval Taylor compatibility from differentiability at the base point.
For the selected scalar line, the order-one Taylor polynomial on `Icc 0 r`
agrees at the endpoint `r` with the `Set.univ` Taylor poly |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:16983 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect
- Signed first-order selected-line Taylor remainder without a reflected Taylor premise.
Cycle 167 removes the explicit `hNegTaylorReflect` input from the cycle-166
signed interval theorem. The only remaining negative-s |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17023 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff
- Signed first-order selected-line Taylor remainder from interval data and base differentiability.
This narrows the cycle-167 signed interval boundary by discharging the
interval-to-`Set.univ` Taylor-compatibility hypot |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17074 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn
- Global selected-test regularity supplies global regularity of each scalar line.
Cycle 168 narrows the `hLine` input exposed by the cycle-167 global-line
Taylor bridge. If the selected source test is globally `C^2` on |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17121 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff
- Signed first-order selected-line Taylor remainder from global line regularity.
This lower_2 bridge removes the signed interval `ContDiffOn` inputs and the
two base differentiability inputs from the lower_1 `BaseDiff` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17143 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn
- Signed first-order selected-line Taylor remainder from ambient source-test regularity.
This cycle-168 bridge supplies the cycle-167 `hLine` input from the paper-facing
global selected-test `C^2` hypothesis. It delibe |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17195 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds
- Signed first-order selected-line Taylor remainder from global line second-derivative bounds.
This lower_2 bridge narrows the signed interval second-derivative domination
inputs left by the source-`ContDiffOn` bridge. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17231 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondDerivEqDirectionalFDeriv
- Scalar-line second derivative as an ambient directional Hessian.
This cycle-169 chain-rule bridge is the local Mathlib step below the
selected-test bounded-Hessian source hypothesis. It rewrites the second
ordinary d |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17352 |
AutoSamplingTheory.SALD.selectedWeakTestScalarLineSecondCoeffDefOfTaylorCoeffWithin
- Scalar-line second derivative from Mathlib's Taylor coefficient convention.
This cycle-179 bridge narrows the source-facing scalar Brownian coefficient
boundary one step further. If the paper correspondence supplies |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17391 |
AutoSamplingTheory.SALD.selectedWeakTestSecondTaylorCoeffDefOfScalarLineSecondCoeffDef
- Ambient diagonal second Taylor coefficient from a scalar-line coefficient.
This cycle-177 bridge is the local Mathlib part below the source-facing
`hSecondTaylorCoeffDef` field. Once the source correspondence identif |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17427 |
AutoSamplingTheory.SALD.selectedWeakTestQuadraticVariationNormalizationOfScalarLineSecondCoeffAndNormalizedVarianceDef
- Quadratic-variation normalization from scalar-line second coefficient data.
This cycle-177 lower_2 bridge composes the scalar-line coefficient bridge with
the cycle-176 normalization bridge. It removes the older prim |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17472 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondBoundsOfDirectionalSecondBound
- Selected/reflected global line-second bounds from one ambient directional bound.
The reflected line has direction `-e`; because the ambient second Frechet
derivative is bilinear, applying it to `(-e, -e)` agrees with |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17518 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNorm
- Ambient second-derivative operator norm bound supplies the selected diagonal bound.
This is the local bounded-Hessian leaf below the cycle-169 directional-Hessian
interface: a uniform operator-norm bound on the ambien |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17556 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestSecondFDerivOpNormOfFDerivFDerivOpNorm
- Hessian-as-derivative-of-gradient bound supplies the iterated-Frechet bound.
This is a source-facing reformulation of the selected-test bounded-Hessian
leaf: if the paper supplies the uniform operator-norm bound on
`f |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17582 |
AutoSamplingTheory.SALD.selectedWeakTestHessianOpNormOfSourceHessianField
- A source-backed Hessian field supplies the selected-test Hessian operator bound.
This is the narrow cycle-173 source-contract bridge: once the faithful source
correspondence provides a Hessian representative for `sour |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17605 |
AutoSamplingTheory.SALD.gaussianRealStdOrthonormalBasisUnit
- Mathlib standard orthonormal basis directions are unit directions.
This discharges the Brownian coordinate side condition `heUnit` for the
standard-basis scalar Ito branch. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17624 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestDirectionalSecondBoundOfSecondFDerivOpNormStdOrthonormalBasis
- Operator-norm Hessian bound specialized to Brownian standard-basis directions.
Cycle 170 removes the separate unit-direction hypothesis for
`e = (stdOrthonormalBasis Real E) i`; the remaining analytic boundary is the
|
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17636 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndDirectionalSecondBound
- First-order selected-line Taylor remainder from an ambient directional Hessian bound.
Cycle 169 narrows the remaining `hLineSecond`/`hNegLineSecond` boundary left by
`gaussianRealSelectedTestLineFirstOrderQuadraticRem |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17661 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestStdOrthonormalFirstOrderQuadraticRemainderBoundOfSourceHessianField
- Standard-basis selected-line Taylor remainder from source Hessian fields.
Cycle 175 narrows the selected-line Taylor-domination leaf for the Brownian
coordinate direction. Once the source correspondence supplies a He |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17695 |
AutoSamplingTheory.SALD.gaussianRealSelectedTestLineSecondOrderQuadraticBoundIntegrable
- Gaussian integrability of the quadratic domination bound.
For the concrete selected-test normalized remainder, the source Taylor
domination leaf is expected to use a quadratic Gaussian bound
`fun z => C * z ^ 2`. Thi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17744 |
AutoSamplingTheory.SALD.selectedWeakTestNormalizedRemainderBoundIntOfQuadraticBound
- Normalized remainder-bound integrability from a concrete quadratic bound.
This cycle-196 bridge narrows the normalized-law integrability leaf itself:
once the source correspondence identifies the scalar dominating rem |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17782 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorOfGaussianMomentRemainder
- One-dimensional scalar Brownian Ito generator from a Taylor moment split.
This cycle-162 bridge narrows the remaining one-dimensional Brownian/Ito
generator boundary to three source-facing scalar inputs: a Taylor mome |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17821 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDefOfOneDimTaylor
- Per-coordinate frozen scalar Ito generator from a one-dimensional Taylor term.
This cycle-161 bridge narrows the supplied boundary
`hFrozenScalarBrownianItoCoordinateGeneratorDef` to the smaller source-cited
one-dimen |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17888 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceFieldSourceAndEventFormula
- Total-event generator formula from trace-field source/event identities.
This cycle-149 lower_2 bridge narrows the two standard-basis inputs left by
the lower_1 total-event route. The total-event formula no longer nee |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:17928 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorTraceActionDefOfTraceLawIntegral
- Trace-action definition from the law-space trace integral.
This cycle-150 helper narrows the remaining
`hemGeneratorTraceActionDef` premise in the total-event trace-field route. The
source-cited law-space trace integ |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18005 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceLawIntegralLaplacianField
- Frozen EM generator Laplacian state integral from trace law and trace
Laplacian source fields.
This lower_1 scout bridge feeds the existing trace-action and trace-field
Laplacian narrowings into the cycle-154 state-in |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18042 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianStateIntegralOfTraceStateIntegralLaplacianField
- Frozen EM generator Laplacian state integral from the trace state integral.
This lower_2 continuation narrows the `hemGeneratorTraceLawIntegral` input left
by the lower_1 state-integral bridge. The law-space trace in |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18108 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLawIntegralSourceAndEventFormula
- Total-event generator formula from a trace law-integral source leaf.
This cycle-150 bridge feeds the trace-action law-integral narrowing into the
cycle-149 lower_2 total-event theorem. It removes only the direct
`hem |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18170 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceStateIntegralSourceAndEventFormula
- Total-event generator formula from a trace state-integral source leaf.
This lower_1 cycle-150 continuation narrows the remaining
`hemGeneratorTraceLawIntegral` premise exposed by the trace-law total-event
route. The |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18224 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndEventFormula
- Total-event generator formula from the selected-test Laplacian state integral.
This lower_2 cycle-150 continuation narrows the trace-state total-event route
one step further. The trace-field measurability and trace-s |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18286 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventLawIntegralFormula
- Weak-FP source-action route from the EM law integral and trace-field event
identification.
This cycle-145 lower_2 consumer removes the direct
`hemGeneratorLaplacianEventFieldStdBasisDef` premise from the law-integral
|
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18367 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventFormula
- Weak-FP source-action route from a trace-event total-state formula.
This cycle-146 consumer removes the direct
`hemGeneratorLaplacianLawIntegral` premise from the cycle-145 trace-event
route. The source-facing EM bou |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18443 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianTraceEventTotalEventTraceLaplacianFormula
- Weak-FP source-action route from the trace-event total-state formula and
a Mathlib-Laplacian trace-field definition.
This cycle-146 lower_1 continuation removes `htraceFieldStdBasis` as a
primitive premise from the cu |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18546 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqTraceFieldOfLaplacianFields
- Trace-event equality from pointwise Laplacian identities.
This cycle-146 lower_2 helper narrows the source equality between the named
frozen EM Laplacian event field and the trace field. It is enough to identify
both |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18616 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqSourceFieldOfStdBasisFields
- Event-field source equality from common standard-basis source fields.
This cycle-152 direct-leaf helper narrows
`hemGeneratorLaplacianEventFieldEqSourceField` itself. Instead of identifying
the frozen EM Laplacian ev |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18650 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceFieldStdBasisDefOfLaplacianField
- Weak-FP source-field standard-basis formula from the Mathlib Laplacian field.
This cycle-152 lower_1 scout helper narrows one of the two remaining
standard-basis leaves exposed by
`generalMovingTargetDiscreteEmGenerat |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18693 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceFieldEqLaplacianOfPointwise
- Weak-FP source-field equality from the pointwise source identity.
This cycle-152 lower_2 helper narrows the remaining
`hweakFpSourceFieldEqLaplacian` leaf from the weak-FP source display. The
field-level equality is |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18733 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianEventFieldEqLaplacianOfWeakFpSourceField
- Event-field Laplacian identity from the named weak-FP source field.
This lower_2 cycle-151 direct-leaf theorem narrows
`hemGeneratorLaplacianEventFieldEqLaplacian` itself. It is enough to identify
the frozen EM Lapla |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18759 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianTotalEventIntegralOfTraceLaplacianStateIntegralSourceAndPointwiseEventFormula
- Total-event generator formula from pointwise event-field and trace-field
Laplacian identities.
This cycle-151 worker packet removes
`hemGeneratorLaplacianEventFieldEqTraceField` from the current cycle-150
trace-Laplac |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18796 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventTotalEventTraceLaplacianFormula
- Weak-FP source-action route from pointwise event-field and trace-field
Laplacian identities.
This cycle-146 lower_2 continuation removes
`hemGeneratorLaplacianEventFieldEqTraceField` as a primitive premise from the
la |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18860 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventActionDefTraceLaplacianFormula
- Weak-FP source-action route from pointwise event/trace Laplacian identities
and the source action definition.
This cycle-147 continuation removes `hemGeneratorLaplacianTotalEventIntegral`
as a primitive premise from t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:18932 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmGeneratorLaplacianActionDefOfStdBasisActionPointwiseEventFormula
- Frozen EM generator action definition from the source standard-basis
action and the pointwise event-field Laplacian identity.
This lower-1 cycle-147 proof-scout helper narrows the remaining
`hemGeneratorLaplacianActio |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19007 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStdBasisActionTraceLaplacianFormula
- Weak-FP source-action route from the source standard-basis action,
pointwise event-field Laplacian identity, and trace-field Laplacian identity.
This lower-1 cycle-147 scout continuation removes
`hemGeneratorLaplacian |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19061 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventLawIntegralTraceLaplacianFormula
- Weak-FP source-action route from the EM law integral, pointwise event-field
Laplacian identity, and trace-field Laplacian identity.
This lower-2 cycle-147 continuation removes the
`hemGeneratorLaplacianStdBasisActionD |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19138 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpSourceActionDefOfEmGeneratorTraceLaplacianPointwiseEventStateEventTraceLaplacianFormula
- Weak-FP source-action route from the state-event source formula on the
current pointwise event/trace-Laplacian branch.
This cycle-148 middle continuation removes `hemGeneratorLaplacianLawIntegral`
as a primitive premi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19215 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakFpLaplacianDefOfSourceLaplacianField
- Weak-FP source-Laplacian definition from a named source field.
This cycle-139 helper targets the non-circular weak-FP side left by
`generalMovingTargetDiscreteTestLaplacianPointwiseOfTestStdBasisSourceFormula`.
Instea |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19295 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTestLaplacianPointwiseOfWeakFpSourceLaplacianField
- Pointwise test-Laplacian normalization from a non-circular weak-FP source
field.
This cycle-139 downstream bridge feeds the weak-FP source-field split into the
cycle-138 pointwise test-Laplacian route. The test-calcu |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19342 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTestLaplacianOperatorNormalizationOfStdBasisSourceFormula
- Operator-level normalization from the two standard-basis source formulas.
This cycle-135 scout helper composes the weak-FP and test-calculus
standard-basis leaves. It removes the older source-pullback hypotheses from |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19391 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfStdBasisSourceFormula
- Second-Green diffusion-source handoff from standard-basis source formulas.
This lower_2 continuation pushes the cycle-135 standard-basis source formulas
into the downstream second-Green consumer. The older source-pul |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19450 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDiffusionSourceOfSecondGreenBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZeroOfSourceLaplacianPullback
- Second-Green diffusion-source handoff with operator normalization narrowed
to shared source-Laplacian pullback definitions.
This dynamic-leaf worker packet replaces the direct
`htestLaplacianOperator : testRegular → t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19630 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldContinuousOnBox
- Continuity of the concrete product flux `hatRhoS * barB` on a box.
This is the first concrete sub-boundary below the cycle-107 box theorem: once
the density representative for `hatRhoS` and the conditional drift `barB |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19793 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt
- Pointwise Frechet derivative of the concrete product flux
`hatRhoS * barB`.
This is the local Mathlib product-rule component below the cycle-108 box
handoff. It converts separate derivatives of the density representa |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19812 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion
- Off-countable derivative of the concrete product flux from separate
density and drift derivative exception sets.
The remaining cycle-108 box-trace boundary asks for Frechet differentiability
of `x ↦ hatRhoDensity x • |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19839 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox
- Boundary-flux integral for the concrete product flux `hatRhoS * barB`.
Compared with
`generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox`, this
specializes `weightedField` to the Euclidean product
` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19883 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv
- Boundary-flux integral for `hatRhoS * barB` with the product derivative
instantiated from separate density and drift derivatives.
This narrows the cycle-108 remaining box-trace boundary by discharging the
generic off- |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:19986 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundary
- Drift source action with the zero-boundary-flux input narrowed to a
boundary trace-product condition.
This version keeps the cycle-101 product-rule and divergence-theorem inputs,
but replaces the raw `hzeroBoundary : |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20129 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero
- Drift source action with the trace-product input narrowed to zero
admissible-test trace on the boundary.
This lower packet discharges the supplied `htraceProductZero` premise of
`generalMovingTargetDiscreteWeakConditi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20214 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings
- Drift weak-action pairing from conditional-drift component pairings.
This cycle-95 lower theorem narrows the first half of
`ASTIS.SALD.cycle94.remaining_barB_divergence_boundary`. Once `barB` is the
paper component f |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20302 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBComponentPairings
- Drift source action with the `barB` weak action reduced to component
conditional pairings.
This composes the cycle-95 component-pairing theorem with the cycle-94
`barB` drift-source handoff. It removes the direct sup |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20383 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion
- One-component `condDistrib` generator-pairing handoff.
This cycle-96 lower theorem targets the conditional-expectation half of
`appendix.tex:1368-1377`. If a named component field such as
`condC_{k,s}` or `condScore_ |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20458 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfIntegralAction
- Canonical component pairing from the named-law `condDistrib` integral.
This cycle-97 lower theorem is the compiled use of
`AutoSamplingTheory.condDistribIntegralNamedLawIntegral` requested after the
middle disintegrat |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20516 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff
- Mapped-law weak derivative with the drift source action factored through
the conditional drift `barB`.
This cycle-94 refinement composes the cycle-92 direct law-derivative route with
`generalMovingTargetDiscreteWeakCo |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20622 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff
- Source-sign handoff with the drift source action factored through the
conditional drift `barB`.
This cycle-94 lower refinement applies the same `barB` weak-action boundary to
the normalized weak Fokker--Planck source- |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20701 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfReadinessAndGeneratorPiecesHandoff
- Cycle-82 middle bridge from endpoint/conditional weak-FP readiness to the
generator-piece source-sign handoff.
Cycle 81 packaged the endpoint laws, named `hatRhoS` marginal, kernel
orientation, and `barB` regularity i |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20786 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointConditionalWeakFpSourceSignsHandoff
- Cycle-82 lower bridge from endpoint/conditional readiness data all the way
to the weak-test source signs.
This composes the cycle-81 endpoint/conditional `WeakFpPrereq` readiness
wrapper with the cycle-82 readiness-to |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20846 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffScalar
- KL-derivative handoff after weak conditional Fokker--Planck substitution.
This is the cycle-73 proof-producing wrapper for the first handoff in
`appendix.tex:1358-1387`. Once the analytic backend has supplied the
dif |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:20994 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSigns
- KL-derivative handoff from normalized weak-FP source signs.
This lower cycle-78 wrapper isolates the final substitution after the weak
conditional Fokker--Planck backend has already produced the paper-normalized
sourc |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21021 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfSourceSignsWithLogAction
- KL-derivative handoff together with the log-ratio weak-FP action.
This lower cycle-83 companion keeps the two source-cited steps adjacent: the
weak conditional Fokker--Planck source signs evaluated at the admissible
l |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21051 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeMassConservationDropScalar
- Scalar mass-conservation drop for the discrete general KL derivative.
In `appendix.tex:1358-1366`, differentiating
`KL(hat rho_s || tilde pi_s)` first gives the log-ratio action plus the
scalar mass derivative `int pa |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21086 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndSourceSignsWithLogAction
- KL weak-FP handoff from a raw differentiated KL display.
This cycle-87 companion removes the older supplied post-mass-drop `hkl`
display from the weak-FP-to-`dK` equality bookkeeping. It starts instead from
the raw d |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21104 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlMassTermZeroOfLawConstantTestDerivative
- General moving-target mapped-law constant-test mass conservation.
For `appendix.tex:1358-1366`, the source drops
`int partial_s hat rho_s dx` after naming `hat rho_s = Law(hat X_s)`. This
local theorem proves the con |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21142 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfLawConstantTestMassAndSourceSignsWithLogAction
- General moving-target raw KL handoff with mapped-law mass conservation.
This cycle-93 refinement removes the primitive
`hmass : massTerm = 0` input from
`generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlAndS |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21173 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlLogRatioLlrDef
- Mathlib log-ratio convention for the discrete general KL boundary.
The paper writes the weak test as `log(hat rho_s / tilde pi_s)`. In the
Lean-facing backend this is represented by Mathlib's log-likelihood ratio
`ll |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21214 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl
- Log-ratio measurability and integrability from finite KL.
This cycle-87 lower theorem discharges the log-ratio measurability and
integrability side hypotheses in `appendix.tex:1358-1366`, provided the local
KL backend |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21231 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteKlLogRatioAdmissibilityClosure
- Narrow weak-test closure boundary for the discrete general KL log-ratio.
After cycle 87, finite KL supplies the absolute-continuity, measurability, and
integrability side of the Mathlib `llr hatRho tildePi` representa |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21255 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlLogRatioAdmissibleOfFiniteKlClosure
- Finite-KL handoff into the narrowed log-ratio admissibility boundary.
This cycle-88 lower theorem removes the old broad supplied
`hlog : Admissible logRatioTest` when the log-ratio test is the Mathlib
`llr hatRho tild |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21295 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfFiniteKlLlrLawConstantTestMassWithLogAction
- Raw KL weak-FP handoff for the Mathlib `llr` test and mapped-law mass.
This cycle-93 lower refinement composes the two accepted KL/log-ratio
backfills for `appendix.tex:1358-1366`: finite KL plus the named cycle-88
cl |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21324 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlr
- Source-cited raw KL differentiability package at the finite-KL `llr` test.
This is the cycle-99 boundary for `appendix.tex:1358-1366`. It is narrower
than a primitive scalar `hklRaw`: finite KL fixes the Mathlib
log- |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21378 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteRawKlDerivativeAtFiniteKlLlrHklRaw
- Extract the raw KL display from the source-cited finite-KL `llr` package.
The only proof done here is the Mathlib finite-KL handoff to absolute
continuity, measurability, and integrability of `llr`; the analytic
param |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21421 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfRawKlBoundaryAtFiniteKlLlrWithLogAction
- KL weak-FP handoff from the narrowed raw-KL finite-KL `llr` boundary.
This cycle-99 theorem removes the primitive `hklRaw` and `hmassDeriv` inputs
from the exact `llr` route by consuming
`GeneralMovingTargetDiscreteRa |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21453 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlr
- Source-cited no-mass raw KL package at the finite-KL `llr` test.
This lower refinement uses the already formalized mapped-law constant-test
calculus instead of keeping the mass derivative inside the raw-KL boundary.
T |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21495 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlrHkl
- Extract the no-mass raw KL display from the finite-KL `llr` package. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21527 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction
- KL weak-FP handoff from the no-mass raw-KL finite-KL `llr` boundary.
This theorem removes the `massTermDerivative` field from the exact finite-KL
`llr` route. The derivative of the mapped-law constant weak-test pairi |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21555 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr
- Pure source-cited no-mass KL differentiability package at the finite-KL
`llr` test.
Cycle 105 removes the sample-space and mapped-law mass data from the no-mass
KL boundary. At this point the remaining theorem is pur |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21615 |
AutoSamplingTheory.SALD.generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl
- Extract the no-mass KL display from the pure finite-KL `llr` package. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21643 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfPureNoMassRawKlBoundaryAtFiniteKlLlrWithLogAction
- KL weak-FP handoff from the pure no-mass raw-KL finite-KL `llr` boundary.
Unlike the cycle-99 no-mass handoff, this theorem does not route through a
zero mass derivative or a sample-space law. It consumes the pure no |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21669 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated
- Dominated target-time derivative for the general discrete KL boundary.
In `appendix.tex:1358-1366`, the target-time term is
`int (hat rho_s / tilde pi_s) * partial_s tilde pi_s dx`. This theorem is the
Mathlib parame |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21712 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr
- Transfer the target-time theorem from a chosen weight to the paper's
source density-ratio representative.
The dominated theorem above intentionally keeps the fixed weight abstract.
This lower bridge isolates the remai |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21778 |
AutoSamplingTheory.SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated
- Feed the dominated target-time theorem into the pure finite-KL `llr`
KL-differentiability package fields.
This narrows the remaining cycle-105 boundary without adding sample-space law
data. Finite KL still supplies t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21834 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfAdmissibleSourceSigns
- KL-derivative handoff composed through the admissible weak-FP source signs.
This lower wrapper makes the cycle-73 dependency on the cycle-72 admissible
weak-test source-sign theorem explicit. It first normalizes the |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21905 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffOfGeneratorPieces
- KL-derivative handoff composed through the generator-level weak-FP pieces.
This cycle-78 wrapper connects the cycle-77 generator/source-sign refinement
directly to the cycle-73 KL derivative substitution. The analyti |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:21944 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoff
- Cycle-83 handoff from endpoint/conditional weak-FP source signs to the
KL-derivative display.
This wrapper composes the cycle-82 endpoint/conditional source-sign handoff
with the normalized weak-FP-to-KL substitution. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22010 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointConditionalKlDerivativeWeakFpHandoffWithLogAction
- Endpoint/conditional KL handoff retaining the log-ratio weak-FP action.
Cycle 84 keeps the active EM backend on the same source block
`appendix.tex:1358-1387`, but asks lower work to consume the accepted
endpoint-read |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22167 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalFpSourceSignsHandoff
- Source-sign handoff for the discrete general EM Fokker--Planck equation.
In `appendix.tex:1379-1387`, the source writes the conditional-drift
Fokker--Planck equation with drift sign `-div(hat rho_s*bar b_{k,s})` and
d |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22326 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalFpSigmaLaplacianSplitHandoff
- Lower handoff algebra for the sigma-weighted conditional Fokker--Planck
split in the discrete general VA-SALD proof.
This is the proof-producing part of `appendix.tex:1380-1387`: once the weak
conditional-drift Fokker |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22344 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteResidualCoefficientRewriteScalar
- Scalar coefficient rewrite for the doubled residual term in the discrete
general VA-SALD Gronwall bridge.
This formalizes the paper algebra turning
`dot{s}(t) * (2*sigma_eta(t)^(-2)*dot t(s(t))^2*alpha^(-1))`
into `2* |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22375 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGammaCoefficientRewriteScalar
- Scalar coefficient rearrangement for the frozen `Gamma` term after the
`s`-to-`t` time change in the discrete general VA-SALD proof. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22393 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDeltaCoefficientRewriteScalar
- Scalar coefficient rearrangement for the frozen `Delta` term after the
`s`-to-`t` time change in the discrete general VA-SALD proof. |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22401 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteFrozenResidualAlgebraVector
- Module-level algebra for the frozen/residual decomposition.
In appendix lines 1469-1478, after the analytic identifications
`delta = dotT • c + score - frozen`, `tildeV = dotT • v`, and
`m = v - c` have been supplied, |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22415 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteYoungFisherShareScalar
- Source coefficient identity for one `sigma_eta^2/8` Young share.
In appendix lines 1493-1511 the two cross terms each consume one quarter of the
available Fisher dissipation `(sigma_eta^2/2)*FI`. This lemma closes on |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22436 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTwoYoungFisherBudgetScalar
- Scalar budget after the two `sigma_eta^2/8` Young splits.
Instantiating `fisherDissipation` with `(sigma_eta^2/2)*FI`, the two Young
cross-term bounds leave exactly one half of that dissipation, namely the source
coef |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22448 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteResidualYoungCoefficientScalar
- Scalar residual coefficient produced by Young with
`epsilon = sigma_eta^2/4`.
The source's residual cross term has `b^2=dot t(s)^2*||m||^2`. Once the
analytic Young inequality has supplied the coefficient `1/(2*epsil |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22467 |
AutoSamplingTheory.SALD.generalMovingTargetDiscretePostYoungDerivativeBoundScalar
- Scalar post-Young handoff for the discrete general VA-SALD derivative.
This packages appendix lines 1469-1517 after the analytic KL derivative,
frozen/residual decomposition, residual Young inequality, and frozen-delt |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22485 |
AutoSamplingTheory.SALD.generalMovingTargetDiscretePostLsiDerivativeBoundScalar
- Scalar LSI handoff for the discrete general VA-SALD derivative.
Once `eq:LSI-KL-FI` supplies `C_LSI*K <= (1/2)*FI`, this converts the
post-Young term `-(sigma_eta^2/4)*FI` into the source
`-(sigma_eta^2/2)*C_LSI*K` da |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22511 |
AutoSamplingTheory.SALD.generalMovingTargetDiscretePostDvDerivativeBoundScalar
- Scalar post-DV handoff for the discrete general VA-SALD derivative.
After DV supplies
`||m||^2 <= alphaInv*K + E_alpha`, this rewrites the residual energy term
into the exact `s`-time damping and residual coefficients |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22548 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteTimeChangedDerivativeBoundScalar
- Scalar time-change handoff for the discrete general VA-SALD derivative.
This is the real/order part of appendix lines 1573-1583: multiply the
`s`-time inequality by `dot{s}(t)`, use
`dot t(s(t)) = dot{s}(t)^(-1)`, and |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22587 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDerivativeDvTimeChangedScalar
- Source-shaped scalar handoff for the discrete general VA-SALD derivative.
This composes the compiled post-Young, LSI, DV, and time-change scalar steps
for appendix lines 1469-1583. All analytic inputs remain explicit |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22635 |
AutoSamplingTheory.SALD.generalMovingTargetDiscretePointwiseGronwallInputOfPostDvTimeChanged
- Pointwise Gronwall-input wrapper for the discrete general VA-SALD time
change.
The scalar theorem above handles one fixed time after the EM/KL derivative,
LSI, residual DV, and constant inverse-schedule inputs have su |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22700 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallNamedCoefficientInput
- Named-coefficient handoff for the final discrete general VA-SALD
Gronwall step.
Cycle 58 supplies the pointwise derivative inequality with the source
coefficient expression. This wrapper lets the final side-condition |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22738 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallEndpointRewriteScalar
- Endpoint rewrite for the final discrete general VA-SALD Gronwall bound.
After `lem:gronwall` is applied to the stitched function `K`, this closes the
pure endpoint-rewrite step from `K(T)` and `K(0)` to the theorem en |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22771 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallDisplayBridgeScalar
- Final scalar/display bridge for the discrete general VA-SALD theorem.
This is the cycle-68 lower proof-producing wrapper for the selected
`sald.unified_discrete_general.cycle68_discrete_general_bridge` packet. It
sta |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22795 |
AutoSamplingTheory.SALD.piVelocityNormMeanZeroH1UpperScalar
- Scalar upper-bound core for the PI norm-equivalence step.
In `appendix.tex:104-112`, the analytic obligations identify `l2Sq` with the
mean-zero variance term and `dotSq` with the gradient norm squared. Once PI
has s |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22837 |
AutoSamplingTheory.SALD.piVelocityNormBoundedFunctionalScalar
- Scalar bounded-functional core for the first PI velocity-norm lower slice.
The source obtains `T_mu(psi) <= ||psi||_L2 ||g||_L2` by Cauchy--Schwarz and
then uses PI to replace `||psi||_L2` by `C_PI^{-1/2}||psi||_{dot |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22855 |
AutoSamplingTheory.SALD.lsiKlFiCoefficientAuditScalar
- Scalar coefficient audit for the source LSI-to-KL/FI display.
After the analytic obligations identify the LSI Dirichlet term with
`(1/4) * FI(rho||pi)`, this lemma preserves the paper's constant
`1/(2*C_LSI)`. It doe |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22870 |
AutoSamplingTheory.SALD.lsiKlFiDensityTestBridgeScalar
- Scalar bridge for applying the normalized LSI test `phi=sqrt(rho/pi)`.
This packages the source handoff in `main_body.tex:208-215` after the analytic
backend has supplied the LSI test normalization, entropy-to-KL iden |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22889 |
AutoSamplingTheory.SALD.lsiKlFiHalfFisherScalar
- Convert the displayed KL/FI comparison into the half-Fisher form used later.
The forward-KL proof consumes the LSI output as `C_LSI*K <= (1/2)*FI`
before substituting it into the derivative inequality. This lemma pro |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22910 |
AutoSamplingTheory.SALD.lsiKlFiDensityTestHalfFisherScalar
- Normalized density-test bridge directly in the half-Fisher derivative form.
After the analytic density-test backend supplies normalization, the LSI test
inequality, entropy-to-KL, and Dirichlet-to-FI identities, this |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22930 |
AutoSamplingTheory.SALD.discreteForwardKlDerivativeSplitOfRawIbpsScalar
- Scalar raw-derivative split for discrete forward-KL.
This is the cycle-89 lower core for the first blocker found by the
`thm:forward-KL-discrete` pressure test. It replaces the older opaque input
`dK = -FI + frozenCr |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22952 |
AutoSamplingTheory.SALD.discreteForwardKlMassTermZeroOfTotalMassDerivative
- Mass term is zero when it is the derivative of a locally constant total mass.
For `eq:KL-derivative-0-discrete`, the paper uses
`int partial_s hat rho_s dx = 0`. This lemma isolates the local calculus part:
once the |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22971 |
AutoSamplingTheory.SALD.discreteForwardKlDerivativeSplitOfMassDerivativeScalar
- Discrete derivative split with mass conservation derived from total mass.
This removes the primitive `hmass : massTerm = 0` input from the cycle-89 raw
IBP scalar route. The remaining analytic boundary is the source- |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:22989 |
AutoSamplingTheory.SALD.discreteForwardKlLawConstantTestTotalMassOne
- Constant weak-test mass of a mapped probability law.
For `eq:KL-derivative-0-discrete`, this is the law-normalization part of
`int partial_s hat rho_s dx = 0`: if `hat rho_s` is represented as
`Measure.map (hatX s) P` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23013 |
AutoSamplingTheory.SALD.discreteForwardKlLawConstantTestHasDerivAtZero
- The mapped-law constant weak test has zero derivative.
This closes the elementary derivative side of the mass-conservation sentence in
`eq:KL-derivative-0-discrete`: after rewriting the law integral to the
sample-spac |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23034 |
AutoSamplingTheory.SALD.discreteForwardKlMassTermZeroOfLawConstantTestDerivative
- Mass term is zero for the concrete mapped-law constant weak test.
This lower-cycle refinement removes the abstract `totalMass`/local-normalization
inputs from the cycle-90 middle handoff. It still requires the source |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23055 |
AutoSamplingTheory.SALD.discreteForwardKlDerivativeSplitOfLawConstantTestMassScalar
- Discrete derivative split with mass conservation from the mapped law.
Compared with `discreteForwardKlDerivativeSplitOfMassDerivativeScalar`, this
specializes the total-mass function to the source law
`hat rho_s = Mea |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23080 |
AutoSamplingTheory.SALD.discreteForwardKlPostLsiDerivativeBoundScalar
- Scalar derivative handoff for discrete forward-KL before the DV step.
This is the lower-cycle proof-producing core for `appendix.tex:388-491`.
It starts after the EM conditional Fokker--Planck and integration-by-parts |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23113 |
AutoSamplingTheory.SALD.discreteForwardKlPostLsiDerivativeBoundOfKlFiScalar
- Discrete forward-KL derivative handoff using the source KL/FI comparison.
This packages the `eq:LSI-KL-FI` scalar half-Fisher bridge into
`discreteForwardKlPostLsiDerivativeBoundScalar`. The density-test proof of
`KL |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23150 |
AutoSamplingTheory.SALD.discreteForwardKlPostLsiDerivativeBoundOfRawIbpsScalar
- Discrete forward-KL derivative handoff from raw KL and named IBP pieces.
This composes the cycle-89 raw derivative split with the existing LSI scalar
handoff. The theorem proves only Real/order bookkeeping once the a |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23177 |
AutoSamplingTheory.SALD.discreteForwardKlPostLsiDerivativeBoundOfMassDerivativeScalar
- Discrete post-LSI handoff with mass conservation derived from total mass.
This is the cycle-90 middle route for `eq:KL-derivative-0-discrete`: it feeds
the total-mass derivative lemma into the cycle-89 raw IBP route, |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23210 |
AutoSamplingTheory.SALD.discreteForwardKlPostLsiDerivativeBoundOfLawConstantTestMassScalar
- Discrete post-LSI handoff with mapped-law constant-test mass conservation.
This is the cycle-90 lower route for `eq:KL-derivative-0-discrete`: the raw
derivative split no longer needs a standalone `hmass` hypothesis o |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23246 |
AutoSamplingTheory.SALD.forwardKlFirstTermFisherSubstitutionScalar
- Scalar substitution for the first derivative term in continuous forward-KL.
In appendix lines 168-185, the analytic obligations first produce the KL
derivative identity and then identify the SALD Fokker--Planck/integr |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23287 |
AutoSamplingTheory.SALD.forwardKlMassConservationDropScalar
- Scalar mass-conservation drop in the continuous forward-KL derivative.
In `appendix.tex:168-174`, differentiating the KL integrand first produces the
extra scalar term corresponding to `int partial_s rho_s dx`. The s |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23301 |
AutoSamplingTheory.SALD.forwardKlMassConservationFirstTermFisherScalar
- First continuous forward-KL derivative scalar handoff after mass conservation.
This composes the source mass-conservation drop with the already isolated
`-FI` first-term substitution from `appendix.tex:176-185`. It d |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23316 |
AutoSamplingTheory.SALD.forwardKlTargetTransportYoungBoundScalar
- Scalar Young bound for the target-side transport term in forward-KL.
In appendix lines 199-208, Cauchy--Schwarz first gives the target-side term
bounded by `sqrt(FI) * ||tilde v_s||`. This lemma formalizes only the f |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23335 |
AutoSamplingTheory.SALD.forwardKlPostYoungDerivativeBoundScalar
- Scalar post-Young derivative bound for continuous forward-KL.
This is the theorem-independent arithmetic after the analytic source steps in
`appendix.tex:168-208` have supplied the KL derivative display, the first-ter |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23358 |
AutoSamplingTheory.SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar
- Post-Young derivative bound using the target-side Cauchy input directly.
This composes `forwardKlTargetTransportYoungBoundScalar` with the existing
post-Young derivative bookkeeping. It still starts after the source |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23374 |
AutoSamplingTheory.SALD.forwardKlLsiDerivativeBoundScalar
- Scalar LSI substitution for the continuous forward-KL derivative bound.
After `appendix.tex:199-208` gives the post-Young bound, the source applies
LSI in `appendix.tex:210-217`. This lemma records only the real-orde |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23394 |
AutoSamplingTheory.SALD.forwardKlLsiDerivativeBoundOfKlFiScalar
- LSI substitution using the source KL/FI comparison directly.
The paper cites `eq:LSI-KL-FI` in the form
`KL <= FI/(2*C_LSI)` and then uses it as
`C_LSI*KL <= (1/2)*FI` in the derivative estimate. This lemma closes on |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23409 |
AutoSamplingTheory.SALD.forwardKlTimeChangedDerivativeBoundScalar
- Scalar inverse-schedule handoff for the continuous forward-KL derivative.
After `appendix.tex:210-217` gives the `s`-time LSI derivative inequality,
`appendix.tex:218-228` changes variables from `s` to `t`. This lemm |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23425 |
AutoSamplingTheory.SALD.forwardKlInverseScheduleDerivativeScalar
- Scalar inverse-derivative handoff for the continuous forward-KL schedule.
The analytic inverse-function theorem is still part of
`sald.forward_kl.schedule_time_change`. This lemma only proves the Real
algebra used af |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23457 |
AutoSamplingTheory.SALD.forwardKlTimeChangeSquareCoefficientRewriteOfProductScalar
- Source-shaped square-coefficient rewrite for the forward-KL time change.
This version starts from the inverse-derivative product identity rather than a
pre-rewritten `dotT = dotS⁻¹`. It is pure scalar algebra for
`ap |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23481 |
AutoSamplingTheory.SALD.forwardKlVelocitySquareScalingScalar
- Scalar velocity-square scaling used by the slowed target.
In `appendix.tex:191-197`, the paper defines
`\tilde v_s = dot{t}(s) v_{t(s)}`. Once the analytic L2 backend has reduced
that identity to scalar norm-square i |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23496 |
AutoSamplingTheory.SALD.forwardKlTimeChangedDerivativeBoundOfProductScalar
- Time-changed forward-KL derivative bound from source-shaped schedule data.
This composes `forwardKlTimeChangedDerivativeBoundScalar` with the scalar
inverse-derivative handoff from `dotS * dotT = 1`. It still assumes |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23511 |
AutoSamplingTheory.SALD.forwardKlPreDvDerivativeBoundScalar
- Scalar pipeline for the continuous forward-KL pre-DV derivative bound.
This theorem composes the already-formalized scalar steps for
`appendix.tex:168-228`: first-term Fisher substitution, target-side
Cauchy/Young, LS |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23533 |
AutoSamplingTheory.SALD.forwardKlPreDvDerivativeBoundOfProductScalar
- Pre-DV derivative pipeline with source-shaped inverse-schedule input.
This is the same scalar pipeline as `forwardKlPreDvDerivativeBoundScalar`, but
the schedule side starts from the product identity
`dot{s}(t) * dot{ |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23567 |
AutoSamplingTheory.SALD.forwardKlPreDvDerivativeBoundOfVelocityScalingScalar
- Pre-DV derivative pipeline with the slowed-velocity square scaling exposed.
This theorem matches the source `appendix.tex:191-228` bookkeeping most
closely among the scalar lemmas: it derives the nonnegativity and squ |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23601 |
AutoSamplingTheory.SALD.forwardKlPreDvDerivativeBoundOfKlFiVelocityScalingScalar
- Pre-DV derivative pipeline using the source KL/FI comparison.
This is the lower-cycle theorem-specific bridge for `appendix.tex:168-228`.
It threads the supplied KL derivative display, first-term Fisher identity,
targ |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23640 |
AutoSamplingTheory.SALD.forwardKlPreDvDerivativeBoundOfRawKlFiVelocityScalingScalar
- Pre-DV derivative pipeline from the raw KL derivative split.
This is the cycle-60 lower scalar wrapper for `appendix.tex:168-228`. It
starts from the source derivative display before the mass-conservation term is
dro |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23674 |
AutoSamplingTheory.SALD.forwardKlPointwisePreDvDerivativeBoundOfRawKlFiVelocityScaling
- Pointwise continuous forward-KL pre-DV derivative handoff.
This is the cycle-65 lower wrapper for `appendix.tex:168-228`. The scalar
lemma above handles one fixed time after the analytic KL derivative,
mass-conservat |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23709 |
AutoSamplingTheory.SALD.forwardKlDerivativeDvGronwallCoefficientOfKlFiVelocityScalingScalar
- Source-shaped handoff from the KL derivative backend and DV to Gronwall.
This is the cycle-50 lower scalar bridge for `appendix.tex:168-241`. It
starts from the explicit analytic inputs owned by `sald.forward_kl.kl_d |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23750 |
AutoSamplingTheory.SALD.generalMovingTargetKlDerivativeResidualSplitScalar
- Scalar residual split for the continuous general VA-SALD KL derivative.
This is the cycle-57 lower proof-producing core for `appendix.tex:765-835`.
After the analytic backend supplies the raw KL derivative split, the |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23795 |
AutoSamplingTheory.SALD.generalMovingTargetKlDerivativeScaledResidualDisplayScalar
- Scaled residual display for the continuous general VA-SALD KL derivative.
This is a cycle-62 lower scalar core for `appendix.tex:813-835`. Once the
analytic backend has supplied the target-transport contribution with |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23817 |
AutoSamplingTheory.SALD.generalMovingTargetPostYoungDerivativeBoundScalar
- Scalar post-Young derivative bound for continuous general VA-SALD.
In `appendix.tex:835-864`, after the Fokker--Planck and target-transport
identities have combined the `c_t` and `v_t` terms into the residual
`m_t=v_t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23841 |
AutoSamplingTheory.SALD.generalMovingTargetLsiDerivativeBoundScalar
- Scalar LSI handoff for continuous general VA-SALD.
The source uses `eq:LSI-KL-FI` after the residual Young step. Once the LSI
backend has supplied `C_LSI*K <= (1/2)*FI`, this lemma converts
`-(sigma_t^2/4)*FI` into t |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23863 |
AutoSamplingTheory.SALD.generalMovingTargetTimeChangedDerivativeBoundScalar
- Scalar time-change handoff for the continuous general VA-SALD derivative.
This is the `appendix.tex:865-884` real/order step after the analytic schedule
backend has supplied `dK/dt=dot{s}(t)*dK/ds` and
`dot t(s(t))=do |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23900 |
AutoSamplingTheory.SALD.generalMovingTargetPreDvDerivativeBoundScalar
- Source-shaped scalar pre-DV derivative pipeline for continuous general VA-SALD.
This composes the compiled scalar pieces for `appendix.tex:835-884`: residual
Young bookkeeping, LSI half-Fisher substitution, and invers |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23940 |
AutoSamplingTheory.SALD.generalMovingTargetKlDerivativePreDvBoundOfSplitScalar
- Source-shaped pre-DV derivative handoff from the raw KL split.
This composes the cycle-57 residual split for `appendix.tex:765-835` with the
existing scalar pipeline for `appendix.tex:835-884`. All analytic inputs re |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:23981 |
AutoSamplingTheory.SALD.generalMovingTargetPostDvGronwallCoefficientScalar
- Post-DV scalar handoff for the continuous general VA-SALD coefficient.
This is the `appendix.tex:885-907` real/order step after the analytic DV
backend has supplied
`alpha * energy <= K + log E_pi exp(alpha * energy)` |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24019 |
AutoSamplingTheory.SALD.generalMovingTargetPostDvGronwallCoefficientOfSigmaScheduleScalar
- Source-shaped post-DV handoff for `thm:general-moving-target-SALD`.
This specializes `generalMovingTargetPostDvGronwallCoefficientScalar` to the
sigma-weighted damping and residual prefactor in `appendix.tex:897-907`: |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24044 |
AutoSamplingTheory.SALD.generalMovingTargetDerivativeDvGronwallCoefficientScalar
- Source-shaped scalar handoff from the general moving-target derivative and
residual DV inputs to the Gronwall differential inequality.
This composes the compiled pre-DV derivative pipeline for
`appendix.tex:765-884` w |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24072 |
AutoSamplingTheory.SALD.generalMovingTargetResidualToGronwallBridgeScalar
- Cycle-67 scalar bridge from the residual KL split to the Gronwall input.
This is the proof-producing lower wrapper for `appendix.tex:765-907` inside
the selected `sald.general_moving_target.cycle67_residual_to_gronwal |
theorem | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24120 |
AutoSamplingTheory.SALD.LsiKlFiBridgeContract
- Lean-facing bridge for the source step `LSI + phi=sqrt(rho/pi)`.
This is contract data for `eq:LSI-KL-FI`, not a proof. It keeps the exact
paper route from LSI to the KL/FI comparison visible before theorem-specific
|
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24162 |
AutoSamplingTheory.SALD.LsiKlFiDensityTestContract
- Narrow proof-obligation interface for the LSI test function
`phi=sqrt(rho/pi)`.
This record keeps the density, finite-quantity, and smooth-test-function
requirements explicit instead of adding them silently to theorem |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24182 |
AutoSamplingTheory.SALD.DvFiniteLogMgfContract
- Middle-layer audit for applying the cited DV variational formula.
The source lemma is cited from Boucheron et al.; this record does not prove it.
It names the local interfaces required before the SALD theorem blocks c |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24207 |
AutoSamplingTheory.SALD.PiVelocityNormDependencyContract
- Middle-layer audit for the PI vocabulary as used by the appendix
velocity-norm lemma.
The PI definition itself is contract data. The subsequent Sobolev, weak-PDE,
and Riesz-representation route remains an analytic ba |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24226 |
AutoSamplingTheory.SALD.FirstAppendixVocabularyPacket
- Upper-role packet for the first appendix/vocabulary re-audit.
This is workflow data, not mathematical proof content. It records the chosen
faithful-paper objective, lower packet, and reviewer checklist for returning |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24249 |
AutoSamplingTheory.SALD.FirstAppendixSourceIndexAuditContract
- Source-index audit for the first appendix/vocabulary layer.
This is upper-role workflow data. It keeps `SALD_original.jsonl`, the first
proof-DAG labels, and the Lean-facing contracts synchronized without changing
an |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24265 |
AutoSamplingTheory.SALD.FirstAppendixMiddleAuditContract
- Middle-role source-to-Lean audit for the first appendix/vocabulary layer.
This workflow contract refines the upper source-index packet into a lower-ready
map: every focused source step is classified as a Lean contract |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24284 |
AutoSamplingTheory.SALD.ForwardKlUpperPacket
- Upper-role packet for returning to the continuous forward-KL proof route.
This is workflow data for `thm:forward-KL`. It records the chosen faithful
objective and review constraints for the moving-target, LSI, DV, an |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24304 |
AutoSamplingTheory.SALD.DiscreteForwardKlUpperPacket
- Upper-role packet for returning to the discrete forward-KL proof route.
This is workflow data for `thm:forward-KL-discrete`. It records one
faithful-paper objective and lower packet while keeping the theorem statement |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24320 |
AutoSamplingTheory.SALD.GeneralVaSaldUpperPacket
- Upper-role packet for guided/general VA-SALD proof routing.
This is workflow data for the guided residual proposition, continuous general
VA-SALD theorem, unified specialization, and discrete general theorem. It
reco |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24337 |
AutoSamplingTheory.SALD.GeneralVaSaldGuidedPathMiddleContract
- Middle-role source-to-Lean packet for the guided/general VA-SALD path.
This workflow contract keeps the cycle focus synchronized across the guided
residual proposition, continuous general theorem, unified specializati |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24354 |
AutoSamplingTheory.SALD.ForwardKlStatementContract
- Faithful data record for the source theorem `thm:forward-KL`.
This pins the statement and appendix proof shape without claiming any of the
measure-theoretic or calculus steps as formalized. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24374 |
AutoSamplingTheory.SALD.AlphaComplexityContract
- Contract for the paper's alpha-complexity vocabulary.
The statement is definitional data. Finiteness, measurability, and monotonicity
facts needed by theorem proofs remain obligations of the relevant theorem
blocks. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24398 |
AutoSamplingTheory.SALD.ForwardKlDerivativeCandidateContract
- Lean-facing interface for the derivative part of `thm:forward-KL`.
This records the exact analytic route used in the appendix before DV and
Gronwall enter the proof. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24414 |
AutoSamplingTheory.SALD.ForwardKlDerivativeSideConditionContract
- Explicit side-condition interface for the derivative block of `thm:forward-KL`.
The source proof uses these conditions in the KL derivative and time-change
steps, but the theorem statement does not state them as stand |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24437 |
AutoSamplingTheory.SALD.ForwardKlDvEnergyCandidateContract
- Lean-facing interface for the DV velocity-energy step in `thm:forward-KL`. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24454 |
AutoSamplingTheory.SALD.ForwardKlDvFiniteLogMgfWitnessContract
- Theorem-specific finite-log-mgf witness needed before applying DV in
`thm:forward-KL`.
The source proof applies `lem:dv_variation` directly at
`appendix.tex:230-241`. This record isolates the missing Lean interface: |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24474 |
AutoSamplingTheory.SALD.ForwardKlDvAlphaMonotonicityContract
- Narrow alpha-monotonicity interface for the forward-KL DV test function.
The source theorem assumes finite alpha0-complexity and then applies DV for
every `0 < alpha <= alpha0`. This record isolates the needed expone |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24499 |
AutoSamplingTheory.SALD.ForwardKlGronwallInstantiationContract
- Lean-facing interface for the final Gronwall instantiation in `thm:forward-KL`. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24516 |
AutoSamplingTheory.SALD.ForwardKlMovingTargetDependencyContract
- Source-facing audit of the moving-target assumptions used by `thm:forward-KL`.
This contract keeps the theorem statement fixed while identifying which
assumptions are stated in the main body, which interfaces are impo |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24535 |
AutoSamplingTheory.SALD.ForwardKlDependencyChainAuditContract
- Line-by-line coefficient audit for the LSI/DV/Gronwall chain in
`thm:forward-KL`.
This is narrower than `ForwardKlMovingTargetDependencyContract`: it records how
the source proof transforms the derivative inequality i |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24559 |
AutoSamplingTheory.SALD.ForwardKlGronwallSideConditionContract
- Endpoint and exponent side conditions for the final `thm:forward-KL`
Gronwall display.
This is a narrow ledger for the last source step: identifying the endpoints
of `K(t)` with the theorem statement and justifying th |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24588 |
AutoSamplingTheory.SALD.ForwardKlEndpointScheduleContract
- Narrow endpoint-schedule ledger for `thm:forward-KL`.
This is the cycle-14 lower slice. It isolates the source's inverse-schedule
endpoint rewrites from the derivative, DV, and Gronwall analytic backends. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24608 |
AutoSamplingTheory.SALD.ForwardKlMiddleSourceToLeanContract
- Middle-role source-to-Lean map for the continuous `thm:forward-KL` proof.
This workflow contract classifies each source step in `appendix.tex:168-252`
as an existing Lean-facing contract, source-cited result, or named |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24630 |
AutoSamplingTheory.SALD.DiscreteForwardKlStatementContract
- Faithful data record for the source theorem `thm:forward-KL-discrete`.
The record pins the main-body theorem, its EM implementation, and the exact
discrete error terms. It does not assert that the analytic estimates |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24652 |
AutoSamplingTheory.SALD.DiscreteForwardKlEmInterpolationSideConditionContract
- Side-condition interface for the EM interpolation used in `thm:forward-KL-discrete`.
The source proof uses three facts at different points: endpoint law matching for
the interpolation, a conditional-drift Fokker--Plan |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24678 |
AutoSamplingTheory.SALD.DiscreteForwardKlEmConditionalFpLowerContract
- Lower-ready line ledger for the conditional Fokker--Planck slice.
The cycle-15 middle packet selects `appendix.tex:347-385` as the first lower
slice. This record keeps that slice narrower than the whole discrete theo |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24698 |
AutoSamplingTheory.SALD.DiscreteForwardKlConditionalDriftDensityContract
- Narrow lower interface for defining the frozen conditional drift.
Before the interpolation Fokker--Planck equation can be stated in Lean, the
conditional expectation in `bar b_{k,s}` has to be represented as a measura |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24721 |
AutoSamplingTheory.SALD.FrozenDeltaCrossLipSaldContract
- Lean-facing interface for the omitted SALD frozen-defect lemma.
The source states that this lemma follows from the later general frozen-defect
lemma by taking c identically zero and sigma_eta(t)=sqrt(2). Until that
s |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24743 |
AutoSamplingTheory.SALD.DiscreteForwardKlDerivativeCandidateContract
- Lean-facing interface for the discrete forward-KL derivative block. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24757 |
AutoSamplingTheory.SALD.DiscreteForwardKlDvFiniteLogMgfWitnessContract
- Theorem-specific finite-log-mgf witness for the discrete forward-KL DV step.
The discrete proof applies `lem:dv_variation` with `nu=hat rho_s` and
`mu=tilde pi_s`. This record isolates the extra EM-interpolation inte |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24781 |
AutoSamplingTheory.SALD.DiscreteForwardKlGronwallInstantiationContract
- Lean-facing interface for the final Gronwall step in `thm:forward-KL-discrete`. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24802 |
AutoSamplingTheory.SALD.DiscreteForwardKlAccumulatedErrorBridgeContract
- Final bridge from the appendix discrete Gronwall display to the theorem
statement.
The appendix ends with a general-schedule bound. The main body states the
linear-slowdown theorem with accumulated `barGamma` and `bar |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24825 |
AutoSamplingTheory.SALD.DiscreteForwardKlEmDefectAccumulationMiddleContract
- Middle-role source-to-Lean packet for the discrete forward-KL route.
This contract does not add a new theorem statement. It records how the cycle
focus spans the EM interpolation, one-step frozen defect, DV velocity |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24849 |
AutoSamplingTheory.SALD.DiscreteForwardKlCoefficientChainAuditContract
- Coefficient audit for the discrete forward-KL proof.
This keeps the one-step `Gamma`/`Delta` coefficients synchronized from the
frozen defect lemma through the derivative inequality, the `s` to `t` time
change, the Gr |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24871 |
AutoSamplingTheory.SALD.GuidedResidualIdentityContract
- Lean-facing interface for the guided-path residual proposition.
This is algebraic contract data for the appendix computation. The derivative
of the normalizer, integration by parts, and mean-zero statement stay as
ob |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24898 |
AutoSamplingTheory.SALD.GeneralMovingTargetStatementContract
- Faithful data record for `thm:general-moving-target-SALD`.
The source theorem is the continuous general VA-SALD bound. It differs from
`thm:forward-KL` by using an implementable velocity `c_t`, diffusion scale
`sigma |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24919 |
AutoSamplingTheory.SALD.GeneralMovingTargetDerivativeCandidateContract
- Lean-facing interface for the derivative block of the continuous general VA-SALD theorem. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24938 |
AutoSamplingTheory.SALD.GeneralMovingTargetDvEnergyCandidateContract
- Lean-facing interface for the DV residual-energy step in the continuous general VA-SALD theorem. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24957 |
AutoSamplingTheory.SALD.GeneralMovingTargetDvFiniteLogMgfWitnessContract
- Theorem-specific finite-log-mgf witness for the residual DV step in
`thm:general-moving-target-SALD`.
The source proof applies `lem:dv_variation` directly with
`Z=alpha*||m_t||^2`. This record isolates the common-spa |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:24976 |
AutoSamplingTheory.SALD.GeneralMovingTargetDvPositiveAlphaScalingContract
- Lower-level backend for the positive-alpha scaling step in the residual DV
bound for `thm:general-moving-target-SALD`.
After the cited DV formula is instantiated with
`Z=alpha*||m_t||^2`, the appendix divides by `alph |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25003 |
AutoSamplingTheory.SALD.GeneralMovingTargetGronwallInstantiationContract
- Lean-facing interface for the Gronwall step in the continuous general VA-SALD theorem. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25019 |
AutoSamplingTheory.SALD.GeneralMovingTargetGronwallSideConditionContract
- Endpoint, exponent, and pure-contraction side conditions for the final
`thm:general-moving-target-SALD` Gronwall display.
The appendix applies Gronwall and then states that the displayed theorem bound
follows. This c |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25041 |
AutoSamplingTheory.SALD.UnifiedForwardKlSpecializationContract
- Source-to-general-theorem bridge for `thm:unified-forward-KL`.
The paper proves the unified VA-SALD theorem by one specialization line:
set `c_t <- u_t` in the general moving-target theorem. This contract expands
onl |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25065 |
AutoSamplingTheory.SALD.UnifiedForwardKlTransportBridgeMiddleContract
- Middle-role packet for the cycle-16 unified transport bridge.
This contract narrows `thm:unified-forward-KL` to the paper's transport
algebra before any lower proof search: combine the centered guided residual
identit |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25091 |
AutoSamplingTheory.SALD.UnifiedForwardKlTransportBridgeLowerContract
- Lower interface for the cycle-16 unified transport bridge.
This record isolates the only algebra selected for lower work: the signed
cancellation between the guided residual identity and the correction-field
divergenc |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25113 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteStatementContract
- Faithful data record for `thm:general-moving-target-SALD-discrete`.
The source theorem is the discrete-time general VA-SALD bound. It reuses the
continuous general theorem hypotheses and the general frozen-delta lemm |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25136 |
AutoSamplingTheory.SALD.GeneralFrozenDeltaCrossLipContract
- Lean-facing interface for the general VA-SALD frozen-delta lemma. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25155 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteDerivativeCandidateContract
- Lean-facing interface for the derivative block of the discrete general VA-SALD theorem. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25170 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteDerivativeSideConditionContract
- Side-condition ledger for the discrete general VA-SALD derivative block.
This keeps the source's interval-wise EM law, conditional drift,
frozen/residual algebra, Young coefficient bookkeeping, and final time-change
i |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25195 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteConditionalDriftContract
- Regular conditional-drift interface for the discrete general VA-SALD
Euler--Maruyama interpolation.
This isolates the source line defining
`bar b_{k,s}(x)` from the later weak Fokker--Planck identity. It records the
|
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25221 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteConditionalLawMeasurabilityContract
- Conditional-law and regularity interface for the named frozen drift
`bar b_{k,s}` in the discrete general VA-SALD EM interpolation.
This is a source-facing ledger, not a construction of disintegration. It
separates t |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25247 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteEndpointConditionalCompatibilityContract
- Endpoint-to-conditional compatibility interface for the named EM law.
This sits between endpoint/common-space `Measure.map` bookkeeping and the
regular conditional kernel required for `bar b_{k,s}`. It records that t |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25274 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteWeakConditionalFpSourceSignContract
- Weak-test source-sign interface for the conditional Fokker--Planck line.
This records the analytic statement invoked at `appendix.tex:1379-1387`: after
the regular conditional drift `bar b_{k,s}` has been constructed, |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25298 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteKlDerivativeWeakFpHandoffContract
- Handoff from weak conditional Fokker--Planck to the discrete KL derivative.
This interface starts at the differentiated KL display
`eq:general_KL_derivative_0_discrete` and records the single analytic bridge
needed be |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25324 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteDvFiniteLogMgfWitnessContract
- Theorem-specific finite-log-mgf witness for the discrete general
VA-SALD residual DV step.
The discrete proof applies `lem:dv_variation` under the EM interpolation law
with `nu=hat rho_s`, `mu=tilde pi_s`, and `Z=alph |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25350 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteGronwallInstantiationContract
- Lean-facing interface for the final Gronwall step in the discrete general VA-SALD theorem. |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25370 |
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteGronwallSideConditionContract
- Side-condition ledger for the final discrete general VA-SALD Gronwall step.
The appendix derives the `t`-time differential inequality on stitched
Euler--Maruyama intervals and then says that applying Gronwall finishes |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25392 |
AutoSamplingTheory.SALD.MainSkeletonAnalyticInterfaceLedger
- Upper-level ledger for the SALD main proof-skeleton sprint.
This is assignment and source-to-Lean route data, not a theorem. It records
which slow analytic interfaces are allowed to remain source-cited or
obligation- |
structure | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25416 |
AutoSamplingTheory.SALD.saldGronwallCandidateContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25430 |
AutoSamplingTheory.SALD.saldGronwallEndpointCalculusContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25453 |
AutoSamplingTheory.SALD.saldGronwallExponentRewriteContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25500 |
AutoSamplingTheory.SALD.saldKLContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25526 |
AutoSamplingTheory.SALD.saldFIContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25533 |
AutoSamplingTheory.SALD.saldLSIContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25540 |
AutoSamplingTheory.SALD.saldLsiKlFiBridgeContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25547 |
AutoSamplingTheory.SALD.saldLsiKlFiDensityTestContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25568 |
AutoSamplingTheory.SALD.saldDvFiniteLogMgfContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25608 |
AutoSamplingTheory.SALD.saldPiVelocityNormDependencyContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25628 |
AutoSamplingTheory.SALD.cycle9FirstAppendixVocabularyPacket
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25651 |
AutoSamplingTheory.SALD.saldAlphaComplexityContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25683 |
AutoSamplingTheory.SALD.continuousForwardKlStatementContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25693 |
AutoSamplingTheory.SALD.forwardKlDerivativeCandidateContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25725 |
AutoSamplingTheory.SALD.forwardKlDerivativeSideConditionContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25757 |
AutoSamplingTheory.SALD.forwardKlDvEnergyCandidateContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25778 |
AutoSamplingTheory.SALD.forwardKlDvFiniteLogMgfWitnessContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25798 |
AutoSamplingTheory.SALD.forwardKlDvAlphaMonotonicityContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25834 |
AutoSamplingTheory.SALD.forwardKlGronwallInstantiationContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25858 |
AutoSamplingTheory.SALD.forwardKlMovingTargetDependencyContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25882 |
AutoSamplingTheory.SALD.forwardKlDependencyChainAuditContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25916 |
AutoSamplingTheory.SALD.cycle10ForwardKlUpperPacket
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:25973 |
AutoSamplingTheory.SALD.forwardKlGronwallSideConditionContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26010 |
AutoSamplingTheory.SALD.forwardKlEndpointScheduleContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26047 |
AutoSamplingTheory.SALD.cycle14ForwardKlUpperPacket
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26072 |
AutoSamplingTheory.SALD.cycle14ForwardKlMiddleContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26109 |
AutoSamplingTheory.SALD.cycle11DiscreteForwardKlUpperPacket
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26169 |
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlUpperPacket
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26206 |
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlMiddleContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26243 |
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlEmConditionalFpLowerContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26308 |
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26347 |
AutoSamplingTheory.SALD.cycle19DiscreteForwardKlUpperPacket
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26375 |
AutoSamplingTheory.SALD.cycle19DiscreteForwardKlMiddleContract
- Cycle-19 middle packet for the discrete forward-KL accumulated-error bridge.
This translates the upper-selected accumulated-error target into a lower-ready
source-to-Lean map. It keeps the final scalar bridge separat |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26422 |
AutoSamplingTheory.SALD.cycle23DiscreteForwardKlUpperPacket
- Cycle-23 upper packet for the discrete forward-KL proof spine.
This returns to `thm:forward-KL-discrete` after the continuous forward-KL
coefficient work. It keeps the full source route visible for middle, but
choose |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26484 |
AutoSamplingTheory.SALD.cycle23DiscreteForwardKlMiddleContract
- Cycle-23 middle packet for the discrete forward-KL coefficient chain.
This translates the upper-selected coefficient audit into a lower-ready
source-to-Lean map. It keeps the first lower slice on appendix lines 454-5 |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26532 |
AutoSamplingTheory.SALD.cycle27DiscreteForwardKlUpperPacket
- Cycle-27 upper packet for the discrete forward-KL accumulated-error bridge.
This returns to `thm:forward-KL-discrete` after the coefficient-chain audit and
selects the next faithful lower slice inside the final Gronwa |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26601 |
AutoSamplingTheory.SALD.cycle27DiscreteForwardKlMiddleContract
- Cycle-27 middle packet for the discrete forward-KL accumulated collection.
This translates the upper-selected accumulated-error slice into a lower-ready
source-to-Lean map. It keeps the first lower target on endpoint |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26649 |
AutoSamplingTheory.SALD.discreteSaldEulerMaruyamaContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26710 |
AutoSamplingTheory.SALD.discreteForwardKlStatementContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26717 |
AutoSamplingTheory.SALD.discreteForwardKlEmInterpolationSideConditionContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26768 |
AutoSamplingTheory.SALD.frozenDeltaCrossLipSaldContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26789 |
AutoSamplingTheory.SALD.discreteForwardKlDerivativeCandidateContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26815 |
AutoSamplingTheory.SALD.discreteForwardKlDvFiniteLogMgfWitnessContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26854 |
AutoSamplingTheory.SALD.discreteForwardKlGronwallInstantiationContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26891 |
AutoSamplingTheory.SALD.discreteForwardKlAccumulatedErrorBridgeContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26919 |
AutoSamplingTheory.SALD.cycle11DiscreteForwardKlMiddleContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:26973 |
AutoSamplingTheory.SALD.discreteForwardKlCoefficientChainAuditContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27032 |
AutoSamplingTheory.SALD.guidedResidualIdentityContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27093 |
AutoSamplingTheory.SALD.generalMovingTargetStatementContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27120 |
AutoSamplingTheory.SALD.generalMovingTargetDerivativeCandidateContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27156 |
AutoSamplingTheory.SALD.generalMovingTargetDvEnergyCandidateContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27199 |
AutoSamplingTheory.SALD.generalMovingTargetDvFiniteLogMgfWitnessContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27218 |
AutoSamplingTheory.SALD.generalMovingTargetDvPositiveAlphaScalingContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27255 |
AutoSamplingTheory.SALD.generalMovingTargetGronwallInstantiationContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27281 |
AutoSamplingTheory.SALD.generalMovingTargetGronwallSideConditionContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27308 |
AutoSamplingTheory.SALD.unifiedForwardKlSpecializationContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27349 |
AutoSamplingTheory.SALD.cycle12GeneralVaSaldUpperPacket
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27390 |
AutoSamplingTheory.SALD.cycle12GeneralVaSaldMiddleContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27426 |
AutoSamplingTheory.SALD.cycle16GeneralVaSaldUpperPacket
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27492 |
AutoSamplingTheory.SALD.cycle20GeneralVaSaldUpperPacket
- Cycle-20 upper packet returning to the guided/general VA-SALD path.
The selected lower target is the final Gronwall/display bridge for
`thm:general-moving-target-SALD-discrete`. This packet is workflow data only:
it |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27536 |
AutoSamplingTheory.SALD.cycle20GeneralVaSaldMiddleContract
- Cycle-20 middle packet for the discrete general VA-SALD Gronwall bridge.
This translates the upper-selected target into a lower-ready source-to-Lean map
for `sald.general_moving_target_discrete.gronwall_side_condition |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27582 |
AutoSamplingTheory.SALD.cycle24GeneralVaSaldUpperPacket
- Cycle-24 upper packet for the continuous general VA-SALD Gronwall bridge.
This returns to `thm:general-moving-target-SALD` after the discrete and
forward-KL coefficient audits. It selects only the endpoint/exponent
s |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27646 |
AutoSamplingTheory.SALD.cycle24GeneralVaSaldMiddleContract
- Cycle-24 middle packet for the continuous general VA-SALD Gronwall bridge.
This translates the upper-selected target into a lower-ready source-to-Lean map
for `sald.general_moving_target.gronwall_side_conditions`. It |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27695 |
AutoSamplingTheory.SALD.cycle28GeneralVaSaldUpperPacket
- Cycle-28 upper packet for the discrete general VA-SALD derivative side conditions.
This returns to the guided/general path after the discrete forward-KL accumulated
collection work. It selects the pre-Gronwall deriva |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27761 |
AutoSamplingTheory.SALD.cycle28GeneralVaSaldMiddleContract
- Cycle-28 middle packet for the discrete general VA-SALD derivative side.
This translates the upper-selected source slice `appendix.tex:1469-1511` into a
lower-ready source-to-Lean map. It keeps the theorem display fi |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27809 |
AutoSamplingTheory.SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27862 |
AutoSamplingTheory.SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27907 |
AutoSamplingTheory.SALD.cycle13FirstAppendixVocabularyPacket
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27937 |
AutoSamplingTheory.SALD.cycle13FirstAppendixSourceIndexAuditContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:27971 |
AutoSamplingTheory.SALD.cycle13FirstAppendixMiddleAuditContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28014 |
AutoSamplingTheory.SALD.cycle17FirstAppendixVocabularyPacket
- Cycle-17 upper packet for rebaselining the first appendix/vocabulary layer.
This returns to the source-index focus after the cycle-16 unified transport
bridge work. It is workflow data only: the four source labels re |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28073 |
AutoSamplingTheory.SALD.cycle17FirstAppendixMiddleAuditContract
- Cycle-17 middle source-to-Lean rebaseline for the first appendix layer.
This translates the cycle-17 upper source-index packet into a lower-ready
source map. It deliberately reuses the existing first-layer contracts |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28112 |
AutoSamplingTheory.SALD.cycle21FirstAppendixVocabularyPacket
- Cycle-21 upper packet for the first appendix/vocabulary source-index layer.
This returns to the original first-DAG labels after the cycle-20 discrete
general VA-SALD scalar coefficient work. It is an upper-role selec |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28172 |
AutoSamplingTheory.SALD.cycle21FirstAppendixMiddleAuditContract
- Cycle-21 middle source-to-Lean audit for the first appendix layer.
This is the middle-role transcript for the cycle-21 upper packet. It rereads
the exact TeX windows, maps each proof step to an existing Lean-facing
c |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28212 |
AutoSamplingTheory.SALD.cycle25FirstAppendixVocabularyPacket
- Cycle-25 upper packet for the first appendix/vocabulary layer.
This returns to the source-index focus after the cycle-24 continuous general
VA-SALD Gronwall coefficient work. It chooses the PI velocity-norm dependenc |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28271 |
AutoSamplingTheory.SALD.cycle25FirstAppendixMiddleAuditContract
- Cycle-25 middle source-to-Lean map for the first appendix layer.
This translates the upper-selected PI velocity-norm backend into a lower-ready
sub-slice while keeping the Gronwall, DV, PI, and LSI/KL/FI statuses fixe |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28309 |
AutoSamplingTheory.SALD.cycle25FirstAppendixPiVelocityNormMiddleObligation
- Cycle-25 middle obligation for the selected PI velocity-norm sub-slice. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28364 |
AutoSamplingTheory.SALD.cycle25PiVelocityNormLowerObligation
- Cycle-25 lower obligation after compiling the scalar PI/Cauchy--Schwarz core. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28381 |
AutoSamplingTheory.SALD.cycle29FirstAppendixVocabularyPacket
- Cycle-29 upper packet for the first appendix/vocabulary layer.
This returns to the first-DAG source-index layer after the cycle-28 guided
general VA-SALD derivative-side algebra. It selects the LSI/KL/FI
density-test |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28404 |
AutoSamplingTheory.SALD.cycle29FirstAppendixMiddleAuditContract
- Cycle-29 middle source-to-Lean map for the first appendix layer.
This translates the upper-selected LSI/KL/FI density-test bridge into a
lower-ready source map. It keeps Gronwall, DV, PI, and the later SALD theorem
s |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28444 |
AutoSamplingTheory.SALD.cycle29LsiKlFiDensityTestMiddleObligation
- Cycle-29 middle obligation for the selected LSI/KL/FI density-test sub-slice. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28497 |
AutoSamplingTheory.SALD.cycle29LsiKlFiDensityTestLowerObligation
- Cycle-29 lower obligation for the LSI/KL/FI density-test coefficient slice. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28517 |
AutoSamplingTheory.SALD.cycle33LsiKlFiDensityTestMiddleObligation
- Cycle-33 middle obligation for the proof-producing density-test scalar slice. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28534 |
AutoSamplingTheory.SALD.cycle33LsiKlFiDensityTestLowerObligation
- Cycle-33 lower obligation for the normalized LSI-test scalar bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28550 |
AutoSamplingTheory.SALD.cycle38LsiKlFiUpperPacket
- Cycle-38 upper packet for the LSI/KL/FI proof-closure target.
This packet follows the current proof-closure order after cycle 36 advanced
Gronwall assembly and cycle 37 advanced the one-sided Donsker--Varadhan
backend |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28575 |
AutoSamplingTheory.SALD.cycle38LsiKlFiUpperObligation
- Cycle-38 upper workflow obligation for the LSI/KL/FI density-test bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28614 |
AutoSamplingTheory.SALD.cycle38LsiKlFiMiddleObligation
- Cycle-38 middle obligation for the LSI/KL/FI Fisher-chain scalar slice. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28642 |
AutoSamplingTheory.SALD.cycle38LsiKlFiLowerObligation
- Cycle-38 lower obligation after compiling a finite-coordinate Fisher-chain handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28662 |
AutoSamplingTheory.SALD.cycle43LsiKlFiUpperPacket
- Cycle-43 upper packet for the remaining LSI/KL/FI density-test backend.
This packet follows the current proof-closure sprint after cycle 41 narrowed
Gronwall endpoint calculus and cycle 42 narrowed the selected-test
D |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28690 |
AutoSamplingTheory.SALD.cycle43LsiKlFiUpperObligation
- Cycle-43 upper workflow obligation for the LSI/KL/FI density-test backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28729 |
AutoSamplingTheory.SALD.cycle43LsiKlFiMiddleObligation
- Cycle-43 middle density-normalization and entropy-transport obligation.
The accompanying declarations in `AutoSamplingTheory/Probability.lean`
formalize the Mathlib-backed Radon-Nikodym mass and entropy transport piec |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28773 |
AutoSamplingTheory.SALD.cycle43LsiKlFiLowerObligation
- Cycle-43 lower finite-coordinate integral Fisher-chain obligation.
The accompanying declarations in `AutoSamplingTheory/Probability.lean` push
the cycle-38 finite-coordinate Fisher-chain identity through an arbitrary
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28802 |
AutoSamplingTheory.SALD.cycle44MainSkeletonAnalyticInterfaceLedger
- Cycle-44 upper ledger for the main SALD theorem-skeleton sprint.
The cycle focus is not another isolated scalar lemma. This packet checks the
five slow analytic interfaces, keeps their unproved backends below formali |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28830 |
AutoSamplingTheory.SALD.cycle44MainSkeletonAnalyticInterfaceObligation
- Cycle-44 upper obligation for the main skeleton analytic interface ledger. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28906 |
AutoSamplingTheory.SALD.cycle44MainSkeletonAnalyticInterfaceDag
- Cycle-44 proof-DAG pane for the five analytic interfaces and theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:28931 |
AutoSamplingTheory.SALD.cycle49MainSkeletonAnalyticReadinessLedger
- Cycle-49 upper readiness check for the five slow analytic interfaces.
This sharpens the cycle-44 ledger after the theorem-level route wrappers from
cycles 45--48 are in place. It is intentionally route data: the unre |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29036 |
AutoSamplingTheory.SALD.cycle49MainSkeletonAnalyticReadinessObligation
- Cycle-49 upper obligation selecting the next theorem-level backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29112 |
AutoSamplingTheory.SALD.cycle49MainSkeletonAnalyticMiddleContract
- Cycle-49 middle audit for the analytic-readiness ledger.
This is the source-to-Lean synchronization layer after the upper readiness
packet. It checks the five analytic interfaces against the current theorem
contracts |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29137 |
AutoSamplingTheory.SALD.cycle49MainSkeletonAnalyticMiddleObligation
- Cycle-49 middle obligation tying the analytic-readiness audit to lower
work and the Markdown conversion window. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29209 |
AutoSamplingTheory.SALD.cycle49MainSkeletonAnalyticReadinessDag
- Cycle-49 proof-DAG pane for the post-route analytic readiness check. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29238 |
AutoSamplingTheory.SALD.cycle45ForwardKlSkeletonUpperPacket
- Cycle-45 upper packet for the continuous forward-KL theorem skeleton.
This keeps the cycle-44 global interface ledger in place and wires those
interfaces into the specific `thm:forward-KL` route requested for main ske |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29313 |
AutoSamplingTheory.SALD.cycle45ForwardKlSkeletonObligation
- Cycle-45 obligation tying the continuous forward-KL theorem skeleton to the
five source-cited analytic interfaces. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29356 |
AutoSamplingTheory.SALD.cycle45ForwardKlSkeletonDag
- Cycle-45 proof-DAG pane for the continuous forward-KL theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29385 |
AutoSamplingTheory.SALD.cycle45ForwardKlSkeletonMiddleContract
- Cycle-45 middle audit for the continuous forward-KL theorem skeleton.
This is the middle-role source-to-Lean synchronization layer for the upper
route wrapper. It checks that the theorem statement and appendix proof |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29419 |
AutoSamplingTheory.SALD.cycle45ForwardKlSkeletonMiddleObligation
- Cycle-45 middle obligation tying the forward-KL route audit to lower work. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29480 |
AutoSamplingTheory.SALD.cycle50ForwardKlSkeletonUpperPacket
- Cycle-50 upper packet for the continuous forward-KL theorem skeleton.
After the cycle-49 post-route readiness audit, this packet re-enters the
specific continuous `thm:forward-KL` route. It records the upper-role che |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29516 |
AutoSamplingTheory.SALD.cycle50ForwardKlSkeletonObligation
- Cycle-50 obligation tying the post-readiness audit back to `thm:forward-KL`. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29558 |
AutoSamplingTheory.SALD.cycle50ForwardKlSkeletonMiddleContract
- Cycle-50 middle audit for the continuous forward-KL theorem skeleton.
This is the middle-role synchronization layer after the cycle-49 readiness
audit and the cycle-50 upper route wrapper. It keeps the theorem statem |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29600 |
AutoSamplingTheory.SALD.cycle50ForwardKlSkeletonMiddleObligation
- Cycle-50 middle obligation selecting the continuous KL derivative backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29663 |
AutoSamplingTheory.SALD.cycle50ForwardKlDerivativeLowerObligation
- Cycle-50 lower obligation for the continuous derivative/DV scalar handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29698 |
AutoSamplingTheory.SALD.cycle50ForwardKlSkeletonDag
- Cycle-50 proof-DAG pane for the continuous forward-KL theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29726 |
AutoSamplingTheory.SALD.cycle51DiscreteForwardKlSkeletonUpperPacket
- Cycle-51 upper packet for the discrete forward-KL theorem skeleton.
This is the post-cycle-50 return to `thm:forward-KL-discrete`. It checks the
five slow analytic interfaces again, then records the discrete theorem |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29804 |
AutoSamplingTheory.SALD.cycle51DiscreteForwardKlSkeletonObligation
- Cycle-51 obligation tying the post-cycle-50 discrete route back to the
source-cited EM/Fokker--Planck interfaces. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29852 |
AutoSamplingTheory.SALD.cycle51DiscreteForwardKlSkeletonMiddleContract
- Cycle-51 middle audit for the discrete forward-KL theorem route.
This is the middle-role synchronization layer for the current sprint. It keeps
the upper theorem route fixed, records the appendix line map around the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29903 |
AutoSamplingTheory.SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation
- Cycle-51 middle obligation tying the discrete route audit to the derivative lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:29968 |
AutoSamplingTheory.SALD.cycle32DvVariationUpperPacket
- Cycle-32 upper packet for the DV proof-closure sprint.
This packet explicitly checks the proof-closure order and selects
`lem:dv_variation` only after the cycle-31 reviewer left `lem:gronwall` as a
partial local proof |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30014 |
AutoSamplingTheory.SALD.cycle32DvVariationInterfaceObligation
- Cycle-32 source-cited interface obligation for the cited DV formula. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30051 |
AutoSamplingTheory.SALD.cycle32DvVariationMiddleAuditContract
- Cycle-32 middle source-to-Lean map for the cited DV formula.
The local Mathlib audit found KL and tilted-measure infrastructure, but no
ready theorem matching the Boucheron/SALD entropy-duality display. This
contract |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30074 |
AutoSamplingTheory.SALD.cycle32DvVariationMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30120 |
AutoSamplingTheory.SALD.cycle32DvVariationLowerObligation
- Cycle-32 lower scalar bridge for the cited DV formula.
This records the proof-producing lower slice: from a bounded set of admissible
variational values, membership of the selected test, and the source-cited
supremum |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30144 |
AutoSamplingTheory.SALD.cycle37DvVariationUpperPacket
- Cycle-37 upper packet for the cited Donsker--Varadhan proof target.
This packet follows the current proof-closure order after cycle 36 advanced
the Gronwall assembly under explicit Mathlib side conditions. It selects |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30171 |
AutoSamplingTheory.SALD.cycle37DvVariationUpperObligation
- Cycle-37 upper workflow obligation for the cited DV interface target. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30209 |
AutoSamplingTheory.SALD.cycle37DvVariationMiddleAuditContract
- Cycle-37 middle source-to-Lean map for the cited DV theorem.
This records the proof-producing Mathlib-backed sublemma now available for the
one-sided admissible-test inequality. It does not promote the paper-cited
su |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30238 |
AutoSamplingTheory.SALD.cycle37DvVariationMiddleObligation
- Cycle-37 middle obligation tracking the new one-sided tilted backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30289 |
AutoSamplingTheory.SALD.cycle37DvVariationLowerObligation
- Cycle-37 lower obligation tracking the composed one-sided DV consequence. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30313 |
AutoSamplingTheory.SALD.cycle42DvVariationMiddleAuditContract
- Cycle-42 middle source-to-Lean map for selected DV tests.
This records the proof-producing middle slice for this cycle: `alpha0` finite
exponential integrability implies the finite-log-mgf hypothesis for the
selected |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30341 |
AutoSamplingTheory.SALD.cycle42DvVariationMiddleObligation
- Cycle-42 obligation for the selected scaled-test DV interface. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30392 |
AutoSamplingTheory.SALD.cycle42DvVariationLowerObligation
- Cycle-42 lower obligation tracking the post-DV scaled energy bound. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30412 |
AutoSamplingTheory.SALD.cycle18ForwardKlUpperPacket
- Cycle-18 upper packet returning to the continuous forward-KL chain.
This packet uses the accepted cycle-17 scalar Gronwall algebra only as a
dependency marker for the continuous theorem's final Gronwall bookkeeping. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30441 |
AutoSamplingTheory.SALD.cycle18ForwardKlMiddleContract
- Cycle-18 middle packet for the continuous forward-KL Gronwall side conditions.
This refines the upper packet into a lower-ready source-to-Lean map for the
last Gronwall display of `thm:forward-KL`. It records how the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30486 |
AutoSamplingTheory.SALD.cycle22ForwardKlUpperPacket
- Cycle-22 upper packet for the continuous forward-KL Gronwall side conditions.
This packet follows the cycle-21 Gronwall outer-integral congruence refinement.
It selects only the theorem-specific coefficient regularity |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30552 |
AutoSamplingTheory.SALD.cycle22ForwardKlMiddleContract
- Cycle-22 middle packet for the continuous forward-KL coefficient bridge.
This converts the upper coefficient-regularity objective into a lower-ready
source-to-Lean map. The packet is deliberately narrower than the fu |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30597 |
AutoSamplingTheory.SALD.cycle26ForwardKlUpperPacket
- Cycle-26 upper packet for the continuous forward-KL DV witness.
This returns to `thm:forward-KL` after the first-appendix cycle-25 PI work and
selects only the theorem-specific Donsker--Varadhan finite-log-mgf/common- |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30664 |
AutoSamplingTheory.SALD.cycle26ForwardKlMiddleContract
- Cycle-26 middle packet for the continuous forward-KL DV witness.
This converts the upper-selected finite-log-mgf/common-space target into a
lower-ready source-to-Lean map. It does not prove the Donsker--Varadhan
form |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30711 |
AutoSamplingTheory.SALD.cycle30ForwardKlUpperPacket
- Cycle-30 upper packet for the continuous forward-KL derivative side.
This packet returns to the front of the `thm:forward-KL` proof after the
cycle-26 DV witness and cycle-29 LSI density-test refinements. It selects |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30779 |
AutoSamplingTheory.SALD.cycle30ForwardKlMiddleContract
- Cycle-30 middle packet for the continuous forward-KL derivative side.
This translates the upper-selected derivative-side target into a lower-ready
source-to-Lean map. The first lower slice is only `appendix.tex:168-1 |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30829 |
AutoSamplingTheory.SALD.cycle34ForwardKlDerivativeUpperPacket
- Cycle-34 upper packet for the continuous forward-KL derivative closure sprint.
This packet explicitly checks the proof-closure order and assigns only the
next proof-producing derivative slice inside `appendix.tex:168- |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30893 |
AutoSamplingTheory.SALD.cycle34ForwardKlDerivativeMiddleContract
- Cycle-34 middle map for the continuous forward-KL derivative scalar closure.
This translates the upper packet into the specific Lean handoff for
`appendix.tex:218-228`. The compiled theorem is pure real arithmetic; t |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30940 |
AutoSamplingTheory.SALD.cycle39ForwardKlDerivativeUpperPacket
- Cycle-39 upper packet for the continuous forward-KL derivative sprint.
This packet follows the current proof-closure focus: keep the source theorem
fixed and translate `appendix.tex:168-228` into the forward-KL
Fokker |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:30994 |
AutoSamplingTheory.SALD.cycle39ForwardKlDerivativeMiddleContract
- Cycle-39 middle map for the source-shaped derivative schedule handoff.
This translates the upper packet into proof-producing scalar targets for
`appendix.tex:191-228`: the inverse-schedule product identity, the
slowed |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31041 |
AutoSamplingTheory.SALD.cycle35DiscreteForwardKlEmFpUpperPacket
- Cycle-35 upper packet for the discrete EM interpolation Fokker--Planck sprint.
The earlier proof-closure items have current scalar or source-cited slices, so
this packet returns to item (5): the Euler--Maruyama interp |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31099 |
AutoSamplingTheory.SALD.cycle35DiscreteForwardKlEmFpMiddleContract
- Cycle-35 middle packet for the EM interpolation Fokker--Planck sprint.
This translates `appendix.tex:260-385` into lower-ready Lean targets while
keeping the analytic endpoint-law and conditional-drift Fokker--Planck
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31147 |
AutoSamplingTheory.SALD.cycle40DiscreteForwardKlEmFpMiddleContract
- Cycle-40 middle packet for the EM endpoint and conditional-FP backend.
This keeps the proof-closure priority on item (5) and refines the cycle-35 EM
spine with law-level endpoint handoffs. The conditional-drift
Fokke |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31204 |
AutoSamplingTheory.SALD.generalVaSaldEulerMaruyamaContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31256 |
AutoSamplingTheory.SALD.generalFrozenDeltaCrossLipContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31263 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteStatementContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31297 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDerivativeCandidateContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31336 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalDriftContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31389 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31429 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31475 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31510 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31578 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDerivativeSideConditionContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31637 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31703 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallInstantiationContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31738 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallSideConditionContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31763 |
AutoSamplingTheory.SALD.saldPIContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31805 |
AutoSamplingTheory.SALD.lsiKlFiDensityTestObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31812 |
AutoSamplingTheory.SALD.dvFiniteLogMgfInterfaceObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31855 |
AutoSamplingTheory.SALD.piVelocityNormBackendObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31863 |
AutoSamplingTheory.SALD.cycle36GronwallUpperPacket
- Cycle-36 upper packet for returning to the Gronwall proof-closure target.
This packet deliberately selects proof-closure priority item (1),
`lem:gronwall`, after cycle 35 finished a local EM interpolation algebra pass |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31888 |
AutoSamplingTheory.SALD.cycle36GronwallUpperObligation
- Cycle-36 upper workflow obligation for the Gronwall proof-closure packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31926 |
AutoSamplingTheory.SALD.cycle36GronwallMiddleObligation
- Cycle-36 middle proof-producing Gronwall assembly record. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31949 |
AutoSamplingTheory.SALD.cycle41GronwallMiddleObligation
- Cycle-41 middle proof-producing Gronwall derivative-source wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31973 |
AutoSamplingTheory.SALD.cycle41GronwallLowerObligation
- Cycle-41 lower endpoint-safe Gronwall interior-derivative assembly record. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:31991 |
AutoSamplingTheory.SALD.gronwallAnalyticObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32010 |
AutoSamplingTheory.SALD.gronwallEndpointCalculusObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32039 |
AutoSamplingTheory.SALD.gronwallExponentRewriteObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32072 |
AutoSamplingTheory.SALD.firstAppendixSourceIndexAuditObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32089 |
AutoSamplingTheory.SALD.firstAppendixMiddleAuditObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32113 |
AutoSamplingTheory.SALD.forwardKlMiddleSourceToLeanMapObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32169 |
AutoSamplingTheory.SALD.cycle30ForwardKlDerivativeSideUpperObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32225 |
AutoSamplingTheory.SALD.cycle30ForwardKlDerivativeSideMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32249 |
AutoSamplingTheory.SALD.forwardKlDensityBoundaryObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32274 |
AutoSamplingTheory.SALD.cycle30ForwardKlDensityBoundaryLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32293 |
AutoSamplingTheory.SALD.cycle34ForwardKlDerivativeScalarObligation
- Cycle-34 upper/lower scalar obligation for the derivative closure sprint. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32315 |
AutoSamplingTheory.SALD.cycle34ForwardKlTargetYoungLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32342 |
AutoSamplingTheory.SALD.cycle34ForwardKlDerivativeMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32362 |
AutoSamplingTheory.SALD.cycle39ForwardKlDerivativeUpperObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32383 |
AutoSamplingTheory.SALD.cycle39ForwardKlDerivativeMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32418 |
AutoSamplingTheory.SALD.forwardKlScheduleTimeChangeObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32450 |
AutoSamplingTheory.SALD.forwardKlDerivativeObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32478 |
AutoSamplingTheory.SALD.forwardKlDvEnergyObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32527 |
AutoSamplingTheory.SALD.forwardKlDvAlphaMonotonicityObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32541 |
AutoSamplingTheory.SALD.forwardKlDvFiniteLogMgfWitnessObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32553 |
AutoSamplingTheory.SALD.cycle26ForwardKlDvWitnessMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32573 |
AutoSamplingTheory.SALD.cycle26ForwardKlDvPositiveAlphaLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32595 |
AutoSamplingTheory.SALD.forwardKlGronwallApplicationObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32612 |
AutoSamplingTheory.SALD.forwardKlGronwallSideConditionObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32620 |
AutoSamplingTheory.SALD.forwardKlEndpointScheduleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32647 |
AutoSamplingTheory.SALD.forwardKlMovingTargetDependencyObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32662 |
AutoSamplingTheory.SALD.forwardKlCoefficientChainObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32684 |
AutoSamplingTheory.SALD.discreteForwardKlEmEndpointObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32702 |
AutoSamplingTheory.SALD.discreteForwardKlEmConditionalFpObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32710 |
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlConditionalDriftDensityObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32718 |
AutoSamplingTheory.SALD.discreteForwardKlStitchedIntervalRegularityObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32731 |
AutoSamplingTheory.SALD.discreteForwardKlEmInterpolationObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32739 |
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlMiddleEmSpineObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32747 |
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlEmConditionalFpLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32766 |
AutoSamplingTheory.SALD.cycle35DiscreteForwardKlEmFpUpperObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32784 |
AutoSamplingTheory.SALD.cycle35DiscreteForwardKlEmFpMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32806 |
AutoSamplingTheory.SALD.cycle35DiscreteForwardKlEmFpLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32831 |
AutoSamplingTheory.SALD.cycle40DiscreteForwardKlEmFpMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32852 |
AutoSamplingTheory.SALD.cycle40DiscreteForwardKlEmEndpointLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32876 |
AutoSamplingTheory.SALD.discreteForwardKlFrozenDeltaObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32892 |
AutoSamplingTheory.SALD.discreteForwardKlDerivativeObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32900 |
AutoSamplingTheory.SALD.cycle51DiscreteForwardKlDerivativeLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32908 |
AutoSamplingTheory.SALD.discreteForwardKlDvFiniteLogMgfWitnessObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32927 |
AutoSamplingTheory.SALD.discreteForwardKlDvVelocityObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32944 |
AutoSamplingTheory.SALD.discreteForwardKlGronwallAccumulationObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32952 |
AutoSamplingTheory.SALD.discreteForwardKlLinearSlowdownObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32960 |
AutoSamplingTheory.SALD.discreteForwardKlResidualExponentBoundObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32968 |
AutoSamplingTheory.SALD.discreteForwardKlAccumulatedErrorBridgeObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:32983 |
AutoSamplingTheory.SALD.cycle19DiscreteForwardKlAccumulatedErrorMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33008 |
AutoSamplingTheory.SALD.discreteForwardKlEmDefectAccumulationMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33027 |
AutoSamplingTheory.SALD.cycle23DiscreteForwardKlCoefficientChainMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33048 |
AutoSamplingTheory.SALD.discreteForwardKlCoefficientChainObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33074 |
AutoSamplingTheory.SALD.cycle27DiscreteForwardKlAccumulatedCollectionUpperObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33097 |
AutoSamplingTheory.SALD.cycle27DiscreteForwardKlAccumulatedCollectionMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33121 |
AutoSamplingTheory.SALD.cycle27DiscreteForwardKlAccumulatedCollectionLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33148 |
AutoSamplingTheory.SALD.cycle46DiscreteForwardKlSkeletonUpperPacket
- Cycle-46 upper packet for the discrete forward-KL theorem skeleton.
This keeps the cycle-44 slow analytic backend ledger in force and wires those
interfaces into `thm:forward-KL-discrete` at theorem-route level. It i |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33177 |
AutoSamplingTheory.SALD.cycle46DiscreteForwardKlSkeletonObligation
- Cycle-46 obligation tying the discrete forward-KL theorem skeleton to the
five source-cited analytic interfaces. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33228 |
AutoSamplingTheory.SALD.cycle46DiscreteForwardKlSkeletonDag
- Cycle-46 proof-DAG pane for the discrete forward-KL theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33266 |
AutoSamplingTheory.SALD.cycle46DiscreteForwardKlSkeletonMiddleContract
- Cycle-46 middle audit for the discrete forward-KL theorem skeleton.
This is the middle-role source-to-Lean synchronization layer for the discrete
route wrapper. It checks the exact statement and appendix proof order, |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33301 |
AutoSamplingTheory.SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation
- Cycle-46 middle obligation tying the discrete route audit to lower work. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33371 |
AutoSamplingTheory.SALD.cycle47GuidedGeneralSkeletonUpperPacket
- Cycle-47 upper packet for the guided residual and continuous general
moving-target theorem skeleton.
This keeps the cycle-44 slow analytic backend ledger in force and wires the
source window `appendix.tex:619-951` int |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33419 |
AutoSamplingTheory.SALD.cycle47GuidedGeneralSkeletonObligation
- Cycle-47 obligation tying the guided residual and continuous general
moving-target theorem skeleton to the already named analytic interfaces. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33466 |
AutoSamplingTheory.SALD.cycle47GuidedGeneralSkeletonMiddleContract
- Cycle-47 middle audit for the guided residual and continuous general
moving-target theorem skeleton.
This is the source-to-Lean synchronization layer for the upper route wrapper.
It checks the exact appendix proof ord |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33505 |
AutoSamplingTheory.SALD.cycle47GuidedGeneralSkeletonMiddleObligation
- Cycle-47 middle obligation tying the guided/general route audit to lower
work. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33572 |
AutoSamplingTheory.SALD.cycle47GuidedGeneralSkeletonDag
- Cycle-47 proof-DAG pane for the guided/general theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33607 |
AutoSamplingTheory.SALD.cycle52GuidedGeneralSkeletonUpperPacket
- Cycle-52 upper packet for the guided residual and continuous general
moving-target theorem after the forward and discrete forward-KL routes.
This is a route-closure check for the current skeleton sprint. It records t |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33663 |
AutoSamplingTheory.SALD.cycle52GuidedGeneralSkeletonObligation
- Cycle-52 obligation tying the guided residual and continuous general
moving-target theorem route to the five explicit analytic backends. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33713 |
AutoSamplingTheory.SALD.cycle52GuidedGeneralSkeletonMiddleContract
- Cycle-52 middle audit for the guided residual and continuous general
moving-target theorem route.
This synchronizes the cycle-52 upper route with the conversion window and
proof-obligation ledger. It keeps the theore |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33773 |
AutoSamplingTheory.SALD.cycle52GuidedGeneralSkeletonMiddleObligation
- Cycle-52 middle obligation tying the guided/general route audit to lower
work. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33841 |
AutoSamplingTheory.SALD.cycle52GuidedGeneralDerivativeDvLowerObligation
- Cycle-52 lower obligation for the general moving-target derivative/DV scalar
handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33895 |
AutoSamplingTheory.SALD.cycle52GuidedGeneralSkeletonDag
- Cycle-52 proof-DAG pane for the guided/general route closure check. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:33922 |
AutoSamplingTheory.SALD.cycle57GuidedGeneralSkeletonUpperPacket
- Cycle-57 upper packet returning to the guided residual and continuous
general moving-target theorem after the cycle-56 discrete forward-KL recovery.
This packet records the required upper phase judgment, rechecks the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34017 |
AutoSamplingTheory.SALD.cycle57GuidedGeneralSkeletonObligation
- Cycle-57 obligation for the upper guided/general route recheck. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34070 |
AutoSamplingTheory.SALD.cycle57GuidedGeneralSkeletonMiddleContract
- Cycle-57 middle audit for the guided residual and continuous general
moving-target theorem route.
This is the middle-role source-to-Lean synchronization layer for the cycle-57
upper route. It verifies the appendix pr |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34128 |
AutoSamplingTheory.SALD.cycle57GuidedGeneralSkeletonMiddleObligation
- Cycle-57 middle obligation tying the guided/general route audit to lower
work. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34197 |
AutoSamplingTheory.SALD.cycle57GuidedGeneralDerivativeSplitLowerObligation
- Cycle-57 lower obligation for the first continuous general derivative split.
The compiled scalar lemmas here begin the selected lower packet for
`sald.general_moving_target.kl_derivative` by reducing the raw derivativ |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34253 |
AutoSamplingTheory.SALD.cycle57GuidedGeneralSkeletonDag
- Cycle-57 proof-DAG pane for the guided/general upper route and selected lower
backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34277 |
AutoSamplingTheory.SALD.cycle48UnifiedDiscreteSkeletonUpperPacket
- Cycle-48 upper packet for closing the unified and discrete general theorem
skeleton route.
This keeps the cycle-44 slow analytic backend ledger in force and wires the
last two theorem-level nodes requested by the task |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34382 |
AutoSamplingTheory.SALD.cycle48UnifiedDiscreteSkeletonObligation
- Cycle-48 obligation tying the unified and discrete general theorem
skeletons to the already named continuous/general and EM analytic interfaces. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34436 |
AutoSamplingTheory.SALD.cycle48UnifiedDiscreteSkeletonMiddleContract
- Cycle-48 middle audit for the unified and discrete general theorem
route.
This source-to-Lean synchronization layer checks the upper route wrapper
against the exact TeX paragraphs and selects the next discrete KL-deri |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34487 |
AutoSamplingTheory.SALD.cycle48GeneralMovingTargetDiscreteEmEndpointFpAuditObligation
- Cycle-48 narrow measure-theory audit for the discrete general EM
endpoint and conditional-law Fokker--Planck backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34561 |
AutoSamplingTheory.SALD.cycle48UnifiedDiscreteSkeletonMiddleObligation
- Cycle-48 middle obligation tying the route audit and the narrow EM
endpoint/conditional-law interface to lower work. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34583 |
AutoSamplingTheory.SALD.cycle48UnifiedDiscreteSkeletonDag
- Cycle-48 proof-DAG pane for the unified and discrete general theorem
route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34618 |
AutoSamplingTheory.SALD.cycle53UnifiedDiscreteGeneralUpperPacket
- Cycle-53 upper packet for the unified and discrete general theorem route.
This consumes the cycle-52 continuous guided/general route and the cycle-48
unified/discrete route, then records the first narrow measure-level |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34718 |
AutoSamplingTheory.SALD.cycle53UnifiedDiscreteGeneralSkeletonObligation
- Cycle-53 obligation tying the final unified/discrete general theorem route
to explicit source-cited interfaces and the narrow Measure.map endpoint
backfill. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34772 |
AutoSamplingTheory.SALD.cycle53UnifiedDiscreteGeneralMiddleContract
- Cycle-53 middle audit for the final unified/discrete general route.
This synchronizes the upper route packet with the conversion window and
proof-obligation ledger. It keeps the source proof order fixed and leaves th |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34827 |
AutoSamplingTheory.SALD.cycle53UnifiedDiscreteGeneralMiddleObligation
- Cycle-53 middle obligation tying the final route audit to lower work. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34895 |
AutoSamplingTheory.SALD.cycle53UnifiedDiscreteGeneralDag
- Cycle-53 proof-DAG pane for the final unified/discrete general route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:34945 |
AutoSamplingTheory.SALD.cycle54MainSkeletonAnalyticInterfaceLedger
- Cycle-54 upper packet for the repeated analytic-interface sprint.
Cycle 54 returns to the sprint-1 focus after the full theorem route has been
threaded once. The objective is to check that the five slow analytic back |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35010 |
AutoSamplingTheory.SALD.cycle54MainSkeletonAnalyticInterfaceObligation
- Cycle-54 upper obligation for the repeated analytic-interface ledger. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35091 |
AutoSamplingTheory.SALD.cycle54MainSkeletonAnalyticMiddleContract
- Cycle-54 middle audit of the repeated analytic-interface sprint.
This is source-to-Lean synchronization for the middle role. It checks that the
upper ledger's five interfaces are actually consumed by the theorem DAGs |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35130 |
AutoSamplingTheory.SALD.cycle54MainSkeletonAnalyticMiddleObligation
- Cycle-54 middle obligation tying the analytic-interface audit to the six
theorem contracts and the lower EM conditional-FP packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35208 |
AutoSamplingTheory.SALD.cycle54MainSkeletonAnalyticInterfaceDag
- Cycle-54 proof-DAG pane for the repeated analytic-interface ledger. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35236 |
AutoSamplingTheory.SALD.cycle55ForwardKlSkeletonUpperPacket
- Cycle-55 upper packet for returning the global analytic-interface audit to
the continuous forward-KL skeleton.
The cycle focus is deliberately narrow: consume the cycle-54 five-backend
check, then re-wire `thm:forward |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35334 |
AutoSamplingTheory.SALD.cycle55ForwardKlSkeletonObligation
- Cycle-55 obligation tying the re-checked analytic interfaces to the
continuous forward-KL theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35381 |
AutoSamplingTheory.SALD.cycle55ForwardKlSkeletonMiddleContract
- Cycle-55 middle audit for the focused continuous forward-KL route.
This synchronizes the cycle-55 upper route with the conversion window,
proof-obligation ledger, and lower packet. It keeps `thm:forward-KL` on the
pa |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35430 |
AutoSamplingTheory.SALD.cycle55ForwardKlSkeletonMiddleObligation
- Cycle-55 middle obligation tying the continuous forward-KL route audit to
the lower derivative/Fokker--Planck packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35494 |
AutoSamplingTheory.SALD.cycle55ForwardKlDerivativeMassLowerObligation
- Cycle-55 lower obligation for the first continuous derivative scalar slice. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35535 |
AutoSamplingTheory.SALD.cycle55ForwardKlSkeletonDag
- Cycle-55 proof-DAG pane for the focused continuous forward-KL route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35558 |
AutoSamplingTheory.SALD.cycle56DiscreteForwardKlSkeletonUpperPacket
- Cycle-56 upper packet for returning the main skeleton sprint to the
discrete forward-KL theorem.
The focus is the theorem route, not a new analytic proof: consume the existing
source-cited EM/Fokker-Planck interfaces, |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35666 |
AutoSamplingTheory.SALD.cycle56DiscreteForwardKlSkeletonObligation
- Cycle-56 obligation tying the discrete theorem route to the existing
source-cited EM/Fokker-Planck and Gronwall/accumulated-error interfaces. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35715 |
AutoSamplingTheory.SALD.cycle56DiscreteForwardKlSkeletonMiddleContract
- Cycle-56 middle audit for the discrete forward-KL theorem route.
This synchronizes the upper cycle-56 route with the source-to-Lean ledger. It
keeps the already wired derivative and DV interfaces as inputs and select |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35769 |
AutoSamplingTheory.SALD.cycle56DiscreteForwardKlSkeletonMiddleObligation
- Cycle-56 middle obligation tying the discrete route audit to the selected
Gronwall lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35837 |
AutoSamplingTheory.SALD.cycle56DiscreteForwardKlGronwallLowerObligation
- Cycle-56 lower obligation for the discrete Gronwall accumulation backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35880 |
AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralUpperPacket
- Cycle-58 upper packet for the final unified/discrete general theorem
refresh.
This packet consumes the now-clean cycle-56 discrete forward-KL route and the
cycle-57 guided/general route. It keeps the cycle focus on t |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35914 |
AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralSkeletonObligation
- Cycle-58 obligation tying the final unified/discrete general theorem refresh
to explicit source-cited interfaces. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:35975 |
AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralMiddleContract
- Cycle-58 middle audit for the unified/discrete general route.
This source-to-Lean synchronization layer checks the upper route against the
paper order and hands lower work to the discrete general Gronwall/display
back |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36032 |
AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralMiddleObligation
- Cycle-58 middle obligation tying the route audit to lower Gronwall/display
work. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36102 |
AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralDag
- Cycle-58 proof-DAG pane for the final unified/discrete general route
refresh and selected lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36150 |
AutoSamplingTheory.SALD.cycle59MainSkeletonAnalyticInterfaceLedger
- Cycle-59 upper ledger for the post-cycle-58 analytic-interface sprint.
The previous cycle closed the unified/discrete general route through reviewer
and build. This ledger records the upper-level phase judgment requi |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36234 |
AutoSamplingTheory.SALD.cycle59MainSkeletonAnalyticInterfaceObligation
- Cycle-59 upper obligation for the analytic-interface recheck. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36323 |
AutoSamplingTheory.SALD.cycle59MainSkeletonAnalyticMiddleContract
- Cycle-59 middle audit for the analytic-interface ledger.
This source-to-Lean synchronization layer checks the upper cycle-59 ledger
against the six theorem consumers and keeps lower work on the theorem-level
discrete |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36364 |
AutoSamplingTheory.SALD.cycle59MainSkeletonAnalyticMiddleObligation
- Cycle-59 middle obligation tying the analytic-interface audit to the six
theorem consumers and the selected lower Gronwall/display packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36444 |
AutoSamplingTheory.SALD.cycle59GeneralMovingTargetDiscreteGronwallLowerObligation
- Cycle-59 lower obligation for the discrete general Gronwall/display
side-condition packet.
The proof-producing part of this lower packet is local: it introduces named
Gronwall coefficients from the cycle-58 pointwise |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36485 |
AutoSamplingTheory.SALD.cycle59MainSkeletonAnalyticInterfaceDag
- Cycle-59 proof-DAG pane for the post-cycle-58 analytic-interface check. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36505 |
AutoSamplingTheory.SALD.cycle60ForwardKlSkeletonUpperPacket
- Cycle-60 upper packet for the continuous forward-KL skeleton after the
accepted cycle-59 route audit.
The previous reviewer/build gate accepted the cycle-59 ledger. This packet
therefore returns to the cycle focus: m |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36617 |
AutoSamplingTheory.SALD.cycle60ForwardKlSkeletonObligation
- Cycle-60 upper obligation tying the accepted cycle-59 analytic ledger back
to the focused continuous forward-KL theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36666 |
AutoSamplingTheory.SALD.cycle60ForwardKlSkeletonMiddleContract
- Cycle-60 middle audit for the post-cycle-59 continuous forward-KL route.
This source-to-Lean synchronization layer checks that the upper cycle-60 route
is consumed by the continuous theorem contract in the same order |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36719 |
AutoSamplingTheory.SALD.cycle60ForwardKlSkeletonMiddleObligation
- Cycle-60 middle obligation tying the post-cycle-59 route audit to the
selected continuous derivative/Fokker--Planck lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36783 |
AutoSamplingTheory.SALD.cycle60ForwardKlDerivativeRawLowerObligation
- Cycle-60 lower obligation for the raw continuous derivative scalar wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36830 |
AutoSamplingTheory.SALD.cycle60ForwardKlSkeletonDag
- Cycle-60 proof-DAG pane for the post-cycle-59 continuous forward-KL
skeleton route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36857 |
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlSkeletonUpperPacket
- Cycle-61 upper packet for the recovered discrete forward-KL skeleton.
Cycle 60 passed the continuous forward-KL reviewer/build gate. This packet
returns to the interrupted cycle-56 discrete route and keeps the next w |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:36996 |
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlSkeletonObligation
- Cycle-61 upper obligation tying the recovered discrete route to the next
Gronwall/accumulated-error lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37052 |
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlSkeletonMiddleContract
- Cycle-61 middle audit for the recovered discrete forward-KL route.
This synchronizes the upper packet with the source transcript and moves lower
work from the cycle-56 pointwise Gronwall input to the accumulated-error |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37112 |
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlSkeletonMiddleObligation
- Cycle-61 middle obligation tying the recovered theorem route to the
accumulated-error bridge lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37181 |
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation
- Cycle-61 lower obligation for the residual integral display wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37245 |
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlSkeletonDag
- Cycle-61 proof-DAG pane for the recovered discrete forward-KL route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37268 |
AutoSamplingTheory.SALD.cycle62GuidedGeneralSkeletonUpperPacket
- Cycle-62 upper packet for the guided residual and continuous general
moving-target theorem route after the accepted cycle-61 discrete recovery.
This is workflow data only. It records the upper phase judgment, checks |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37405 |
AutoSamplingTheory.SALD.cycle62GuidedGeneralSkeletonObligation
- Cycle-62 upper obligation tying the guided/general theorem route to the
accepted cycle-61 recovery and the five explicit analytic backend interfaces. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37463 |
AutoSamplingTheory.SALD.cycle62GuidedGeneralSkeletonMiddleContract
- Cycle-62 middle audit for the guided residual and continuous general
moving-target theorem route.
This synchronizes the upper route wrapper with the conversion window and
proof-obligation ledger. It keeps the source |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37533 |
AutoSamplingTheory.SALD.cycle62GuidedGeneralSkeletonMiddleObligation
- Cycle-62 middle obligation tying the guided/general route audit to the next
lower backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37601 |
AutoSamplingTheory.SALD.cycle62GuidedGeneralScaledResidualLowerObligation
- Cycle-62 lower scalar handoff for the continuous general KL derivative. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37655 |
AutoSamplingTheory.SALD.cycle62GuidedGeneralSkeletonDag
- Cycle-62 proof-DAG pane for the guided/general upper route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37677 |
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralSkeletonUpperPacket
- Cycle-63 upper packet for the unified and discrete general theorem route.
This is workflow data only. It records the required upper phase judgment,
rechecks the five slow analytic interfaces before assigning follow-u |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37816 |
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation
- Cycle-63 upper obligation for the unified/discrete general route refresh. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37877 |
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation
- Cycle-63 narrow measure-theory backfill below the discrete general EM
endpoint-law interface. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37933 |
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralMiddleContract
- Cycle-63 middle audit for the unified/discrete general route.
This source-to-Lean synchronization layer checks the upper route and the local
paired endpoint-law backfill against the paper order, then narrows lower wor |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:37960 |
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralMiddleObligation
- Cycle-63 middle obligation tying the route audit to the next lower
conditional-law/Fokker--Planck packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38034 |
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralDag
- Cycle-63 proof-DAG pane for the unified/discrete general route and one
narrow endpoint-law backfill. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38090 |
AutoSamplingTheory.SALD.cycle64MainSkeletonAnalyticInterfaceLedger
- Cycle-64 upper ledger for the analytic-interface sprint.
This is route data only. It records the required upper phase judgment,
re-checks the five slow analytic backends, and routes them through the six
faithful theo |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38207 |
AutoSamplingTheory.SALD.cycle64MainSkeletonAnalyticInterfaceObligation
- Cycle-64 upper obligation for the refreshed analytic-interface ledger. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38289 |
AutoSamplingTheory.SALD.cycle64MainSkeletonAnalyticMiddleContract
- Cycle-64 middle audit for the analytic-interface ledger.
This synchronizes the upper ledger with the source transcript and narrows the
lower packet to the EM interpolation conditional-law/Fokker--Planck backend
that r |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38329 |
AutoSamplingTheory.SALD.cycle64MainSkeletonAnalyticMiddleObligation
- Cycle-64 middle obligation tying the analytic-interface audit to the
selected EM conditional-law/Fokker--Planck lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38419 |
AutoSamplingTheory.SALD.cycle64MainSkeletonAnalyticInterfaceDag
- Cycle-64 proof-DAG pane for the analytic-interface ledger and selected
conditional-law/Fokker--Planck lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38463 |
AutoSamplingTheory.SALD.cycle65ForwardKlSkeletonUpperPacket
- Cycle-65 upper packet for the continuous forward-KL skeleton after the
accepted cycle-64 analytic-interface and conditional-drift pass.
This is upper-route data only. It returns the main skeleton sprint to
`thm:forwa |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38599 |
AutoSamplingTheory.SALD.cycle65ForwardKlSkeletonObligation
- Cycle-65 upper obligation tying the accepted cycle-64 analytic ledger back
to the focused continuous forward-KL theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38651 |
AutoSamplingTheory.SALD.cycle65ForwardKlSkeletonMiddleContract
- Cycle-65 middle audit for the continuous forward-KL route after the
post-cycle-64 upper packet.
This source-to-Lean synchronization layer checks the exact theorem statement
and proof order used by `thm:forward-KL`, th |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38708 |
AutoSamplingTheory.SALD.cycle65ForwardKlSkeletonMiddleObligation
- Cycle-65 middle obligation tying the post-cycle-64 forward-KL route audit
to the selected continuous derivative/Fokker--Planck lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38774 |
AutoSamplingTheory.SALD.cycle65ForwardKlDerivativePointwiseLowerObligation
- Cycle-65 lower proof-producing obligation for the pointwise continuous
derivative wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38827 |
AutoSamplingTheory.SALD.cycle65ForwardKlSkeletonDag
- Cycle-65 proof-DAG pane for the post-cycle-64 continuous forward-KL
skeleton route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:38853 |
AutoSamplingTheory.SALD.cycle66DiscreteForwardKlSkeletonUpperPacket
- Cycle-66 upper packet for the discrete forward-KL skeleton after the
accepted cycle-65 continuous route.
This is upper-route data only. It returns the main skeleton sprint to
`thm:forward-KL-discrete`, checks that th |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39009 |
AutoSamplingTheory.SALD.cycle66DiscreteForwardKlSkeletonObligation
- Cycle-66 upper obligation tying the accepted cycle-65 continuous route back
to the focused discrete forward-KL theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39068 |
AutoSamplingTheory.SALD.cycle66DiscreteForwardKlSkeletonMiddleContract
- Cycle-66 middle audit for the post-cycle-65 discrete forward-KL route.
This keeps the theorem-level transcript fixed while moving lower work to the
Gronwall output and accumulated-error bridge selected by the upper pa |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39139 |
AutoSamplingTheory.SALD.cycle66DiscreteForwardKlSkeletonMiddleObligation
- Cycle-66 middle obligation tying the post-cycle-65 discrete route to the
accumulated-error bridge lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39209 |
AutoSamplingTheory.SALD.cycle66DiscreteForwardKlAccumulatedDisplayLowerObligation
- Cycle-66 lower obligation for the final scalar display wrapper in the
discrete forward-KL accumulated-error bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39279 |
AutoSamplingTheory.SALD.cycle66DiscreteForwardKlSkeletonDag
- Cycle-66 proof-DAG pane for the post-cycle-65 discrete forward-KL skeleton
route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39305 |
AutoSamplingTheory.SALD.cycle67GuidedGeneralSkeletonUpperPacket
- Cycle-67 upper packet for returning to the guided residual and continuous
general moving-target theorem after the accepted cycle-66 discrete route.
This is a theorem-skeleton route packet only. It rechecks the five s |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39444 |
AutoSamplingTheory.SALD.cycle67GuidedGeneralSkeletonObligation
- Cycle-67 upper obligation for the guided residual and continuous general
moving-target theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39501 |
AutoSamplingTheory.SALD.cycle67GuidedGeneralSkeletonMiddleContract
- Cycle-67 middle audit for the guided residual and continuous general
moving-target theorem route.
This is the middle-role synchronization layer after the cycle-67 upper route.
It checks the source proof in order, reco |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39556 |
AutoSamplingTheory.SALD.cycle67GuidedGeneralSkeletonMiddleObligation
- Cycle-67 middle obligation tying the guided/general route audit to the
residual-to-Gronwall lower packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39626 |
AutoSamplingTheory.SALD.cycle67GuidedGeneralResidualGronwallBridgeObligation
- Cycle-67 source-cited interface for the residual-to-Gronwall bridge in the
continuous general moving-target proof. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39682 |
AutoSamplingTheory.SALD.cycle67GuidedGeneralResidualGronwallLowerObligation
- Cycle-67 lower proof-producing obligation for the residual-to-Gronwall
scalar bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39726 |
AutoSamplingTheory.SALD.cycle67GuidedGeneralSkeletonDag
- Cycle-67 proof-DAG pane for the guided residual and continuous general
moving-target route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39758 |
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralSkeletonUpperPacket
- Cycle-68 upper packet for the unified forward-KL theorem and the
discrete-time general moving-target theorem.
This is a theorem-skeleton route packet only. It reuses the accepted
continuous guided/general route from |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39896 |
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralSkeletonObligation
- Cycle-68 upper obligation for routing the unified and discrete general
theorems through the accepted continuous/general skeletons and explicit slow
interfaces. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:39957 |
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation
- Cycle-68 source-cited bridge for the final unified/discrete general
theorem route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40008 |
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleContract
- Cycle-68 middle audit for the unified and discrete general theorem route.
This source-to-Lean synchronization layer verifies the upper packet in the
paper order and keeps lower work on the source-cited discrete genera |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40066 |
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralSkeletonMiddleObligation
- Cycle-68 middle obligation tying the route audit to the selected
unified/discrete general source-cited bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40141 |
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation
- Cycle-68 lower proof-producing obligation for the final discrete general
Gronwall/display bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40209 |
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralDag
- Cycle-68 proof-DAG pane for the unified and discrete general theorem
route refresh. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40235 |
AutoSamplingTheory.SALD.cycle69MainSkeletonAnalyticInterfaceLedger
- Cycle-69 upper ledger after the full theorem-route sprint.
This is source-to-Lean route data only. It records the required upper phase
judgment, rechecks the five slow analytic interfaces after the cycle-68
unified/d |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40366 |
AutoSamplingTheory.SALD.cycle69MainSkeletonAnalyticInterfaceObligation
- Cycle-69 upper obligation for the post-route analytic-interface ledger. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40448 |
AutoSamplingTheory.SALD.cycle69MainSkeletonAnalyticMiddleContract
- Cycle-69 middle audit for the post-route analytic-interface ledger.
This synchronizes the upper ledger with the source transcript after all six
theorem skeletons have been wired. It keeps the selected lower packet on |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40488 |
AutoSamplingTheory.SALD.cycle69MainSkeletonAnalyticMiddleObligation
- Cycle-69 middle obligation tying the post-route analytic-interface audit
to the selected shared EM conditional-law/Fokker--Planck backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40576 |
AutoSamplingTheory.SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation
- Cycle-69 lower obligation for the source-sign EM FP handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40616 |
AutoSamplingTheory.SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation
- Cycle-70 middle obligation for the conditional-law/measurability slice. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40636 |
AutoSamplingTheory.SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation
- Cycle-70 lower obligation for the named conditional drift handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40654 |
AutoSamplingTheory.SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation
- Cycle-71 middle obligation for endpoint-law-to-conditional-law compatibility. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40674 |
AutoSamplingTheory.SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation
- Cycle-71 local wrapper obligation for the endpoint-to-conditional bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40693 |
AutoSamplingTheory.SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalDag
- Cycle-71 proof-DAG pane for endpoint-law-to-conditional-law compatibility. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40713 |
AutoSamplingTheory.SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation
- Cycle-72 middle obligation for the weak conditional Fokker--Planck
source-sign interface. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40773 |
AutoSamplingTheory.SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation
- Cycle-72 local wrapper obligation for weak-FP source signs. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40790 |
AutoSamplingTheory.SALD.cycle72GeneralMovingTargetDiscreteWeakFpDag
- Cycle-72 proof-DAG pane for weak conditional Fokker--Planck source signs. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40808 |
AutoSamplingTheory.SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperPacket
- Cycle-73 upper packet for the KL-derivative handoff from weak FP.
This is the fourth single-backend backfill packet after the post-route cycle-69
ledger. It keeps the active backend fixed at
`sald.general_moving_targ |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40873 |
AutoSamplingTheory.SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation
- Cycle-73 upper obligation for the KL-derivative handoff packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40937 |
AutoSamplingTheory.SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation
- Cycle-73 middle obligation for the weak-FP-to-KL derivative source map. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40955 |
AutoSamplingTheory.SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation
- Cycle-73 local wrapper obligation for weak-FP-to-KL derivative substitution. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40974 |
AutoSamplingTheory.SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpDag
- Cycle-73 proof-DAG pane for weak-FP-to-KL derivative handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:40994 |
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface
- Cycle-74 source-cited Mathlib measure interface for the blocked
conditional-kernel layer.
This is intentionally narrow: it records the Mathlib conditional-expectation
kernel that lower work should audit before attempt |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41061 |
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperPacket
- Cycle-74 upper packet selecting the minimal cited measure interface after
cycle-73 weak-FP-to-KL substitution. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41083 |
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation
- Cycle-74 upper obligation for the minimal cited measure interface. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41141 |
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation
- Cycle-74 middle obligation for the conditional-kernel source-to-Lean map. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41158 |
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation
- Cycle-74 lower obligation for the supplied-kernel regularity handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41182 |
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceDag
- Cycle-74 proof-DAG pane for the conditional-kernel measure blocker. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41204 |
AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperPacket
- Cycle-75 upper packet returning to the EM conditional-law interface.
Cycle 74 recorded the blocked conditional-kernel theorem as a precise
source-cited interface. Cycle 75 keeps the same source window and asks lower
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41295 |
AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation
- Cycle-75 upper obligation for the focused conditional-law backfill. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41357 |
AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation
- Cycle-75 middle source-to-Lean map for the conditional-law orientation and
measurability handoff.
Mathlib's `condDistrib Y X μ` is oriented by the conditioning variable first:
for `X_k^eta | hat X_s`, the generated jo |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41386 |
AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation
- Cycle-75 lower obligation for the swapped-orientation supplied-kernel
regularity wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41417 |
AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillDag
- Cycle-75 proof-DAG pane for the conditional-law construction backfill. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41440 |
AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperPacket
- Cycle-76 upper packet for endpoint-law to conditional compatibility.
Cycle 75 established the swapped `condDistrib` orientation wrapper. Cycle 76
returns to the endpoint-law bookkeeping and packages it with that swap |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41517 |
AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation
- Cycle-76 upper obligation for the endpoint-to-conditional backfill. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41579 |
AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation
- Cycle-76 middle map for endpoint-law to conditional-law compatibility. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41595 |
AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation
- Cycle-76 lower obligation for the endpoint Measure.map to swapped
conditional-kernel wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41616 |
AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalDag
- Cycle-76 proof-DAG pane for endpoint-law to conditional compatibility. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41637 |
AutoSamplingTheory.SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation
- Cycle-77 middle obligation for the generator-level weak FP source-sign
handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41704 |
AutoSamplingTheory.SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation
- Cycle-77 lower obligation for the generator-level weak FP source-sign
wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41724 |
AutoSamplingTheory.SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorDag
- Cycle-77 proof-DAG pane for the generator-level weak conditional
Fokker--Planck source-sign handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41746 |
AutoSamplingTheory.SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperPacket
- Cycle-78 upper packet for the generator-to-KL derivative handoff.
Cycle 77 sharpened the weak conditional Fokker--Planck source signs down to
generator pieces. Cycle 78 keeps the same EM backend and asks lower work t |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41812 |
AutoSamplingTheory.SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation
- Cycle-78 upper obligation for the generator-to-KL handoff packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41875 |
AutoSamplingTheory.SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation
- Cycle-78 middle obligation for the source-to-Lean KL handoff map. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41893 |
AutoSamplingTheory.SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation
- Cycle-78 local wrapper obligation for generator-piece weak-FP to KL
derivative substitution. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41915 |
AutoSamplingTheory.SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorDag
- Cycle-78 proof-DAG pane for the generator-piece weak-FP to KL derivative
handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:41937 |
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureInterface
- Cycle-79 source-cited measure/calculus interface for the weak generator
time-derivative theorem behind the EM interpolation Fokker-Planck line.
Cycles 77 and 78 compiled the equality packaging after a generator identi |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42009 |
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperPacket
- Cycle-79 upper packet for the minimal cited weak-FP generator interface. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42035 |
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperObligation
- Cycle-79 upper obligation for the weak generator-to-law cited interface. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42094 |
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureMiddleObligation
- Cycle-79 middle obligation for the weak generator-to-law source map. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42111 |
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation
- Cycle-79 lower obligation for the local Measure.map weak-test handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42133 |
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureDag
- Cycle-79 proof-DAG pane for the weak generator-to-law cited interface. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42151 |
AutoSamplingTheory.SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperPacket
- Cycle-80 upper packet returning to the conditional-law/measurability layer.
Cycle 79 exposed the weak generator-to-law theorem boundary, but that theorem
still depends on the conditional law and named drift field from |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42240 |
AutoSamplingTheory.SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperObligation
- Cycle-80 upper obligation for the conditional-law/measurability packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42303 |
AutoSamplingTheory.SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityMiddleObligation
- Cycle-80 middle source-to-Lean map for the EM conditional-law interface.
This records the exact theorem boundary still missing after cycles 74, 75, and
79: the regular conditional kernel for `X_k^eta | hat X_s=x`, the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42331 |
AutoSamplingTheory.SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation
- Cycle-80 lower obligation for the endpoint/conditional drift-regularity
wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42362 |
AutoSamplingTheory.SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityDag
- Cycle-80 proof-DAG pane for the conditional-law/measurability backfill. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42382 |
AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperPacket
- Cycle-81 upper packet for the endpoint-law-to-conditional-law bridge.
Cycle 80 compiled only a supplied-hypothesis wrapper around endpoint/orientation
and conditional-drift regularity facts. This packet keeps the nex |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42485 |
AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperObligation
- Cycle-81 upper obligation for the endpoint-to-conditional packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42549 |
AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation
- Cycle-81 middle obligation for the endpoint-to-conditional weak-FP
readiness handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42572 |
AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation
- Cycle-81 lower obligation for the endpoint-only weak-FP prerequisite
handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42593 |
AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalDag
- Cycle-81 proof-DAG pane for the endpoint-to-conditional upper packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42612 |
AutoSamplingTheory.SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperPacket
- Cycle-82 upper packet for the weak conditional Fokker--Planck source-sign
backend.
Cycle 81 supplied the endpoint/conditional readiness package consumed before
weak FP. This upper packet returns to the paper's associ |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42712 |
AutoSamplingTheory.SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsUpperObligation
- Cycle-82 upper obligation selecting the weak conditional FP source-sign
packet after the endpoint/conditional readiness work. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42784 |
AutoSamplingTheory.SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation
- Cycle-82 middle obligation for the readiness-to-source-sign bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42805 |
AutoSamplingTheory.SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation
- Cycle-82 lower obligation for the endpoint-readiness-to-source-sign bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42826 |
AutoSamplingTheory.SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsDag
- Cycle-82 proof-DAG pane for the weak conditional FP source-sign packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42848 |
AutoSamplingTheory.SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperPacket
- Cycle-83 upper packet for the endpoint weak-FP to KL-derivative handoff.
Cycle 82 accepted the endpoint/conditional source-sign wrapper. This packet
connects that accepted weak-FP source-sign output to the differenti |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:42947 |
AutoSamplingTheory.SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation
- Cycle-83 upper obligation selecting the endpoint weak-FP to KL handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43017 |
AutoSamplingTheory.SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation
- Cycle-83 middle obligation for the endpoint source-signs to KL source map. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43034 |
AutoSamplingTheory.SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation
- Cycle-83 lower obligation for the endpoint source-signs to KL wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43053 |
AutoSamplingTheory.SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpDag
- Cycle-83 proof-DAG pane for endpoint weak-FP source signs to KL
derivative handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43075 |
AutoSamplingTheory.SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperPacket
- Cycle-84 upper packet for the active EM interpolation backend after the
cycle-83 KL-derivative handoff.
The cycle focus allows a new source-cited Mathlib/measure interface only when
proof-producing work is blocked. C |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43163 |
AutoSamplingTheory.SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendUpperObligation
- Cycle-84 upper obligation selecting active EM-backend proof work before any
minimal cited measure-interface fallback. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43238 |
AutoSamplingTheory.SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation
- Cycle-84 middle obligation for the active EM-backend source map. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43260 |
AutoSamplingTheory.SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation
- Cycle-84 lower obligation for the endpoint log-action active-backend
handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43290 |
AutoSamplingTheory.SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendDag
- Cycle-84 proof-DAG pane for the active EM-backend handoff/fallback
decision. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43313 |
AutoSamplingTheory.SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperPacket
- Cycle-85 upper packet for the post-cycle-84 conditional-kernel theorem
boundary.
Cycle 84 compiled another endpoint-level handoff under supplied hypotheses. In
cycle 85 the lower work must stop adding wrappers of the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43424 |
AutoSamplingTheory.SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryUpperObligation
- Cycle-85 upper obligation selecting the conditional-kernel theorem
boundary instead of another supplied-hypothesis wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43498 |
AutoSamplingTheory.SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryMiddleObligation
- Cycle-85 middle boundary narrowing using local Mathlib conditional-kernel
helpers.
This is not another supplied-hypothesis wrapper. The local declarations in
`AutoSamplingTheory/Probability.lean` compile the Mathlib |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43532 |
AutoSamplingTheory.SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryLowerObligation
- Cycle-85 lower boundary reduction for named conditional-integral fields.
The lower packet compiles the law-space conditional-integral regularity facts
needed after the middle sample-space orientation work. The theore |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43571 |
AutoSamplingTheory.SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryDag
- Cycle-85 proof-DAG pane for the conditional-kernel theorem boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43594 |
AutoSamplingTheory.SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperPacket
- Cycle-86 upper packet returning to the generator-to-law weak FP boundary.
Cycle 85 discharged the generic named-field regularity part of the conditional
law backend under explicit Mathlib hypotheses. The next lower p |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43752 |
AutoSamplingTheory.SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation
- Cycle-86 upper obligation selecting the generator-to-law weak FP boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43823 |
AutoSamplingTheory.SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryMiddleObligation
- Cycle-86 middle source map for the generator-to-law weak FP boundary.
This is not another source-sign wrapper. It translates the paper's invocation
of the Fokker--Planck equation associated with the frozen EM interpol |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43850 |
AutoSamplingTheory.SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryLowerObligation
- Cycle-86 lower obligation for the sample-space derivative to law weak-FP
generator handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43875 |
AutoSamplingTheory.SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryDag
- Cycle-86 proof-DAG pane for the weak generator-to-law boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:43895 |
AutoSamplingTheory.SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperPacket
- Cycle-87 upper packet for the KL/log-ratio analytic boundary.
Cycle 86 removed the abstract generator equality from the weak-FP source-sign
route by transporting a supplied sample-space derivative to the law integral. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44008 |
AutoSamplingTheory.SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation
- Cycle-87 upper obligation selecting the KL/log-ratio analytic boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44084 |
AutoSamplingTheory.SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryMiddleObligation
- Cycle-87 middle source map for the KL/log-ratio analytic boundary.
This translates the source line `since int partial_s hat rho_s dx = 0` into a
lower-ready split: prove the raw differentiated KL formula with an expli |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44110 |
AutoSamplingTheory.SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation
- Cycle-87 lower scalar handoff for the KL/log-ratio mass-conservation drop. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44130 |
AutoSamplingTheory.SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryDag
- Cycle-87 proof-DAG pane for the KL/log-ratio analytic boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44154 |
AutoSamplingTheory.SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddlePacket
- Cycle-88 middle packet for the log-ratio weak-test admissibility boundary.
Cycle 87 discharged the generic finite-KL log-ratio regularity side
conditions. The next non-wrapper boundary is the supplied
`hlog : Admissi |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44310 |
AutoSamplingTheory.SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation
- Cycle-88 middle source map for the log-ratio admissibility boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44388 |
AutoSamplingTheory.SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation
- Cycle-88 lower handoff for log-ratio weak-test admissibility. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44411 |
AutoSamplingTheory.SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityDag
- Cycle-88 proof-DAG pane for the log-ratio admissibility boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44432 |
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlClosurePressureUpperPacket
- Cycle-89 upper pressure test for the discrete forward-KL theorem route.
This packet does not introduce a new theorem wrapper. It records the requested
post-cycle-84 pressure test: route `thm:forward-KL-discrete` thro |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44526 |
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlClosurePressureUpperObligation
- Cycle-89 obligation recording the pressure-test blocker. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44572 |
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlClosurePressureMiddleObligation
- Cycle-89 middle source map for the discrete forward-KL pressure test. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44606 |
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlClosurePressureLowerObligation
- Cycle-89 lower handoff for the discrete derivative IBP/FI split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44642 |
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlClosurePressureDag
- Cycle-89 proof-DAG pane for the discrete theorem closure pressure test. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44673 |
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationUpperPacket
- Cycle-90 upper packet for the reviewed discrete KL mass-conservation blocker.
The active EM backend remains the shared `appendix.tex:1358-1387` route, but
cycle 89's reviewer accepted a theorem-route blocker for
`thm: |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44819 |
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationUpperObligation
- Cycle-90 obligation selecting the mass-conservation lower boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44864 |
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationMiddleObligation
- Cycle-90 middle obligation for the compiled mass-derivative route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44890 |
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationLowerObligation
- Cycle-90 lower obligation for mapped-law constant-test mass conservation. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44914 |
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationDag
- Cycle-90 proof-DAG pane for the discrete KL mass-conservation blocker. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:44942 |
AutoSamplingTheory.SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation
- Cycle-91 middle obligation for the conditional-kernel theorem boundary.
This records the post-cycle-90 return to the active EM backend:
`appendix.tex:1368-1377`, where `bar b_{k,s}` is defined by conditioning on
`\hat |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45072 |
AutoSamplingTheory.SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation
- Cycle-91 lower/backfill obligation for the compiled concrete
`condDistrib` drift-regularity theorem.
The new theorem removes the older supplied component regularity hypotheses for
the canonical conditional-integral ro |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45099 |
AutoSamplingTheory.SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelDag
- Cycle-91 proof-DAG pane for the conditional-kernel component-field
backfill. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45123 |
AutoSamplingTheory.SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitMiddleObligation
- Cycle-92 middle obligation for the weak generator-to-law boundary.
This keeps the conversion window on `appendix.tex:1379-1387`: the paper invokes
the Fokker--Planck equation for the frozen EM interpolation, and the c |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45212 |
AutoSamplingTheory.SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation
- Cycle-92 lower obligation for the split-generator law-transport handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45232 |
AutoSamplingTheory.SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitDag
- Cycle-92 proof-DAG pane for the split-generator weak-FP boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45253 |
AutoSamplingTheory.SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeMiddleObligation
- Cycle-93 middle obligation for the KL/log-ratio mass derivative boundary.
This returns to `appendix.tex:1358-1366` after cycle 92's accepted weak-FP
split-generator handoff. The selected middle packet removes one sup |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45347 |
AutoSamplingTheory.SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation
- Cycle-93 lower obligation for the compiled mapped-law mass handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45368 |
AutoSamplingTheory.SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeDag
- Cycle-93 proof-DAG pane for the KL/log-ratio mass derivative boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45393 |
AutoSamplingTheory.SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionMiddleObligation
- Cycle-94 middle obligation for the conditional-drift source-action
boundary.
This returns to the weak Fokker--Planck invocation at `appendix.tex:1379-1387`
after the cycle-93 KL/log-ratio mass handoff. The selected s |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45498 |
AutoSamplingTheory.SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation
- Cycle-94 lower obligation for the compiled `barB` drift-action handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45521 |
AutoSamplingTheory.SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionDag
- Cycle-94 proof-DAG pane for the conditional-drift weak action boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45546 |
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlClosurePressureUpperPacket
- Cycle-95 upper packet for the discrete forward-KL closure pressure test.
This is an upper-role route record only. It rechecks
`thm:forward-KL-discrete` after the cycle-94 `barB` weak-action handoff and
selects the ne |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45622 |
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlClosurePressureUpperObligation
- Cycle-95 obligation recording the discrete theorem pressure-test blocker. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45670 |
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlClosurePressureMiddleObligation
- Cycle-95 middle source map for the discrete forward-KL pressure test. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45701 |
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlClosurePressureLowerObligation
- Cycle-95 lower obligation for the component-pairing reduction of the
`barB` drift-action blocker. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45735 |
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlClosurePressureDag
- Cycle-95 proof-DAG pane for the post-cycle-94 pressure test. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45761 |
AutoSamplingTheory.SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingMiddleObligation
- Cycle-96 middle obligation for the active EM conditional-law backend.
The upper packet rejected the non-EM LSI/DV/Gronwall fallback because the EM
backend still has named work at `appendix.tex:1368-1387`. This middle |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45894 |
AutoSamplingTheory.SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation
- Cycle-96 lower obligation for the compiled one-component pairing handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45925 |
AutoSamplingTheory.SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingDag
- Cycle-96 proof-DAG pane for the condexp generator-pairing middle packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:45947 |
AutoSamplingTheory.SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingMiddleObligation
- Cycle-97 middle obligation for the canonical conditional-integral pairing.
This cycle stops adding supplied-hypothesis wrappers and proves the
Mathlib-style map-law disintegration theorem that lower needs for the
cano |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46043 |
AutoSamplingTheory.SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingLowerObligation
- Cycle-97 lower-ready obligation after the compiled disintegration theorem. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46069 |
AutoSamplingTheory.SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingDag
- Cycle-97 proof-DAG pane for the canonical `condDistrib` pairing packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46089 |
AutoSamplingTheory.SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation
- Cycle-98 middle obligation for the `barB` weak-divergence boundary.
The active source span is the Fokker--Planck source-sign line
`appendix.tex:1379-1387`. Cycle 98 keeps the packet on the divergence half of
`ASTIS.S |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46175 |
AutoSamplingTheory.SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation
- Cycle-98 lower-ready obligation for the compiled integral no-boundary
handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46198 |
AutoSamplingTheory.SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryDag
- Cycle-98 proof-DAG pane for the `barB` no-boundary integral packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46222 |
AutoSamplingTheory.SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeMiddleObligation
- Cycle-99 middle obligation for the KL/log-ratio analytic boundary.
This returns to `appendix.tex:1358-1366` and narrows the remaining primitive
`hklRaw` display to a source-cited raw-KL theorem at the exact Mathlib
`l |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46299 |
AutoSamplingTheory.SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeLowerObligation
- Cycle-99 lower-ready obligation for the finite-KL `llr` raw-KL package. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46325 |
AutoSamplingTheory.SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeDag
- Cycle-99 proof-DAG pane for the raw KL finite-KL `llr` boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46352 |
AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefMiddleObligation
- Cycle-100 middle obligation for the `barB` weak-pairing definition
alignment inside the no-boundary drift source route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46447 |
AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation
- Cycle-100 lower-ready obligation for the compiled weak-pairing definition
alignment handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46467 |
AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefDag
- Cycle-100 proof-DAG pane for the `barB` weak-pairing definition
alignment packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46488 |
AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation
- Cycle-100 lower-ready obligation for the inner-gradient contraction
handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46549 |
AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundDag
- Cycle-100 proof-DAG pane for the inner-gradient contraction packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46570 |
AutoSamplingTheory.SALD.cycle101DiscreteForwardKlClosurePressureMiddleObligation
- Cycle-101 middle synchronization for the discrete forward-KL closure pressure
test.
This is intentionally not another broad theorem-route wrapper. Cycles 89 and
95 already record the route through the discrete theore |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46626 |
AutoSamplingTheory.SALD.cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation
- Cycle-101 lower product-rule handoff for the no-boundary `barB` drift
boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46656 |
AutoSamplingTheory.SALD.cycle101DiscreteForwardKlClosurePressureDag
- Cycle-101 proof-DAG pane for the pressure test after the cycle-100
inner-gradient handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46679 |
AutoSamplingTheory.SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryMiddleObligation
- Cycle-102 middle/lower-ready handoff for the remaining zero-boundary-flux
piece of the `hatRhoS * barB` no-boundary theorem.
The cycle stays on the active EM backend and does not open the non-EM
LSI/DV/Gronwall fallba |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46756 |
AutoSamplingTheory.SALD.cycle102DiscreteForwardKlTraceZeroLowerObligation
- Cycle-102 lower handoff reducing the trace-product condition to zero test
trace on the boundary.
This removes the supplied `htraceProductZero` premise from the trace-boundary
route when admissible tests have zero boun |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46784 |
AutoSamplingTheory.SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryDag
- Cycle-102 proof-DAG pane for the trace-product zero-flux packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46801 |
AutoSamplingTheory.SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionMiddleObligation
- Cycle-103 middle obligation for the `condC` conditional-kernel
component-version theorem.
This returns to the conditional drift definition at `appendix.tex:1368-1377`
and narrows the cycle-91 remaining boundary to one |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46879 |
AutoSamplingTheory.SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionLowerObligation
- Cycle-103 lower obligation after the compiled `condExpKernel.map` bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46906 |
AutoSamplingTheory.SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionDag
- Cycle-103 proof-DAG pane for the one-component conditional-kernel
versioning packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:46929 |
AutoSamplingTheory.SALD.cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportUpperObligation
- Cycle-104 upper packet for the named-law generator-to-law weak-FP
transport boundary.
This cycle stays on the active EM conditional-law/Fokker--Planck backend over
`appendix.tex:1358-1387`, narrowed to `appendix.tex:1 |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47022 |
AutoSamplingTheory.SALD.cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportLowerObligation
- Cycle-104 lower obligation for the compiled named-law weak derivative
transport theorem. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47042 |
AutoSamplingTheory.SALD.cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportDag
- Cycle-104 proof-DAG pane for the named-law generator-to-law weak-FP
transport packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47063 |
AutoSamplingTheory.SALD.cycle105GeneralMovingTargetDiscretePureRawKlDerivativeMiddleObligation
- Cycle-105 middle obligation for the pure no-mass KL/log-ratio boundary.
This cycle returns to `appendix.tex:1358-1366` and narrows the cycle-99
no-mass package. Once the mass term has been removed, the remaining KL
d |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47144 |
AutoSamplingTheory.SALD.cycle105GeneralMovingTargetDiscretePureRawKlDerivativeLowerObligation
- Cycle-105 lower obligation for the compiled pure no-mass KL handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47166 |
AutoSamplingTheory.SALD.cycle105GeneralMovingTargetDiscretePureRawKlDerivativeDag
- Cycle-105 proof-DAG pane for the pure no-mass KL/log-ratio boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47189 |
AutoSamplingTheory.SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftMiddleObligation
- Cycle-106 middle obligation for canonical conditional-integral drift
regularity.
This cycle returns to the conditional-drift line `appendix.tex:1368-1377` and
chooses the canonical `condDistrib` representative of `bar |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47255 |
AutoSamplingTheory.SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftLowerObligation
- Cycle-106 lower obligation for the compiled canonical `condDistrib` drift
regularity theorem. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47278 |
AutoSamplingTheory.SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftDag
- Cycle-106 proof-DAG pane for canonical conditional-integral drift
regularity. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47296 |
AutoSamplingTheory.SALD.cycle107DiscreteForwardKlBoundaryFluxIntegralLowerObligation
- Cycle-107 lower obligation for the Mathlib box divergence theorem
specialization behind the boundary-flux integral representation.
This packet acts on the `hboundaryFluxIntegral` premise consumed by
`generalMovingTarg |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47382 |
AutoSamplingTheory.SALD.cycle107DiscreteForwardKlBoundaryFluxIntegralDag
- Cycle-107 proof-DAG pane for the boundary-flux integral packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47404 |
AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityLowerObligation
- Cycle-108 lower obligation for the concrete product-flux instantiation of
the cycle-107 Mathlib box theorem.
This packet stays on the active EM backend. It acts only on the continuity
piece of
`ASTIS.SALD.forward_KL_ |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47462 |
AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDag
- Cycle-108 proof-DAG pane for the concrete `hatRhoS * barB` continuity
piece of the box-trace boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47485 |
AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeLowerObligation
- Cycle-108 lower obligation for the concrete product-flux Frechet
derivative sub-boundary.
This packet stays below the active EM no-boundary backend. It acts only on
the Frechet differentiability piece of
`ASTIS.SALD. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47544 |
AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDag
- Cycle-108 proof-DAG pane for the concrete `hatRhoS * barB` derivative
piece of the box-trace boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47569 |
AutoSamplingTheory.SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary
- Cycle-109 exact missing theorem for the named `barB` source definition.
This is the lower-ready boundary behind `appendix.tex:1368-1377` after the
canonical `condDistrib` regularity theorem from cycle 106. The target |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47627 |
AutoSamplingTheory.SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefMiddleObligation
- Cycle-109 middle packet for the named `barB` source-definition bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47651 |
AutoSamplingTheory.SALD.cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation
- Cycle-109 lower packet using Mathlib's product conditional-expectation
identity for the named `barB` source definition. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47675 |
AutoSamplingTheory.SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefDag
- Cycle-109 proof-DAG pane for the named `barB` source-definition
boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47696 |
AutoSamplingTheory.SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasMiddleObligation
- Cycle-110 middle packet for the selected named `barB` representative.
This keeps the post-cycle-109 lower packet on the source conditional-drift
definition. The only supplied side condition discharged here is the
equ |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47828 |
AutoSamplingTheory.SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasLowerObligation
- Cycle-110 lower obligation for equality-set measurability of named `barB`. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47846 |
AutoSamplingTheory.SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasDag
- Cycle-110 proof-DAG pane for the named `barB` equality-set packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47863 |
AutoSamplingTheory.SALD.cycle110GeneralMovingTargetDiscreteWeakFpDominatedGeneratorLowerObligation
- Cycle-110 lower obligation for the dominated parametric-integral
generator-to-law step.
This is the assigned appendix.tex:1379-1387 packet: it does not add another
source-sign wrapper. Instead it derives the sample-s |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47947 |
AutoSamplingTheory.SALD.cycle110GeneralMovingTargetDiscreteWeakFpDominatedGeneratorDag
- Cycle-110 proof-DAG pane for the dominated generator-to-law weak-FP
transport packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:47968 |
AutoSamplingTheory.SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeMiddleObligation
- Cycle-111 middle obligation for the KL target-time subboundary.
The active source slice is still `appendix.tex:1358-1366`. This packet does
not restate the whole pure raw-KL package; it isolates the target-density ti |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48035 |
AutoSamplingTheory.SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeLowerObligation
- Cycle-111 lower obligation for the dominated target-time theorem. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48056 |
AutoSamplingTheory.SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeDag
- Cycle-111 proof-DAG pane for the target-time KL derivative packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48078 |
AutoSamplingTheory.SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeMiddleObligation
- Cycle-112 middle obligation for the selected named `barB` representative.
The active source slice is `appendix.tex:1368-1377`. This packet narrows the
remaining `hbarBCondExp` premise after cycle 110 by replacing it |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48179 |
AutoSamplingTheory.SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeLowerObligation
- Cycle-112 lower obligation for the compiled conditional-expectation
representative handoff. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48200 |
AutoSamplingTheory.SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeDag
- Cycle-112 proof-DAG pane for the named `barB` conditional-expectation
representative boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48225 |
AutoSamplingTheory.SALD.cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityLowerObligation
- Cycle-113 lower obligation for pulling the selected named `barB`
regularity back from the state marginal.
The discrete theorem pressure test reaches the cycle-112 named `barB`
representative boundary. This lower pack |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48334 |
AutoSamplingTheory.SALD.cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityDag
- Cycle-113 proof-DAG pane for the named `barB` regularity pullback. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48354 |
AutoSamplingTheory.SALD.cycle114GeneralMovingTargetDiscreteCanonicalBarBStateEventSetIntegralLowerObligation
- Cycle-114 lower obligation for the canonical state-event set-integral
part of the remaining named `barB` boundary.
This packet stays on the dynamic leaf
`ASTIS.SALD.cycle113.remaining_named_barB_state_event_set_integr |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48427 |
AutoSamplingTheory.SALD.cycle114GeneralMovingTargetDiscreteCanonicalBarBStateEventSetIntegralDag
- Cycle-114 proof-DAG pane for the canonical state-event set-integral
narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48447 |
AutoSamplingTheory.SALD.cycle115GeneralMovingTargetDiscreteNamedBarBSelectedVersionMiddleObligation
- Cycle-115 middle obligation for the selected-version boundary left after
the canonical state-event set-integral theorem.
This packet stays on
`ASTIS.SALD.cycle114.remaining_named_barB_version_after_canonical_state_eve |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48502 |
AutoSamplingTheory.SALD.cycle115GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation
- Cycle-115 lower obligation for the source conditional-expectation version
of the selected `barB` bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48522 |
AutoSamplingTheory.SALD.cycle115GeneralMovingTargetDiscreteNamedBarBSelectedVersionDag
- Cycle-115 proof-DAG pane for the selected named `barB` version boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48541 |
AutoSamplingTheory.SALD.cycle116GeneralMovingTargetDiscreteCanonicalBarBCondExpLowerObligation
- Cycle-116 lower obligation for the canonical conditional-expectation
representative of the named `barB` drift.
This packet stays on
`ASTIS.SALD.cycle115.remaining_named_barB_condExp_source_representative` for
`appendi |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48617 |
AutoSamplingTheory.SALD.cycle116GeneralMovingTargetDiscreteCanonicalBarBCondExpDag
- Cycle-116 proof-DAG pane for the canonical `barB` conditional-expectation
representative. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48639 |
AutoSamplingTheory.SALD.cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionMiddleObligation
- Cycle-117 middle obligation for the selected paper `barB` version step.
The refreshed blueprint illness area is not another conditional-expectation
wrapper. Cycle 116 already proves the old `hbarBCondExp` input for t |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48713 |
AutoSamplingTheory.SALD.cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionLowerObligation
- Cycle-117 lower obligation for the source-supported pointwise canonical
representative choice.
The lower packet compiles the bridge from the source's pointwise definition of
`\bar b_{k,s}` as the canonical conditional |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48743 |
AutoSamplingTheory.SALD.cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionDag
- Cycle-117 proof-DAG pane for the selected paper `barB` version boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48762 |
AutoSamplingTheory.SALD.cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamMiddleObligation
- Cycle-118 middle obligation for direct canonical downstream use.
The refreshed illness area is no longer a request for another
`hbarBCondExp` bridge. The source definition at `appendix.tex:1368-1377`
allows the lower |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48839 |
AutoSamplingTheory.SALD.cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamLowerObligation
- Cycle-118 lower obligation for the direct canonical EM state-event
interface.
This lower packet compiles the concrete representative handoff requested by
the refreshed illness area: the downstream interface may take ` |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48869 |
AutoSamplingTheory.SALD.cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamDag
- Cycle-118 proof-DAG pane for the direct canonical `barB` downstream route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48888 |
AutoSamplingTheory.SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerMiddleObligation
- Cycle-119 middle obligation for consuming the canonical `barB` witness in
the weak-FP generator/source-sign path.
The refreshed illness area after cycle 118 is no longer the representative
choice for `barB`. This pac |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48944 |
AutoSamplingTheory.SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerLowerObligation
- Cycle-119 lower obligation for the canonical `barB` weak-FP consumer.
This is a proof-producing lower packet, not a theorem-status promotion. It
compiles the consumer that takes the canonical conditional-drift/state- |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48974 |
AutoSamplingTheory.SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerDag
- Cycle-119 proof-DAG pane for the canonical `barB` weak-FP consumer. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:48999 |
AutoSamplingTheory.SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationMiddleObligation
- Cycle-120 middle obligation for the EM sample-path derivative and
domination inputs.
After the cycle-119 canonical `barB` weak-FP consumer compiled, the remaining
source-cited theorem was still a bundle. This middle |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49091 |
AutoSamplingTheory.SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationLowerObligation
- Cycle-120 lower-ready obligation for the concrete EM path
derivative/domination package.
The lower theorem should be proved from the frozen interpolation formula and
admissible-test regularity. It should not absorb t |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49117 |
AutoSamplingTheory.SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag
- Cycle-120 proof-DAG pane for the EM sample-path derivative/domination
subboundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49139 |
AutoSamplingTheory.SALD.cycle121GeneralMovingTargetDiscreteEmSampleMeasLowerObligation
- Cycle-121 lower obligation for discharging the sample measurability input
from the EM interval dominated packet.
This packet stays inside the cycle-120 EM path-derivative/domination boundary.
It removes only the suppl |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49246 |
AutoSamplingTheory.SALD.cycle121GeneralMovingTargetDiscreteEmSampleMeasDag
- Cycle-121 proof-DAG pane for sample measurability discharge inside the EM
path-derivative/domination boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49268 |
AutoSamplingTheory.SALD.cycle122GeneralMovingTargetDiscreteEmSampleIntLowerObligation
- Cycle-122 lower obligation for discharging the sample integrability input
from the EM interval measurable dominated packet.
This packet stays inside the cycle-121 EM path-derivative/domination boundary.
It removes onl |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49334 |
AutoSamplingTheory.SALD.cycle122GeneralMovingTargetDiscreteEmSampleIntDag
- Cycle-122 proof-DAG pane for sample integrability discharge inside the EM
path-derivative/domination boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49357 |
AutoSamplingTheory.SALD.cycle123GeneralMovingTargetDiscreteEmSampleDerivMeasLowerObligation
- Cycle-123 lower obligation for discharging the sample derivative
measurability input from the EM interval measurable/integrable dominated
packet.
This packet stays inside the cycle-122 EM path-derivative/domination bo |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49428 |
AutoSamplingTheory.SALD.cycle123GeneralMovingTargetDiscreteEmSampleDerivMeasDag
- Cycle-123 proof-DAG pane for derivative measurability discharge inside the
EM path-derivative/domination boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49450 |
AutoSamplingTheory.SALD.cycle124GeneralMovingTargetDiscreteEmSampleDerivBoundLowerObligation
- Cycle-124 lower obligation for discharging the sample derivative bound input
from the EM interval measurable/integrable/derivative-measurable dominated
packet.
This packet stays inside the cycle-123 EM path-derivative |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49518 |
AutoSamplingTheory.SALD.cycle124GeneralMovingTargetDiscreteEmSampleDerivBoundDag
- Cycle-124 proof-DAG pane for derivative-bound discharge inside the EM
path-derivative/domination boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49538 |
AutoSamplingTheory.SALD.cycle124GeneralMovingTargetDiscreteEmBoundIntLowerObligation
- Cycle-124 lower obligation for discharging the bound-integrability input
from the EM interval derivative-bound dominated packet.
This packet stays inside the EM path-derivative/domination dynamic leaf. It
removes onl |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49602 |
AutoSamplingTheory.SALD.cycle124GeneralMovingTargetDiscreteEmBoundIntDag
- Cycle-124 proof-DAG pane for bound-integrability transport inside the EM
path-derivative/domination boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49624 |
AutoSamplingTheory.SALD.cycle125GeneralMovingTargetDiscreteEmPathDerivLowerObligation
- Cycle-125 lower obligation for discharging the pointwise path-derivative
input from the EM interval bound-integrability dominated packet.
This packet stays inside the EM path-derivative/domination dynamic leaf. It
re |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49691 |
AutoSamplingTheory.SALD.cycle125GeneralMovingTargetDiscreteEmPathDerivDag
- Cycle-125 proof-DAG pane for path-derivative discharge inside the EM
path-derivative/domination boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49712 |
AutoSamplingTheory.SALD.cycle126GeneralMovingTargetDiscreteEmDerivValueLowerObligation
- Cycle-126 lower obligation for discharging the derivative-value input from
the EM interval path-derivative dominated packet.
This packet stays inside the EM conditional-law/Fokker--Planck backend. It
removes only the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49777 |
AutoSamplingTheory.SALD.cycle126GeneralMovingTargetDiscreteEmDerivValueDag
- Cycle-126 proof-DAG pane for derivative-value discharge inside the EM
conditional-drift/weak-Fokker--Planck backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49797 |
AutoSamplingTheory.SALD.cycle127GeneralMovingTargetDiscreteEmDriftActionLowerObligation
- Cycle-127 lower obligation for discharging the canonical `barB` drift
weak-action input from the EM interval path-value dominated packet.
This packet stays inside the EM conditional-law/Fokker--Planck backend. It
rem |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49864 |
AutoSamplingTheory.SALD.cycle127GeneralMovingTargetDiscreteEmPairMeasLowerObligation
- Cycle-127 lower obligation for discharging the raw canonical `barB`
pairing-measurability input from the drift-action packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49884 |
AutoSamplingTheory.SALD.cycle127GeneralMovingTargetDiscreteEmDriftActionDag
- Cycle-127 proof-DAG pane for the canonical `barB` drift-action discharge
inside the EM conditional-drift/weak-Fokker--Planck backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49902 |
AutoSamplingTheory.SALD.cycle128GeneralMovingTargetDiscreteEmNoBoundaryTraceLowerObligation
- Cycle-128 lower obligation for narrowing the direct canonical `barB`
no-boundary input in the EM weak-FP consumer. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:49982 |
AutoSamplingTheory.SALD.cycle128GeneralMovingTargetDiscreteEmCanonicalBarBMeasLowerObligation
- Cycle-128 lower obligation for discharging the canonical `barB`
measurability input after the no-boundary trace refiner. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50003 |
AutoSamplingTheory.SALD.cycle128GeneralMovingTargetDiscreteEmNoBoundaryTraceDag
- Cycle-128 proof-DAG pane for the canonical `barB` no-boundary trace
refinement inside the EM conditional-drift/weak-Fokker--Planck backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50023 |
AutoSamplingTheory.SALD.cycle129GeneralMovingTargetDiscreteEmDiffusionSourceLowerObligation
- Cycle-129 lower-ready obligation for narrowing the remaining diffusion
source-action input in the canonical EM weak-FP consumer. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50106 |
AutoSamplingTheory.SALD.cycle129GeneralMovingTargetDiscreteEmDiffusionSourceDag
- Cycle-129 proof-DAG pane for the remaining diffusion source-action
boundary inside the canonical EM weak-FP backend. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50125 |
AutoSamplingTheory.SALD.cycle130GeneralMovingTargetDiscreteEmLaplacianIbPLowerObligation
- Cycle-130 lower-ready obligation for narrowing the remaining weak
Laplacian action input in the EM diffusion-source helper. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50180 |
AutoSamplingTheory.SALD.cycle130GeneralMovingTargetDiscreteEmGreenLaplacianIbPScoutObligation
- Cycle-130 lower_1 scout obligation for narrowing the weak Laplacian IBP
identity to the two Green-identity steps visible in the source proof. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50198 |
AutoSamplingTheory.SALD.cycle130GeneralMovingTargetDiscreteEmFirstGreenNoBoundaryFluxLowerObligation
- Cycle-130 lower_2 obligation narrowing the first Green identity to
no-boundary flux algebra. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50219 |
AutoSamplingTheory.SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenNoBoundaryFluxLowerObligation
- Cycle-131 middle/lower-ready obligation narrowing the second Green identity
to no-boundary flux algebra. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50240 |
AutoSamplingTheory.SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenTraceBoundaryLowerObligation
- Cycle-131 lower_1 scout/lower handoff narrowing the second-Green zero
boundary flux input to a trace-product no-boundary condition. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50260 |
AutoSamplingTheory.SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenBoxBoundaryFluxLowerObligation
- Cycle-131 lower_2 obligation narrowing the second-Green boundary-flux
integral input to the existing Mathlib box-divergence interface. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50277 |
AutoSamplingTheory.SALD.cycle131GeneralMovingTargetDiscreteEmSecondGreenNoBoundaryFluxDag
- Cycle-131 proof-DAG pane for removing the direct second Green premise from
the EM diffusion-source action path. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50296 |
AutoSamplingTheory.SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTestTraceZeroLowerObligation
- Cycle-132 lower packet narrowing the remaining second-Green trace-product
zero input to the source-facing zero-test-trace condition for admissible weak
tests. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50400 |
AutoSamplingTheory.SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTraceEqTestTraceZeroScoutObligation
- Cycle-132 lower_1 scout continuation narrowing the second-Green zero-test
trace input to trace identification plus the existing admissible-test zero
trace boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50418 |
AutoSamplingTheory.SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenPointwiseTraceEqLowerObligation
- Cycle-132 lower_2 continuation narrowing second-Green trace identification
from an a.e. statement to pointwise selected-trace equality. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50437 |
AutoSamplingTheory.SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTestTraceZeroDag
- Cycle-132 proof-DAG pane for the second-Green zero-test-trace narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50454 |
AutoSamplingTheory.SALD.cycle133GeneralMovingTargetDiscreteEmSecondGreenPointwiseTestTraceZeroLowerObligation
- Cycle-133 middle packet narrowing the admissible-test zero boundary trace
input from an a.e. statement to a pointwise source theorem. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50558 |
AutoSamplingTheory.SALD.cycle133GeneralMovingTargetDiscreteEmTestLaplacianNormalizationScoutObligation
- Cycle-133 lower_1 scout packet narrowing the test-Laplacian normalization
input from the broad weak-FP context to a test-local theorem. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50576 |
AutoSamplingTheory.SALD.cycle133GeneralMovingTargetDiscreteEmTestLaplacianOperatorNormalizationLowerObligation
- Cycle-133 lower_2 packet narrowing the test-local Laplacian normalization
to an operator-level source identity for the regular test calculus. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50593 |
AutoSamplingTheory.SALD.cycle133GeneralMovingTargetDiscreteEmSecondGreenPointwiseTestTraceZeroDag
- Cycle-133 proof-DAG pane for the pointwise admissible-test trace boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50609 |
AutoSamplingTheory.SALD.cycle134GeneralMovingTargetDiscreteEmTestLaplacianSourcePullbackLowerObligation
- Cycle-134 dynamic-leaf packet narrowing the operator-level
test-Laplacian normalization to shared source-pullback definitions. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50717 |
AutoSamplingTheory.SALD.cycle134GeneralMovingTargetDiscreteEmSourceLaplacianStdBasisScoutObligation
- Cycle-134 lower_1 proof-scout obligation for the Mathlib source formula
below the shared source-pullback Laplacian boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50736 |
AutoSamplingTheory.SALD.cycle134GeneralMovingTargetDiscreteEmWeakFpLaplacianStdBasisLowerObligation
- Cycle-134 lower_2 obligation narrowing the weak-FP Laplacian definition
leaf to the standard-basis source formula. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50754 |
AutoSamplingTheory.SALD.cycle134GeneralMovingTargetDiscreteEmTestLaplacianSourcePullbackDag
- Cycle-134 proof-DAG pane for the source-pullback test-Laplacian boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50771 |
AutoSamplingTheory.SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisLowerObligation
- Cycle-135 lower obligation narrowing the test-calculus Laplacian action
definition leaf to the standard-basis source formula. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50876 |
AutoSamplingTheory.SALD.cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation
- Cycle-135 lower_2 downstream consumer for the standard-basis
test-Laplacian source formulas. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50896 |
AutoSamplingTheory.SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisDag
- Cycle-135 proof-DAG pane for the test-calculus standard-basis boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:50916 |
AutoSamplingTheory.SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityLowerObligation
- Cycle-136 lower obligation narrowing the weak-FP standard-basis source
formula to a density-Laplacian action formula. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51023 |
AutoSamplingTheory.SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityDag
- Cycle-136 proof-DAG pane for the weak-FP standard-basis source-density
boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51045 |
AutoSamplingTheory.SALD.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionLowerObligation
- Cycle-137 lower obligation narrowing the weak-FP density-Laplacian action
boundary to pointwise weak Laplacian integration by parts. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51151 |
AutoSamplingTheory.SALD.cycle137GeneralMovingTargetDiscreteEmPointwiseGreenIbPScoutObligation
- Cycle-137 lower_1 scout obligation splitting pointwise weak Laplacian IBP
into the three Green/test-calculus identities from the source proof. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51171 |
AutoSamplingTheory.SALD.cycle137GeneralMovingTargetDiscreteEmSecondGreenPointwiseBoxBoundaryFluxLowerObligation
- Cycle-137 lower_2 narrowing of the second-Green pointwise leaf to the
box-divergence and pointwise-trace boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51190 |
AutoSamplingTheory.SALD.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionDag
- Cycle-137 proof-DAG pane for the weak-FP density-Laplacian pointwise
integration-by-parts boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51210 |
AutoSamplingTheory.SALD.cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxLowerObligation
- Cycle-138 narrowing of the first-Green pointwise leaf to boundary-flux
cancellation facts. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51321 |
AutoSamplingTheory.SALD.cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseSourcePullbackScoutObligation
- Cycle-138 lower_1 scout bridge narrowing pointwise test-Laplacian
normalization to shared source-pullback definitions. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51340 |
AutoSamplingTheory.SALD.cycle138GeneralMovingTargetDiscreteEmTestLaplacianPointwiseStdBasisLowerObligation
- Cycle-138 lower_2 bridge from the test-calculus standard-basis source
formula to the pointwise test-Laplacian normalization leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51358 |
AutoSamplingTheory.SALD.cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxDag
- Cycle-138 proof-DAG pane for the first-Green pointwise boundary-flux
sub-boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51377 |
AutoSamplingTheory.SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation
- Cycle-139 weak-FP source-Laplacian field split for the pointwise
test-Laplacian route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51486 |
AutoSamplingTheory.SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation
- Cycle-139 lower_1 state-integral scout for `hweakFpSourceActionDef`. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51505 |
AutoSamplingTheory.SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation
- Cycle-139 lower_2 source-Laplacian state-integral narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51520 |
AutoSamplingTheory.SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianDag
- Cycle-139 proof-DAG pane for the weak-FP source-Laplacian field split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51539 |
AutoSamplingTheory.SALD.cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralLowerObligation
- Cycle-140 narrowing of the source-Laplacian state-integral leaf to the
frozen EM generator component. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51667 |
AutoSamplingTheory.SALD.cycle140GeneralMovingTargetDiscreteEmGeneratorLawIntegralScoutObligation
- Cycle-140 lower_1 law-integral scout for the frozen EM generator
Laplacian component. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51687 |
AutoSamplingTheory.SALD.cycle140GeneralMovingTargetDiscreteEmGeneratorSourceFunctionalLowerObligation
- Cycle-140 lower_2 source-functional narrowing for the frozen EM generator
Laplacian law integral. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51707 |
AutoSamplingTheory.SALD.cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralDag
- Cycle-140 proof-DAG pane for the source-Laplacian state-integral boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51725 |
AutoSamplingTheory.SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceLowerObligation
- Cycle-141 narrowing of the frozen EM generator source-action leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51858 |
AutoSamplingTheory.SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceFieldSourceScoutObligation
- Cycle-141 lower_1 scout narrowing below the EM generator standard-basis leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51877 |
AutoSamplingTheory.SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceLawIntegralLowerObligation
- Cycle-141 lower_2 law-integral narrowing below the trace-action leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51894 |
AutoSamplingTheory.SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralMiddleObligation
- Cycle-142 state-integral narrowing below the trace-law leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51911 |
AutoSamplingTheory.SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianStateIntegralScoutObligation
- Cycle-142 lower_1 scout narrowing below the trace-state integral leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51929 |
AutoSamplingTheory.SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation
- Cycle-142 lower_2 law-integral narrowing below the EM Laplacian
state-integral leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51950 |
AutoSamplingTheory.SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceDag
- Cycle-141 proof-DAG pane for the EM generator source-action split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:51969 |
AutoSamplingTheory.SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralDag
- Cycle-142 proof-DAG pane for the EM generator trace state-integral split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52118 |
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventLowerObligation
- Cycle-143 state-event narrowing for the frozen EM generator Laplacian law
integral. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52297 |
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventDag
- Cycle-143 proof-DAG pane for the state-event law-integral split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52316 |
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventLowerObligation
- Cycle-143 lower-1 pointwise narrowing for the frozen EM generator
Laplacian state-event formula. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52402 |
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventDag
- Cycle-143 lower-1 proof-DAG pane for pointwise event-field narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52421 |
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefLowerObligation
- Cycle-143 lower-2 action-definition narrowing for the frozen EM generator
Laplacian total-event formula. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52505 |
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefDag
- Cycle-143 lower-2 proof-DAG pane for the action-definition narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52523 |
AutoSamplingTheory.SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventLowerObligation
- Cycle-144 standard-basis event-field narrowing for the frozen EM generator
Laplacian pointwise event formula. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52609 |
AutoSamplingTheory.SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventDag
- Cycle-144 proof-DAG pane for the standard-basis frozen EM event-field
narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52630 |
AutoSamplingTheory.SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionLowerObligation
- Cycle-144 lower-2 standard-basis action narrowing for the frozen EM
generator Laplacian action definition. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52719 |
AutoSamplingTheory.SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDag
- Cycle-144 lower-2 proof-DAG pane for the standard-basis action
narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52739 |
AutoSamplingTheory.SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawLowerObligation
- Cycle-145 law-integral narrowing for the frozen EM generator
standard-basis Laplacian action definition. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52828 |
AutoSamplingTheory.SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawDag
- Cycle-145 proof-DAG pane for the law-integral to standard-basis action
narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52849 |
AutoSamplingTheory.SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventLowerObligation
- Cycle-145 lower_2 narrowing for the frozen EM generator Laplacian
event-field standard-basis definition. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52938 |
AutoSamplingTheory.SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventDag
- Cycle-145 lower_2 proof-DAG pane for the trace-field event narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:52958 |
AutoSamplingTheory.SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventLowerObligation
- Cycle-146 narrowing for the law-space generator Laplacian integral under
the latest trace-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53045 |
AutoSamplingTheory.SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventDag
- Cycle-146 proof-DAG pane for the trace-event total-event narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53068 |
AutoSamplingTheory.SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianLowerObligation
- Cycle-146 lower_1 narrowing for the trace-field standard-basis formula
inside the latest trace-event total-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53136 |
AutoSamplingTheory.SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianDag
- Cycle-146 lower_1 proof-DAG pane for the trace-field Laplacian
narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53157 |
AutoSamplingTheory.SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventLowerObligation
- Cycle-146 lower_2 narrowing for the event-field/trace-field equality
inside the latest trace-event total-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53241 |
AutoSamplingTheory.SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventDag
- Cycle-146 lower_2 proof-DAG pane for the pointwise trace-event
narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53261 |
AutoSamplingTheory.SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianLowerObligation
- Cycle-147 narrowing for the total-event formula inside the current
pointwise trace-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53344 |
AutoSamplingTheory.SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianDag
- Cycle-147 proof-DAG pane for the pointwise action-definition
trace-Laplacian narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53363 |
AutoSamplingTheory.SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseStdBasisActionTraceLaplacianLowerObligation
- Cycle-147 lower-1 proof-scout narrowing for the action-definition premise
inside the current pointwise event/trace-Laplacian route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53442 |
AutoSamplingTheory.SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseStdBasisActionTraceLaplacianDag
- Cycle-147 lower-1 proof-DAG pane for the standard-basis action narrowing
inside the pointwise event/trace-Laplacian route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53463 |
AutoSamplingTheory.SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseLawIntegralTraceLaplacianLowerObligation
- Cycle-147 lower-2 narrowing for the standard-basis action premise inside
the current pointwise event/trace-Laplacian route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53546 |
AutoSamplingTheory.SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseLawIntegralTraceLaplacianDag
- Cycle-147 lower-2 proof-DAG pane for the law-integral narrowing inside the
pointwise event/trace-Laplacian route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53567 |
AutoSamplingTheory.SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorPointwiseStateEventTraceLaplacianMiddleObligation
- Cycle-148 middle narrowing for the law-integral premise inside the current
pointwise event/trace-Laplacian route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53651 |
AutoSamplingTheory.SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorPointwiseStateEventTraceLaplacianDag
- Cycle-148 proof-DAG pane for the state-event narrowing inside the
pointwise event/trace-Laplacian route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53670 |
AutoSamplingTheory.SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorStateEventPointwiseScoutObligation
- Cycle-148 lower_1 scout for the state-event equality left by the current
state-event trace-Laplacian route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53752 |
AutoSamplingTheory.SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorStateEventPointwiseScoutDag
- Cycle-148 lower_1 proof-DAG pane for reducing the state-event equality to
the pointwise event-field identity. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53771 |
AutoSamplingTheory.SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorTotalEventSourceFunctionalLowerObligation
- Cycle-148 lower_2 narrowing for the total-event action formula left by
the state-event trace-Laplacian route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53814 |
AutoSamplingTheory.SALD.cycle148GeneralMovingTargetDiscreteEmGeneratorTotalEventSourceFunctionalDag
- Cycle-148 lower_2 proof-DAG pane for reducing the total-event action
formula to the source-functional action definition. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53834 |
AutoSamplingTheory.SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceFunctionalLowerObligation
- Cycle-149 narrowing for the source-functional action definition left by
the cycle-148 total-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53882 |
AutoSamplingTheory.SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceFunctionalDag
- Cycle-149 proof-DAG pane for replacing the source-functional action
definition by the standard-basis source formula in the total-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53905 |
AutoSamplingTheory.SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceEventLowerObligation
- Cycle-149 lower_1 narrowing for the event-field Laplacian identity left
by the standard-basis source-functional total-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53953 |
AutoSamplingTheory.SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventStdBasisSourceEventDag
- Cycle-149 lower_1 proof-DAG pane for replacing the pointwise event-field
Laplacian identity by the standard-basis event-field source formula. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:53976 |
AutoSamplingTheory.SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceFieldSourceEventLowerObligation
- Cycle-149 lower_2 narrowing for the standard-basis source and event-field
premises left by the lower_1 total-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54024 |
AutoSamplingTheory.SALD.cycle149GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceFieldSourceEventDag
- Cycle-149 lower_2 proof-DAG pane for replacing the standard-basis
source/event premises by trace-field source identities. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54049 |
AutoSamplingTheory.SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLawIntegralSourceEventMiddleObligation
- Cycle-150 middle narrowing for the trace-action definition left by the
current total-event trace-field route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54102 |
AutoSamplingTheory.SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceStateIntegralSourceEventLowerObligation
- Cycle-150 lower_1 narrowing for the trace law-integral premise left by
the current total-event trace-law route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54125 |
AutoSamplingTheory.SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralSourceEventLowerObligation
- Cycle-150 lower_2 narrowing for the trace-state premise left by the
current total-event trace-state route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54151 |
AutoSamplingTheory.SALD.cycle150GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLawIntegralSourceEventDag
- Cycle-150 proof-DAG pane for reducing the total-event trace-action leaf to
the law-space trace integral. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54179 |
AutoSamplingTheory.SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventObligation
- Cycle-151 direct-leaf narrowing for the event-field/trace-field equality
left by the current trace-Laplacian state total-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54357 |
AutoSamplingTheory.SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventDag
- Cycle-151 proof-DAG pane for the direct event-field/trace-field equality
leaf in the trace-Laplacian state total-event route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54381 |
AutoSamplingTheory.SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorPointwiseEventSourceFieldLowerObligation
- Cycle-151 lower_2 direct-leaf narrowing for the pointwise EM event-field
Laplacian identity. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54446 |
AutoSamplingTheory.SALD.cycle151GeneralMovingTargetDiscreteEmGeneratorPointwiseEventSourceFieldDag
- Cycle-151 lower_2 proof-DAG pane for the pointwise event-field source-field
leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54464 |
AutoSamplingTheory.SALD.cycle152GeneralMovingTargetDiscreteEmGeneratorEventSourceFieldStdBasisLowerObligation
- Cycle-152 direct-leaf narrowing for the EM event-field/source-field
equality. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54508 |
AutoSamplingTheory.SALD.cycle152GeneralMovingTargetDiscreteWeakFpSourceFieldStdBasisScoutObligation
- Cycle-152 lower_1 scout narrowing for the weak-FP source-field
standard-basis leaf exposed by the direct event/source equality split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54527 |
AutoSamplingTheory.SALD.cycle152GeneralMovingTargetDiscreteWeakFpSourceFieldPointwiseLowerObligation
- Cycle-152 lower_2 direct narrowing for the weak-FP source-field/Laplacian
equality. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54545 |
AutoSamplingTheory.SALD.cycle152GeneralMovingTargetDiscreteEmGeneratorEventSourceFieldStdBasisDag
- Cycle-152 proof-DAG pane for the source-field equality narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54561 |
AutoSamplingTheory.SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalLowerObligation
- Cycle-153 direct-leaf narrowing for the selected-test Laplacian state integral. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54651 |
AutoSamplingTheory.SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralStdBasisSourceFunctionalLowerObligation
- Cycle-153 lower_2 narrowing from the standard-basis source formula. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54673 |
AutoSamplingTheory.SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalDag
- Cycle-153 proof-DAG pane for the state-integral/source-functional leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54697 |
AutoSamplingTheory.SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalLowerObligation
- Cycle-154 narrowing of the state-integral standard-basis source premise. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54793 |
AutoSamplingTheory.SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceLawIntegralLaplacianFieldScoutObligation
- Cycle-154 lower_1 scout narrowing of the trace-field state-integral
inputs. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54821 |
AutoSamplingTheory.SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceStateIntegralLaplacianFieldLowerObligation
- Cycle-154 lower_2 narrowing of the trace-law state-integral input. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54846 |
AutoSamplingTheory.SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalDag
- Cycle-154 proof-DAG pane for the trace-field source-functional state-integral split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:54871 |
AutoSamplingTheory.SALD.cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldLowerObligation
- Cycle-155 narrowing of the trace-state integral source boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55006 |
AutoSamplingTheory.SALD.cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldDag
- Cycle-155 proof-DAG pane for the trace-state sample-integral split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55031 |
AutoSamplingTheory.SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseLowerObligation
- Cycle-156 narrowing of the trace-field/Laplacian source boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55097 |
AutoSamplingTheory.SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseDag
- Cycle-156 proof-DAG pane for the trace-field pointwise split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55123 |
AutoSamplingTheory.SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisLowerObligation
- Cycle-157 narrowing of the EM Laplacian event-field standard-basis boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55240 |
AutoSamplingTheory.SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseLaplacianScoutObligation
- Cycle-157 lower_1 scout split for the remaining pointwise event-field display. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55258 |
AutoSamplingTheory.SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarLowerObligation
- Cycle-157 lower_2 narrowing of the pointwise event-field Laplacian leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55277 |
AutoSamplingTheory.SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDag
- Cycle-157 proof-DAG pane for the event-field pointwise standard-basis split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55294 |
AutoSamplingTheory.SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarAuditObligation
- Cycle-158 middle audit for the remaining scalar event-field Delta identity.
This is an explicit wrapper-churn rejection for the current illness area. Lean
still sees the named frozen EM Laplacian event field as an ab |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55409 |
AutoSamplingTheory.SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefLowerObligation
- Cycle-158 lower_2 source-definition boundary after the Brownian event-field split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55429 |
AutoSamplingTheory.SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarAuditDag
- Cycle-158 proof-DAG pane for the scalar event-field Delta blocker. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55448 |
AutoSamplingTheory.SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseLowerObligation
- Cycle-159 middle narrowing of the Brownian event-field definition.
The remaining cycle-158 source boundary was the function equality
`hEmGeneratorLaplacianEventFieldBrownianDef`. This packet exposes the smaller
point |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55540 |
AutoSamplingTheory.SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisLowerObligation
- Cycle-159 lower_2 standard-basis split of the Brownian pointwise boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55559 |
AutoSamplingTheory.SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDag
- Cycle-159 proof-DAG pane for the Brownian event-field pointwise split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55579 |
AutoSamplingTheory.SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLowerObligation
- Cycle-160 middle narrowing of the Brownian coordinate-trace boundary.
The remaining cycle-159 source boundary was the pointwise coordinate
Hessian-trace display
`hEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasi |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55678 |
AutoSamplingTheory.SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseScoutObligation
- Cycle-160 lower_1 pointwise scout split of the frozen scalar Ito generator boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55698 |
AutoSamplingTheory.SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseCoordinateLowerObligation
- Cycle-160 lower_2 coordinate-generator split of the pointwise frozen scalar
Ito generator boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55718 |
AutoSamplingTheory.SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDag
- Cycle-160 proof-DAG pane for the frozen scalar Brownian Ito generator split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55738 |
AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorLowerObligation
- Cycle-161 narrowing of the per-coordinate frozen scalar Brownian Ito generator.
The remaining cycle-160 coordinate-generator pair included the supplied
per-coordinate theorem `hFrozenScalarBrownianItoCoordinateGenerat |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55866 |
AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation
- Cycle-161 lower_1 scout route below the one-dimensional Brownian/Ito Taylor boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55891 |
AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation
- Cycle-161 lower_2 moment-algebra narrowing below the one-dimensional Taylor boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55914 |
AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDag
- Cycle-161 proof-DAG pane for the one-dimensional coordinate Taylor split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:55936 |
AutoSamplingTheory.SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderLowerObligation
- Cycle-162 narrowing below the one-dimensional Brownian/Ito Taylor boundary.
The remaining cycle-161 source theorem
`hFrozenScalarBrownianItoTaylorRemainderGeneratorLimit` is no longer treated as
one opaque generator e |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56060 |
AutoSamplingTheory.SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderScoutObligation
- Cycle-162 lower_1 scout route for the normalized scalar Taylor remainder. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56089 |
AutoSamplingTheory.SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderDctLowerObligation
- Cycle-162 lower_2 compiled DCT theorem for the normalized scalar Taylor remainder. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56113 |
AutoSamplingTheory.SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderDag
- Cycle-162 proof-DAG pane for the scalar Taylor remainder split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56139 |
AutoSamplingTheory.SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorLowerObligation
- Cycle-163 lower packet for the selected-test scalar Taylor pointwise limit.
The refreshed blueprint after cycle 162 names the source-specific `hPoint`
input of `gaussianRealNormalizedTaylorRemainderIntegralTendstoZero |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56282 |
AutoSamplingTheory.SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderSourceEqLowerObligation
- Cycle-163 lower_2 packet for the normalized-remainder source equality. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56311 |
AutoSamplingTheory.SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorDag
- Cycle-163 proof-DAG pane for the source Taylor pointwise limit. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56343 |
AutoSamplingTheory.SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasLowerObligation
- Cycle-164 lower packet for the concrete selected-test remainder `hMeas`.
The refreshed blueprint and upper handoff select only the eventual
`AEStronglyMeasurable` input below the concrete DCT theorem. This packet
dis |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56472 |
AutoSamplingTheory.SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundIntLowerObligation
- Cycle-164 lower_2 packet for the concrete selected-test quadratic bound
`hBoundInt`.
Once the source Taylor domination leaf is stated with the quadratic Gaussian
bound `fun z => C * z ^ 2`, this packet discharges the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56513 |
AutoSamplingTheory.SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasDag
- Cycle-164 proof-DAG pane for the concrete normalized-remainder measurability leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56541 |
AutoSamplingTheory.SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundLowerObligation
- Cycle-165 lower packet for the concrete selected-test remainder `hBound`.
The refreshed blueprint after cycle 164 leaves the pointwise domination input
as the next dynamic leaf. This packet compiles the local algebra |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56641 |
AutoSamplingTheory.SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation
- Cycle-165 lower_2 packet splitting the deterministic Taylor quotient bound. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56671 |
AutoSamplingTheory.SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundDag
- Cycle-165 proof-DAG pane for the concrete normalized-remainder domination leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56697 |
AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation
- Cycle-166 dynamic-leaf packet for the selected-line first-order remainder.
The cycle-165 split left `hFirst` as a supplied deterministic quotient bound.
This packet narrows that supplied quotient to the source-facing |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56798 |
AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation
- Cycle-166 lower_1 nonnegative interval Taylor proof-scout packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56824 |
AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation
- Cycle-166 lower_2 signed interval Taylor combination packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56849 |
AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderDag
- Cycle-166 proof-DAG pane for the first-order selected-line remainder split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56877 |
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectMiddleObligation
- Cycle-167 reflected Taylor compatibility discharge packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:56993 |
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectDag
- Cycle-167 proof-DAG pane for the reflected Taylor compatibility discharge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57022 |
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Obligation
- Cycle-167 lower_1 Taylor-compatibility narrowing packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57096 |
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineContDiffLower2Obligation
- Cycle-167 lower_2 global line regularity narrowing packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57125 |
AutoSamplingTheory.SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffMiddleObligation
- Cycle-168 ambient selected-test regularity narrowing packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57155 |
AutoSamplingTheory.SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation
- Cycle-168 lower_2 global line-second-derivative narrowing packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57182 |
AutoSamplingTheory.SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondMiddleObligation
- Cycle-169 ambient directional-Hessian narrowing packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57211 |
AutoSamplingTheory.SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondLower2Obligation
- Cycle-169 lower_2 second-Frechet-derivative operator-norm narrowing packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57239 |
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Dag
- Cycle-167 proof-DAG pane for the Taylor-data narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57259 |
AutoSamplingTheory.SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDag
- Cycle-168 proof-DAG pane for ambient source-test regularity to selected line. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57358 |
AutoSamplingTheory.SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDag
- Cycle-169 proof-DAG pane for the ambient directional-Hessian line-second split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57459 |
AutoSamplingTheory.SALD.cycle170GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormLower2Obligation
- Cycle-170 lower_2 narrowing from the Lean iterated-Frechet bound to the
source-facing Hessian operator-norm bound. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57577 |
AutoSamplingTheory.SALD.cycle170GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormDag
- Cycle-170 lower_2 proof-DAG pane for the selected-test Hessian source
interface. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57600 |
AutoSamplingTheory.SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation
- Cycle-171 middle refiner rejecting an opaque `testRegular` wrapper for the
remaining selected-test Hessian source contract. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57651 |
AutoSamplingTheory.SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag
- Cycle-171 proof-DAG pane for the rejected wrapper and remaining source
contract. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57675 |
AutoSamplingTheory.SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation
- Cycle-171 lower_1 source audit for the selected-test Hessian contract. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57760 |
AutoSamplingTheory.SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation
- Cycle-171 lower_2 rejection of the unsourced selected-test Hessian
projection route. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57784 |
AutoSamplingTheory.SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation
- Cycle-172 middle refiner after blueprint refresh: keep the live target on
the exact selected-test Hessian source contract and reject same-field wrappers. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57812 |
AutoSamplingTheory.SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag
- Cycle-172 proof-DAG pane for the refreshed Hessian source-contract
illness area. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57840 |
AutoSamplingTheory.SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation
- Cycle-172 lower_1 proof-scout audit for the selected-test Hessian source
contract. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57946 |
AutoSamplingTheory.SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation
- Cycle-172 lower_2 rejection of the only admissible Hessian projection route
after checking that no source-backed selected weak-test bounded-Hessian field
is available. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:57976 |
AutoSamplingTheory.SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceMiddleObligation
- Cycle-173 middle refiner: blueprint-guided source-contract recovery for
the selected-test Hessian operator norm, with wrapper churn rejected. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58009 |
AutoSamplingTheory.SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation
- Cycle-173 lower_1 proof-scout route for the remaining selected-test
Hessian source contract. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58040 |
AutoSamplingTheory.SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation
- Cycle-173 lower_2 compiled bridge from source-backed Hessian fields to the
selected-test Hessian operator-norm bound. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58070 |
AutoSamplingTheory.SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag
- Cycle-173 proof-DAG pane for the middle source-contract recovery packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58096 |
AutoSamplingTheory.SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationMiddleObligation
- Cycle-174 middle packet after the source-Hessian field audit.
The selected weak-test Hessian representative fields left by cycle 173 are kept
as a source-contract gap. This packet moves only to the connected scalar
B |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58205 |
AutoSamplingTheory.SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationLower1Obligation
- Cycle-174 lower_1 proof-scout packet for the Brownian
quadratic-variation normalization leaf.
This records the source route and the exact lower_2 algebraic theorem shape.
It does not close the normalization theorem or |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58239 |
AutoSamplingTheory.SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationLower2Obligation
- Cycle-174 lower_2 compiled bridge from the source-backed coefficient and
variance fields to the Brownian quadratic-variation normalization. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58265 |
AutoSamplingTheory.SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationDag
- Cycle-174 proof-DAG pane for the Brownian quadratic-variation leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58291 |
AutoSamplingTheory.SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationMiddleObligation
- Cycle-175 dynamic-leaf worker packet for the standard-basis selected-line
Taylor-domination leaf.
The cycle-174 reviewer accepted the Brownian quadratic-variation algebraic
bridge. This packet returns to the connecte |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58416 |
AutoSamplingTheory.SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower1Obligation
- Cycle-175 lower_1 proof-scout packet for the quadratic-coefficient source
boundary.
After the selected-line Taylor-domination bridge is compiled, the connected
Brownian/Ito coefficient leaf should not be closed by a b |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58454 |
AutoSamplingTheory.SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticCoeffLower2Obligation
- Cycle-175 lower_2 compiled bridge from the source second-Taylor
coefficient identity to the downstream quadratic-coefficient definition. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58481 |
AutoSamplingTheory.SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDag
- Cycle-175 proof-DAG pane for the selected-line Taylor-domination bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58508 |
AutoSamplingTheory.SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneMiddleObligation
- Cycle-176 dynamic-leaf worker packet for the normalized scalar Brownian
variance field.
The cycle-175 reviewer accepted the quadratic-coefficient bridge, leaving
`hSecondTaylorCoeffDef` plus the separate `hVarianceOne |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58637 |
AutoSamplingTheory.SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower1Obligation
- Cycle-176 lower_1 proof-scout route for the normalized scalar Brownian
variance field. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58668 |
AutoSamplingTheory.SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower2Obligation
- Cycle-176 lower_2 compiled bridge through the normalized variance field.
The direct variance-one theorem was already available when lower_2 arrived, so
this packet composes it with the cycle-175 second-Taylor coeffici |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58707 |
AutoSamplingTheory.SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneDag
- Cycle-176 proof-DAG pane for the normalized Brownian variance field. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58742 |
AutoSamplingTheory.SALD.cycle177GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSecondTaylorCoeffMiddleObligation
- Cycle-177 middle packet for the remaining Brownian/Ito coefficient side.
The selected weak-test Hessian fields are kept as source-contract gaps after
the source audit. This packet narrows the connected coefficient le |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58879 |
AutoSamplingTheory.SALD.cycle177GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSecondTaylorCoeffDag
- Cycle-177 proof-DAG pane for the scalar-line coefficient narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:58916 |
AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceMiddleObligation
- Cycle-178 middle packet for the normalized Brownian variance source field.
This packet follows the accepted source-Hessian decision: the Hessian fields
remain source-contract gaps, so the connected Brownian/Ito varian |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59019 |
AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation
- Cycle-178 lower_1 proof-scout packet for the normalized coordinate law. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59053 |
AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation
- Cycle-178 lower_2 compiled bridge for the normalized Brownian coordinate law. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59083 |
AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDag
- Cycle-178 proof-DAG pane for normalized Brownian variance law narrowing. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59112 |
AutoSamplingTheory.SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffMiddleObligation
- Cycle-179 middle packet after the source-Hessian audit.
The selected weak-test Hessian fields are not derivable from the checked
original-source anchors. This packet keeps those fields as source-contract
gaps and ass |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59260 |
AutoSamplingTheory.SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffDag
- Cycle-179 proof-DAG pane for the source-Hessian audit decision and next
scalar-line coefficient packet. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59300 |
AutoSamplingTheory.SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower1Obligation
- Cycle-179 lower_1 route for the remaining scalar-line coefficient boundary.
This is a proof-scout packet, not a theorem-status promotion. It keeps the
selected weak-test Hessian fields as source-contract gaps and nar |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59446 |
AutoSamplingTheory.SALD.cycle179GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoScalarLineCoeffLower2Obligation
- Cycle-179 lower_2 compiled bridge from the scalar Taylor coefficient
convention to the scalar-line second-derivative coefficient boundary. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59481 |
AutoSamplingTheory.SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentMiddleObligation
- Cycle-180 middle packet for the Taylor moment decomposition leaf.
The source-Hessian fields remain source-contract gaps. This packet follows the
cycle-180 upper assignment and narrows the sibling Brownian/Ito Taylor |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59522 |
AutoSamplingTheory.SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPolynomialIntegrabilityLower1Obligation
- Cycle-180 lower_1 compiled polynomial-integrability bridge.
This proof-scout packet discharges the linear and quadratic Gaussian summand
integrability inputs in the Taylor moment split. The remaining source-facing
Ta |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59571 |
AutoSamplingTheory.SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDominatedRemainderLower2Obligation
- Cycle-180 lower_2 dominated-remainder integrability bridge.
This worker packet discharges the `hRemainderInt` input in the Taylor moment
split from the normalized-remainder measurability/domination package already
tra |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59611 |
AutoSamplingTheory.SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDag
- Cycle-180 proof-DAG pane for the Taylor moment decomposition split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59646 |
AutoSamplingTheory.SALD.cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralMiddleObligation
- Cycle-183 middle packet for the Brownian coordinate Taylor integral leaf.
The bridge is only the `MeasureTheory.integral_congr_ae` transport from the
paper's source scalar Taylor integrand to the local Taylor-sum inte |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59777 |
AutoSamplingTheory.SALD.cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegrandLower2Obligation
- Cycle-183 lower_2 packet for the Brownian source Taylor integrand leaf.
The compiled theorem turns the source-facing pointwise scalar Taylor identity
into the a.e. equality consumed by the middle integral-congruence b |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59820 |
AutoSamplingTheory.SALD.cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDag
- Cycle-183 proof-DAG pane for the Brownian coordinate source-integral leaf. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59846 |
AutoSamplingTheory.SALD.cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawMiddleObligation
- Cycle-184 middle packet for the Brownian coordinate source-integral leaf.
The compiled bridge narrows `hBrownianCoordinateGeneratorSourceIntegralDef` to
the actual normalized scalar-coordinate law definition plus the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59943 |
AutoSamplingTheory.SALD.cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedLawLower2Obligation
- Cycle-184 lower_2 packet for the normalized scalar-coordinate law leaf.
The compiled theorem transports the sample-space expectation of the paper's
source Taylor integrand to the normalized scalar-coordinate law by
`M |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:59986 |
AutoSamplingTheory.SALD.cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDag
- Cycle-184 proof-DAG pane for the Brownian coordinate source-integral law split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60013 |
AutoSamplingTheory.SALD.cycle185GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitLower2Obligation
- Cycle-185 lower_2 packet for the remainder-generator law leaf.
The compiled theorem narrows `hRemainderGeneratorLimitDef` to the source
definition of the normalized remainder integral under the actual normalized
scala |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60123 |
AutoSamplingTheory.SALD.cycle185GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitDag
- Cycle-185 proof-DAG pane for the remainder-generator law split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60156 |
AutoSamplingTheory.SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandPointwiseMiddleObligation
- Cycle-186 middle packet for the pointwise source Taylor integrand leaf.
The active Brownian/Ito frozen-interpolation backend now narrows
`hSourceTaylorIntegrandPointwise`, the pointwise identity consumed by the
cycle- |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60245 |
AutoSamplingTheory.SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandPointwiseDag
- Cycle-186 proof-DAG pane for the source Taylor integrand pointwise split. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60274 |
AutoSamplingTheory.SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceLinearTermLower2Obligation
- Cycle-186 lower_2 packet for the source linear term leaf.
This narrows `hSourceLinearTermDef` to the two smaller source-cited fields
identified by the lower_1 proof scout: the first-order Taylor source term along
the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60355 |
AutoSamplingTheory.SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceLinearTermLower2Dag
- Cycle-186 proof-DAG pane for the source linear term bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60381 |
AutoSamplingTheory.SALD.cycle187GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceQuadraticTermMiddleObligation
- Cycle-187 packet for the source quadratic term leaf.
This narrows the remaining quadratic side of the cycle-186 source Taylor
integrand split. The compiled theorem reduces `hSourceQuadraticTermDef` to
the paper-facin |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60463 |
AutoSamplingTheory.SALD.cycle187GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceQuadraticTermDag
- Cycle-187 proof-DAG pane for the source quadratic term bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60492 |
AutoSamplingTheory.SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDefMiddleObligation
- Cycle-188 packet for the source Taylor integrand definition leaf.
This narrows `hSourceTaylorIntegrandDef` to the raw selected-line increment
definition and the selected-line Taylor split into source linear term, sour |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60575 |
AutoSamplingTheory.SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDefDag
- Cycle-188 proof-DAG pane for the source Taylor integrand definition bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60604 |
AutoSamplingTheory.SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorSplitLower2Obligation
- Cycle-188 lower_2 packet for the selected-line Taylor split leaf.
This narrows `hSelectedLineTaylorSplitDef` one step further. The source-facing
Taylor expansion itself remains explicit as `hSelectedLineTaylorRawSpli |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60687 |
AutoSamplingTheory.SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorSplitLower2Dag
- Cycle-188 proof-DAG pane for the selected-line Taylor split bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60719 |
AutoSamplingTheory.SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsMiddleObligation
- Cycle-189 packet for the Brownian coordinate Taylor integral leaf.
This composes the cycle-183 source-integral/a.e. bridge with the cycle-186,
cycle-187, and cycle-188 source Taylor bridges. The older top-level
`hBro |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60784 |
AutoSamplingTheory.SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsDag
- Cycle-189 proof-DAG pane for the raw-term coordinate Taylor integral bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60828 |
AutoSamplingTheory.SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandRawLower2Obligation
- Cycle-189 lower_2 packet for the raw source Taylor integrand leaf.
This narrows `hSourceTaylorIntegrandRawDef` to two smaller source-cited
definition fields. The compiled theorem only composes those fields: the
paper |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60908 |
AutoSamplingTheory.SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandRawLower2Dag
- Cycle-189 lower_2 proof-DAG pane for the raw source Taylor integrand bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:60933 |
AutoSamplingTheory.SALD.cycle190GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementCoordinateLineMiddleObligation
- Cycle-190 middle packet for the selected-increment coordinate-line leaf.
This narrows `hSelectedIncrementCoordinateLineDef` to two smaller source-cited
definition fields: the selected increment is the selected weak-te |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61003 |
AutoSamplingTheory.SALD.cycle190GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementCoordinateLineDag
- Cycle-190 proof-DAG pane for the selected-increment endpoint bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61029 |
AutoSamplingTheory.SALD.cycle190GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitLower2Obligation
- Cycle-190 lower_2 packet for the normalized-remainder law leaf.
The compiled theorem discharges `hRemainderGeneratorNormalizedLawDef` as a
primitive supplied hypothesis in the `hRemainderGeneratorLimitDef` route. It
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61125 |
AutoSamplingTheory.SALD.cycle190GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitLower2Dag
- Cycle-190 lower_2 proof-DAG pane for the remainder-limit scalar-pushforward bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61160 |
AutoSamplingTheory.SALD.cycle191GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLower2Obligation
- Cycle-191 middle packet for the Brownian source-integral law leaf.
The compiled theorem discharges `hBrownianCoordinateGeneratorNormalizedLawDef`
as a primitive supplied hypothesis in the
`hBrownianCoordinateGenerator |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61268 |
AutoSamplingTheory.SALD.cycle191GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLower2Dag
- Cycle-191 proof-DAG pane for the source-integral scalar-pushforward bridge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61303 |
AutoSamplingTheory.SALD.cycle192GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralLower2Obligation
- Cycle-192 lower_2 packet for the Brownian Taylor-integral leaf.
The compiled theorem discharges `hBrownianCoordinateGeneratorSourceIntegralDef`
as a primitive supplied hypothesis in the
`hBrownianCoordinateGeneratorTa |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61411 |
AutoSamplingTheory.SALD.cycle192GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralLower2Dag
- Cycle-192 proof-DAG pane for the Taylor-integral source-integral discharge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61455 |
AutoSamplingTheory.SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentMiddleObligation
- Cycle-193 middle packet for the Taylor moment split consumer.
The compiled theorem discharges the primitive
`hBrownianCoordinateGeneratorTaylorIntegralDef` and
`hRemainderGeneratorLimitDef` supplied hypotheses inside |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61571 |
AutoSamplingTheory.SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentRemainderLimitLower2Obligation
- Cycle-193 lower_2 packet for the one-hypothesis Taylor moment consumer.
The compiled theorem discharges only `hRemainderGeneratorLimitDef` inside the
dominated Taylor moment decomposition. It keeps
`hBrownianCoordina |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61631 |
AutoSamplingTheory.SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentRemainderLimitLower2Dag
- Cycle-193 lower_2 proof-DAG pane for the one-hypothesis remainder-limit discharge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61672 |
AutoSamplingTheory.SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentMiddleDag
- Cycle-193 proof-DAG pane for the Taylor moment scalar-pushforward discharge. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61734 |
AutoSamplingTheory.SALD.cycle197GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderBoundDefLower2Obligation
- Cycle-197 lower_2 packet for the normalized remainder bound definition.
The existing Lean declarations treat `remainderBound` and `remainderBoundC`
as parameters. Therefore the source-facing equality
`remainderBound |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61810 |
AutoSamplingTheory.SALD.cycle197GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderBoundDefLower2Dag
- Cycle-197 proof-DAG pane for the normalized remainder bound-definition gap. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61839 |
AutoSamplingTheory.SALD.cycle198GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumLower2Obligation
- Cycle-198 lower_2 packet for the frozen Brownian event-field coordinate sum.
The existing cycle-160 assembly theorem consumes
`hFrozenScalarBrownianItoEventFieldCoordinateSum` as a source input; it does
not define the |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61884 |
AutoSamplingTheory.SALD.cycle198GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumLower2Dag
- Cycle-198 proof-DAG pane for the frozen Brownian coordinate-sum gap. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61911 |
AutoSamplingTheory.SALD.cycle201GeneralMovingTargetDiscreteEmInterpolationSelectedTestLaplacianContinuityLower2Obligation
- Cycle-201 lower_2 packet for selected-test Laplacian regularity.
The cycle-199 local bridge already reduces the law-dependent
`hsourceLaplacianFieldMeas` premise to ordinary measurability of
`Laplacian.laplacian (sele |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61952 |
AutoSamplingTheory.SALD.cycle201GeneralMovingTargetDiscreteEmInterpolationSelectedTestLaplacianContinuityLower2Dag
- Cycle-201 proof-DAG pane for the selected-test Laplacian continuity gap. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:61983 |
AutoSamplingTheory.SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Obligation
- Cycle-202 lower_2 packet for the selected endpoint coordinate-line field.
The cycle-190 local bridge already reduces the selected-increment coordinate
line and raw source Taylor integrand leaves to endpoint naming plu |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62026 |
AutoSamplingTheory.SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Dag
- Cycle-202 proof-DAG pane for the selected endpoint coordinate-line gap. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62066 |
AutoSamplingTheory.SALD.cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Obligation
- Cycle-203 lower_2 packet for the selected-increment endpoint field.
The cycle-190 local bridge already reduces the selected-increment coordinate
line and raw source Taylor integrand leaves to two endpoint-facing sourc |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62113 |
AutoSamplingTheory.SALD.cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Dag
- Cycle-203 proof-DAG pane for the selected-increment endpoint gap. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62152 |
AutoSamplingTheory.SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Obligation
- Cycle-204 lower_2 packet for the source Taylor integrand naming field.
The cycle-189 and cycle-190 local bridges already reduce the raw source
Taylor integrand leaf to the source naming field below plus endpoint-facin |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62198 |
AutoSamplingTheory.SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Dag
- Cycle-204 proof-DAG pane for the source Taylor integrand naming gap. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62239 |
AutoSamplingTheory.SALD.cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Obligation
- Cycle-205 lower_2 packet for the selected-line raw Taylor split.
The cycle-188 bridge already reduces `hSelectedLineTaylorSplitDef` to the raw
one-dimensional selected-line Taylor identity plus source-term naming fiel |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62285 |
AutoSamplingTheory.SALD.cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Dag
- Cycle-205 proof-DAG pane for the selected-line raw Taylor split gap. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62334 |
AutoSamplingTheory.SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleObligation
- Cycle-206 middle packet for the normalized-remainder pullback field.
The cycle-185 scalar-pushforward bridge already consumes
`hRemainderPullbackDef` to derive the normalized-law remainder generator
identity. This pa |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62384 |
AutoSamplingTheory.SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleDag
- Cycle-206 proof-DAG pane for the normalized-remainder pullback gap. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62430 |
AutoSamplingTheory.SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackLower2Obligation
- Cycle-206 lower_2 packet for the normalized-remainder pullback field.
Lower_2 inspected the local Lean interfaces and found
`remainderGeneratorLimit`, `normalizedRemainder`, and
`scalarBrownianCoordinate` only as para |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62478 |
AutoSamplingTheory.SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackLower2Dag
- Cycle-206 proof-DAG pane for the lower_2 remainder-pullback gap. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62528 |
AutoSamplingTheory.SALD.cycle130GeneralMovingTargetDiscreteEmLaplacianIbPDag
- Cycle-130 proof-DAG pane for the weak Laplacian integration-by-parts
sub-boundary inside the EM diffusion source action. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62568 |
AutoSamplingTheory.SALD.cycle70GeneralMovingTargetDiscreteConditionalLawDag
- Cycle-70 proof-DAG pane for the EM conditional-law/measurability backfill. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62668 |
AutoSamplingTheory.SALD.cycle69MainSkeletonAnalyticInterfaceDag
- Cycle-69 proof-DAG pane for the post-route analytic-interface ledger. |
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62728 |
AutoSamplingTheory.SALD.guidedResidualNormalizerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62856 |
AutoSamplingTheory.SALD.guidedResidualIdentityObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62864 |
AutoSamplingTheory.SALD.generalMovingTargetDerivativeObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62872 |
AutoSamplingTheory.SALD.generalMovingTargetDvEnergyObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62896 |
AutoSamplingTheory.SALD.generalMovingTargetDvPositiveAlphaScalingObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62910 |
AutoSamplingTheory.SALD.generalMovingTargetDvFiniteLogMgfWitnessObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62925 |
AutoSamplingTheory.SALD.generalMovingTargetGronwallApplicationObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62941 |
AutoSamplingTheory.SALD.cycle24GeneralVaSaldGronwallMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62949 |
AutoSamplingTheory.SALD.generalMovingTargetGronwallSideConditionObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62971 |
AutoSamplingTheory.SALD.generalMovingTargetPureContractionObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62991 |
AutoSamplingTheory.SALD.cycle16UnifiedForwardKlTransportBridgeMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:62999 |
AutoSamplingTheory.SALD.cycle16UnifiedForwardKlTransportBridgeLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63014 |
AutoSamplingTheory.SALD.unifiedForwardKlTransportBridgeObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63029 |
AutoSamplingTheory.SALD.unifiedForwardKlSpecializationObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63045 |
AutoSamplingTheory.SALD.generalVaSaldGuidedPathMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63063 |
AutoSamplingTheory.SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63092 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmInterpolationObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63115 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConstantScheduleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63123 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteFrozenDeltaObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63131 |
AutoSamplingTheory.SALD.cycle28GeneralVaSaldDerivativeSideMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63139 |
AutoSamplingTheory.SALD.cycle28GeneralVaSaldDerivativeSideLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63159 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDerivativeSideConditionObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63174 |
AutoSamplingTheory.SALD.cycle54GeneralMovingTargetDiscreteEmFpLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63182 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDerivativeObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63197 |
AutoSamplingTheory.SALD.cycle53GeneralMovingTargetDiscreteDerivativeDvLowerObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63268 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDvMEnergyObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63292 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63306 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallApplicationObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63323 |
AutoSamplingTheory.SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63331 |
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallSideConditionObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63354 |
AutoSamplingTheory.SALD.discreteUnifiedVaSaldSpecializationObligation
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63381 |
AutoSamplingTheory.SALD.gronwallContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63389 |
AutoSamplingTheory.SALD.dvContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63408 |
AutoSamplingTheory.SALD.piDefinitionContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63430 |
AutoSamplingTheory.SALD.lsiKlFiVocabularyContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63445 |
AutoSamplingTheory.SALD.continuousSaldContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63469 |
AutoSamplingTheory.SALD.forwardKlProofDag
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:63530 |
AutoSamplingTheory.SALD.discreteForwardKlProofDag
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:64011 |
AutoSamplingTheory.SALD.discreteSaldContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:64708 |
AutoSamplingTheory.SALD.generalVaSaldProofDag
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:64861 |
AutoSamplingTheory.SALD.generalVaSaldDiscreteProofDag
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65136 |
AutoSamplingTheory.SALD.guidedResidualContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65432 |
AutoSamplingTheory.SALD.generalVaSaldContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65466 |
AutoSamplingTheory.SALD.unifiedForwardKlContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65517 |
AutoSamplingTheory.SALD.generalVaSaldDiscreteContract
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65570 |
AutoSamplingTheory.SALD.saldTheoremContracts
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65707 |
AutoSamplingTheory.SALD.saldSourceForLabel
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65721 |
AutoSamplingTheory.SALD.saldLeanTargetForLabel
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65734 |
AutoSamplingTheory.SALD.cycle49MainSkeletonDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65738 |
AutoSamplingTheory.SALD.cycle50ForwardKlDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65751 |
AutoSamplingTheory.SALD.cycle51DiscreteForwardKlDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65767 |
AutoSamplingTheory.SALD.cycle52GuidedGeneralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65787 |
AutoSamplingTheory.SALD.cycle53UnifiedDiscreteGeneralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65805 |
AutoSamplingTheory.SALD.cycle54MainSkeletonDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65832 |
AutoSamplingTheory.SALD.cycle55ForwardKlDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65859 |
AutoSamplingTheory.SALD.cycle56DiscreteForwardKlDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65901 |
AutoSamplingTheory.SALD.cycle57GuidedGeneralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65948 |
AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:65992 |
AutoSamplingTheory.SALD.cycle59MainSkeletonDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66030 |
AutoSamplingTheory.SALD.cycle60ForwardKlDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66059 |
AutoSamplingTheory.SALD.cycle65ForwardKlDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66103 |
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66157 |
AutoSamplingTheory.SALD.cycle66DiscreteForwardKlDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66227 |
AutoSamplingTheory.SALD.cycle62GuidedGeneralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66301 |
AutoSamplingTheory.SALD.cycle67GuidedGeneralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66361 |
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66424 |
AutoSamplingTheory.SALD.cycle69MainSkeletonDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66486 |
AutoSamplingTheory.SALD.cycle70EmConditionalLawDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66532 |
AutoSamplingTheory.SALD.cycle71EmEndpointConditionalDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66550 |
AutoSamplingTheory.SALD.cycle72EmWeakFpDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66571 |
AutoSamplingTheory.SALD.cycle73EmKlDerivativeWeakFpDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66591 |
AutoSamplingTheory.SALD.cycle74EmConditionalKernelMeasureDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66614 |
AutoSamplingTheory.SALD.cycle75EmConditionalLawBackfillDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66640 |
AutoSamplingTheory.SALD.cycle76EmEndpointConditionalDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66668 |
AutoSamplingTheory.SALD.cycle77EmWeakFpSourceSignsDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66692 |
AutoSamplingTheory.SALD.cycle78EmKlDerivativeGeneratorDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66712 |
AutoSamplingTheory.SALD.cycle79EmWeakFpGeneratorMeasureDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66735 |
AutoSamplingTheory.SALD.cycle80EmConditionalLawMeasurabilityDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66764 |
AutoSamplingTheory.SALD.cycle81EmEndpointConditionalDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66792 |
AutoSamplingTheory.SALD.cycle82EmWeakFpSourceSignsDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66823 |
AutoSamplingTheory.SALD.cycle83EmKlDerivativeEndpointWeakFpDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66854 |
AutoSamplingTheory.SALD.cycle84ActiveEmBackendDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66879 |
AutoSamplingTheory.SALD.cycle85EmConditionalKernelBoundaryDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66907 |
AutoSamplingTheory.SALD.cycle86EmWeakFpGeneratorBoundaryDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66949 |
AutoSamplingTheory.SALD.cycle87EmKlLogRatioBoundaryDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:66977 |
AutoSamplingTheory.SALD.cycle88EmKlLogRatioAdmissibilityDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67013 |
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlPressureDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67043 |
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67085 |
AutoSamplingTheory.SALD.cycle91EmConditionalKernelDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67128 |
AutoSamplingTheory.SALD.cycle92EmWeakFpGeneratorSplitDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67154 |
AutoSamplingTheory.SALD.cycle93EmKlMassDerivativeDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67178 |
AutoSamplingTheory.SALD.cycle94EmWeakFpDriftActionDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67207 |
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlPressureDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67232 |
AutoSamplingTheory.SALD.cycle96EmCondexpGeneratorPairingDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67270 |
AutoSamplingTheory.SALD.cycle97EmCanonicalCondDistribPairingDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67300 |
AutoSamplingTheory.SALD.cycle98EmBarBDivergenceNoBoundaryDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67328 |
AutoSamplingTheory.SALD.cycle99EmRawKlDerivativeDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67351 |
AutoSamplingTheory.SALD.cycle100EmBarBWeakGradDefDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67380 |
AutoSamplingTheory.SALD.cycle101DiscreteForwardKlPressureDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67408 |
AutoSamplingTheory.SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67431 |
AutoSamplingTheory.SALD.cycle103EmConditionalKernelComponentVersionDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67455 |
AutoSamplingTheory.SALD.cycle104EmWeakFpNamedLawTransportDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67483 |
AutoSamplingTheory.SALD.cycle105EmPureRawKlDerivativeDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67509 |
AutoSamplingTheory.SALD.cycle106EmCanonicalCondDistribDriftDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67537 |
AutoSamplingTheory.SALD.cycle107DiscreteForwardKlBoundaryFluxDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67562 |
AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67580 |
AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67599 |
AutoSamplingTheory.SALD.cycle109EmNamedBarBSourceDefDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67620 |
AutoSamplingTheory.SALD.cycle110EmNamedBarBEqMeasDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67649 |
AutoSamplingTheory.SALD.cycle110EmWeakFpDominatedGeneratorDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67670 |
AutoSamplingTheory.SALD.cycle111EmKlTargetTimeDerivativeDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67688 |
AutoSamplingTheory.SALD.cycle112EmNamedBarBCondExpRepresentativeDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67718 |
AutoSamplingTheory.SALD.cycle113EmNamedBarBStateFieldRegularityDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67750 |
AutoSamplingTheory.SALD.cycle114EmCanonicalBarBStateEventSetIntegralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67770 |
AutoSamplingTheory.SALD.cycle115EmNamedBarBSelectedVersionDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67792 |
AutoSamplingTheory.SALD.cycle116EmCanonicalBarBCondExpDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67820 |
AutoSamplingTheory.SALD.cycle117EmNamedBarBVersionSelectionDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67845 |
AutoSamplingTheory.SALD.cycle118EmCanonicalBarBDownstreamDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67872 |
AutoSamplingTheory.SALD.cycle119EmCanonicalBarBWeakFpConsumerDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67897 |
AutoSamplingTheory.SALD.cycle120EmPathDerivativeDominationDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67925 |
AutoSamplingTheory.SALD.cycle121EmSampleMeasDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67950 |
AutoSamplingTheory.SALD.cycle122EmSampleIntDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67970 |
AutoSamplingTheory.SALD.cycle123EmSampleDerivMeasDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:67992 |
AutoSamplingTheory.SALD.cycle124EmSampleDerivBoundDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68014 |
AutoSamplingTheory.SALD.cycle124EmBoundIntDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68033 |
AutoSamplingTheory.SALD.cycle125EmPathDerivDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68053 |
AutoSamplingTheory.SALD.cycle126EmDerivValueDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68071 |
AutoSamplingTheory.SALD.cycle127EmDriftActionDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68089 |
AutoSamplingTheory.SALD.cycle128EmNoBoundaryTraceDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68111 |
AutoSamplingTheory.SALD.cycle129EmDiffusionSourceDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68137 |
AutoSamplingTheory.SALD.cycle130EmLaplacianIbPDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68156 |
AutoSamplingTheory.SALD.cycle131EmSecondGreenNoBoundaryFluxDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68182 |
AutoSamplingTheory.SALD.cycle132EmSecondGreenTestTraceZeroDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68208 |
AutoSamplingTheory.SALD.cycle133EmSecondGreenPointwiseTestTraceZeroDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68231 |
AutoSamplingTheory.SALD.cycle134EmTestLaplacianSourcePullbackDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68254 |
AutoSamplingTheory.SALD.cycle135EmTestLaplacianStdBasisDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68280 |
AutoSamplingTheory.SALD.cycle136EmWeakFpStdBasisSourceDensityDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68304 |
AutoSamplingTheory.SALD.cycle137EmWeakFpDensityLaplacianActionDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68327 |
AutoSamplingTheory.SALD.cycle138EmFirstGreenPointwiseBoundaryFluxDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68355 |
AutoSamplingTheory.SALD.cycle139EmWeakFpSourceLaplacianDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68383 |
AutoSamplingTheory.SALD.cycle140EmLaplacianSourceStateIntegralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68410 |
AutoSamplingTheory.SALD.cycle141EmGeneratorStdBasisSourceDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68437 |
AutoSamplingTheory.SALD.cycle142EmGeneratorTraceStateIntegralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68465 |
AutoSamplingTheory.SALD.cycle143EmGeneratorLaplacianStateEventDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68498 |
AutoSamplingTheory.SALD.cycle144EmGeneratorLaplacianStdBasisEventDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68535 |
AutoSamplingTheory.SALD.cycle145EmGeneratorLaplacianStdBasisLawDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68565 |
AutoSamplingTheory.SALD.cycle146EmGeneratorTraceEventTotalEventDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68595 |
AutoSamplingTheory.SALD.cycle147EmGeneratorPointwiseActionDefTraceLaplacianDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68634 |
AutoSamplingTheory.SALD.cycle148EmGeneratorPointwiseStateEventTraceLaplacianDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68673 |
AutoSamplingTheory.SALD.cycle149EmGeneratorTotalEventStdBasisSourceFunctionalDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68728 |
AutoSamplingTheory.SALD.cycle150EmGeneratorTotalEventTraceLawIntegralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68769 |
AutoSamplingTheory.SALD.cycle151EmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68814 |
AutoSamplingTheory.SALD.cycle152EmGeneratorEventSourceFieldStdBasisDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68845 |
AutoSamplingTheory.SALD.cycle153EmGeneratorLaplacianStateIntegralSourceFunctionalDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68877 |
AutoSamplingTheory.SALD.cycle154EmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68909 |
AutoSamplingTheory.SALD.cycle155EmGeneratorTraceStateIntegralLaplacianFieldDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68954 |
AutoSamplingTheory.SALD.cycle156EmGeneratorTraceFieldPointwiseDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:68982 |
AutoSamplingTheory.SALD.cycle157EmGeneratorLaplacianEventFieldPointwiseStdBasisDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69016 |
AutoSamplingTheory.SALD.cycle158EmGeneratorLaplacianEventFieldPointwiseScalarAuditDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69051 |
AutoSamplingTheory.SALD.cycle159EmGeneratorLaplacianEventFieldBrownianPointwiseDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69077 |
AutoSamplingTheory.SALD.cycle160EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69105 |
AutoSamplingTheory.SALD.cycle161EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69139 |
AutoSamplingTheory.SALD.cycle162EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69173 |
AutoSamplingTheory.SALD.cycle163EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69211 |
AutoSamplingTheory.SALD.cycle164EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoMeasDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69250 |
AutoSamplingTheory.SALD.cycle165EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHBoundDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69291 |
AutoSamplingTheory.SALD.cycle166EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHFirstDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69330 |
AutoSamplingTheory.SALD.cycle167EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69378 |
AutoSamplingTheory.SALD.cycle168EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69455 |
AutoSamplingTheory.SALD.cycle169EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69499 |
AutoSamplingTheory.SALD.cycle170EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69539 |
AutoSamplingTheory.SALD.cycle171EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69559 |
AutoSamplingTheory.SALD.cycle172EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69585 |
AutoSamplingTheory.SALD.cycle173EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69618 |
AutoSamplingTheory.SALD.cycle174EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69655 |
AutoSamplingTheory.SALD.cycle175EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69688 |
AutoSamplingTheory.SALD.cycle176EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69726 |
AutoSamplingTheory.SALD.cycle177EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSecondTaylorCoeffDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69763 |
AutoSamplingTheory.SALD.cycle178EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69802 |
AutoSamplingTheory.SALD.cycle179EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69848 |
AutoSamplingTheory.SALD.cycle180EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69897 |
AutoSamplingTheory.SALD.cycle183EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:69957 |
AutoSamplingTheory.SALD.cycle184EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70002 |
AutoSamplingTheory.SALD.cycle185EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70056 |
AutoSamplingTheory.SALD.cycle186EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70109 |
AutoSamplingTheory.SALD.cycle187EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceQuadraticTermDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70161 |
AutoSamplingTheory.SALD.cycle188EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDefDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70205 |
AutoSamplingTheory.SALD.cycle189EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70256 |
AutoSamplingTheory.SALD.cycle190EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementCoordinateLineDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70320 |
AutoSamplingTheory.SALD.cycle191EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralScalarPushforwardDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70366 |
AutoSamplingTheory.SALD.cycle192EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralScalarPushforwardDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70413 |
AutoSamplingTheory.SALD.cycle193EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentScalarPushforwardDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70460 |
AutoSamplingTheory.SALD.cycle197EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderBoundDefDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70518 |
AutoSamplingTheory.SALD.cycle198EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70549 |
AutoSamplingTheory.SALD.cycle199EmInterpolationWeakFpSourceLaplacianFieldMeasDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70576 |
AutoSamplingTheory.SALD.cycle201EmInterpolationSelectedTestLaplacianContinuityDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70598 |
AutoSamplingTheory.SALD.cycle202EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70629 |
AutoSamplingTheory.SALD.cycle203EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70665 |
AutoSamplingTheory.SALD.cycle204EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70700 |
AutoSamplingTheory.SALD.cycle205EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70737 |
AutoSamplingTheory.SALD.cycle206EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70779 |
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70826 |
AutoSamplingTheory.SALD.cycle64MainSkeletonDependencyNames
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70875 |
AutoSamplingTheory.SALD.saldDependenciesForLabel
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:70921 |
AutoSamplingTheory.SALD.saldReusedByForLabel
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:71284 |
AutoSamplingTheory.SALD.saldStatusForLabel
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:71294 |
AutoSamplingTheory.SALD.saldFirstProofDag
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:71301 |
AutoSamplingTheory.SALD.saldExcludedFiles
|
def | Partial | Not mapped | AutoSamplingTheory.SALD |
AutoSamplingTheory/SALD.lean:71312 |
AutoSamplingTheory.ItoDiffusionContract
|
structure | Partial | Not mapped | AutoSamplingTheory.SDE |
AutoSamplingTheory/SDE.lean:9 |
AutoSamplingTheory.FokkerPlanckContract
|
structure | Partial | Not mapped | AutoSamplingTheory.SDE |
AutoSamplingTheory/SDE.lean:19 |
AutoSamplingTheory.EulerMaruyamaContract
|
structure | Partial | Not mapped | AutoSamplingTheory.SDE |
AutoSamplingTheory/SDE.lean:27 |
AutoSamplingTheory.DiscretizationErrorContract
|
structure | Partial | Not mapped | AutoSamplingTheory.SDE |
AutoSamplingTheory/SDE.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Algebra.linear_growth_of_step_growth
- A uniform lower bound on every one-step increment telescopes linearly. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Algebra.LinearGrowthOfStep |
AutoSamplingTheory/TechnicalLemmas/Algebra/LinearGrowthOfStep.lean:17 |
AutoSamplingTheory.TechnicalLemmas.Algebra.reciprocal_growth_implies_inverse_time_bound
- Reciprocal growth converts to the usual `A / t` upper bound. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Algebra.ReciprocalGrowthRate |
AutoSamplingTheory/TechnicalLemmas/Algebra/ReciprocalGrowthRate.lean:16 |
AutoSamplingTheory.TechnicalLemmas.Analysis.BoundedQuadraticCostIntegrability.norm_sub_lt_center_distance_add_two_radius
- Points lying in two radius-`r` balls have pairwise distance bounded by the
center distance plus `2r`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.BoundedQuadraticCostIntegrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/BoundedQuadraticCostIntegrability.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Analysis.BoundedQuadraticCostIntegrability.integrable_norm_sub_sq_of_prob_one_bounded_rectangle
- A probability law concentrated on a bounded product rectangle has
integrable squared displacement. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.BoundedQuadraticCostIntegrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/BoundedQuadraticCostIntegrability.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Analysis.BoundedQuadraticCostIntegrability.integrable_cross_norm_sub_sq_of_prob_one_bounded_rectangles
- If two probability laws on joint pairs are respectively concentrated on
bounded rectangles, then the cross squared displacement formed from the first
source coordinate and second target coordinate is integrable under t |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.BoundedQuadraticCostIntegrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/BoundedQuadraticCostIntegrability.lean:85 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff
- A smooth real cutoff equal to one on `[-1, 1]` and zero outside `(-2, 2)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_eq_smoothTransition
- The unit cutoff written using Mathlib's smooth transition function. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_contDiff
- The unit cutoff is infinitely differentiable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_eq_one_of_abs_le_one
- The unit cutoff is one when `|x| <= 1`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:73 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_eq_zero_of_two_le_abs
- The unit cutoff vanishes when `2 <= |x|`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:80 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_mem_Icc
- The unit cutoff takes values in `[0, 1]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:87 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_hasCompactSupport
- The one-dimensional unit cutoff has compact support. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:91 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_deriv_bounded
- The derivative of the one-dimensional unit cutoff is bounded by one
positive constant. The constant is chosen before any radial scale. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:113 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_secondDeriv_continuous
- The second derivative of the unit cutoff is continuous. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:124 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_secondDeriv_hasCompactSupport
- The second derivative of the unit cutoff has compact support. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:135 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff_secondDeriv_bounded
- One positive constant bounds the second derivative of the fixed unit
cutoff. The constant is chosen before any radial scale, which is the compact
support input needed for a later `C / R^2` radial Hessian bound. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:142 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff
- The radial cutoff at scale `R`, given by `x ↦ smoothUnitCutoff (‖x‖ / R)`. |
def | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:155 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_eq_one_of_norm_le
- The radial cutoff is one on the closed ball of radius `R`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:159 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_eq_zero_of_two_mul_le_norm
- The radial cutoff vanishes when `2 * R <= ||x||`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:166 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_mem_Icc
- Every radial cutoff value lies in `[0, 1]`, for any scale `R`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:175 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.fderiv_norm_div_bound
- Scaling the norm by a positive radius gives an operator-norm derivative
bound of `1 / R`. Mathlib's totalized `fderiv` makes the statement valid at
the origin as well. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:182 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_contDiff
- For positive scale, the radial cutoff is infinitely differentiable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:198 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_fderiv_bound
- A single positive constant controls the first derivative of every
positive-scale radial cutoff by `C / R`. The quantifier order records the
scale-uniformity needed by cutoff exhaustion arguments. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:235 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_fderiv_eq_zero_of_two_mul_le_norm
- The totalized derivative of the radial cutoff vanishes throughout the
outer zero region, including its boundary sphere. At the boundary the cutoff
is a global minimum rather than locally constant; `IsLocalMin.fderiv_e |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:296 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_support_subset_closedBall
- The support of the radial cutoff lies in the closed ball of radius `2 * R`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:305 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_tsupport_subset_closedBall
- The topological support of the radial cutoff lies in the same closed ball. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:317 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_hasCompactSupport
- In finite dimension, a positive-scale radial cutoff has compact support. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:324 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_iteratedFDeriv_two_bound
- A single positive constant controls the second iterated Fréchet
derivative of every positive-scale radial cutoff by `C / R^2`.
The proof first bounds the second derivative of the unit-scale radial cutoff
using continu |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:339 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.radialSmoothCutoff_tendsto_one
- At each fixed point, the positive-scale radial cutoffs tend to one as the scale diverges. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:390 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.exists_contDiff_eq_one_tsupport_subset
- A compact subset of an open set admits a smooth compactly supported plateau in that set.
The function takes values in `[0, 1]` and is identically one on the compact set. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Cutoff.lean:408 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence
- Pointwise coordinate divergence of a finite-dimensional Euclidean vector
field.
For `F : EuclideanSpace ℝ ι → EuclideanSpace ℝ ι`, this is the coordinate sum
`∑ᵢ ∂ᵢ Fᵢ`, expressed using Mathlib's `lineDeriv` and the c |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_eq_sum_lineDeriv
- Unfold the ASTIS pointwise coordinate-divergence definition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_eq_sum_fderiv_apply_of_hasFDerivAt
- If a vector field has Frechet derivative `F'` at `x`, then the ASTIS
coordinate divergence is the coordinate trace-style sum `∑ᵢ (F' eᵢ)ᵢ`.
This matches the pointwise divergence summand shape used by Mathlib's
box-int |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:60 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_eq_sum_fderiv_apply_of_differentiableAt
- If a vector field is differentiable at `x`, then the ASTIS coordinate
divergence is the Mathlib divergence-theorem summand with `fderiv ℝ F x`.
This is the pointwise bridge needed before instantiating Mathlib's integr |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:85 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.continuousLinearEquiv_apply_euclideanSpace_single
- The `PiLp` continuous linear equivalence sends the Euclidean coordinate
unit to the corresponding Pi-space coordinate function. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:96 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.hasFDerivAt_radialSmoothCutoff_comp_toLp
- The derivative of the Euclidean radial cutoff transports to raw finite Pi
space through `WithLp.toLp 2` by the chain rule.
This is the cutoff-side `HasFDerivAt` producer consumed by the finite-box
cutoff-smul route. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:114 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.tendsto_integral_norm_fderiv_radialSmoothCutoff_comp_toLp_apply
- For an integrable finite Pi-space vector field, the `L¹` norm of the
radial-cutoff gradient applied to that field vanishes as the cutoff scale tends
to infinity.
The domination retains the operator norm of the inverse |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:157 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.tendsto_integral_radialSmoothCutoff_comp_toLp_smul
- Multiplication by the PiLp-wrapped radial cutoff converges to the identity
under integration for every integrable real normed-space-valued source field.
The statement is measure-generic and uses only integrability of |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:307 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.tendsto_setIntegral_norm_norm_ge_comp_toLp
- The `L¹` norm of an integrable field on the complement of an expanding
Euclidean ball tends to zero, expressed in raw finite-Pi coordinates.
The tail sets are `R ≤ ‖WithLp.toLp 2 x‖`. They form an antitone family wit |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:365 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.sum_smulRight_apply_pi_single_eq_apply
- The trace contribution of `χ'.smulRight G` over the standard finite Pi
basis is exactly the scalar derivative `χ'` applied to `G`.
This is pure finite-dimensional linear algebra. It identifies the cutoff
cross term u |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:403 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt
- Pointwise bridge from Mathlib's Pi-space derivative to ASTIS
`EuclideanSpace` coordinate divergence for a wrapped vector field.
This is the pointwise core needed to discharge the `hdiv_ae` assumption in the
box face-t |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:426 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.eventuallyEq_restrict_Icc_of_eqOn_univ_pi_Ioo_diff_countable
- If two functions on a finite-dimensional box agree on the open box away
from a countable exceptional set, then they agree a.e. on the closed box with
respect to restricted volume.
This is a reusable measure-theoretic |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:476 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_wrapped_toPi_trace_ae_of_ae_hasFDerivAt
- A.e. bridge from ASTIS wrapped coordinate divergence to Mathlib's Pi-space
trace summand, assuming the Pi-space derivative exists a.e. on the restricted
box.
This theorem intentionally does not derive the a.e. differe |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:499 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.coordinateDivergence_wrapped_toPi_trace_ae_of_hasFDerivAt_off_countable
- A.e. bridge from an open-box/off-countable `HasFDerivAt` hypothesis to the
`hdiv_ae` shape required by the finite-box face-term wrapper.
This discharges only the a.e. equality assumption. It does not prove
integrabil |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:522 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integrableOn_coordinateDivergence_wrapped_of_integrableOn_trace_of_hasFDerivAt_off_countable
- Transfer box integrability from Mathlib's Pi-space trace summand to the
ASTIS wrapped coordinate-divergence integrand.
This closes only the representation mismatch between the two integrands. The
trace integrability |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:550 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_of_hasFDerivAt_off_countable
- Box-level signed-face divergence theorem wrapper for ASTIS coordinate
divergence.
Mathlib's Bochner divergence theorem is stated on `Fin (n + 1) → ℝ`; ASTIS
finite Euclidean pointwise calculations use `EuclideanSpace |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:584 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_of_integrableOn_trace_of_hasFDerivAt_off_countable
- Box-level signed-face divergence theorem wrapper for ASTIS coordinate
divergence, using Mathlib's trace-integrability hypothesis directly.
Compared with `integral_coordinateDivergence_toPi_box_of_hasFDerivAt_off_count |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:651 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_eq_zero_of_boundary_component_eq_zero
- If the normal component of a Pi-space vector field vanishes on every
lower and upper face of a finite box, then Mathlib's signed face-term sum is
zero.
This is a boundary-value producer for the finite-box divergence r |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:689 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_eq_zero_of_update_boundary_component_eq_zero
- Version of `signedFaceTermSum_eq_zero_of_boundary_component_eq_zero`
with boundary values expressed by `Function.update`.
This is often the more convenient shape for later support or cutoff lemmas:
if replacing coordi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:712 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.update_boundary_component_eq_zero_of_eq_zero_off_univ_pi_Ioo
- If a Pi-space vector field vanishes outside the open box
`Set.univ.pi (fun i => Set.Ioo (a i) (b i))`, then its normal components
vanish after updating any coordinate to either endpoint.
This is a direct boundary prod |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:741 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_eq_zero_of_eq_zero_off_univ_pi_Ioo
- Off-open-box vanishing implies Mathlib's finite-box signed face-term sum
is zero.
This composes `update_boundary_component_eq_zero_of_eq_zero_off_univ_pi_Ioo`
with the update-shaped face-term producer. It still does |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:775 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.eq_zero_off_univ_pi_Ioo_of_support_subset_univ_pi_Ioo
- If the support of a Pi-space vector field is contained in the open box,
then the field vanishes outside that open box.
This is a support-to-boundary staging leaf. It uses plain
`Function.support`; it does not assert |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:794 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_cutoff_tsupport_subset_univ_pi_Ioo
- Smooth finite-dimensional cutoff localized inside a Pi-open box.
For any point of `Set.univ.pi (fun i => Set.Ioo (a i) (b i))`, Mathlib's
finite-dimensional bump theorem supplies a smooth real-valued cutoff whose
topo |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:814 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo
- Topological-support containment implies plain function-support containment
inside a finite Pi-open box.
This is the bridge needed by the finite-box cutoff route: Mathlib's smooth
cutoff theorem naturally returns `tsup |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:835 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_cutoff_support_subset_univ_pi_Ioo
- Smooth finite-dimensional cutoff localized inside a Pi-open box, with both
topological-support and plain function-support conclusions.
This packages `exists_contDiff_cutoff_tsupport_subset_univ_pi_Ioo` with
`support_s |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:850 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_support_eq_univ_pi_Ioo
- Smooth nonnegative bump whose plain support is exactly a finite Pi-open box.
This is the finite-box specialization of Mathlib's
`IsOpen.exists_contDiff_support_eq`. It is useful when a later cutoff argument
needs non |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:874 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.positive_on_univ_pi_Ioo_of_support_eq_univ_pi_Ioo
- A `[0,1]`-valued function whose support is exactly a finite Pi-open box is
strictly positive at every point of that box.
This is only a support/range consequence. It does not construct a compactly
supported cutoff, p |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:890 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.Icc_subset_univ_pi_Ioo_of_strict_bounds
- A closed inner Pi-box is contained in a strictly larger open Pi-box.
This is a bookkeeping leaf for exhaustion arguments. It only proves the
coordinate set inclusion needed to feed local cutoff construction; it does |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:909 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_cutoff_eq_one_on_Icc_tsupport_subset_outer_univ_pi_Ioo
- Smooth plateau for a finite closed Pi-box inside a strictly larger open
Pi-box.
The extra hypothesis `a ≤ b` records that the inner box is nonempty in the
intended exhaustion use. The construction comes from the gene |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:930 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.exists_contDiff_cutoff_support_subset_outer_univ_pi_Ioo_of_mem_Icc
- Local smooth cutoff for a point in an inner closed Pi-box, supported in a
strictly larger open Pi-box.
This packages the closed-box-to-open-box inclusion with
`exists_contDiff_cutoff_support_subset_univ_pi_Ioo`. It i |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:956 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_eq_zero_of_support_subset_univ_pi_Ioo
- Support contained in the open box implies Mathlib's finite-box signed
face-term sum is zero.
This is still only a finite-box support-to-face producer. It does not prove
that a concrete Langevin/cutoff vector field ha |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:977 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.support_smul_subset_univ_pi_Ioo_of_eq_zero_off_univ_pi_Ioo
- If a scalar cutoff vanishes outside the open Pi-box, then multiplying any
Pi-space vector field by this cutoff gives a vector field supported in the open
Pi-box.
This is a plain support-containment leaf for finite-box |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:998 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.support_smul_subset_univ_pi_Ioo_of_scalar_support_subset_univ_pi_Ioo
- If a scalar cutoff is supported in the open Pi-box, then multiplying any
Pi-space vector field by this cutoff gives a vector field supported in the open
Pi-box.
This only uses `Function.support`; it is not a `HasCompa |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1018 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.support_smul_subset_univ_pi_Ioo_of_scalar_tsupport_subset_univ_pi_Ioo
- If the topological support of a scalar cutoff is contained in the open
Pi-box, then multiplying any vector field by that cutoff is plain-supported in
the same open box.
This is the direct consumer-facing bridge from M |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1041 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.continuousOn_smul_vectorField_of_continuousOn
- Closed-box continuity for a scalar cutoff times a Pi-space vector field.
This packages Mathlib's `ContinuousOn.smul` in the exact finite-box shape used
by the cutoff-smul divergence-theorem route. It does not prove s |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1058 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.hasFDerivAt_smul_vectorField_of_hasFDerivAt
- Pointwise Frechet derivative for a scalar cutoff times a Pi-space vector
field.
The derivative is exactly the Mathlib product-rule derivative
`χ x • G' + χ'.smulRight (G x)`. This is only a pointwise derivative leaf; |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1075 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.hasFDerivAt_smul_vectorField_off_countable
- Open-box/off-countable Frechet derivative wrapper for a scalar cutoff times
a Pi-space vector field.
This derives the `Hd` shape required by the finite-box divergence-theorem
handoffs from separate derivative hypothes |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1095 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.continuousOn_smul_vectorField_trace_of_component_continuousOn
- Closed-box continuity of the cutoff-smul product-rule trace from only the
coordinate component continuity needed by the trace summand.
The expanded summand is
`χ x * (G' x eᵢ)ᵢ + (χ' x eᵢ) * (G x)ᵢ`. This leaf theref |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1123 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.continuousOn_smul_vectorField_trace_of_components
- Closed-box continuity of the cutoff-smul product-rule trace from component
continuity of the cutoff, cutoff derivative field, vector field, and vector
field derivative.
This only assembles continuity of the trace expr |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1160 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integrableOn_smul_vectorField_trace_of_continuousOn
- Closed-box integrability for the trace of the cutoff-smul product-rule
derivative, assuming that trace expression is continuous on the closed box.
This is a compact-box integrability handoff only. It does not prove c |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1189 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_smul_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo
- Scalar cutoff vanishing outside the open Pi-box implies Mathlib's finite-box
signed face-term sum is zero for the cutoff-smul vector field.
Regularity of the cutoff-smul field is not addressed here; this is only the
f |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1212 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_smul_eq_zero_of_scalar_support_subset_univ_pi_Ioo
- Scalar cutoff support contained in the open Pi-box implies Mathlib's
finite-box signed face-term sum is zero for the cutoff-smul vector field.
This is still a finite-box support-to-face producer, not a smooth-cutoff
c |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1231 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.signedFaceTermSum_smul_eq_zero_of_scalar_tsupport_subset_univ_pi_Ioo
- Scalar cutoff topological support contained in the open Pi-box implies
Mathlib's finite-box signed face-term sum is zero for the cutoff-smul vector
field.
This is a direct `tsupport`-API handoff for the local smooth-c |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1253 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_integrableOn_trace_of_hasFDerivAt_off_countable
- Finite-box zero-face corollary for ASTIS coordinate divergence.
This is the smallest finite-box integration-by-parts handoff: once the signed
face term from Mathlib's divergence theorem is explicitly known to vanish, |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1276 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_boundary_component_eq_zero
- Finite-box coordinate-divergence integral vanishes when the vector field's
normal component is explicitly zero on every lower and upper face.
This composes the finite-box signed-face divergence theorem with the
compon |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1311 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_update_boundary_component_eq_zero
- Finite-box coordinate-divergence integral vanishes from `Function.update`
boundary-value hypotheses.
This is the `Function.update`-shaped companion to
`integral_coordinateDivergence_toPi_box_eq_zero_of_boundary_compon |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1344 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_eq_zero_off_univ_pi_Ioo
- Finite-box coordinate-divergence integral vanishes when the vector field
vanishes outside the open Pi-box.
This is still a finite-box conditional theorem: it assumes the trace
integrability and open-box/off-countable |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1376 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_support_subset_univ_pi_Ioo
- Finite-box coordinate-divergence integral vanishes when the vector field's
support is contained in the open Pi-box.
This composes the support-to-face producer with the finite-box divergence
wrapper. It is not a compa |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1404 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo
- Finite-box coordinate-divergence integral vanishes for a cutoff-smul vector
field when the scalar cutoff vanishes outside the open Pi-box.
This preserves the existing divergence-theorem hypotheses for the cutoff-smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1432 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo
- Finite-box coordinate-divergence integral vanishes for a cutoff-smul vector
field when the scalar cutoff support is contained in the open Pi-box.
This is not a compact-support or whole-space IBP result; it simply feed |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1460 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_tsupport_subset_univ_pi_Ioo
- Finite-box coordinate-divergence integral vanishes for a cutoff-smul vector
field when the scalar cutoff's topological support is contained in the open
Pi-box.
This is a `tsupport`-API variant of the scalar-support fi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1492 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_regularity
- Finite-box coordinate-divergence integral vanishes for a cutoff-smul vector
field, deriving the continuity and off-countable Frechet differentiability
hypotheses from separate cutoff and vector-field regularity assumpt |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1523 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_trace_continuous
- Finite-box coordinate-divergence integral vanishes for a cutoff-smul vector
field when the scalar cutoff vanishes outside the open Pi-box, deriving the
regularity hypotheses from separate cutoff/vector-field assumption |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1564 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_component_continuous
- Component-continuity version of
`integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_trace_continuous`.
It derives closed-box trace continuity from separate continuity assumptions on
`χ |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1602 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_regularity
- Scalar-support version of
`integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_regularity`.
It derives the cutoff-smul continuity and open-box/off-countable derivative
hypotheses, but s |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1639 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_fderiv
- Canonical-`fderiv` scalar-support version of the cutoff-smul finite-box
zero integral handoff.
This removes only the supplied derivative-field parameter `G'`, replacing it
by `fderiv ℝ G` under open-box differentiabil |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1679 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_trace_continuous
- Scalar-support version of
`integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_trace_continuous`.
It uses closed-box continuity of the product-rule trace to discharge the
compact-box tr |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1713 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_component_continuous
- Component-continuity scalar-support version of the cutoff-smul finite-box
trace handoff.
It derives the trace-continuity input from separate continuity assumptions on
`χ`, `χ'`, `G`, and `G'`, then applies the scalar- |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1750 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence.integral_coordinateDivergence_wrapped_eq_zero_of_contDiff_of_hasCompactSupport
- The whole-space coordinate-divergence integral of a compactly supported
`C¹` vector field is zero.
The field is represented in raw finite-Pi coordinates, while
`coordinateDivergence` is evaluated after the canonical ` |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Divergence |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1794 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.hasGradientAt_expNegPotential_of_hasGradientAt
- Chain rule for the Gibbs weight `exp (-V)` in Mathlib's gradient API.
This is the pointwise gradient identity behind the Langevin supplied hypothesis
`∇rho = -rho • ∇V` when `rho x = exp (-V x)`. It does not prove an |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.gradient_expNegPotential_eq_of_hasGradientAt
- Mathlib-gradient form of the Gibbs weight chain rule from a supplied
potential gradient.
This is the same local chain rule as
`hasGradientAt_expNegPotential_of_hasGradientAt`, followed by Mathlib's
uniqueness theorem |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:55 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.gradient_expNegPotential_coordinate_eq_of_hasGradientAt
- Coordinate form of
`gradient_expNegPotential_eq_of_hasGradientAt` on finite Euclidean space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:65 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.gradient_expNegPotential_eq_of_differentiableAt
- Pointwise Mathlib-gradient form of the Gibbs weight chain rule.
If `V` is differentiable at `x`, then Mathlib's total `gradient` of
`fun y => exp (-V y)` agrees with the expected vector
`-exp (-V x) • gradient V x`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:84 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.gradient_expNegPotential_coordinate_eq_of_differentiableAt
- Coordinate form of
`gradient_expNegPotential_eq_of_differentiableAt` on finite Euclidean space.
This is the narrow reusable leaf that supplies the Gibbs-weight chain-rule
coordinate equality used by the Langevin algeb |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:97 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.continuous_gradient_of_contDiff_one
- A globally `C¹` real-valued function has a continuous Mathlib gradient.
This is the reusable regularity handoff from a test-function class to the
component-continuity hypothesis used in the Langevin finite-box trace l |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:114 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_of_hasGradientAt
- Apply the Frechet derivative to a vector when a gradient representative is
supplied.
This is the basic bridge from Mathlib's `fderiv` to the inner-product
gradient convention. It is pointwise only: it does not choose |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:128 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt
- Apply the Frechet derivative to a vector and rewrite the result using
Mathlib's total `gradient`.
This is the pointwise `fderiv`/`gradient` bridge used before finite-coordinate
Langevin displays. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:144 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_coordinate_eq_gradient_coordinate_of_differentiableAt
- Euclidean coordinate form of the pointwise `fderiv`/`gradient` bridge.
For the coordinate unit `eᵢ`, applying `fderiv ℝ f x` is the corresponding
coordinate of Mathlib's `gradient f x`. This removes only the local
gr |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:158 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.hasGradientAt_coordinateUnit_hasLineDerivAt
- A supplied Mathlib gradient gives the line derivative in a coordinate unit
direction, with value equal to that coordinate of the gradient.
This is a pointwise Euclidean coordinate bridge. It does not define divergenc |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:179 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra.HasGradientAt.mul
- Product rule for gradients of real-valued functions on a real Hilbert
space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/GradientAlgebra.lean:36 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra.HasGradientAt.comp_real
- Scalar outer-function chain rule for gradients. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/GradientAlgebra.lean:48 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra.HasGradientAt.inv
- Reciprocal rule for gradients on the nonzero locus.
The coefficient is kept in Mathlib's native derivative form
`-(f x ^ 2)⁻¹`; downstream algebra may rewrite it as `-1 / f(x)^2` when
needed. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/GradientAlgebra.lean:69 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra.HasGradientAt.div
- Quotient rule for gradients on the nonzero-denominator locus.
It is stated in the product-with-reciprocal form produced directly by the two
reusable rules above:
`grad(f/g) = f * (-(g^2)⁻¹ grad g) + g⁻¹ grad f`.
Thi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/GradientAlgebra.lean:86 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra.gradient_mul_eq_of_differentiableAt
- Total-gradient form of the product rule under pointwise differentiability. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/GradientAlgebra.lean:98 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra.gradient_comp_real_eq_of_differentiableAt
- Total-gradient form of the scalar chain rule. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/GradientAlgebra.lean:107 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra.gradient_inv_eq_of_differentiableAt
- Total-gradient reciprocal rule under pointwise differentiability and a
nonzero value. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/GradientAlgebra.lean:118 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra.gradient_div_eq_of_differentiableAt
- Total-gradient quotient rule under pointwise differentiability and a
nonzero denominator. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.GradientAlgebra |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/GradientAlgebra.lean:128 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.laplacian_eq_sum_stdOrthonormalBasis
- Mathlib's finite-dimensional standard-orthonormal-basis formula for the
Laplacian, exposed as an ASTIS calculus leaf.
For Ch.1 Langevin this is the coordinate bridge behind a supplied Laplacian
identifier `lapF = ∑ᵢ ∂ |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Laplacian.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.laplacianFunctional_eq_of_stdOrthonormalBasis_sum
- Handoff form of `laplacian_eq_sum_stdOrthonormalBasis` for source-defined
Laplacian functionals.
This is useful when a paper defines a weak-generator or test-function action by
the coordinate second-derivative sum and |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Laplacian.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.continuous_laplacian_of_contDiff_two
- A globally `C²` real-valued function has a continuous Mathlib Laplacian.
This packages the standard finite-dimensional route: expand the Laplacian in a
standard orthonormal basis, use continuity of the second iterated |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Laplacian.lean:64 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.norm_laplacian_le_finrank_mul_norm_iteratedFDeriv_two
- The Laplacian is bounded by dimension times the operator norm of the
second iterated Fréchet derivative.
The dimension factor comes only from summing the diagonal evaluations in a
standard orthonormal basis. This is |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Laplacian.lean:83 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian.radialSmoothCutoff_laplacian_bound
- Positive-scale radial cutoff Laplacians have the expected `R^-2` bound,
with the finite-dimensional trace factor shown explicitly. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Laplacian |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Laplacian.lean:105 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.hasLineDerivAt_mul
- Product rule for algebra-valued line derivatives.
This is a direct wrapper around Mathlib's `HasDerivAt.mul` applied to the
one-dimensional curve `t ↦ x + t • v`. In the Langevin tree, the real-valued
specialization |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.hasLineDerivAt_rho_mul
- Real-valued `rho * g` specialization of `hasLineDerivAt_mul`, in the
summand order used by finite-coordinate weighted-divergence algebra. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_rho_mul_eq_of_hasLineDerivAt
- Line-derivative equality form of the real-valued `rho * g` product rule.
This is the form needed by coordinate divergence displays, where a source
calculation usually names `lineDeriv ℝ (fun y => rho y * g y) x v` rat |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:69 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_expNegPotential_mul_eq_of_differentiableAt
- Coordinate-unit product rule for the Gibbs weight `exp (-V)`.
Given differentiability of the potential and a supplied coordinate derivative
of `g`, this computes the line derivative of
`fun y => exp (-V y) * g y` in t |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:87 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.hasLineDerivAt_fderiv_apply_const_of_hasFDerivAt_fderiv
- A supplied derivative of the first derivative gives the coordinate line
derivative of `fun y => fderiv ℝ f y v`.
This is a small Hessian-wiring leaf for the Langevin tree. It does not assert
that the supplied second- |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:141 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_fderiv_apply_const_eq_of_hasFDerivAt_fderiv
- Line-derivative equality form of
`hasLineDerivAt_fderiv_apply_const_of_hasFDerivAt_fderiv`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:154 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_fderiv_apply_const_eq_iteratedFDeriv_two
- If the total `fderiv` map is differentiable at `x`, then the line derivative
of the fixed slice `fun y => fderiv ℝ f y v` is the two-fold iterated
derivative.
This is the direct bridge from a coordinate derivative of |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:169 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_fderiv_apply_coordinate_eq_iteratedFDeriv_two
- Coordinate-unit version of
`lineDeriv_fderiv_apply_const_eq_iteratedFDeriv_two` on finite Euclidean
space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:190 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv.lineDeriv_expNegPotential_mul_fderiv_coordinate_eq
- Coordinate product rule for the explicit Gibbs weight multiplied by the
coordinate derivative represented as `fderiv ℝ f y eᵢ`.
Compared with `lineDeriv_expNegPotential_mul_eq_of_differentiableAt`, this
also discharge |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv |
AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:218 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexAEDifferentiable.ae_differentiableAt_of_convexOn_univ
- A finite-valued globally convex potential on a finite-dimensional real
normed space is Frechet differentiable almost everywhere with respect to any
additive Haar measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexAEDifferentiable |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexAEDifferentiable.lean:34 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexDomainACAEDifferentiable.ae_differentiableAt_of_convexOn_of_absolutelyContinuous
- A real-valued convex function is `μ`-a.e. Frechet differentiable whenever
`μ` is absolutely continuous with respect to additive Haar measure and is
almost everywhere concentrated on the convex domain on which convexity |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexDomainACAEDifferentiable |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexDomainACAEDifferentiable.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexGradientGapSharpness.quadratic_gap_lower_bound
- A horizon-dependent positive quadratic has an actual gradient-descent gap
of order `β / (N + 1)`. The smoothness bound `β` need not be tight. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexGradientGapSharpness |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexGradientGapSharpness.lean:20 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexLocalSubgradient.SupportsOn
- `y` supports `phi` at `x` relative to a set `s`. Downstream proper-potential
arguments will use `s` equal to the finite effective domain. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexLocalSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexLocalSubgradient.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexLocalSubgradient.supportsOn_of_supportsAt
- A global support inequality restricts to any set. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexLocalSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexLocalSubgradient.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexLocalSubgradient.eq_gradient_of_supportsOn_of_hasFDerivAt
- If the relative support set is a neighborhood of `x`, then at a Frechet
point the supporting vector is exactly the Riesz representative of the
derivative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexLocalSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexLocalSubgradient.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexLocalSubgradient.supportsOn_unique_of_hasFDerivAt
- Consequently, two vectors supporting the same real function on a
neighborhood of a differentiability point coincide. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexLocalSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexLocalSubgradient.lean:67 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexOpenAEDifferentiable.ae_differentiableAt_of_convexOn_isOpen
- A finite-valued convex function on an open convex domain of a
finite-dimensional real normed space is Frechet differentiable almost
everywhere at points of that domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexOpenAEDifferentiable |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexOpenAEDifferentiable.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSmoothGradient.gradient_gap_sq_le_bregman
- The positive-modulus Bregman gap controls the squared gradient difference.
Dividing by `2 * β > 0` recovers the source's reciprocal formula (3.4). |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSmoothGradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexSmoothGradient.lean:34 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSmoothGradient.gradient_cocoercive
- Convex gradients are cocoercive in the division-free normalization.
The zero-modulus case follows by relaxing the upper model to `β + ε` and
letting positive `ε` tend to zero, without assigning meaning to a zero denomi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSmoothGradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexSmoothGradient.lean:64 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSmoothGradient.gradient_lipschitz
- A C¹ convex function with the global quadratic upper model has a
`β`-Lipschitz gradient, including `β = 0`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSmoothGradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexSmoothGradient.lean:93 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSubgradient.SupportsAt
- `y` is a global Hilbert-space subgradient/supporting vector of `phi` at
`x`. This is deliberately a pointwise relation; an OT coupling can later be
required to be concentrated on this graph without first choosing a tra |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexSubgradient.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSubgradient.eq_gradient_of_supportsAt_of_hasFDerivAt
- At a Frechet differentiability point, a supporting vector is the unique
Riesz representative of the derivative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexSubgradient.lean:44 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSubgradient.supportsAt_unique_of_hasFDerivAt
- In particular, two supporting vectors at the same differentiability point
coincide. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexSubgradient.lean:65 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC1.sub_eq_integral_gradient
- The fundamental theorem of calculus along an affine segment. Continuity
and interval integrability of the genuine gradient pairing follow from C¹. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC1 |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexityC1.lean:30 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC1.strongConvexOn_univ_of_gradient_mono_integral
- Quantitative gradient monotonicity implies the chord inequality by the
source's two affine-segment FTC identities and integration of their difference.
The modulus can be signed; no Hessian or extra integrability is ass |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC1 |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexityC1.lean:47 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC1.convexity_equivalences
- Proposition 1.6, part 1: on all of Euclidean space, the C¹ chord,
quadratic lower-model and gradient-monotonicity conditions are equivalent.
The nonnegative modulus is retained exactly as in the source. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC1 |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexityC1.lean:111 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC2.gradient_sub_inner_eq_integral_fderiv2
- The genuine Hessian integrates to the gradient difference paired with the
segment direction. C² supplies continuity and integrability. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC2 |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexityC2.lean:26 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC2.gradient_mono_iff_fderiv2_lower
- Global quantitative gradient monotonicity is equivalent to the genuine
Hessian diagonal lower bound, by a right derivative limit and the FTC. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC2 |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexityC2.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC2.strongConvexOn_iff_fderiv2_lower
- Proposition 1.6 part 2, with the source's whole Euclidean domain,
nonnegative modulus and C² regularity. Together with the C¹ theorem this
connects all four source conditions. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexityC2 |
AutoSamplingTheory/TechnicalLemmas/Analysis/ConvexityC2.lean:88 |
AutoSamplingTheory.TechnicalLemmas.Analysis.CycleSuccessorDistinct.cycleLength_pos_of_cycleValue_pos
- A strictly positive pairing-cycle value cannot occur on the one-point
cycle. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.CycleSuccessorDistinct |
AutoSamplingTheory/TechnicalLemmas/Analysis/CycleSuccessorDistinct.lean:25 |
AutoSamplingTheory.TechnicalLemmas.Analysis.CycleSuccessorDistinct.cycleSuccessorPerm_ne_self_of_pos
- On a cycle with at least two coordinates, translation by one has no fixed
points. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.CycleSuccessorDistinct |
AutoSamplingTheory/TechnicalLemmas/Analysis/CycleSuccessorDistinct.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicCostExpectation.integral_cyclicCost_lt_diagonal_of_uniform_cycleValue
- If every coordinate law is concentrated on a local set and the whole local
box carries one uniform positive cycle-value margin, then the normalized cyclic
re-pairing has strictly smaller expected quadratic cost.
Integ |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicCostExpectation |
AutoSamplingTheory/TechnicalLemmas/Analysis/CyclicCostExpectation.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost.cycleSuccessorPerm
- Modular successor permutation on the nonempty finite cycle. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/CyclicQuadraticCost.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost.cycleSuccessorPerm_apply
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/CyclicQuadraticCost.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost.permutedPairingGap_cycleSuccessor_eq_cycleValue
- The permutation-generic pairing gap becomes exactly the existing
`cycleValue` for the modular successor. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/CyclicQuadraticCost.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost.diagonal_sub_cyclicCost_eq_two_cycleValue
- Exact quadratic cost identity for the cyclic successor. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/CyclicQuadraticCost.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost.cyclicQuadraticCost_lt_of_cycleValue_pos
- A strict positive pairing-cycle violation produces a strictly cheaper
quadratic cyclic re-pairing. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.CyclicQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/CyclicQuadraticCost.lean:55 |
AutoSamplingTheory.TechnicalLemmas.Analysis.DiagonalProductCostIntegral.integral_diagonalQuadraticCost_eq_sum
- Under a finite product of probability laws on pairs, the expected diagonal
quadratic cost is the sum of the expected one-coordinate quadratic costs. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.DiagonalProductCostIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/DiagonalProductCostIntegral.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment.absorb_quadratic
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment |
AutoSamplingTheory/TechnicalLemmas/Analysis/GibbsGradientMoment.lean:34 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment.weighted_square
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment |
AutoSamplingTheory/TechnicalLemmas/Analysis/GibbsGradientMoment.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment.weighted_gradient
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment |
AutoSamplingTheory/TechnicalLemmas/Analysis/GibbsGradientMoment.lean:78 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment.directional_ibp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment |
AutoSamplingTheory/TechnicalLemmas/Analysis/GibbsGradientMoment.lean:132 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment.gibbs_gradient_moment
- Probability, actual Gibbs L1, the score-square/diagonal-Hessian identity,
and its sharp curvature-dimension upper bound for the normalized Gibbs law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment |
AutoSamplingTheory/TechnicalLemmas/Analysis/GibbsGradientMoment.lean:209 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentBasic.gradient_step_descent_of_quadratic_upper_bound
- Actual gradient-step descent from a global quadratic upper model.
Includes a zero step and signed upper-model coefficient. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentBasic |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentBasic.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentComplexity.distance_le_of_log_bound
- The source logarithmic iteration threshold suffices for distance accuracy;
zero initial distance needs no logarithm or positive iteration count. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentComplexity |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentComplexity.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentContraction.gradient_step_contraction
- One gradient step contracts by `sqrt (1 - α * h)` under the global curvature
and quadratic upper models, with explicit nonnegative step. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentContraction |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentContraction.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentContraction.gradient_descent_distance_bound
- The actual Nth gradient-descent iterate has geometric, then exponential,
distance control about a supplied global minimizer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentContraction |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentContraction.lean:62 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.gradient_step_endpoint_bound
- Actual gradient maps satisfy the endpoint spectral bound for nonnegative steps.
The curvature moduli may be signed; all Hessians are derived from the C² objective. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentOptimalStep.lean:25 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.optimal_gradient_step
- The step `2/(α+β)` gives the sharp uniform curvature-envelope contraction.
Its factor minimizes the endpoint max-envelope over every real step.
The `α=0` boundary is nonexpansive, and `α=β` is retained. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentOptimalStep.lean:102 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentPL.gradient_descent_pl_value_bound
- Final-value bound for actual gradient iterates under a PL model.
No additional nonnegative-coefficient restriction or convexity is required. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentPL |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentPL.lean:24 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentRates.convex_value_le
- Convex function-value rate at every positive iteration, for any comparator. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentRates |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentRates.lean:20 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentRates.strongly_convex_value_le
- Strongly convex rational rate, and exact inverse-power form on its positive-base domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentRates |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentRates.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentSharpness.exists_quadratic_worst_case
- Every fixed step has a positive scalar quadratic attaining the endpoint
max-envelope at every iteration count; the balanced step attains its minimax
factor on the same witness. The curvature parameters are class bounds |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentSharpness |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentSharpness.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentStationarity.gradient_descent_sum_sq_bound
- Telescoping the actual gradient-step decrease bounds the accumulated squared gradients.
This unnormalized inequality includes zero steps and empty sums. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentStationarity |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentStationarity.lean:25 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentStationarity.exists_gradient_descent_norm_le
- Among the first `N` actual gradient iterates, one has small gradient norm.
This is a best-iterate guarantee, not a last-iterate or global optimality guarantee. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentStationarity |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentStationarity.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentValue.gradient_step_energy_bound
- One-step energy inequality for every comparator, with the necessary
nonnegative step explicit. No sign restriction on the comparator gap. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentValue |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentValue.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentValue.gradient_descent_weighted_value_bound
- Weighted final function gap for actual gradient-descent iterates. The
coefficient domain is explicit and includes both zero and one. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentValue |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentValue.lean:54 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowContraction.norm_sub_le
- Two supplied gradient trajectories contract on every finite forward interval.
Zero curvature gives nonexpansiveness; positive curvature gives exponential decay.
No existence of a trajectory or of a minimizer is asserte |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowContraction |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientFlowContraction.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowLastIterate.lyapunov_and_rates
- The source Lyapunov decreases along an actual convex gradient trajectory,
yielding last-time gradient and improved objective upper bounds. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowLastIterate |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientFlowLastIterate.lean:22 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowPL.dissipation_and_decay
- Actual gradient dynamics imply energy dissipation and the PL objective rate,
including the initial and final endpoints of a finite nonnegative time interval. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowPL |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientFlowPL.lean:22 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowStationarity.exists_min_norm_le
- A minimum gradient norm is attained on the supplied time interval, and is
bounded by the square root of the initial objective gap divided by elapsed time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowStationarity |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientFlowStationarity.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowValue.value_le
- The objective gap along an actual convex gradient trajectory, including the
zero-curvature rate, at every positive time of a supplied forward interval. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowValue |
AutoSamplingTheory/TechnicalLemmas/Analysis/GradientFlowValue.lean:22 |
AutoSamplingTheory.TechnicalLemmas.Analysis.HessianStrongConvexity.strongConvexOn_univ_of_fderiv2_lower
- An everywhere C² potential whose genuine second Fréchet derivative is
bounded below on diagonal directions is strongly convex with the same modulus.
The `ContDiff` hypothesis prevents totalized derivatives from serving |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.HessianStrongConvexity |
AutoSamplingTheory/TechnicalLemmas/Analysis/HessianStrongConvexity.lean:28 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_ofReal_ne_top_of_integrable_nonneg
- A nonnegative integrable real function has finite `ℝ≥0∞` lintegral after
`ENNReal.ofReal`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_mul_norm_sq
- Finite-dimensional Gaussian quadratic tails are Lebesgue-integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_add_mul_norm_sq
- A shifted finite-dimensional Gaussian quadratic tail is
Lebesgue-integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:63 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_norm_sq_ne_top
- The `ℝ≥0∞` integral of a shifted finite-dimensional Gaussian quadratic
tail is finite. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:73 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_add_mul_norm_sub_sq
- A centered finite-dimensional Gaussian quadratic tail is
Lebesgue-integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:81 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_norm_sub_sq_ne_top
- The `ℝ≥0∞` integral of a centered finite-dimensional Gaussian quadratic
tail is finite. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:88 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_add_mul_abs
- One-dimensional Laplace tails are Lebesgue-integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:97 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_abs_ne_top
- The `ℝ≥0∞` integral of a one-dimensional Laplace tail is finite. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:130 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integral_exp_neg_add_mul_abs_eq
- Exact normalizer for one-dimensional absolute-linear Laplace tails. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:137 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_abs_eq
- Exact `ℝ≥0∞` normalizer for one-dimensional absolute-linear Laplace
tails. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:194 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_mul_norm_sq_eq
- Exact `ℝ≥0∞` normalizer for the finite-dimensional quadratic Gaussian
tail. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:206 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_norm_sq_eq
- Exact `ℝ≥0∞` normalizer for a shifted finite-dimensional quadratic
Gaussian tail. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:216 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_exp_neg_add_mul_norm_sub_sq_eq
- Exact `ℝ≥0∞` normalizer for a centered finite-dimensional quadratic
Gaussian tail. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:234 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_exp_neg_add_mul_norm_sq
- The explicitly normalized finite-dimensional quadratic Gibbs density is a
probability measure on Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:252 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_exp_neg_add_mul_norm_sub_sq
- The explicitly normalized centered finite-dimensional quadratic Gibbs
density is a probability measure on Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:298 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_exp_neg_add_mul_abs
- The explicitly normalized one-dimensional absolute-linear Laplace Gibbs
density is a probability measure on Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:344 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_gibbsDensityENNReal_ne_top_of_ae_quadratic_lower_bound
- A quadratic lower bound on a potential gives a finite Gibbs normalization
constant on finite-dimensional Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:388 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_gibbsDensityENNReal_ne_top_of_ae_centered_quadratic_lower_bound
- A centered quadratic lower bound on a potential gives a finite Gibbs
normalization constant on finite-dimensional Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:399 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_gibbsDensityENNReal_ne_top_of_ae_abs_linear_lower_bound
- A one-dimensional absolute-linear lower bound on a potential gives a finite
Gibbs normalization constant on Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:410 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_gibbsDensityENNReal_ne_top_of_strongConvexOn_minimizer
- A strongly convex potential with an exposed global minimizer has a finite
Gibbs normalization constant on finite-dimensional Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:421 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_quadratic_lower_bound
- A measurable potential with a quadratic lower bound defines a normalized
Gibbs probability measure on finite-dimensional Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:438 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_centered_quadratic_lower_bound
- A measurable potential with a centered quadratic lower bound defines a
normalized Gibbs probability measure on finite-dimensional Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:454 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_abs_linear_lower_bound
- A measurable one-dimensional potential with an absolute-linear lower bound
defines a normalized Gibbs probability measure on Lebesgue space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:470 |
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.isProbabilityMeasure_withDensity_normalized_gibbs_of_strongConvexOn_minimizer
- A measurable strongly convex potential with an exposed global minimizer
defines a normalized Gibbs probability measure on finite-dimensional Lebesgue
space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/Integrability.lean:487 |
AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage.leftAverageError
- Mean pointwise error over the left interval `[t-h,t]`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage |
AutoSamplingTheory/TechnicalLemmas/Analysis/LeftLebesgueAverage.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage.tendsto_sub_nhdsGT_zero_nhdsLT
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage |
AutoSamplingTheory/TechnicalLemmas/Analysis/LeftLebesgueAverage.lean:26 |
AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage.ae_tendsto_leftAverageError_real
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage |
AutoSamplingTheory/TechnicalLemmas/Analysis/LeftLebesgueAverage.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage.ae_tendsto_leftAverageError
- For almost every time, the left average error converges to zero as a
strictly positive nonnegative window shrinks to zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage |
AutoSamplingTheory/TechnicalLemmas/Analysis/LeftLebesgueAverage.lean:56 |
AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage.ae_tendsto_leftAverageError_two_mul
- Sequential form used by dyadic meshes: any positive real mesh tending to
zero gives vanishing left average error along twice that mesh. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage |
AutoSamplingTheory/TechnicalLemmas/Analysis/LeftLebesgueAverage.lean:74 |
AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage.abs_normalized_setIntegral_sub_le_two_mul_leftAverageError
- A normalized average on a subinterval of the left neighborhood has error
at most twice the full-neighborhood average when its mass is half the mass of
the neighborhood. This is the deterministic estimate behind one-ce |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage |
AutoSamplingTheory/TechnicalLemmas/Analysis/LeftLebesgueAverage.lean:104 |
AutoSamplingTheory.TechnicalLemmas.Analysis.MeasurableGradient.measurable_gradient
- Mathlib's total Hilbert gradient of any real-valued function is measurable.
At nondifferentiability points both the underlying total derivative and the
gradient use their canonical default value. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.MeasurableGradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/MeasurableGradient.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain.chainValue
- Affine value of a finite chain at a terminal source point `z`.
For `[p₀,...,pₖ]`, this is
`Σ_{j<k} <p_j.2, p_{j+1}.1-p_j.1> + <p_k.2, z-p_k.1>`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChain.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain.PairingClosedChainMonotone
- The rooted list condition consumed by the Rockafellar potential: closing a
nonempty chain back at the source coordinate of its head has nonpositive value. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChain.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain.chainValue_singleton
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChain.lean:47 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain.chainValue_append_cons
- Concatenating a nonempty second chain at `q` splits the value at the join. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChain.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain.exists_split_of_mem
- Membership of an element in a list gives a prefix/suffix decomposition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChain.lean:66 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain.exists_duplicate_split
- A duplicated element gives two explicit occurrences with an intermediate loop. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChain.lean:78 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain.chainValue_duplicate_split
- Removing the segment between two identical pair points decomposes the
original affine chain value into the shortened outer chain plus the closed
inner loop. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChain.lean:91 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity.tuplePathValue
- Split form of the pairing value of a nonempty finite tuple: all ordinary
successor edges followed by a final edge to the chosen terminal source point. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChainMonotonicity.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity.chainValue_ofFn_eq_tuplePathValue
- `chainValue` of `List.ofFn p` is exactly the split tuple path value.
The proof peels the final tuple entry using `List.ofFn_succ'`; therefore no
cyclic index arithmetic appears here. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChainMonotonicity.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity.tuplePathValue_at_first_eq_cycle_sum
- The split tuple path closed at its first source coordinate is the canonical
modular pairing-cycle sum. The wraparound edge is discharged algebraically by
`Fin.neg_last`, rather than by fragile modular arithmetic autom |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChainMonotonicity.lean:72 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity.chainValue_ofFn_at_first_eq_cycle_sum
- Closing `List.ofFn p` at its first source coordinate recovers the exact
pairing-cycle sum used by `PairingDistinctCycleMonotone`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChainMonotonicity.lean:93 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity.chainValue_nonpos_of_nodup
- Nodup rooted chains are exactly the direct finite-cycle case. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChainMonotonicity.lean:102 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity.pairingClosedChainMonotone_of_distinct_aux
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChainMonotonicity.lean:137 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity.pairingClosedChainMonotone_of_distinct
- The standard transport endpoint on injective finite cycles already implies
the closed rooted-list condition consumed by the Rockafellar construction. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingClosedChainMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingClosedChainMonotonicity.lean:182 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCycleNeighborhood.cycleValue
- The signed pairing increment around a finite nonempty cycle. Positive
values are strict violations of `PairingCycleMonotone`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCycleNeighborhood |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCycleNeighborhood.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCycleNeighborhood.continuous_cycleValue
- The pairing cycle value varies continuously with all points of the finite
tuple. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCycleNeighborhood |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCycleNeighborhood.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCycleNeighborhood.exists_pairwiseDisjoint_open_rectangles_of_cycleValue_pos
- A strict violation at distinct points persists on pairwise disjoint open
rectangles around those points. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCycleNeighborhood |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCycleNeighborhood.lean:57 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCycleQuantitativeNeighborhood.exists_pairwiseDisjoint_bounded_open_rectangles_of_cycleValue_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCycleQuantitativeNeighborhood |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCycleQuantitativeNeighborhood.lean:25 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity.PairingCycleMonotone
- Standard finite-cycle pairing monotonicity on a relation `Gamma`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCyclicMonotonicity.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity.PairingDistinctCycleMonotone
- The distinct-point cycle contract expected directly from the future
finite-neighborhood perturbation proof of optimality. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCyclicMonotonicity.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity.PairingShiftedCycleMonotone
- The standard finite-cycle condition in diagonal-versus-shifted pairing
form. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCyclicMonotonicity.lean:59 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity.cycle_increment_sum_eq_shifted_sub_diag
- Pointwise algebra behind the increment and shifted cycle conventions. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCyclicMonotonicity.lean:66 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity.pairingCycleMonotone_iff_shifted
- Rockafellar increment form and diagonal/shifted form are exactly
equivalent. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCyclicMonotonicity.lean:76 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity.PairingCycleMonotone.distinct
- The standard all-cycle property immediately implies the distinct-cycle
contract used by perturbation arguments. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCyclicMonotonicity.lean:90 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity.PairingCycleMonotone.mono
- Restricting a pairing-cyclically-monotone relation preserves the property. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCyclicMonotonicity.lean:98 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity.PairingDistinctCycleMonotone.mono
- Restriction also preserves the distinct-cycle contract. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCyclicMonotonicity.lean:107 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity.pairingCycleMonotone_empty
- The empty relation is vacuously pairing cyclically monotone. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingCyclicMonotonicity |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingCyclicMonotonicity.lean:116 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain.chainValue_convexCombination
- Every fixed rooted-chain functional is affine in its terminal point. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarConvexDomain.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain.bddAbove_properValueSet
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarConvexDomain.lean:53 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain.coe_chainValue_le_potential
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarConvexDomain.lean:58 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain.properRockafellarPotential_combo_le
- If the proper potential is finite at `x` and `y`, then its value at any
convex combination is bounded above by the same convex combination of the two
finite endpoint values. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarConvexDomain.lean:69 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain.properRockafellarPotential_combo_lt_top
- Convex combinations of finite-domain points remain finite. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarConvexDomain.lean:115 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain.convex_effectiveDomain
- The effective domain of the list-based proper Rockafellar candidate is a
convex set. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarConvexDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarConvexDomain.lean:128 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.rockafellarValueSet
- Real values generated at `x` by nonempty chains rooted at `base` whose pair
points all belong to `Gamma`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.properRockafellarValueSet
- The same rooted finite-chain values embedded in `WithTop ℝ`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.properRockafellarPotential
- Extended-real Rockafellar candidate. `⊤` records target points at which the
rooted affine chain values are unbounded above. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:47 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.EffectiveDomain
- Finite/effective domain of an extended real potential. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.singleton_mem_rockafellarValueSet
- The singleton root chain contributes its supporting affine functional. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:56 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.rockafellarValueSet_nonempty
- Hence every rooted real value set is nonempty once the root belongs to the
relation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:66 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.properRockafellarValueSet_nonempty
- The extended value set is nonempty for the same reason. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:73 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.rockafellarValueSet_at_root_nonpos
- Closed-chain nonpositivity bounds every real rooted chain value at the root
by zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:82 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.isGreatest_zero_properRockafellarValueSet_at_root
- At the root, zero is the greatest extended rooted-chain value: the singleton
chain attains zero and every closed chain is nonpositive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:92 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.properRockafellarPotential_at_root_eq_zero
- The proper Rockafellar candidate is normalized to zero at its chosen root. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:105 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.root_mem_effectiveDomain
- Therefore the root belongs to the finite domain of the extended potential. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:115 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.effectiveDomain_nonempty
- In particular, the finite domain is nonempty. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:125 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential.properRockafellarPotential_at_root_eq_zero_of_distinct
- Direct composition from the transport-produced distinct-cycle condition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarPotential |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarPotential.lean:133 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain.finitePart
- A total real representative of an extended-real potential. Its value at
`⊤` is an arbitrary default and is never used as mathematical data. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarRealDomain.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain.finitePart_eq_untop_of_lt_top
- On a finite point, `finitePart` is exactly the unique real value represented
by the extended-real potential. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarRealDomain.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain.coe_finitePart_of_lt_top
- Re-embedding the finite real representative recovers the original
extended-real value. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarRealDomain.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain.coe_finitePart_on_effectiveDomain
- The total representative agrees with the proper potential throughout its
finite/effective domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarRealDomain.lean:59 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain.convexOn_finitePart_effectiveDomain
- The finite real representative of the proper list-based Rockafellar
potential is convex on its effective domain. This is the domain-aware real
convexity interface needed by the later a.e.-differentiability step. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealDomain |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarRealDomain.lean:68 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealSupport.finitePart_support_on_effectiveDomain
- An extended-real supporting vector becomes an ordinary supporting
inequality for `finitePart` after restricting the comparison point to the
finite effective domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealSupport |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarRealSupport.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealSupport.finitePart_support_on_effectiveDomain_of_mem
- Specialized consumer-facing form for points of a closed-chain-monotone
relation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarRealSupport |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarRealSupport.lean:54 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient.chainValue_concat_singleton
- Appending one pair point to a finite chain adds exactly the final affine
increment. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSubgradient.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient.add_inner_mem_rockafellarValueSet
- A chain value at `x` can be extended by any relation point `(x,y)`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSubgradient.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient.rockafellarValueSet_le_inner_of_mem
- Closed-chain nonpositivity supplies an explicit finite upper bound for every
rooted chain value at the source coordinate of a relation point. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSubgradient.lean:67 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient.bddAbove_properRockafellarValueSet
- The extended value set is always bounded above by `⊤`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSubgradient.lean:99 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient.coe_le_properRockafellarPotential_of_mem
- Every explicit rooted-chain value lies below the extended Rockafellar
supremum. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSubgradient.lean:106 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient.properRockafellarPotential_lt_top_of_mem
- At every source coordinate appearing in `Gamma`, the proper Rockafellar
potential is finite. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSubgradient.lean:116 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient.ProperSupportsAt
- Extended-real supporting-vector relation. The existential real value makes
finiteness at the contact point part of the proposition instead of a hidden
precondition. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSubgradient.lean:136 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient.properSupportsAt_of_mem
- Every point of a closed-chain-monotone relation supports the proper
Rockafellar potential at its source coordinate. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSubgradient.lean:145 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient.properSupportsAt_of_mem_of_distinct
- Direct bridge from the transport-produced distinct-cycle condition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSubgradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSubgradient.lean:189 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSupportGradient.eq_gradient_of_properSupportsAt_of_mem_interior
- At an interior differentiability point of an extended-real potential's
finite domain, every honest extended-real supporting vector is the gradient of
the finite real representative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSupportGradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSupportGradient.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSupportGradient.snd_eq_gradient_of_mem_of_mem_interior
- Relation-point specialization for the proper list-based Rockafellar
potential. Closed-chain monotonicity supplies the support relation; interior
membership and differentiability perform the final analytic collapse. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PairingRockafellarSupportGradient |
AutoSamplingTheory/TechnicalLemmas/Analysis/PairingRockafellarSupportGradient.lean:55 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedProductCostIntegral.measurePreserving_pair_eval
- Evaluating two distinct coordinates of a finite product probability is a
measure-preserving map to the corresponding two-coordinate product law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedProductCostIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PermutedProductCostIntegral.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedProductCostIntegral.integral_permutedQuadraticCost_eq_sum
- For a fixed-point-free permutation, the expected permuted quadratic cost
under the finite product law is the sum of the corresponding two-coordinate
cross-cost expectations. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedProductCostIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PermutedProductCostIntegral.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost.diagonalQuadraticCost
- Diagonal quadratic cost of a finite family of pairs. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/PermutedQuadraticCost.lean:42 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost.permutedQuadraticCost
- Quadratic cost after permuting only the source coordinates. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/PermutedQuadraticCost.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost.permutedPairingGap
- Pairing increment associated with the same permutation. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/PermutedQuadraticCost.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost.quadraticCostDifference_pointwise
- Pointwise square expansion used before summing over the permutation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/PermutedQuadraticCost.lean:54 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost.diagonal_sub_permuted_eq_two_pairingGap
- Exact algebraic identity behind the quadratic-cost perturbation argument:
re-pairing by `σ` changes the finite quadratic cost by twice the corresponding
pairing increment. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/PermutedQuadraticCost.lean:64 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost.permutedQuadraticCost_lt_of_pairingGap_pos
- A positive pairing gap gives a strictly cheaper quadratic re-pairing. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PermutedQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Analysis/PermutedQuadraticCost.lean:93 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.prefixIntegral
- Prefix integral on the finite nonnegative-time horizon. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:24 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.prefixIntegral_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:28 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.prefixIntegrand_eq_indicator
- The moving-prefix integrand is the indicator of an initial interval. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.prefixIntegral_eq_setIntegral
- Prefix integration is an ordinary set integral over the active initial
interval. This representation exposes the exact measure restriction needed
for cross-horizon consistency. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.prefixIntegral_eq_of_le_horizons
- An earlier prefix integral is independent of which larger finite horizon is
used as the ambient truncation container. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.continuous_prefixIntegral
- The prefix integral is continuous in its upper time argument. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:64 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.prefixIntegral_mono
- Prefix integration is monotone in time for pointwise nonnegative
integrands. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:122 |
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.prefixIntegral_eq_terminal_of_le
- Prefix integration stabilizes once the observation time passes the
terminal horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral |
AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:151 |
AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticGradientDescent.quadratic_gradient_iterate
- Actual quadratic-gradient iterates equal the powers of the linear update. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticGradientDescent |
AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticGradientDescent.lean:26 |
AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticGradientDescent.quadratic_eigenmode
- A supplied eigenmode gives exact iterates, distances to zero and quadratic values.
No existence of an eigenvector or stability of the chosen step is assumed. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticGradientDescent |
AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticGradientDescent.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularization.strongConvexOn_and_lipschitzWith_gradient_add_quadratic
- A nonnegative quadratic regularization shifts the strong-convexity and
actual gradient-Lipschitz constants by its precision, including zero precision.
All derivatives are genuine because the input potential is everywhe |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularization |
AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticRegularization.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationFirstOrder.curvature_gradient_and_smoothness
- Quadratic regularization of a differentiable convex objective: actual
curvature, gradient and smoothness, including zero regularization. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationFirstOrder |
AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticRegularizationFirstOrder.lean:21 |
AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationOracle.run
- A query-fuel interpreter. `next` either halts with an output or requests a
query. Each reply updates the state and costs one query. Halting is inspected
even at zero fuel. `none` marks exhaustion, never a successful ou |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationOracle |
AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticRegularizationOracle.lean:18 |
AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationOracle.simulate_regularized
- Correct each original value/gradient reply using known quadratic data.
For every adaptive program and fuel, this preserves the actual regularized
execution, including its state, halt/exhaustion outcome and query count. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationOracle |
AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticRegularizationOracle.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationTransfer.exists_minimizer_radius_and_accuracy
- A continuous objective with an attained minimum has an actual regularized
minimum in a proper normed group; its radius and approximate values transfer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationTransfer |
AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticRegularizationTransfer.lean:22 |
AutoSamplingTheory.TechnicalLemmas.Analysis.RestartLogComplexity.logarithmic_accuracy_and_cost
- A rounded logarithmic horizon gives actual restart accuracy and explicit
cost, including zero rounds; budget absorption and small-error order are conditional. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.RestartLogComplexity |
AutoSamplingTheory/TechnicalLemmas/Analysis/RestartLogComplexity.lean:22 |
AutoSamplingTheory.TechnicalLemmas.Analysis.RestartReduction.radius_accuracy_and_cost
- Actual scheduled restart states halve their certified radius; one final
polishing call reaches the requested accuracy with the exact finite budget. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.RestartReduction |
AutoSamplingTheory/TechnicalLemmas/Analysis/RestartReduction.lean:21 |
AutoSamplingTheory.TechnicalLemmas.Analysis.SmoothnessEquivalences.upper_model_iff_gradient_upper
- The global quadratic upper model is equivalent to a one-sided gradient
bound. The parameter is allowed to be signed. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.SmoothnessEquivalences |
AutoSamplingTheory/TechnicalLemmas/Analysis/SmoothnessEquivalences.lean:21 |
AutoSamplingTheory.TechnicalLemmas.Analysis.SmoothnessEquivalences.upper_model_iff_fderiv2_upper
- With genuine C² regularity the same upper model is equivalent to the
Hessian diagonal upper bound, without assuming convexity. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.SmoothnessEquivalences |
AutoSamplingTheory/TechnicalLemmas/Analysis/SmoothnessEquivalences.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn
- A strongly convex function lies above its first-order model by the
quadratic term `m / 2 * ‖y - x‖²`.
This is the ASTIS-owned shared port of Optlib's pinned
`Strong_Convex_second_lower`. The statement is deliberately |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexFirstOrder |
AutoSamplingTheory/TechnicalLemmas/Analysis/StrongConvexFirstOrder.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexFirstOrder.gradient_inner_lower_bound_of_strongConvexOn
- The gradient of a strongly convex function has the corresponding inner-product
lower bound on its domain.
Mathematical provenance: Optlib `Strong_Convex_lower`, commit
`5da27c5f95aa6a8a45b8c14b968ade4c13ff18c3`,
`Optl |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexFirstOrder |
AutoSamplingTheory/TechnicalLemmas/Analysis/StrongConvexFirstOrder.lean:97 |
AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexGibbsIntegrability.integrable_exp_neg_of_strongConvexOn
- A differentiable strongly convex potential has an integrable Gibbs weight
for canonical volume. Positive modulus is essential to the Gaussian-envelope
argument. No minimizer, gradient field, or normalizer is supplied a |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexGibbsIntegrability |
AutoSamplingTheory/TechnicalLemmas/Analysis/StrongConvexGibbsIntegrability.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexGradientConverse.strongConvexOn_of_gradient_inner_lower_bound
- Quantitative monotonicity of genuine ambient gradients on a convex domain
implies strong convexity, with the same (possibly signed) modulus. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexGradientConverse |
AutoSamplingTheory/TechnicalLemmas/Analysis/StrongConvexGradientConverse.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexPLPullback.exists_minimizer_and_pl
- Surjectivity lifts the attained minimum, and the adjoint derivative's
coercivity transports strongly convex gap control to the actual composite gradient. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexPLPullback |
AutoSamplingTheory/TechnicalLemmas/Analysis/StrongConvexPLPullback.lean:28 |
AutoSamplingTheory.TechnicalLemmas.Analysis.UniformRegularization.uniform_accuracy_and_query_bound
- Uniform success and actual query bounds for the corrected program. The
callbacks and initial state are fixed before either objective is quantified.
No monotonicity of the natural-valued budget function is required. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Analysis.UniformRegularization |
AutoSamplingTheory/TechnicalLemmas/Analysis/UniformRegularization.lean:22 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.ClosedGraphResolvent.weak_resolvent
- Orthogonal projection onto the scaled closed graph constructs the unique
weak solution. Norms of domain elements below are their ambient H norms;
the explicit bounds depend on epsilon and do not persist uniformly as it |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.ClosedGraphResolvent |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/ClosedGraphResolvent.lean:24 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.dirichletForm
- The generator Dirichlet form `E(f,g) = integral f (-L)g d mu`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:22 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.variance
- Variance as the squared centered `L2(mu)` norm. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:28 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.PoincareAdmissible
- Domain conditions needed to read both sides of the generator Poincare
inequality as genuine finite integrals. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:33 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.SatisfiesPoincare
- Chewi Definition 1.2.19: the generator Poincare inequality
`Var_mu(f) <= C * E(f,f)` for every admissible observable. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:42 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.densityEntropy
- Relative entropy of a density `rho` with respect to its reference
probability measure. Mathlib's totalized `Real.log 0 = 0` gives the standard
zero-density convention in the product `rho * log rho`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:52 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.LogSobolevAdmissible
- Domain conditions for the density formulation of log-Sobolev. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:56 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.SatisfiesLogSobolev
- Chewi Definition 1.2.25: the density log-Sobolev inequality
`KL(rho mu || mu) <= (C/2) E(rho, log rho)`. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:67 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance
- Variance written as the integral of the squared centered observable.
Admissibility is deliberately separate because the Bochner integral is
totalized outside its integrable domain. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Poincare.lean:28 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.dirichletEnergy
- The Euclidean/inner-product Dirichlet energy of a test function. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Poincare.lean:32 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.Admissible
- Exact integrability domain used by the local Poincare interface. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Poincare.lean:36 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.Satisfies
- A measure satisfies the Poincare inequality with constant `C` on an
explicit test class.
The convention is `Var_μ(f) ≤ C * E_μ(f)`. Probability normalization is
part of the contract rather than an implicit convention |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Poincare.lean:46 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance_nonneg
- Variance is nonnegative on its stated integral representation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Poincare.lean:52 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.dirichletEnergy_nonneg
- Dirichlet energy is nonnegative on its stated integrability domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Poincare.lean:59 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.mono_constant
- Increasing a nonnegative Poincare constant preserves the inequality on
the same test class and admissibility domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Poincare.lean:67 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance_le
- The inequality component can be consumed without unpacking the probability
and nonnegative-constant fields manually. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Poincare.lean:77 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.mono_tests
- Restricting the test class preserves a Poincare inequality and all of its
measure and constant data. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Poincare.lean:85 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_lemma_1_2_20
- Chewi's differential Gronwall lemma on `[0, T]`.
The source assumes a differentiable scalar function satisfying
`g' t ≤ c * g t`. Mathlib's one-sided Gronwall theorem accepts the weaker
right-slope formulation; ordina |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:38 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.DissipationCurve
- A scalar energy/dissipation curve with an exact right-derivative identity.
`scale` records the coefficient in
`d/dt energy(t) = -scale * dissipation(t)`. The derivative is taken within
`[t, ∞)`, matching semigroups de |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:69 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.exponential_decay_of_scaled_dissipation_from
- Coercivity plus exact dissipation gives exponential decay between any two
times `s ≤ t`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:79 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.exponential_decay_of_scaled_dissipation
- A coercive inequality along a dissipation curve implies exponential decay
from time zero. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:114 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.scaled_dissipation_of_exponential_decay
- Exponential decay from every starting time forces the instantaneous
coercivity inequality.
The proof compares the energy with its exponential envelope on `[s, ∞)`. Their
difference has a local maximum at `s`; the one- |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:130 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_21_forward_from
- Forward direction of Chewi, Theorem 1.2.21, between arbitrary times. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:202 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_21_forward
- Forward direction of Chewi, Theorem 1.2.21, from time zero. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:221 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_21_backward
- Backward scalar direction of Chewi, Theorem 1.2.21. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:232 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_21_scalar_equivalence
- Scalar equivalence behind Chewi, Theorem 1.2.21. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:253 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_22_forward_from
- Forward direction of Chewi, Theorem 1.2.22, between arbitrary times. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:268 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_22_forward
- Forward direction of Chewi, Theorem 1.2.22, from time zero. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:278 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_22_backward
- Backward scalar direction of Chewi, Theorem 1.2.22. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:288 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_22_scalar_equivalence
- Scalar equivalence behind Chewi, Theorem 1.2.22. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:298 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_26_forward_from
- Forward direction of Chewi, Theorem 1.2.26, between arbitrary times. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:308 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_26_forward
- Forward direction of Chewi, Theorem 1.2.26, from time zero. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:327 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_26_backward
- Backward scalar direction of Chewi, Theorem 1.2.26. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:338 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay.chewi_theorem_1_2_26_scalar_equivalence
- Scalar equivalence behind Chewi, Theorem 1.2.26. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:359 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedBochner.integrated_bochner_identity
- Actual weighted Bochner identity, all compact-test integrability facts, and
its curvature-energy consequence. This does not establish Poincare. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedBochner |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/WeightedBochner.lean:34 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradient.compact_gradient_closable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradient |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/WeightedGradient.lean:34 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientDistribution.lp_locallyIntegrable_volume
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientDistribution |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/WeightedGradientDistribution.lean:22 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientDistribution.closed_gradient_distributional
- Elements of the same closed weighted gradient graph have locally integrable
volume representatives and satisfy the ordinary weak gradient identity, with
both test products integrable. No differentiability of L2 represe |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientDistribution |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/WeightedGradientDistribution.lean:43 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientWeak.compact_directional_ibp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientWeak |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/WeightedGradientWeak.lean:22 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientWeak.closed_gradient_weighted_ibp
- Every member of the same closed gradient graph satisfies weighted compact
test integration by parts, with both integrands genuinely integrable. The exact
original graph is the already constructed smooth core, not an as |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientWeak |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/WeightedGradientWeak.lean:66 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedLocalL2.lp_locallyMemLp_volume
- A Gibbs L2 representative has locally integrable squared norm for volume,
and is L2 on each compact volume restriction. Only continuity of the potential
is required; the target needs no scalar structure or completeness |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedLocalL2 |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/WeightedLocalL2.lean:24 |
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedResolvent.weak_resolvent_distributional
- The genuine positive-epsilon resolvent has an ordinary weak gradient and
satisfies the weighted divergence-form equation, with all three test products
integrable for volume. The same partial operator is retained throug |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedResolvent |
AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/WeightedResolvent.lean:22 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussianPi
- Product standard Gaussian measure on coordinate functions. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussianPi_isProbabilityMeasure
|
instance | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussianPi_isFiniteMeasure
|
instance | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.map_eval_stdGaussianPi
- Coordinate projections under the ASTIS product Gaussian are standard normal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integral_id_gaussianReal_zero
- The centered real Gaussian has zero mean. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integral_exp_mul_gaussianReal
- The one-dimensional Gaussian moment-generating function as a plain integral. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:57 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integral_exp_mul_gaussianReal_zero_one
- The standard real Gaussian moment-generating function. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:66 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integrable_eval_stdGaussianPi
- Coordinate projections under the ASTIS product Gaussian are integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:72 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integrable_const_mul_eval_stdGaussianPi
- Scalar multiples of product-Gaussian coordinate projections are integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:109 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integrable_linearForm_stdGaussianPi
- Finite linear forms in product-Gaussian coordinates are integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:114 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integrable_const_mul_sq_gaussianReal_zero
- A centered real Gaussian integrates every scalar quadratic bound.
This is the reusable Gaussian integrability fact needed by the SALD normalized
remainder bound `fun z => C * z ^ 2`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:128 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integrable_sq_eval_stdGaussianPi
- Coordinate squares under the ASTIS product Gaussian are integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:150 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integral_eval_stdGaussianPi
- Coordinate projections under the ASTIS product Gaussian have zero mean. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:172 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integral_const_mul_eval_stdGaussianPi
- Scalar multiples of product-Gaussian coordinate projections have zero mean. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:192 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integral_linearForm_stdGaussianPi
- Finite linear forms in product-Gaussian coordinates have zero mean. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:197 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integral_exp_linearForm_stdGaussianPi
- Moment-generating function of a finite product standard Gaussian linear form. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:211 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.integral_exp_centered_linearForm_stdGaussianPi
- Centered Esscher normalizer for a finite product standard Gaussian linear form. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:231 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.gaussianReal_withDensity_exp_shift
- One-dimensional standard Gaussian Esscher density shift. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:254 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.pi_gaussianReal_withDensity_exp_shift
- Finite product standard Gaussian Esscher density shift. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:275 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.pi_gaussianReal_shift_integral
- Finite product Gaussian Esscher change of measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:314 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussianPi_withDensity_exp_shift
- `stdGaussianPi` spelling of the finite product Gaussian Esscher density shift. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:340 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussianPi_shift_integral
- `stdGaussianPi` spelling of finite product Gaussian Esscher change of measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:348 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.pi_gaussianReal_shift_integral_map_toLp
- Pushforward-to-`EuclideanSpace` spelling of finite product Gaussian Esscher
change of measure.
This is the bridge from coordinate-product Gaussian statements to Mathlib's
finite-dimensional Hilbert-space Gaussian inte |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:364 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussianPi_shift_integral_map_toLp
- `stdGaussianPi` spelling of the pushforward-to-`EuclideanSpace` finite
product Gaussian Esscher change of measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:379 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.inner_toLp_toLp_eq_sum_mul
- Inner product of two coordinate functions after the `EuclideanSpace`
`WithLp.toLp 2` embedding. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:393 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.norm_sq_toLp_eq_sum_sq
- Squared norm after the `EuclideanSpace` `WithLp.toLp 2` embedding. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:404 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussian_shift_integral_map_toLp
- `stdGaussian` inner-product spelling of the finite-dimensional Gaussian
Esscher change of measure after pushing product coordinates to
`EuclideanSpace`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:411 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.variance_id_gaussianReal_zero_one
- The unit real Gaussian has unit variance in the real-valued Mathlib variance convention. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:446 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.nnrealVarianceOneOfGaussianRealUnitLaw
- Package a scalar-coordinate Gaussian law and a variance-field definition
into the `NNReal` unit-variance field used by Brownian/Ito normalizations. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:453 |
AutoSamplingTheory.TechnicalLemmas.Gaussian.realVarianceOneOfNNRealVarianceOne
- Convert an `NNReal` unit-variance field into the real-valued unit field
that often appears after algebraic normalization. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Gaussian |
AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:477 |
AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates.euclideanSpace_inner_toLp_toLp_eq_sum_mul
- Inner product of two real coordinate functions after the `EuclideanSpace`
`WithLp.toLp 2` embedding. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates |
AutoSamplingTheory/TechnicalLemmas/Geometry/EuclideanSpaceCoordinates.lean:21 |
AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates.euclideanSpace_inner_eq_sum_mul
- Inner product of two real `EuclideanSpace` vectors in coordinates. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.EuclideanSpaceCoordinates |
AutoSamplingTheory/TechnicalLemmas/Geometry/EuclideanSpaceCoordinates.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity.IsAlphaGeodesicallyConvex
- Chewi Definition 1.3.26, condition 1: `F` is alpha-geodesically convex
along every selected geodesic, with the source normalization
`alpha * t * (1-t) / 2`. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity |
AutoSamplingTheory/TechnicalLemmas/Geometry/GeodesicConvexity.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity.firstOrder_geodesicConvexity
- The chord formulation of geodesic alpha-convexity implies its first-order
form along a differentiable selected geodesic. The scalar
`gradientPairing` is the derivative of `F` along the path at its initial
point; ident |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity |
AutoSamplingTheory/TechnicalLemmas/Geometry/GeodesicConvexity.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn
- A positive real-valued function is log-concave on `s` when its logarithm is
concave on `s`.
The positivity condition is explicit because Chewi-style density arguments
usually need it separately from the convex-analysi |
def | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_iff
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_of_concave_log
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.concaveOn_log
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.convexOn_neg_log
- The negative logarithm of a positive log-concave function is convex. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:59 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.convex_sublevel_neg_log
- Sublevel sets of the negative-log potential of a positive log-concave
function are convex. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:68 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.convex_domain
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:74 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.convex_superlevel
- Superlevel sets of a positive log-concave function are convex within the
log-concavity domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:82 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.quasiconcaveOn
- Positive log-concave functions are quasiconcave: all superlevel sets are convex. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:116 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.subset
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:122 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.restrict_superlevel
- Restricting a positive log-concave function to one of its superlevel sets
preserves log-concavity. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:130 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.comp_linearMap
- Precomposition by a linear map preserves log-concavity on the preimage domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:137 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.comp_affineMap
- Precomposition by an affine map preserves log-concavity on the preimage domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:146 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.mul
- The pointwise product of two positive log-concave functions on the same
domain is log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:156 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.rpow
- A nonnegative real power of a positive log-concave function is log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:169 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.prod
- Product-domain tensorization: the product of log-concave factors on
convex domains is log-concave on the Cartesian product. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:182 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.const_mul
- Multiplication by a positive constant preserves log-concavity. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:210 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const
- A positive constant function is log-concave on every convex domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:225 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_of_convexOn
- If `V` is convex, then the unnormalized Gibbs shape `exp (-V)` is
log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:232 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_of_convexOn
- A positive multiple of the Gibbs shape of a convex potential is
log-concave. This is the convex-analytic part of normalized Gibbs-density
bookkeeping; the measure/integral normalization proof is a separate leaf. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:241 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_abs
- The absolute value is convex on the real line. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:249 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_const_mul_abs_add
- Nonnegative absolute-linear real potentials are convex. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:255 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_abs_linear
- The Gibbs shape of a nonnegative absolute-linear real potential is
log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:263 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_abs_linear
- Positive scalar normalization preserves log-concavity of absolute-linear
Laplace shapes. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:271 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_explicit_abs_linear_normalized_density
- The explicitly normalized one-dimensional absolute-linear Laplace density
is log-concave as a real-valued density shape. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:280 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_norm_sq
- The squared norm is convex on any real normed vector space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:290 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_const_mul_norm_sq_add
- Nonnegative quadratic norm potentials are convex. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:314 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_quadratic_norm
- The Gibbs shape of a nonnegative quadratic norm potential is log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:323 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_quadratic_norm
- Positive multiples of nonnegative quadratic Gibbs shapes are log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:331 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_explicit_quadratic_normalized_density
- The explicitly normalized finite-dimensional quadratic Gibbs density is log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:340 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_const_mul_norm_sub_sq_add
- Shifted nonnegative quadratic norm potentials are convex. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:353 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_shifted_quadratic_norm
- The Gibbs shape of a shifted nonnegative quadratic norm potential is log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:371 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_shifted_quadratic_norm
- Positive multiples of shifted nonnegative quadratic Gibbs shapes are log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:380 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_explicit_shifted_quadratic_normalized_density
- The explicitly normalized shifted finite-dimensional quadratic Gibbs density is log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:389 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.convexOn_univ_const_mul_norm_fst_sub_snd_sq_add
- The two-point quadratic potential `(x, y) ↦ a‖x-y‖^2+b` is convex. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:402 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_pair_sub_quadratic_norm
- The two-point quadratic Gibbs kernel shape `(x, y) ↦ exp (-(a‖x-y‖^2+b))`
is log-concave on the product space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:417 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_const_mul_exp_neg_pair_sub_quadratic_norm
- Positive multiples of two-point quadratic Gibbs kernel shapes are log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:426 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_explicit_pair_sub_quadratic_kernel
- The finite-dimensional Gaussian-kernel normalizing constant times
`exp (-(a‖x-y‖^2+b))` is log-concave as a function of `(x, y)`.
This is a geometry/kernel-shape leaf. It does not claim that the function is a
probabi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:439 |
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_id_Ioi
- The identity density on the positive ray is log-concave. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity |
AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:452 |
AutoSamplingTheory.TechnicalLemmas.Geometry.MetricCurve.HasMetricDerivativeAt
- A curve has metric derivative `speed` at `t` when its distance quotient
converges to that finite nonnegative real along punctured times. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.MetricCurve |
AutoSamplingTheory/TechnicalLemmas/Geometry/MetricCurve.lean:21 |
AutoSamplingTheory.TechnicalLemmas.Geometry.MetricCurve.IsAbsolutelyContinuousMetricCurve
- Chewi Definition 1.3.16 (informal): a measure-valued curve is absolutely
continuous when a finite metric derivative exists for almost every time.
The name is intentionally source-facing: the stronger standard metric-s |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Geometry.MetricCurve |
AutoSamplingTheory/TechnicalLemmas/Geometry/MetricCurve.lean:34 |
AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.convexOn_of_strongConvexOn_nonneg
- A nonnegatively strongly convex function is convex.
This is the small Mathlib-facing bridge from Chewi's strong-convexity
assumptions to ordinary convex-potential density geometry. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity |
AutoSamplingTheory/TechnicalLemmas/Geometry/StrongConvexity.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.logConcaveOn_exp_neg_of_strongConvexOn
- A strongly convex potential with nonnegative modulus gives a log-concave
unnormalized Gibbs density shape. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity |
AutoSamplingTheory/TechnicalLemmas/Geometry/StrongConvexity.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.logConcaveOn_const_mul_exp_neg_of_strongConvexOn
- Positive scalar normalization preserves the log-concavity of a Gibbs shape
whose potential is strongly convex with nonnegative modulus. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity |
AutoSamplingTheory/TechnicalLemmas/Geometry/StrongConvexity.lean:42 |
AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.centered_quadratic_lower_bound_of_strongConvexOn_minimizer
- A strongly convex function with a global minimizer has a centered quadratic
lower bound.
The constant `k / 4` is the midpoint consequence of Mathlib's
`StrongConvexOn` convention. It is intentionally not the sharp `k |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity |
AutoSamplingTheory/TechnicalLemmas/Geometry/StrongConvexity.lean:57 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher.State
|
abbrev | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalDirichletFisher.lean:50 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher.DirichletPairDomain
- The canonical density/log-ratio pair is in the generator integration-by-
parts domain needed to invoke Chewi Theorem 1.2.14.
The fields are exactly the three integrability terms, stationarity of the
product observable |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalDirichletFisher.lean:58 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher.HasCanonicalFisherGamma
- The concrete carré-du-champ identification required to turn the abstract
Dirichlet form into canonical relative Fisher information.
For overdamped Langevin this is the measure-domain version of
`Gamma(f,g)=inner (grad |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalDirichletFisher.lean:91 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher.dirichletForm_density_logRatio_eq_information
- Chewi Theorem 1.2.14 plus the concrete Gamma/score identification gives
exactly the canonical relative Fisher information:
`E_pi(dmu/dpi, log(dmu/dpi)) = FI(mu || pi)`.
This is the reusable Dirichlet--Fisher edge con |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalDirichletFisher.lean:108 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher.dirichletForm_density_logRatio_eq_integral_scoreSq
- Equivalent source-facing integral form of the same bridge. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalDirichletFisher.lean:132 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher.hasDerivAt_eq_neg_information_of_eq_neg_dirichlet
- A supplied KL derivative written as minus the canonical density/log-ratio
Dirichlet form immediately becomes the standard dissipation rate `-FI`.
This theorem intentionally starts *after* the analytic law-evolution /
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalDirichletFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalDirichletFisher.lean:154 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalFisherTransportPairing.memLp_score_and_displacement
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalFisherTransportPairing |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalFisherTransportPairing.lean:33 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalFisherTransportPairing.integrable_pairing_of_isCoupling
- The canonical relative score paired with displacement is integrable under
any coupling of finite-second-moment marginals. Optimality is not needed. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalFisherTransportPairing |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalFisherTransportPairing.lean:53 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalFisherTransportPairing.abs_integral_pairing_le_sqrt_information_mul_wasserstein
- For an optimal coupling, the absolute canonical score/displacement pairing
is bounded by the square root of canonical Fisher information times the
actual Wasserstein distance. The moment hypotheses give integrable cost |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalFisherTransportPairing |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalFisherTransportPairing.lean:69 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation.State
|
abbrev | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalKLDissipation.lean:50 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation.FlowDerivativeDomain
- Explicit analytic contract for differentiating the *actual* Mathlib KL
curve at one time and identifying the density velocity.
`kl_finite_near` prevents the totalized `ENNReal.toReal` convention from being
silently us |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalKLDissipation.lean:59 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation.kl_hasDerivAt_integral_rhoDot_mul_logRatio
- The analytic KL derivative formula loses its `+1` term exactly by mass
conservation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalKLDissipation.lean:85 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation.kl_hasDerivAt_eq_neg_dirichletForm
- The forward equation and the pairwise generator symmetry turn the remaining
KL derivative into minus the canonical density/log-ratio Dirichlet form. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalKLDissipation.lean:119 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation.kl_hasDerivAt_eq_neg_information
- Abstract Chapter 1.2 KL/Fisher dissipation join: once the law-evolution
contract and the Dirichlet--Fisher Gamma contract are both available, the
actual Mathlib KL curve has derivative `-FI`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalKLDissipation |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalKLDissipation.lean:184 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.State
|
abbrev | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:41 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.scoreSq
- The squared canonical relative score. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:44 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.SmoothFiniteScoreDomain
- Explicit regularity contract for the smooth finite branch of relative
Fisher information.
The future Sobolev/Dirichlet-energy extension should map into this contract on
its smooth finite subdomain; it should not weake |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:54 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.information
- Canonical relative Fisher information on the explicit smooth finite score
domain.
It is exactly the existing `RelativeFisher.information` with base measure `mu`
and density `1`, so no second Fisher hierarchy is introd |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:66 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.information_eq_relativeFisher
- The canonical guarded definition is definitionally the shared
`RelativeFisher` object. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:73 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.information_eq_integral_scoreSq
- On the guarded smooth finite domain, the canonical Fisher information has
the expected measure-level formula
`FI(mu || pi) = integral ||grad log(d mu / d pi)||^2 dmu`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:84 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.information_eq_integral_density_mul_scoreSq
- The same canonical Fisher information rewritten against the reference
measure using Mathlib's Radon--Nikodym integral formula:
`FI(mu || pi) = integral density(mu|pi) * scoreSq(mu|pi) dpi`.
Crucially, this changes on |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:98 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.scoreSq_integrable
- The squared score is integrable by the domain contract, rather than by an
implicit convention of the total integral. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:111 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.information_nonneg
- Canonical relative Fisher information is nonnegative on its guarded smooth
finite domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:119 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher.information_proof_irrel
- The guarded value does not depend on the proof witness used to establish the
same smooth finite score domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.CanonicalRelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/CanonicalRelativeFisher.lean:128 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.FisherTransport.sq_div_le_fisher_of_sq_le_mul
- Divide a Fisher--transport inequality by a positive squared transport
radius.
This is the algebraic step used in Chewi Theorem 8.4.1 after Wasserstein
geodesic convexity and Cauchy--Schwarz establish
`kl^2 ≤ fi * R2`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.FisherTransport |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/FisherTransport.lean:28 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.FisherTransport.neg_half_fisher_le_neg_half_sq_div
- Combine Fisher--transport control with the negative one-half factor from
heat-flow KL dissipation.
The conclusion is written as `-((kl^2 / R2) / 2)` to keep this lemma purely
order-algebraic; source-facing assembly ma |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.FisherTransport |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/FisherTransport.lean:42 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.FisherTransport.kl_derivative_upper_bound_of_fisher_transport
- Source-shaped algebraic composition: if the KL derivative is exactly
`-FI/2`, then the Fisher--transport inequality gives the reciprocal-KL style
differential upper bound used in the proximal proof. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.FisherTransport |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/FisherTransport.lean:54 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.GeodesicFisherTransport.value_le_neg_directionalDerivative
- A zero-geodesically-convex functional whose endpoint is a minimizer of
value zero is bounded by the negative initial directional derivative.
This is the direct specialization of Chewi display (1.4.7) with `alpha = 0`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.GeodesicFisherTransport |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/GeodesicFisherTransport.lean:32 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.GeodesicFisherTransport.sq_le_fisher_mul_sq_of_le_sqrt_mul
- Scalar Cauchy--Schwarz closure: a nonnegative value bounded by
`sqrt(fisher) * distance` satisfies the squared Fisher--transport inequality. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.GeodesicFisherTransport |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/GeodesicFisherTransport.lean:50 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.GeodesicFisherTransport.sq_le_fisher_mul_dist_sq_of_geodesic_first_order
- Reusable geodesic Fisher--transport join.
If a nonnegative zero-geodesically-convex functional vanishes at the terminal
point of a selected geodesic, and the negative initial directional derivative
is at most `sqrt(fi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.GeodesicFisherTransport |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/GeodesicFisherTransport.lean:71 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.KLDensity.klPointwiseDerivSimplify
- Pointwise algebra for differentiating `q * log (q / p)`.
This proves only the real-field simplification. Positivity, measurability,
integrability, and dominated differentiation under the integral are separate
regular |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.KLDensity |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/KLDensity.lean:24 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.KLDensity.klDerivativeRemoveMassTerm
- Remove the mass-conservation term from a supplied KL derivative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.KLDensity |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/KLDensity.lean:33 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.density
- The canonical real-valued Radon--Nikodym density representative used by
ASTIS. The underlying measure-theoretic object remains Mathlib's ENNReal-valued
`rnDeriv`; `toReal` is only the real representative needed by cal |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:36 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.logRatio
- The canonical log-density ratio. This is definitionally Mathlib's
log-likelihood ratio, so KL and Fisher layers share one representative. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:41 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.logRatio_apply
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:45 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.measurable_density
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:50 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.measurable_logRatio
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:55 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.density_nonneg
- The real RN density is pointwise nonnegative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:60 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.density_ae_pos_of_absolutelyContinuous
- Absolute continuity makes the canonical real RN density positive `mu`-a.e.
The `rnDeriv < ∞` obligation is explicit because `toReal ∞ = 0`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:66 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.exp_logRatio_ae_eq_density_of_absolutelyContinuous
- Exponentiating the canonical log-density ratio recovers the canonical RN
density `mu`-a.e. under absolute continuity. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:77 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.density_self_ae
- The canonical density of a measure relative to itself is one a.e. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:86 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.logRatio_self_ae
- The canonical log-density ratio of a measure relative to itself is zero
a.e. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:93 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio.toReal_klDiv_eq_integral_logRatio_of_probability
- For probability measures, finite KL has the source-facing integral form
`KL(mu || pi) = integral log(d mu / d pi) dmu`
at the real-valued level. Mathlib's `klDiv` remains the canonical ENNReal
measure divergence; th |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RNLogRatio |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RNLogRatio.lean:104 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher.State
|
abbrev | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RelativeFisher.lean:38 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher.densityEnergy
- Pointwise relative-Fisher energy for a supplied density and a supplied
log-density ratio. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RelativeFisher.lean:42 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher.information
- Relative Fisher information with respect to an explicit base measure.
For the usual Euclidean density representation the base measure will be
Lebesgue measure and `q` will be the density of `mu`; equivalently the inte |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RelativeFisher.lean:52 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher.densityEnergy_nonneg
- Fisher energy density is nonnegative wherever the supplied density is
nonnegative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RelativeFisher.lean:59 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher.densityEnergy_eq_zero_of_gradient_eq_zero
- A vanishing relative score gives zero pointwise Fisher energy. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RelativeFisher.lean:66 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher.information_eq_zero_of_gradient_ae_eq_zero
- If the relative score vanishes almost everywhere, then the relative Fisher
information vanishes. This statement needs no positivity or normalization
assumption on `q`; those belong to the source-facing density bridge. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RelativeFisher.lean:75 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher.information_congr_gradient_ae
- The Fisher functional is insensitive to changing the supplied log-ratio on
an a.e. set *provided its gradients themselves agree a.e.*. This is the exact
representative-level congruence needed after a Sobolev/Radon--Ni |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.RelativeFisher |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/RelativeFisher.lean:89 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.renyiIntegrand
- The real-valued Renyi/Hellinger-style density integrand
`p^a q^(1-a)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:27 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.renyiIntegrandENNReal
- The `ℝ≥0∞` version used for lintegral contracts. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:31 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.renyiIntegrand_nonneg
- Nonnegative input densities give a nonnegative Renyi integrand. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:35 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.renyiIntegrand_pos
- Positive input densities give a positive Renyi integrand. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:41 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.measurable_renyiIntegrand
- A measurable pair of real densities gives a measurable Renyi integrand for
orders `a ∈ [0,1]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:50 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.measurable_renyiIntegrandENNReal
- Measurability of the `ℝ≥0∞` Renyi integrand used in lintegrals. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:62 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.lintegral_renyiIntegrandENNReal_ne_top_of_ae_le
- A finite envelope gives a finite Renyi lintegral. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:70 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.hasDerivAt_renyiIntegrand
- Pointwise derivative rule for the Renyi density integrand. Positivity,
domination, and differentiating under the integral are deliberately outside
this leaf. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:80 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergence.hasDerivAt_weighted_f_divergence_integrand
- Weighted quotient chain rule underlying simultaneous `f`-divergence
flows. The nonzero denominator is the local positivity condition on the
reference density. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergence |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/SimultaneousFDivergence.lean:30 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergence.hasDerivAt_weighted_f_divergence_integrand_of_ratio
- The same identity with the ratio named explicitly. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergence |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/SimultaneousFDivergence.lean:60 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergenceGradient.common_diffusion_pairing_eq
- Pointwise cancellation of the two gradient pairings in a simultaneous
common-diffusion `f`-divergence calculation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergenceGradient |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/SimultaneousFDivergenceGradient.lean:29 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergenceGradient.common_diffusion_pairing_eq_of_gradients
- Source-shaped variant where the three gradient expressions have already
been identified and are supplied by equality hypotheses. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergenceGradient |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/SimultaneousFDivergenceGradient.lean:42 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergenceIntegral.derivativeIntegrand
- The pointwise time-derivative expression supplied by the simultaneous
weighted-quotient chain rule. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergenceIntegral |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/SimultaneousFDivergenceIntegral.lean:31 |
AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergenceIntegral.hasDerivAt_integral_weighted_f_divergence_of_dominated
- Dominated differentiation of a simultaneous `f`-divergence density with
respect to a fixed base measure.
All analytic regularity needed by Mathlib's parametric-integral theorem is kept
explicit. The only derived inpu |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.InformationTheory.SimultaneousFDivergenceIntegral |
AutoSamplingTheory/TechnicalLemmas/InformationTheory/SimultaneousFDivergenceIntegral.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Measure.CanonicalGlobalCompetitor.canonicalGlobalCompetitor
- The canonical ambient competitor associated with positive local blocks and
a target-marginal permutation. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CanonicalGlobalCompetitor |
AutoSamplingTheory/TechnicalLemmas/Measure/CanonicalGlobalCompetitor.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Measure.CanonicalGlobalCompetitor.commonSliceRemoved_toMeasure_le_ambient
- The canonical common-mass removed sum is dominated by the ambient measure
whenever the original local-block sum is dominated by the ambient measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CanonicalGlobalCompetitor |
AutoSamplingTheory/TechnicalLemmas/Measure/CanonicalGlobalCompetitor.lean:53 |
AutoSamplingTheory.TechnicalLemmas.Measure.CanonicalGlobalCompetitor.ambient_eq_remainder_add_commonSliceRemoved
- The ambient measure decomposes into the canonical remainder plus the
canonical removed common-mass slice sum. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CanonicalGlobalCompetitor |
AutoSamplingTheory/TechnicalLemmas/Measure/CanonicalGlobalCompetitor.lean:68 |
AutoSamplingTheory.TechnicalLemmas.Measure.CanonicalGlobalCompetitor.canonicalGlobalCompetitor_preserves_marginals
- The canonical global competitor preserves both marginals of the ambient
joint finite measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CanonicalGlobalCompetitor |
AutoSamplingTheory/TechnicalLemmas/Measure/CanonicalGlobalCompetitor.lean:81 |
AutoSamplingTheory.TechnicalLemmas.Measure.ChewiTheorem1_3_23.wassersteinDistance_displacementInterpolation_eq_abs
- Arbitrary-time form of the constant-speed identity for the displacement
interpolation generated by one quadratic-optimal coupling. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ChewiTheorem1_3_23 |
AutoSamplingTheory/TechnicalLemmas/Measure/ChewiTheorem1_3_23.lean:44 |
AutoSamplingTheory.TechnicalLemmas.Measure.ChewiTheorem1_3_23.chewi_theorem_1_3_23_constant_speed
- Chewi Theorem 1.3.23 in the repository's source-side geodesic interface:
any curve represented by an optimal displacement interpolation has constant
`W₂` speed on `[0,1]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ChewiTheorem1_3_23 |
AutoSamplingTheory/TechnicalLemmas/Measure/ChewiTheorem1_3_23.lean:82 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass.mass_smul_nnreal
- Scaling a finite measure scales its total mass by the same nonnegative
scalar. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMass.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass.commonMassProduct
- Canonical product-space measure associated with two finite measures after
normalizing by the total mass of the first measure. The useful marginal
identities require that the two masses agree and that this common mass |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMass.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass.commonMassProduct_mass
- If the two input finite measures have the same positive mass, the common
mass product has exactly that mass. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMass.lean:47 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass.commonMassProduct_map_fst
- The first marginal of the common mass product is the first input measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMass.lean:56 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass.commonMassProduct_map_snd
- The second marginal of the common mass product is the second input measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMass.lean:67 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass.exists_joint_of_eq_positive_mass
- Existence wrapper: two finite measures with the same positive total mass
admit a finite measure on the product space with exactly those two marginals
and the same total mass. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMass.lean:80 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassNormalizedProduct.commonMassProduct_toMeasure_eq_mass_smul_normalized_prod
- The equal-mass finite product is the common mass times the product of the
normalized probability laws. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassNormalizedProduct |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassNormalizedProduct.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassProductMap.mass_map_eq
- A measurable pushforward of a finite measure preserves its total mass. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassProductMap |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassProductMap.lean:27 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassProductMap.commonMassProduct_map_prodMap
- `commonMassProduct` commutes with applying measurable maps to its two
coordinates. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassProductMap |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassProductMap.lean:36 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSlice.commonMassSlice
- Canonical slice of a finite measure with prescribed target mass. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSlice |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassSlice.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSlice.commonMassSlice_mass
- If the ambient finite measure has positive mass, its canonical slice has
exactly the prescribed total mass. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSlice |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassSlice.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSlice.commonMassSlice_toMeasure_le
- If the target mass is no larger than the ambient mass, the canonical slice
is dominated by the original measure. The domination is stated for the
underlying `Measure`, exactly the order relation consumed by later meas |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSlice |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassSlice.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily.commonMassSliceFamily
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassSliceFamily.lean:30 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily.commonMassSliceFamily_mass
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassSliceFamily.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily.commonMassSliceFamily_toMeasure_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassSliceFamily.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily.commonMassSliceFamily_mass_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassSliceFamily.lean:48 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily.commonMassSliceFamily_mass_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonMassSliceFamily |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonMassSliceFamily.lean:55 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.addNoise
- Law of an independent sample from `μ` plus an independent noise sample from
`κ`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.measurable_addPair
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.synchronousNoiseMap
- Synchronous-noise map on an endpoint coupling and one common noise sample. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.measurable_synchronousNoiseMap
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.synchronousNoiseCoupling
- Add one common independent noise sample to both coordinates of a coupling. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:55 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.pairLeft
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:59 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.pairRight
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:60 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.measurable_pairLeft
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:62 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.measurable_pairRight
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:66 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.map_pairLeft_prod_eq
- The `(left endpoint, noise)` marginal of `γ ⊗ κ` is `μ ⊗ κ` whenever
`γ` couples `μ` and `ν`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:72 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.map_pairRight_prod_eq
- The `(right endpoint, noise)` marginal is `ν ⊗ κ`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:92 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.isCoupling_synchronousNoiseCoupling
- Synchronous common noise sends a coupling of `μ,ν` to a coupling of their
additive-noise laws. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:113 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.quadraticCost_synchronousNoiseMap
- Common translation preserves the pointwise quadratic transport cost. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:148 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.lintegral_quadraticCost_synchronousNoiseCoupling
- The synchronous-noise coupling has exactly the same quadratic cost as the
original endpoint coupling. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:159 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction.wassersteinDistance_addNoise_le
- Wasserstein distance contracts under adding one common independent noise
law. No moment or Gaussian assumption is needed: if the original distance is
infinite the claim is automatic, while the finite branch is obtaine |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonNoiseContraction |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonNoiseContraction.lean:183 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonRemovableMass.commonRemovableMass
- Minimum total mass among a nonempty finite family of local measures. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonRemovableMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonRemovableMass.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonRemovableMass.commonRemovableMass_le
- The common removable mass is bounded above by every local total mass. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonRemovableMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonRemovableMass.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonRemovableMass.exists_mass_eq_commonRemovableMass
- One local piece attains the common removable mass. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonRemovableMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonRemovableMass.lean:44 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonRemovableMass.commonRemovableMass_pos
- If every local finite measure has positive total mass, then their common
removable mass is positive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonRemovableMass |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonRemovableMass.lean:53 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement.commonSliceRemoved
- Sum of the canonical equal-mass slices removed from the local joint blocks. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonSlicePermutationReplacement.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement.commonSlicePermutationReplacement
- Canonical replacement obtained by permuting the target marginals of the
common-mass slices. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonSlicePermutationReplacement.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement.commonSlicePermutationReplacement_map_fst
- The canonical replacement has exactly the first marginal of the removed
slice sum. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonSlicePermutationReplacement.lean:42 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement.commonSlicePermutationReplacement_map_snd
- The canonical replacement has exactly the second marginal of the removed
slice sum. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonSlicePermutationReplacement.lean:56 |
AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement.commonSlicePermutationReplacement_preserves_removed_marginals
- Both marginal identities packaged for the global replacement competitor. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CommonSlicePermutationReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/CommonSlicePermutationReplacement.lean:69 |
AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity.truncatedContinuousNNReal
- The level-`n` bounded continuous truncation of a nonnegative continuous
function. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity |
AutoSamplingTheory/TechnicalLemmas/Measure/ContinuousCostWeakLowerSemicontinuity.lean:36 |
AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity.truncatedContinuousNNReal_apply
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity |
AutoSamplingTheory/TechnicalLemmas/Measure/ContinuousCostWeakLowerSemicontinuity.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity.iSup_coe_min_nat_eq
- Natural truncations increase pointwise to the original finite nonnegative
value, viewed in `ENNReal`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity |
AutoSamplingTheory/TechnicalLemmas/Measure/ContinuousCostWeakLowerSemicontinuity.lean:60 |
AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity.lintegral_eq_iSup_truncated
- Monotone convergence expresses an unbounded continuous nonnegative
integral as the supremum of its bounded continuous truncations. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity |
AutoSamplingTheory/TechnicalLemmas/Measure/ContinuousCostWeakLowerSemicontinuity.lean:73 |
AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity.lowerSemicontinuous_lintegral_continuous_nnreal
- Integration of any continuous `NNReal`-valued cost is lower
semicontinuous for weak convergence of probability measures. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity |
AutoSamplingTheory/TechnicalLemmas/Measure/ContinuousCostWeakLowerSemicontinuity.lean:99 |
AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity.quadraticCostNNReal
- The finite `NNReal` representative of the quadratic displacement cost. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity |
AutoSamplingTheory/TechnicalLemmas/Measure/ContinuousCostWeakLowerSemicontinuity.lean:122 |
AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity.continuous_quadraticCostNNReal
- The `NNReal` quadratic cost is continuous. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity |
AutoSamplingTheory/TechnicalLemmas/Measure/ContinuousCostWeakLowerSemicontinuity.lean:127 |
AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity.coe_quadraticCostNNReal
- The finite representative agrees exactly with Samplinglib's existing
extended nonnegative quadratic cost. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity |
AutoSamplingTheory/TechnicalLemmas/Measure/ContinuousCostWeakLowerSemicontinuity.lean:134 |
AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity.lowerSemicontinuous_quadraticCostFunctional
- The quadratic Kantorovich objective is lower semicontinuous on the weak
space of probability measures on `E × E`.
`SecondCountableTopology E` is the product-Borel bridge required by Mathlib's
weak probability-measure |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ContinuousCostWeakLowerSemicontinuity |
AutoSamplingTheory/TechnicalLemmas/Measure/ContinuousCostWeakLowerSemicontinuity.lean:146 |
AutoSamplingTheory.TechnicalLemmas.Measure.ConvexInteriorAE.ae_mem_interior_of_convex_of_absolutelyContinuous
- If `μ` is absolutely continuous with respect to additive Haar measure and
is almost everywhere supported on a convex set `s`, then `μ` is actually almost
everywhere supported on `interior s`. The only removed points l |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ConvexInteriorAE |
AutoSamplingTheory/TechnicalLemmas/Measure/ConvexInteriorAE.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingAEMarginals.ae_fst_of_isCoupling
- Pull an arbitrary first-marginal almost-everywhere proposition back to the
joint coupling. No measurability assumption on the proposition is required;
`ae_of_ae_map` works directly at the filter level. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingAEMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingAEMarginals.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingAEMarginals.ae_snd_of_isCoupling
- Pull an arbitrary second-marginal almost-everywhere proposition back to the
joint coupling. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingAEMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingAEMarginals.lean:42 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingConvexDomainAE.ae_fst_mem_interior_and_differentiableAt
- If a coupling is almost everywhere concentrated over a convex source
domain, and its first marginal is absolutely continuous with respect to Haar
measure, then the first coordinate lies in the domain interior and a rea |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingConvexDomainAE |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingConvexDomainAE.lean:42 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingGraph.map_eq_of_isCoupling_of_ae_snd_eq
- A measurable map whose graph supports a coupling transports the first
marginal exactly to the second marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingGraph |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingGraph.lean:24 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingGraphIdentity.eq_map_graph_of_isCoupling_of_ae_snd_eq
- A coupling concentrated almost everywhere on the graph of a measurable map
is exactly the graph pushforward of its first marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingGraphIdentity |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingGraphIdentity.lean:27 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability.measurePreserving_fst_of_isCoupling
- The first coordinate of a coupling is measure-preserving onto its first
marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingQuadraticIntegrability.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability.measurePreserving_snd_of_isCoupling
- The second coordinate of a coupling is measure-preserving onto its second
marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingQuadraticIntegrability.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability.integrable_norm_sq_fst_of_isCoupling
- Finite second moment of the first marginal pulls back to the first
coordinate under any coupling. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingQuadraticIntegrability.lean:47 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability.integrable_norm_sq_snd_of_isCoupling
- Finite second moment of the second marginal pulls back to the second
coordinate under any coupling. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingQuadraticIntegrability.lean:57 |
AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability.integrable_norm_sub_sq_of_isCoupling
- Any coupling of two finite-second-moment marginals has integrable squared
displacement. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.CouplingQuadraticIntegrability |
AutoSamplingTheory/TechnicalLemmas/Measure/CouplingQuadraticIntegrability.lean:67 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementChangeOfVariables.integral_image_affineDisplacementMap_eq_integral_det_smul
- Interior displacement change of variables with the absolute Jacobian
removed.
The endpoint derivative may be only positive semidefinite. The strict
`(1-t) I` contribution at `t < 1` makes the displacement derivative |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementChangeOfVariables |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementChangeOfVariables.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexGradientPositive.isPositive_fderiv_of_convex_gradient_field
- At every point where the gradient vector field of a globally convex
potential is Frechet differentiable, its derivative is a positive operator.
This statement is dimension-free. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexGradientPositive |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementConvexGradientPositive.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexGradientPositive.toMatrix_fderiv_posSemidef_of_convex_gradient_field
- In an orthonormal basis, the same local derivative therefore has a PSD
matrix representation. The basis is only a coordinate witness; positivity is
proved before choosing it. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexGradientPositive |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementConvexGradientPositive.lean:65 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexGradientPositive.det_affineDisplacementDerivative_pos_of_convex_gradient_field
- Consequently, the interior affine displacement derivative has strictly
positive determinant at every such differentiability point. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexGradientPositive |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementConvexGradientPositive.lean:77 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexPotentialSupport.isSupportingField_of_convexOn_univ_hasFDerivAt_inner
- A globally convex differentiable potential whose derivative is represented
by `T` through the inner product makes `T` a global supporting field. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexPotentialSupport |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementConvexPotentialSupport.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexPotentialSupport.isMonotoneMap_of_convexOn_univ_hasFDerivAt_inner
- The gradient field of a globally convex differentiable potential is
monotone, obtained by composing the explicit supporting-field theorem with the
already verified algebraic monotonicity edge. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexPotentialSupport |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementConvexPotentialSupport.lean:70 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexPotentialSupport.injective_affineDisplacementMap_of_convexOn_univ_hasFDerivAt_inner
- Consequently, every interior displacement map
`S_t(x) = (1-t)x + t T(x)` is injective for `0 <= t < 1`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementConvexPotentialSupport |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementConvexPotentialSupport.lean:80 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos.toMatrix_affineDisplacementDerivative_posDef
- An SPD matrix representation of the endpoint derivative gives an SPD matrix
representation of the affine displacement derivative on the full segment. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeDetPos.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos.toMatrix_affineDisplacementDerivative_posDef_of_posSemidef
- A PSD endpoint derivative is already enough for strict positivity of the
interior affine derivative matrix. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeDetPos.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos.det_affineDisplacementDerivative_pos
- Under an SPD endpoint derivative, the continuous-linear determinant of the
affine displacement derivative is strictly positive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeDetPos.lean:63 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos.det_affineDisplacementDerivative_pos_of_posSemidef
- Interior-time determinant positivity from the weaker PSD endpoint
hypothesis. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeDetPos.lean:74 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos.abs_det_affineDisplacementDerivative_eq
- The absolute determinant in change of variables is redundant under an SPD
endpoint derivative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeDetPos.lean:85 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos.abs_det_affineDisplacementDerivative_eq_of_posSemidef
- Interior-time absolute-determinant removal from a PSD endpoint derivative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeDetPos |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeDetPos.lean:94 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeLogDet.neg_log_det_affineDisplacementDerivative_le
- Continuous-linear-map form of the affine `-log det` convexity inequality.
This is the direct representation-level bridge from the Fréchet derivative used
by change of variables to the matrix log-determinant theorem. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeLogDet |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeLogDet.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeLogDet.hasFDerivAt_affineDisplacementMap_and_neg_log_det
- Pointwise derivative package: if `T` has derivative `T'` at `x` and the
matrix of `T'` is SPD, then the displacement map has the expected affine
Fréchet derivative and that derivative satisfies the literal log-det ineq |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeLogDet |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeLogDet.lean:63 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeMatrix.toMatrix_affineDisplacementDerivative
- In any finite basis, the matrix of the displacement derivative is the
literal affine matrix `(1-t) I + t A`, where `A` is the matrix of `T'`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeMatrix |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeMatrix.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeMatrix.det_affineDisplacementDerivative_eq_matrix_det
- The determinant used by the Fréchet/change-of-variables layer is exactly
the determinant of the affine matrix in any finite basis. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementDerivativeMatrix |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementDerivativeMatrix.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementEntropyPushforward.integral_logDensity_eq_endpoint_sub_logJacobian
- Push a logarithmic density identity through an exact transport map and
separate the endpoint entropy term from the logarithmic Jacobian term. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementEntropyPushforward |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementEntropyPushforward.lean:36 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementGradientDerivativeSymmetry.isSymmetric_fderiv_of_gradient_field
- The derivative of a globally represented gradient field is symmetric at
every point where that vector field is Frechet differentiable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementGradientDerivativeSymmetry |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementGradientDerivativeSymmetry.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.IsQuadraticOptimalCoupling
- A coupling is quadratic-cost optimal when it attains the Kantorovich
infimum defining the squared 2-Wasserstein distance. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolation.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.displacementInterpolation
- The law at time `t` of `(1 - t) X₀ + t X₁` when the joint law of
`(X₀, X₁)` is `γ`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolation.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.displacementInterpolation_zero
- The displacement interpolation starts at the first marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolation.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.displacementInterpolation_one
- The displacement interpolation ends at the second marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolation.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.IsWassersteinGeodesic
- Chewi Definition 1.3.25: a Wasserstein geodesic, also called the
displacement or McCann interpolation, is the affine-law curve generated by
an optimal coupling of two `P₂,ac` endpoint laws.
The name records the source |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolation.lean:57 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.isWassersteinGeodesic_displacementInterpolation
- An optimal coupling and two `P₂,ac` endpoints generate the source
displacement-interpolation predicate. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolation.lean:68 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.endpoints_of_isWassersteinGeodesic
- Every source displacement interpolation has the prescribed endpoints. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolation.lean:77 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationConstantSpeed.isProbabilityMeasure_displacementInterpolation
- A displacement marginal of a probability coupling is again a probability
measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationConstantSpeed |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationConstantSpeed.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationConstantSpeed.wassersteinDistance_interpolation_le
- The canonical two-time interpolation coupling gives the sharp linear upper
bound when the times are ordered. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationConstantSpeed |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationConstantSpeed.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationConstantSpeed.interpolation_coefficients_sum_one
- Ordered times partition the unit interval into the three nonnegative pieces
`s`, `t-s`, and `1-t`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationConstantSpeed |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationConstantSpeed.lean:76 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationConstantSpeed.wassersteinDistance_interpolation_eq_of_le
- Chewi's constant-speed identity for ordered interpolation times.
The endpoint laws carry the source `P₂,ac` assumptions. Absolute continuity
is not used in the metric argument itself; its role here is to keep the the |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationConstantSpeed |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationConstantSpeed.lean:91 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost.pointMap_sub_pointMap
- Difference of two affine displacement maps. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCost.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost.measurable_quadraticCost
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCost.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost.quadraticCost_pairPointMap_eq
- Pointwise quadratic cost scaling under the two-time displacement map. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCost.lean:55 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost.lintegral_quadraticCost_interpolationCoupling_eq
- The exact quadratic cost of the canonical two-time interpolation coupling
is `|s-t|^2` times the original endpoint-plan cost. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCost.lean:72 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost.wassersteinDistance_sq_interpolation_le
- Any endpoint coupling therefore gives the expected upper bound between two
of its displacement-interpolation marginals. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCost |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCost.lean:86 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling.pointMap
- The affine point map used by displacement interpolation. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCoupling.lean:28 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling.measurable_pointMap
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCoupling.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling.interpolationCoupling
- Push the original endpoint coupling through the interpolation maps at two
times. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCoupling.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling.measurable_pairPointMap
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCoupling.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling.isCoupling_interpolationCoupling
- The two-time pushforward has the displacement law at time `s` as its first
marginal and the displacement law at time `t` as its second marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationCoupling |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationCoupling.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationConstantSpeed.wassersteinDistance_interpolation_eq_of_le_of_integrable_norm_sq
- The ordered constant-speed identity for arbitrary probability endpoints
with finite second moments.
This is the metric-identity component of Statistical Optimal Transport,
Theorem 7.6. Optimal-coupling existence is an |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolationP2ConstantSpeed |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolationP2ConstantSpeed.lean:34 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianAffineSpectrum.affineIdentityMatrix_mulVec_eigenvectorBasis
- Every eigenvector in the canonical Hermitian eigenbasis of a real SPD
matrix `A` remains an eigenvector of `(1-t) I + t A`, with eigenvalue
`(1-t) + t * lambda_i`.
No ordering claim about `Matrix.IsHermitian.eigenvalu |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianAffineSpectrum |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianAffineSpectrum.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.affineIdentityEigenvalue_pos
- The eigenvalue of `(1-t) I + t A` corresponding to a positive eigenvalue
`a` of `A` stays positive for `0 <= t <= 1`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:64 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.log_affineIdentity_ge
- Scalar logarithmic concavity along the segment from the identity eigenvalue
`1` to a positive eigenvalue `a`.
This is the one-dimensional inequality used eigenvalue-by-eigenvalue in the
Jacobian determinant calculatio |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:76 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.spectrumLogDet
- Sum of logarithms of a finite positive spectrum. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:88 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.spectrumLogDet_eigenvalues_eq_log_det
- For a real positive-definite matrix, the abstract finite-spectrum log-det
is exactly the logarithm of the literal matrix determinant.
This is the matrix bridge needed before the Jacobian entropy leaf can consume
an ac |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:97 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.neg_spectrumLogDet_eigenvalues_eq_neg_log_det
- The same SPD determinant bridge in the entropy-sign orientation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:106 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.affineIdentityMatrix_posDef
- The literal affine Jacobian `(1-t) I + t A` stays positive definite along
`0 <= t <= 1` whenever `A` is positive definite.
For `t < 1`, the identity contribution has a strictly positive coefficient and
is positive def |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:118 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.affineIdentityMatrix_posDef_of_posSemidef
- Interior-time form needed for a Brenier Jacobian: positive semidefiniteness
of `A` already suffices because `(1-t) I` is strictly positive for `t < 1`.
The strict endpoint exclusion is intentional. A positive-semidef |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:137 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.det_conjStarAlgAut_eq
- Determinant is invariant under the star-conjugation by a unitary matrix.
This is the determinant-only quotient of a unitary change of basis. Keeping it
separate avoids encoding any choice of eigenvalue enumeration in |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:154 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.affineIdentitySpectrum
- Spectrum of the affine Jacobian `(1-t) I + t A` when `lambda` is the
spectrum of `A`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:174 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.affineIdentityMatrix_diagonal
- Affine interpolation commutes literally with formation of a diagonal
matrix. This is the basis-level calculation used after spectral
diagonalization. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:181 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.det_affineIdentity_diagonal
- Determinant of the affine identity segment against a diagonal matrix. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:191 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.spectrumLogDet_affineIdentity_ge
- Eigenvalue-coordinate form of the log-determinant concavity used in the
entropy half of Chewi Theorem 1.4.5. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:201 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.neg_spectrumLogDet_affineIdentity_le
- Equivalent `-log det` orientation: the Jacobian contribution to entropy is
convex along the identity-to-transport interpolation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:215 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy.integral_neg_spectrumLogDet_affineIdentity_le
- Integral form of the same Jacobian entropy inequality for a measurable
family of positive spectra. Integrability is explicit so Mathlib's totalized
Bochner integral cannot silently certify a non-integrable entropy ter |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianEntropy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianEntropy.lean:229 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianMatrixLogDet.det_affineIdentityMatrix_eq_prod_eigenvalues
- The literal determinant of the affine identity segment is the product of
the affine transforms of the eigenvalues of an SPD matrix.
The proof diagonalizes `A` by the canonical unitary eigenbasis, transports the
affine |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianMatrixLogDet |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianMatrixLogDet.lean:48 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianMatrixLogDet.spectrumLogDet_affineIdentity_eigenvalues_eq_log_det
- The finite-spectrum log-det of the affine eigenvalues is exactly the
literal log determinant of the affine matrix. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianMatrixLogDet |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianMatrixLogDet.lean:87 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianMatrixLogDet.neg_log_det_affineIdentity_le
- Literal SPD matrix form of the `-log det` convexity used in the entropy
half of Chewi Theorem 1.4.5. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementJacobianMatrixLogDet |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementJacobianMatrixLogDet.lean:101 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapDerivative.affineDisplacementMap
- The point map whose pushforward gives displacement interpolation when an
endpoint transport map `T` is available. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapDerivative |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMapDerivative.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapDerivative.affineDisplacementDerivative
- The affine continuous-linear map predicted by differentiating the
displacement point map. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapDerivative |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMapDerivative.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapDerivative.hasFDerivAt_affineDisplacementMap
- Pointwise Fréchet derivative of the displacement map. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapDerivative |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMapDerivative.lean:42 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapDerivative.hasFDerivWithinAt_affineDisplacementMap
- Within-set Fréchet derivative in the form consumed by Mathlib's
change-of-variables API. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapDerivative |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMapDerivative.lean:54 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapInjectivity.IsMonotoneMap
- Hilbert-space monotonicity of a point map. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapInjectivity |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMapInjectivity.lean:36 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapInjectivity.inner_affineDisplacementMap_sub
- Exact inner-product expansion for the affine displacement map. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapInjectivity |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMapInjectivity.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapInjectivity.affineDisplacementMap_inner_lower_bound
- Monotonicity of `T` gives a strong-monotonicity lower bound for the
interior displacement map. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapInjectivity |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMapInjectivity.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapInjectivity.injective_affineDisplacementMap_of_monotone
- For every `0 <= t < 1`, a monotone endpoint map produces an injective
interior displacement map. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMapInjectivity |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMapInjectivity.lean:62 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMonotoneDerivative.inner_fderiv_nonneg_of_monotone
- The Frechet derivative of a monotone Hilbert-space map has nonnegative
quadratic form in every direction. No symmetry of the derivative is used. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMonotoneDerivative |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMonotoneDerivative.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMonotoneDerivative.isPositive_fderiv_of_monotone_of_isSymmetric
- If the derivative of a monotone map is additionally symmetric, then it is
a positive operator. This packages the exact remaining split needed for a
future gradient/Hessian regularity edge: monotonicity supplies the qu |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementMonotoneDerivative |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementMonotoneDerivative.lean:90 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPositiveOperator.toMatrix_posSemidef_of_isPositive
- A positive continuous-linear operator has a PSD matrix in every
orthonormal basis. This is the coordinate bridge needed by the existing
interior determinant theorem. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPositiveOperator |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementPositiveOperator.lean:44 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPositiveOperator.det_affineDisplacementDerivative_pos_of_isPositive
- Positive endpoint operator implies strict positivity of the interior
Jacobian determinant of the displacement derivative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPositiveOperator |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementPositiveOperator.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPositiveOperator.abs_det_affineDisplacementDerivative_eq_of_isPositive
- The absolute Jacobian determinant is redundant at interior times when the
endpoint derivative is a positive operator. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPositiveOperator |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementPositiveOperator.lean:62 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPositiveOperator.integral_image_affineDisplacementMap_eq_integral_det_smul_of_isPositive
- Coordinate-free form of the interior displacement change-of-variables
join: the derivative hypothesis is stated as positivity of the operator, while
an orthonormal basis is used only internally to enter Mathlib's matri |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPositiveOperator |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementPositiveOperator.lean:75 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPotentialEnergy.potential_pointMap_le
- Pointwise strong-convexity estimate along the affine displacement map. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPotentialEnergy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementPotentialEnergy.lean:34 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPotentialEnergy.integral_potential_pointMap_le
- Integrating the pointwise strong-convexity estimate preserves the same
upper bound when both real-valued sides are explicitly integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPotentialEnergy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementPotentialEnergy.lean:63 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPotentialEnergy.integral_displacementInterpolation_eq_integral_pointMap
- Integrating a strongly measurable potential against the displacement
marginal is exactly integrating the potential along the affine point map under
the original coupling. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPotentialEnergy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementPotentialEnergy.lean:81 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPotentialEnergy.integral_potential_displacementInterpolation_le
- Source-oriented potential-energy inequality for a displacement marginal.
The right side is intentionally still written on the endpoint coupling; a
later bookkeeping node may rewrite its three integrals using the two ma |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementPotentialEnergy |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementPotentialEnergy.lean:94 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementRealQuadraticCost.ofReal_integral_norm_sq_eq_lintegral_quadraticCost
- The real Bochner integral of squared displacement and the ENNReal
quadratic-cost lintegral are the same finite quantity, viewed in ENNReal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementRealQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementRealQuadraticCost.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementRealQuadraticCost.ofReal_integral_norm_sq_eq_wassersteinDistance_sq_of_optimal
- For a quadratic-cost optimal coupling, the real squared-displacement
integral is exactly the squared Wasserstein distance after embedding into
ENNReal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementRealQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementRealQuadraticCost.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementRealQuadraticCost.integral_norm_sq_eq_wassersteinDistance_sq_toReal_of_optimal
- Real-valued form convenient for the potential-energy inequality: the
quadratic coupling integral is the `toReal` value of `W₂^2`. Integrability of
the real cost supplies finiteness automatically through the preceding |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementRealQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementRealQuadraticCost.lean:69 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementSupportingPotential.IsSupportingField
- A vector field `T` is a global first-order supporting field for a real
potential `phi` when its affine tangent expression at every `x` lies below
`phi` at every `y`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementSupportingPotential |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementSupportingPotential.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementSupportingPotential.isMonotoneMap_of_isSupportingField
- A global supporting field is monotone in the Hilbert-space sense used by
the interior displacement injectivity theorem. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementSupportingPotential |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementSupportingPotential.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementSupportingPotential.injective_affineDisplacementMap_of_supportingField
- Direct composition of the support inequality with the previously isolated
interior-injectivity leaf. This theorem still does not assume or prove that
`phi` is convex; it only exposes the exact support contract a future |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementSupportingPotential |
AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementSupportingPotential.lean:62 |
AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder.finiteRemainder
- The finite remainder left after removing a finite measure from an ambient
finite measure. The definition makes sense without a domination hypothesis;
exact reconstruction uses domination below. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder |
AutoSamplingTheory/TechnicalLemmas/Measure/FiniteRemainder.lean:30 |
AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder.finiteRemainder_toMeasure
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder |
AutoSamplingTheory/TechnicalLemmas/Measure/FiniteRemainder.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder.finiteRemainder_add_removed_eq
- A dominated removed finite measure can be added back to its canonical
remainder to recover the ambient finite measure exactly. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder |
AutoSamplingTheory/TechnicalLemmas/Measure/FiniteRemainder.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder.ambient_eq_finiteRemainder_add_removed
- Orientation consumed directly by the replacement-competitor algebra. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder |
AutoSamplingTheory/TechnicalLemmas/Measure/FiniteRemainder.lean:53 |
AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder.exists_finiteRemainder_of_le
- Existence form: every dominated finite block admits an explicit additive
remainder decomposition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.FiniteRemainder |
AutoSamplingTheory/TechnicalLemmas/Measure/FiniteRemainder.lean:61 |
AutoSamplingTheory.TechnicalLemmas.Measure.FiniteSumDomination.toMeasure_finsetSum_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.FiniteSumDomination |
AutoSamplingTheory/TechnicalLemmas/Measure/FiniteSumDomination.lean:24 |
AutoSamplingTheory.TechnicalLemmas.Measure.FiniteSumDomination.toMeasure_fintypeSum_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.FiniteSumDomination |
AutoSamplingTheory/TechnicalLemmas/Measure/FiniteSumDomination.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Measure.FiniteSumDomination.toMeasure_fintypeSum_le_ambient
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.FiniteSumDomination |
AutoSamplingTheory/TechnicalLemmas/Measure/FiniteSumDomination.lean:45 |
AutoSamplingTheory.TechnicalLemmas.Measure.GaussianLikelihood.translated_gaussian_likelihood
- Exact likelihood ratio for actual translated isotropic Gaussians.
The explicit likelihood is measurable and the RN identity is almost everywhere. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GaussianLikelihood |
AutoSamplingTheory/TechnicalLemmas/Measure/GaussianLikelihood.lean:22 |
AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing.scaledStdGaussian
- Centered isotropic Gaussian noise obtained by scaling the standard Gaussian
by `sigma`. The definition is valid for every real scale; heat-flow interfaces
below use the nonnegative scale `sqrt t`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing |
AutoSamplingTheory/TechnicalLemmas/Measure/GaussianSmoothing.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing.scaledStdGaussian_isProbabilityMeasure
|
instance | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing |
AutoSamplingTheory/TechnicalLemmas/Measure/GaussianSmoothing.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing.gaussianSmoothing
- Gaussian smoothing of a law by independent centered Gaussian noise of
scale `sigma`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing |
AutoSamplingTheory/TechnicalLemmas/Measure/GaussianSmoothing.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing.wassersteinDistance_gaussianSmoothing_le
- Gaussian smoothing is `W₂`-contractive because it adds the same independent
noise law to both endpoint measures. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing |
AutoSamplingTheory/TechnicalLemmas/Measure/GaussianSmoothing.lean:55 |
AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing.heatSmoothing
- Source-normalized heat smoothing: nonnegative heat time `t` corresponds to
adding centered Gaussian noise with standard-deviation scale `sqrt t`, hence
covariance `t I`.
This is the measure/convolution side of the hea |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing |
AutoSamplingTheory/TechnicalLemmas/Measure/GaussianSmoothing.lean:70 |
AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing.heatSmoothing_eq_gaussianSmoothing_sqrt
- Unfold the source normalization from heat time to Gaussian standard
deviation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing |
AutoSamplingTheory/TechnicalLemmas/Measure/GaussianSmoothing.lean:75 |
AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing.wassersteinDistance_heatSmoothing_le
- Simultaneous heat smoothing is `W₂`-contractive.
The proof is purely the common-noise coupling theorem at Gaussian scale
`sqrt t`; it does not use or assert the heat PDE. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GaussianSmoothing |
AutoSamplingTheory/TechnicalLemmas/Measure/GaussianSmoothing.lean:83 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.gibbsDensityENNReal
- The `ℝ≥0∞` density associated with an unnormalized Gibbs potential. |
def | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:25 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.gibbsDensityENNReal_pos
- Gibbs densities are pointwise positive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.gibbsDensityENNReal_lt_top
- Gibbs densities are pointwise finite as `ℝ≥0∞` values. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:34 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.measurable_gibbsDensityENNReal
- A measurable potential gives a measurable `ℝ≥0∞` Gibbs density. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.aemeasurable_gibbsDensityENNReal
- An a.e.-measurable potential gives an a.e.-measurable `ℝ≥0∞` Gibbs
density. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.lintegral_gibbsDensityENNReal_ne_zero
- Over a nonzero measure, an a.e.-measurable Gibbs density has nonzero
lintegral because it is pointwise positive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:60 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.lintegral_gibbsDensityENNReal_ne_top_of_ae_le
- An a.e. finite envelope gives a finite Gibbs normalization constant. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:74 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.gibbsDensityENNReal_le_of_potential_ge
- If a potential `V` is bounded below by `W` at a point, then the Gibbs
density of `V` is bounded above by the Gibbs density of `W` there. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:84 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.gibbsDensityENNReal_ae_le_of_ae_potential_ge
- A.e. potential lower bounds give a.e. Gibbs-density envelope bounds. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:90 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.lintegral_gibbsDensityENNReal_ne_top_of_ae_potential_ge
- A finite Gibbs integral for a lower potential `W` is an envelope proof for
the larger potential `V`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:98 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.lintegral_gibbsDensityENNReal_ne_top_of_ae_ge_const
- On a finite base measure, an a.e. lower bound on the potential gives a
finite Gibbs normalization constant. This is the compact-domain/truncated-law
envelope leaf; coercive Lebesgue tails are a stronger separate theor |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:109 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.isProbabilityMeasure_withDensity_normalized_gibbs
- A finite nonzero Gibbs normalization constant gives a probability measure
through reciprocal-lintegral normalization and `withDensity`. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:124 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_le
- A nonzero base measure and a finite a.e. envelope are enough to normalize a
Gibbs density into a probability measure. This is the reusable contract that
later coercivity/growth leaves should target. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:137 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_potential_ge
- A measurable potential `V` whose Gibbs density is dominated by the Gibbs
density of a lower potential `W` with finite integral normalizes to a
probability measure. This is the first reusable potential-envelope interfa |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:153 |
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.isProbabilityMeasure_withDensity_normalized_gibbs_of_ae_ge_const
- On a finite nonzero base measure, an a.e. lower bound on a measurable
potential is enough to construct the normalized Gibbs probability measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs |
AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:167 |
AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral.integral_withDensity_inv_mul_gibbsDensityENNReal_eq_integral_inv_mul_exp_smul
- Bochner integrals against a Gibbs `withDensity` measure rewrite to a
base-measure integral weighted by the real Gibbs density.
The only scalar hypothesis needed for this algebraic rewrite is `Z ≠ 0`, which
keeps the ` |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral |
AutoSamplingTheory/TechnicalLemmas/Measure/GibbsIntegral.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral.integral_withDensity_lintegral_inv_mul_gibbsDensityENNReal_eq_integral_lintegral_inv_mul_exp_smul
- Source-facing specialization of the Gibbs integral rewrite where the scalar
is the Gibbs lintegral. The finite-normalizer proof, when needed to obtain a
probability measure, remains a separate input to the probability |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral |
AutoSamplingTheory/TechnicalLemmas/Measure/GibbsIntegral.lean:56 |
AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral.integral_withDensity_lintegral_inv_mul_gibbsDensityENNReal_eq_integral_lintegral_inv_mul_exp_smul_of_neZero
- Nonzero-base-measure specialization of the source-facing Gibbs integral
rewrite. It discharges the nonzero Gibbs normalizer from positivity of
`exp (-V)` and `[NeZero μ]`.
This remains only a Bochner-integral rewrite |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GibbsIntegral |
AutoSamplingTheory/TechnicalLemmas/Measure/GibbsIntegral.lean:76 |
AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_convexOn
- A finite nonzero `ℝ≥0∞` normalizer preserves log-concavity when an
unnormalized Gibbs density is viewed as a real-valued normalized density shape.
The hypotheses `Z ≠ 0` and `Z ≠ ∞` are explicit because otherwise `.to |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity |
AutoSamplingTheory/TechnicalLemmas/Measure/GibbsLogConcavity.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_strongConvexOn
- Strong-convexity wrapper for the real-valued normalized Gibbs-density shape
associated with an `ℝ≥0∞` density and a finite nonzero normalizer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity |
AutoSamplingTheory/TechnicalLemmas/Measure/GibbsLogConcavity.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_lintegral_normalized_gibbsDensityENNReal_toReal_of_convexOn
- Source-facing specialization of
`logConcaveOn_normalized_gibbsDensityENNReal_toReal_of_convexOn` where the
normalizing scalar is the supplied finite nonzero Gibbs integral. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity |
AutoSamplingTheory/TechnicalLemmas/Measure/GibbsLogConcavity.lean:58 |
AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_lintegral_normalized_gibbsDensityENNReal_toReal_of_strongConvexOn
- Strong-convex source-facing specialization where the normalizing scalar is
the supplied finite nonzero Gibbs integral. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity |
AutoSamplingTheory/TechnicalLemmas/Measure/GibbsLogConcavity.lean:72 |
AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_lintegral_normalized_gibbsDensityENNReal_toReal_of_strongConvexOn_minimizer
- A measurable strongly convex potential with an exposed global minimizer has
a real-valued normalized Gibbs-density shape that is log-concave on all of
space.
This combines the strong-convexity shape lemma with the alr |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity |
AutoSamplingTheory/TechnicalLemmas/Measure/GibbsLogConcavity.lean:92 |
AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity.logConcaveOn_normalized_laplace_gibbsDensityENNReal_toReal
- The explicitly normalized one-dimensional absolute-linear Laplace Gibbs
`ℝ≥0∞` density becomes a real-valued log-concave density shape after `.toReal`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity |
AutoSamplingTheory/TechnicalLemmas/Measure/GibbsLogConcavity.lean:113 |
AutoSamplingTheory.TechnicalLemmas.Measure.IsotropicGaussianDensity.map_sqrt_smul_stdGaussian_eq_withDensity
- Scaling the standard Gaussian by `sqrt η` gives the explicit isotropic
Gaussian density relative to the canonical volume measure of `E`. The inverse
square-root normalizer is raised to the natural-number dimension, so |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.IsotropicGaussianDensity |
AutoSamplingTheory/TechnicalLemmas/Measure/IsotropicGaussianDensity.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual.DualFeasible
- A pair of integrable potentials is dual-feasible when its sum is bounded
by the cost almost everywhere under the product of the marginals. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual |
AutoSamplingTheory/TechnicalLemmas/Measure/KantorovichDual.lean:21 |
AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual.dualTransportValue
- Chewi Definition 1.3.6: the value of the Kantorovich dual optimization
problem. At the source's finite-second-moment quadratic cost, the feasible
objectives are nonempty and bounded above; those analytic facts are not |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual |
AutoSamplingTheory/TechnicalLemmas/Measure/KantorovichDual.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual.dualTransportValue_eq_sSup
- Chewi display (1.3.7): source-facing expansion of the dual value. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual |
AutoSamplingTheory/TechnicalLemmas/Measure/KantorovichDual.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation.one
- The constant-one element of `L²(pi)` for a finite measure. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation |
AutoSamplingTheory/TechnicalLemmas/Measure/L2Expectation.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation.expectation
- Expectation on `L²(pi)` as the continuous linear functional
`f ↦ <1,f>_{L²(pi)}`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation |
AutoSamplingTheory/TechnicalLemmas/Measure/L2Expectation.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation.expectation_apply_eq_inner
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation |
AutoSamplingTheory/TechnicalLemmas/Measure/L2Expectation.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation.expectation_apply_eq_integral
- The `L²(pi)` expectation functional is exactly the Bochner integral of the
chosen `Lp` representative. The equality is representative-safe because both
sides are insensitive to `pi`-a.e. changes. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation |
AutoSamplingTheory/TechnicalLemmas/Measure/L2Expectation.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation.inner_one_eq_integral
- Source-facing integral form of the constant-one `L²` pairing. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.L2Expectation |
AutoSamplingTheory/TechnicalLemmas/Measure/L2Expectation.lean:66 |
AutoSamplingTheory.TechnicalLemmas.Measure.OptimalContinuousCost.exists_optimal_coupling
- A continuous nonnegative cost attains its actual infimum over couplings of
probabilities on complete second-countable metric Borel spaces. The minimum
may be infinite; no finite moment or optimizer is assumed. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.OptimalContinuousCost |
AutoSamplingTheory/TechnicalLemmas/Measure/OptimalContinuousCost.lean:14 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.mass_map_of_measurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:27 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.sourceMarginal
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.targetMarginal
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.sourceMarginal_mass
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:42 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.targetMarginal_mass
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:47 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.finiteMeasure_map_finset_sum
- Pushforward commutes with a finite sum of finite measures. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.finiteMeasure_map_fintype_sum
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:68 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.permutedMarginalJoint
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:75 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.permutedMarginalJoint_map_fst
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:81 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.permutedMarginalJoint_map_snd
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:91 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.permutedMarginalJoint_mass
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:101 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.permutedMarginalReplacement
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:116 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.permutedMarginalReplacement_map_fst
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:121 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement.permutedMarginalReplacement_map_snd
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedMarginalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedMarginalReplacement.lean:142 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.realQuadraticCost
- Real quadratic transport cost observable on one joint pair. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.crossQuadraticCost
- Cross-coordinate quadratic observable used when source and target are
sampled from two local joint laws. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:48 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.realQuadraticCost_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.measurePreserving_commonMassProduct_source_target
- Mapping a common-mass product of two joint blocks by `(fst,snd)` produces
exactly the common-mass product of the source marginal of the first block and
the target marginal of the second block. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:59 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.integral_marginal_commonMassProduct_eq_mass_mul_normalized_cross
- One source-target replacement block has cost equal to the common mass times
the cross-cost expectation under the product of the two normalized joint laws. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:73 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.integrable_realQuadraticCost_of_normalize
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:102 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.integrable_marginal_commonMassProduct_of_normalized_cross
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:116 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.integral_sum_eq_mass_mul_diagonalProduct
- The quadratic cost of a finite sum of equal-mass joint blocks is the common
mass times the diagonal quadratic-cost expectation under the product of their
normalized laws. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:135 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.integral_permutedReplacement_symm_eq_mass_mul_permutedProduct
- For a fixed-point-free permutation `sigma`, the true finite cost of the
`σ⁻¹` marginal replacement is the common mass times the normalized permuted
product-law expectation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:173 |
AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost.integral_permutedReplacement_symm_lt_sum_of_productGap
- Any strict normalized product-law cost improvement transfers to a strict
finite-measure improvement of the corresponding marginal replacement. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PermutedReplacementQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/PermutedReplacementQuadraticCost.lean:231 |
AutoSamplingTheory.TechnicalLemmas.Measure.PositiveComponentAE.absolutelyContinuous_smul_add_left
- A measure is absolutely continuous with respect to any mixture containing
it with nonzero `ENNReal` weight. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PositiveComponentAE |
AutoSamplingTheory/TechnicalLemmas/Measure/PositiveComponentAE.lean:25 |
AutoSamplingTheory.TechnicalLemmas.Measure.PositiveComponentAE.absolutelyContinuous_add_smul_right
- Symmetric version for the right component. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PositiveComponentAE |
AutoSamplingTheory/TechnicalLemmas/Measure/PositiveComponentAE.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Measure.PositiveComponentAE.ae_of_ae_smul_add_left
- Any property holding almost everywhere under a positive mixture also holds
almost everywhere under its left component. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PositiveComponentAE |
AutoSamplingTheory/TechnicalLemmas/Measure/PositiveComponentAE.lean:44 |
AutoSamplingTheory.TechnicalLemmas.Measure.PositiveComponentAE.ae_of_ae_add_smul_right
- Any property holding almost everywhere under a positive mixture also holds
almost everywhere under its right component. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PositiveComponentAE |
AutoSamplingTheory/TechnicalLemmas/Measure/PositiveComponentAE.lean:54 |
AutoSamplingTheory.TechnicalLemmas.Measure.PowerPerspective.lintegral_perspective_le
- Holder gives the power-perspective inequality without assuming finite
numerator or right-hand integral. The denominator is positive and finite. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.PowerPerspective |
AutoSamplingTheory/TechnicalLemmas/Measure/PowerPerspective.lean:18 |
AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness.IsProbabilityCoupling
- Probability-measure version of the fixed-marginal coupling predicate. It
is definitionally adapted to the weak topology, whose continuous maps are the
`ProbabilityMeasure.map` operations. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness |
AutoSamplingTheory/TechnicalLemmas/Measure/ProbabilityCouplingCompactness.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness.probabilityCouplingSet
- Fixed-marginal probability couplings as a subset of the weak probability
measure space. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness |
AutoSamplingTheory/TechnicalLemmas/Measure/ProbabilityCouplingCompactness.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness.isProbabilityCoupling_iff_isCoupling_toMeasure
- The topology-facing probability coupling predicate agrees exactly with
Samplinglib's raw-measure `Transport.IsCoupling`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness |
AutoSamplingTheory/TechnicalLemmas/Measure/ProbabilityCouplingCompactness.lean:60 |
AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness.isClosed_probabilityCouplingSet
- Fixed marginal constraints are closed in the weak topology on probability
measures. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness |
AutoSamplingTheory/TechnicalLemmas/Measure/ProbabilityCouplingCompactness.lean:89 |
AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness.isTightMeasureSet_couplingSet
- All raw couplings of two fixed probability measures form a tight family.
This is the reusable fixed-marginal tightness statement behind Prokhorov. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness |
AutoSamplingTheory/TechnicalLemmas/Measure/ProbabilityCouplingCompactness.lean:114 |
AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness.isTightMeasureSet_probabilityCouplingSet
- The underlying raw measures of the probability-coupling set are tight. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness |
AutoSamplingTheory/TechnicalLemmas/Measure/ProbabilityCouplingCompactness.lean:129 |
AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness.isCompact_probabilityCouplingSet
- Probability couplings with two fixed marginals are compact for weak
convergence. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ProbabilityCouplingCompactness |
AutoSamplingTheory/TechnicalLemmas/Measure/ProbabilityCouplingCompactness.lean:140 |
AutoSamplingTheory.TechnicalLemmas.Measure.Product.measurable_update_prod_pi
- The coordinate-replacement map `(y, x) ↦ Function.update x i y` is measurable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Product |
AutoSamplingTheory/TechnicalLemmas/Measure/Product.lean:26 |
AutoSamplingTheory.TechnicalLemmas.Measure.Product.map_update_prod_pi
- Replacing one coordinate of a product sample by an independent sample from
that coordinate preserves the product law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Product |
AutoSamplingTheory/TechnicalLemmas/Measure/Product.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Measure.Product.measurePreserving_update_prod_pi
- Measure-preserving wrapper for coordinate replacement under a product law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Product |
AutoSamplingTheory/TechnicalLemmas/Measure/Product.lean:92 |
AutoSamplingTheory.TechnicalLemmas.Measure.Product.integral_update_prod_pi_eq_integral
- Averaging a function after one-coordinate replacement over the fresh
coordinate and the original product sample recovers its product-law integral. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Product |
AutoSamplingTheory/TechnicalLemmas/Measure/Product.lean:99 |
AutoSamplingTheory.TechnicalLemmas.Measure.Product.integrable_update_slice_ae
- If a function is integrable on a finite product law, then for almost every
base product sample, the one-coordinate replacement slice is integrable in the
fresh coordinate. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Product |
AutoSamplingTheory/TechnicalLemmas/Measure/Product.lean:121 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalBrenierMap.ae_snd_eq_gradient_of_quadraticOptimal_of_base
- Fix any support point as the Rockafellar root. For an optimal quadratic
coupling with absolutely continuous first marginal and finite second moments,
the coupling is almost everywhere concentrated on the graph of the |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalBrenierMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalBrenierMap.lean:63 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalBrenierMap.map_gradient_eq_of_quadraticOptimal_of_base
- The graph concentration above upgrades immediately to the Monge identity:
the gradient of the finite proper Rockafellar representative pushes the first
marginal exactly to the second marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalBrenierMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalBrenierMap.lean:109 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalBrenierMap.exists_base_map_gradient_eq_of_quadraticOptimal
- A probability optimal coupling has nonempty support, so a Rockafellar root
can be chosen internally. This is the source-facing existence form: there is a
support-normalized proper Rockafellar construction whose gradie |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalBrenierMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalBrenierMap.lean:132 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalBrenierMap.exists_base_map_gradient_eq_of_quadraticOptimal_p2ac
- `P₂,ac` wrapper matching the source-facing Wasserstein class already used by
Samplinglib's direct optimal-support theorem. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalBrenierMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalBrenierMap.lean:158 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap.graphCoupling
- Joint law induced by a measurable transport map. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMap.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap.isCoupling_graphCoupling
- A graph pushforward has first marginal `mu` and second marginal `nu` as soon
as `T` is measurable and pushes `mu` to `nu`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMap.lean:36 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap.ae_eq_of_graphCoupling_eq
- Equality of two measurable graph couplings determines the underlying maps
almost everywhere with respect to their common first marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMap.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap.IsQuadraticOptimalMap
- A quadratic-optimal transport map is a measurable map whose pushforward is
the prescribed target and whose induced graph coupling attains the quadratic
Kantorovich optimum. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMap.lean:80 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap.isQuadraticOptimalMap_iff
- Expansion of the optimal-map interface. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMap.lean:88 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap.isQuadraticOptimalMap_of_eq_graphCoupling
- Package a measurable pushforward map as optimal once its graph coupling is
identified with an already optimal coupling. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMap.lean:99 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap.isQuadraticOptimalCoupling_graphCoupling
- Forgetting the map packaging recovers the optimality of its graph coupling. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMap |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMap.lean:112 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMapUniqueness.HasUniqueQuadraticOptimalCoupling
- A uniqueness principle for quadratic-optimal couplings with fixed
marginals. This interface is intentionally proposition-level: any later
Brenier uniqueness theorem can discharge it without the map layer depending on
h |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMapUniqueness |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMapUniqueness.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMapUniqueness.ae_eq_of_quadraticOptimalMap_of_uniqueCoupling
- If the quadratic-optimal coupling between `mu` and `nu` is unique, then any
two quadratic-optimal transport maps are equal `mu`-almost everywhere. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMapUniqueness |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMapUniqueness.lean:58 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMapUniqueness.ae_eq_of_quadraticOptimalMap_of_forall_optimal_eq
- The same bridge with the coupling-uniqueness hypothesis written directly,
useful when a consumer already has a theorem rather than the named predicate. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMapUniqueness |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMapUniqueness.lean:69 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint.midpointMeasure
- Arithmetic midpoint of two measures. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMidpoint.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint.inv_two_add_inv_two
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMidpoint.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint.isCoupling_midpoint
- Couplings with common marginals are closed under the arithmetic midpoint. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMidpoint.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint.lintegral_midpoint
- The lower integral of any nonnegative function over a midpoint measure is
the midpoint of the two lower integrals. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMidpoint.lean:55 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint.isQuadraticOptimalCoupling_midpoint
- The midpoint of two quadratic-optimal couplings with identical marginals
is again a quadratic-optimal coupling. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalMidpoint |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalMidpoint.lean:65 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalRealMinimality.integral_norm_sq_le_of_quadraticOptimal
- A quadratic-optimal coupling minimizes the ordinary real
squared-displacement integral among all couplings with the same marginals,
whenever the compared real costs are integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalRealMinimality |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalRealMinimality.lean:30 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalSupportCyclic.pairingDistinctCycleMonotone_support_of_quadraticOptimal_finite
- Finite-measure core of the direct Brenier perturbation: an optimal
quadratic coupling between two finite-second-moment marginals has support
satisfying every distinct finite pairing-cycle inequality. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalSupportCyclic |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalSupportCyclic.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalSupportCyclic.pairingDistinctCycleMonotone_support_of_quadraticOptimal
- Ordinary-measure wrapper. A probability first marginal forces an optimal
coupling to be a probability measure and hence a finite measure, after which
the finite-measure core applies. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalSupportCyclic |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalSupportCyclic.lean:162 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalSupportCyclic.pairingDistinctCycleMonotone_support_of_quadraticOptimal_p2ac
- Source-facing `P₂,ac` specialization. Absolute continuity is not used by
the support-cyclical-monotonicity perturbation itself, but this is the endpoint
shape consumed by the later Brenier/Rockafellar construction in |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalSupportCyclic |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalSupportCyclic.lean:180 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalUniqueness.eq_of_quadraticOptimal
- Two quadratic-optimal couplings with the same marginals are equal when the
first marginal is absolutely continuous and both marginals have finite second
moments. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalUniqueness |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalUniqueness.lean:48 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalUniqueness.eq_of_quadraticOptimal_p2ac_source
- `P₂,ac` wrapper on the source law. The target is deliberately not assumed
absolutely continuous; only its finite second moment enters the uniqueness
proof. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuadraticOptimalUniqueness |
AutoSamplingTheory/TechnicalLemmas/Measure/QuadraticOptimalUniqueness.lean:98 |
AutoSamplingTheory.TechnicalLemmas.Measure.QuantitativeSupportLocalBlocks.exists_quantitative_positive_local_blocks_of_cycleValue_pos
- A strict positive cycle through distinct support points yields one bounded
rectangle family carrying simultaneously all topology and measure data needed
by the later common-mass perturbation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.QuantitativeSupportLocalBlocks |
AutoSamplingTheory/TechnicalLemmas/Measure/QuantitativeSupportLocalBlocks.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.lintegral_fin_nat_prod_eq_prod
- ENNReal Fubini for products of per-coordinate functions over `Fin n`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.lintegral_fintype_prod_eq_prod
- ENNReal Fubini for products of per-coordinate functions over a finite type. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:76 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.pi_withDensity_prod
- A finite product measure tilted by a product density decomposes coordinatewise. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:92 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.withDensity_univ_eq_lintegral
- The total mass of a `withDensity` measure is the lintegral of the density. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:125 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.isProbabilityMeasure_withDensity_of_lintegral_eq_one
- A density with lintegral one defines a probability measure after
`withDensity`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:131 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.isProbabilityMeasure_withDensity_ofReal_exp_of_integral_eq_one
- A real exponential tilt with Bochner integral one defines a probability
measure through `withDensity`.
This is the small ASTIS-owned version of the exponential-tilt normalization
pattern used in entropy-duality and Gi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:142 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.isFiniteMeasure_withDensity_of_lintegral_ne_top
- A density with finite lintegral defines a finite measure after
`withDensity`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:156 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.lintegral_inv_lintegral_mul_eq_one
- Normalizing a finite nonzero density by the reciprocal of its lintegral
gives lintegral one. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:164 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.isProbabilityMeasure_withDensity_normalized_lintegral
- A finite nonzero density defines a probability measure after reciprocal
lintegral normalization and `withDensity`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:174 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.withDensity_absolutelyContinuous_base
- `withDensity` is always absolutely continuous with respect to its base
measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:186 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.measurableEquiv_map_withDensity
- Transport an explicit `withDensity` measure through a measurable
equivalence. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:193 |
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.withDensity_rnDeriv_eq_of_absolutelyContinuous
- Radon--Nikodym reconstruction of an absolutely continuous measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym |
AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:209 |
AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitor.replacementCompetitor
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitor |
AutoSamplingTheory/TechnicalLemmas/Measure/ReplacementCompetitor.lean:26 |
AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitor.replacementCompetitor_map_fst
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitor |
AutoSamplingTheory/TechnicalLemmas/Measure/ReplacementCompetitor.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitor.replacementCompetitor_map_snd
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitor |
AutoSamplingTheory/TechnicalLemmas/Measure/ReplacementCompetitor.lean:48 |
AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitor.replacementCompetitor_preserves_marginals
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitor |
AutoSamplingTheory/TechnicalLemmas/Measure/ReplacementCompetitor.lean:65 |
AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitorQuadraticCost.integrable_ambient_of_remainder_removed
- Integrability of the unchanged remainder and removed block implies
integrability of the reconstructed ambient measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitorQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/ReplacementCompetitorQuadraticCost.lean:30 |
AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitorQuadraticCost.integrable_replacementCompetitor
- Integrability of the unchanged remainder and replacement block implies
integrability of the global replacement competitor. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitorQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/ReplacementCompetitorQuadraticCost.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitorQuadraticCost.integral_replacementCompetitor_lt_ambient
- A strict cost improvement on the removed/replacement part remains strict
after the same remainder is added to both sides. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.ReplacementCompetitorQuadraticCost |
AutoSamplingTheory/TechnicalLemmas/Measure/ReplacementCompetitorQuadraticCost.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Measure.StrictCycleCheaperLocalReplacement.exists_localBlocks_cyclicReplacement_lt_of_cycleValue_pos
- A strict positive cycle through distinct support points yields positive
local blocks whose canonical inverse-successor marginal replacement is
strictly cheaper than the common-mass slice sum removed from those blocks. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.StrictCycleCheaperLocalReplacement |
AutoSamplingTheory/TechnicalLemmas/Measure/StrictCycleCheaperLocalReplacement.lean:53 |
AutoSamplingTheory.TechnicalLemmas.Measure.SupportLocalBlocks.restrict_mass_pos_of_mem_support_of_isOpen
- A support point gives a positive-mass finite-measure restriction to every
open neighborhood containing it. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.SupportLocalBlocks |
AutoSamplingTheory/TechnicalLemmas/Measure/SupportLocalBlocks.lean:34 |
AutoSamplingTheory.TechnicalLemmas.Measure.SupportLocalBlocks.sum_restrict_le_of_pairwiseDisjoint_open
- A finite family of pairwise-disjoint open restrictions has total measure
bounded by the ambient finite measure. The statement is made on the ordered
underlying `Measure` type. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.SupportLocalBlocks |
AutoSamplingTheory/TechnicalLemmas/Measure/SupportLocalBlocks.lean:52 |
AutoSamplingTheory.TechnicalLemmas.Measure.SupportLocalBlocks.rectangle_restrict_mass_pos
- Restriction of a joint finite measure to an open rectangle around a support
point has positive mass. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.SupportLocalBlocks |
AutoSamplingTheory/TechnicalLemmas/Measure/SupportLocalBlocks.lean:77 |
AutoSamplingTheory.TechnicalLemmas.Measure.SupportLocalBlocks.exists_positive_local_blocks_of_cycleValue_pos
- The topology output of a strict cycle violation, together with membership
of the cycle points in the support of `rho`, yields positive pairwise-disjoint
local finite-measure blocks whose sum is dominated by `rho`.
Thi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.SupportLocalBlocks |
AutoSamplingTheory/TechnicalLemmas/Measure/SupportLocalBlocks.lean:96 |
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.IsCoupling
- A measure on a product space couples two marginals when its first and
second marginals are the specified measures. Probability normalization remains
visible through the marginal measures' typeclass assumptions at consu |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Transport |
AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:22 |
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.couplingSet
- The feasible set of couplings with prescribed marginals. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Transport |
AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:27 |
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.transportCost
- Chewi Definition 1.3.1 and display (1.3.2): the Kantorovich
transport cost for an extended nonnegative cost function.
Measurability and lower semicontinuity of `c` are not needed to state the
extended-real infimum. T |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Measure.Transport |
AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:36 |
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.transportCost_eq_sInf
- Source-facing expansion of the Kantorovich value in display (1.3.2). |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Measure.Transport |
AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:41 |
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.transportCost_le_lintegral_of_isCoupling
- Every feasible coupling gives an upper bound on the Kantorovich optimum.
This is the basic `sInf <= candidate` edge used repeatedly when a concrete
coupling is constructed (for example from a displacement interpolatio |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Transport |
AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:53 |
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.exists_isCoupling_lintegral_lt_of_transportCost_lt
- Strictly above the Kantorovich infimum, one can select an actual feasible
coupling whose cost is already below that threshold.
This is the reusable `sInf` near-optimal-selection edge. It makes no optimizer
existence |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Transport |
AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:66 |
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left
- A prescribed probability marginal forces the joint coupling measure to
have total mass one. This recovers the probability-measure interface required
by expectations and transport costs from the marginal contract. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Transport |
AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:81 |
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isCoupling_prod
- The independent product measure is a coupling of two probability measures.
This supplies the canonical nonemptiness witness for the Kantorovich feasible
set in Chewi, Definition 1.3.1. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Transport |
AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:92 |
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.couplingSet_nonempty
- The feasible set in the Kantorovich problem is nonempty for probability
marginals. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.Transport |
AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:99 |
AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.gluingMeasure
- The three-coordinate measure used in the gluing argument:
`γ₁₂₃(dx,dy,dz) = γ₁₂(dx,dy) γ₂₃(dz | y)`.
The product is encoded as `((x,y),z)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing |
AutoSamplingTheory/TechnicalLemmas/Measure/TransportGluing.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.fst_gluingMeasure
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing |
AutoSamplingTheory/TechnicalLemmas/Measure/TransportGluing.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.map_snd_fst_gluingMeasure
- If the second marginal of the first plan is the first marginal of the
second plan, then the `(y,z)` marginal of the glued measure is exactly the
second plan:
`(π₂₃)♯ γ₁₂₃ = γ₂₃`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing |
AutoSamplingTheory/TechnicalLemmas/Measure/TransportGluing.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing.exists_gluing_of_isCoupling
- Source-shaped gluing statement for two couplings with a shared middle
marginal. It packages the two pair-marginal identities without assuming
optimality of either plan. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing |
AutoSamplingTheory/TechnicalLemmas/Measure/TransportGluing.lean:81 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment.integrable_norm_sq_fst_prod
- A finite second moment in the first coordinate remains integrable under the
independent product law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinFiniteSecondMoment.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment.integrable_norm_sq_snd_prod
- A finite second moment in the second coordinate remains integrable under
the independent product law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinFiniteSecondMoment.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment.integrable_norm_sub_sq_prod
- The squared displacement of two independent finite-second-moment samples is
integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinFiniteSecondMoment.lean:47 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment.lintegral_quadraticCost_prod_lt_top
- The quadratic cost of the independent product coupling is finite. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinFiniteSecondMoment.lean:68 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment.wassersteinDistance_lt_top_of_integrable_norm_sq
- Two probability laws with finite second moments have finite `W₂`
distance. Absolute continuity is not needed for this fact. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinFiniteSecondMoment.lean:83 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment.wassersteinDistance_lt_top_of_p2ac
- In particular, the `P₂,ac` predicate used by the source has finite
Wasserstein distance between any two of its elements. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinFiniteSecondMoment |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinFiniteSecondMoment.lean:101 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.quadraticCost
- Squared Euclidean transport cost as an extended nonnegative function. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:23 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance
- Chewi Definition 1.3.4: the 2-Wasserstein distance is the positive
square root of the quadratic Kantorovich transport cost. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_sq
- Chewi display (1.3.5): the square of `W₂` is the infimum of the
quadratic costs over all couplings. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:36 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_sq_le_lintegral_of_isCoupling
- Every concrete coupling bounds the squared Wasserstein distance from
above by its quadratic transport cost.
This is the source-facing bridge used before proving optimal-plan existence or
constant-speed displacement ge |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.wassersteinDistance_le_sqrt_lintegral_of_isCoupling
- Every concrete coupling also bounds the Wasserstein distance directly by
the square root of its quadratic cost. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:61 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.exists_isCoupling_sqrt_lintegral_lt_of_wassersteinDistance_lt
- Strictly above the Wasserstein distance, one can choose an actual coupling
whose quadratic `L²` cost has square root below the same threshold.
This is the distance-level form of `Transport`'s strict `sInf` selection
l |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:78 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.IsAbsolutelyContinuousFiniteSecondMoment
- Chewi Definition 1.3.12: a probability measure in `P₂,ac` has finite
second moment and is absolutely continuous with respect to Lebesgue volume.
The generic finite-dimensional real inner-product space specializes to
E |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:113 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.isAbsolutelyContinuousFiniteSecondMoment_iff
- Expansion of the three conditions in the `P₂,ac` definition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:122 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry.swapPair
- Coordinate swap on a pair. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSymmetry.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry.measurable_swapPair
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSymmetry.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry.isCoupling_map_swapPair
- Swapping a coupling exchanges its two marginals. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSymmetry.lean:36 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry.lintegral_quadraticCost_map_swapPair
- Quadratic transport cost is invariant under coordinate swap. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSymmetry.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry.wassersteinDistance_le_reverse
- One half of Wasserstein symmetry, obtained from swapped near-optimal
couplings. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSymmetry.lean:64 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry.wassersteinDistance_comm
- The quadratic Wasserstein distance is symmetric. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSymmetry.lean:93 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt
- Strict-threshold form of the Wasserstein triangle argument.
If `r₁₂` and `r₂₃` lie strictly above the two adjacent Wasserstein distances,
then the endpoint distance lies strictly below their sum. No optimal coupling
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangle.lean:42 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.edgeLength12
- First edge length on a triple encoded as `((x,y),z)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:28 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.edgeLength23
- Second edge length on a triple encoded as `((x,y),z)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.edgeLength13
- Endpoint edge length on a triple encoded as `((x,y),z)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.edgeLength12_measurable
- The extended-nonnegative first edge length is measurable on a Borel normed
space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:44 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.edgeLength23_measurable
- Measurability of the middle-to-last edge. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:56 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.edgeLength13_measurable
- Measurability of the endpoint edge. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:68 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.edgeLength13_le_add
- Pointwise triangle inequality for the three edge lengths. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:80 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm
- The ENNReal `L2` seminorm used by the transport proof. The exponent is a
real `rpow`, matching Mathlib's Minkowski theorem. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:94 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_mono
- Monotonicity of the ENNReal `L2` seminorm. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:100 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2Seminorm_add_le
- Minkowski's inequality in the exact `p=2` form used below. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:109 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore.l2_edge_triangle
- The `L2` endpoint displacement of any triple joint law is bounded by the
sum of its two adjacent `L2` displacements. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleCore.lean:120 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleExact.wassersteinDistance_le_add_of_lt_left
- Close the second strict threshold while keeping the first one fixed. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleExact |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleExact.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleExact.wassersteinDistance_triangle
- Chewi's `W₂` triangle inequality, obtained by closing the remaining strict
threshold after the transport/gluing/Minkowski join. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleExact |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleExact.lean:56 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.pair12
- The `(x,y)` projection from triples encoded as `((x,y),z)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.pair23
- The `(y,z)` projection from triples encoded as `((x,y),z)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:40 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.pair13
- The `(x,z)` projection from triples encoded as `((x,y),z)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:43 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_pair12
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_pair23
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_pair13
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:54 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.measurable_quadraticCost
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:58 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.edgeLength12_rpow_two_eq_quadraticCost_pair12
- Squaring the first ENNReal edge length gives the quadratic cost of the
`(x,y)` pair. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:65 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.edgeLength23_rpow_two_eq_quadraticCost_pair23
- Squaring the second edge length gives the quadratic cost of `(y,z)`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:74 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.edgeLength13_rpow_two_eq_quadraticCost_pair13
- Squaring the endpoint edge length gives the quadratic cost of `(x,z)`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:83 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge12_eq_pairCost
- The first-edge `L²` seminorm is the square root of the quadratic cost of
its pair marginal. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:93 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge23_eq_pairCost
- The second-edge `L²` seminorm is the square root of the `(y,z)` pair cost. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:106 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.l2Seminorm_edge13_eq_pairCost
- The endpoint-edge `L²` seminorm is the square root of the `(x,z)` pair
marginal cost. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:120 |
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals.isCoupling_map_pair13_of_pair12_pair23
- If the `(x,y)` and `(y,z)` pair marginals of a triple law are couplings of
`μ₁,μ₂` and `μ₂,μ₃`, then its `(x,z)` pair marginal couples `μ₁,μ₃`.
This is the marginal bookkeeping edge needed after transport-plan gluing. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals |
AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangleMarginals.lean:136 |
AutoSamplingTheory.TechnicalLemmas.Measure.integrable_of_measure_eq
- Integrability is invariant under replacing the ambient measure by an equal measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Measure |
AutoSamplingTheory/TechnicalLemmas/Measure.lean:80 |
AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalKernel.condDistribIntegralNamedFieldIntegral
- Integral identity for a named conditional-integral field.
If `field` is the chosen `hatRho`-a.e. version of the canonical
`condDistrib` integral, then integrating `field` against the named law equals
the original join |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalKernel |
AutoSamplingTheory/TechnicalLemmas/Probability/ConditionalKernel.lean:30 |
AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalResampling.fst_compProd_condDistrib_snd_eq_self
- A joint finite law is recovered by combining its first marginal with the
conditional distribution of the second coordinate given the first.
This is the source-neutral law identity behind one-block Gibbs/heat-bath
resa |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalResampling |
AutoSamplingTheory/TechnicalLemmas/Probability/ConditionalResampling.lean:32 |
AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath.heatBath
- Resample coordinate `i` conditionally on all remaining coordinates.
Only the selected coordinate must be nonempty and Standard Borel. The target
may be any finite measure, including zero; its conditional version is spe |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/CoordinateHeatBath.lean:25 |
AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath.heatBath_isMarkovKernel
- The coordinate update has total mass one at every input. This does not
assert conditional support on marginal-null fibers. |
instance | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/CoordinateHeatBath.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath.heatBath_invariant
- Exact invariance by literal reuse of one-block heat-bath and measurable
kernel transport. No convergence from other initial laws is inferred. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/CoordinateHeatBath.lean:42 |
AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath.heatBath_ae_apply_eq
- At each input `x`, every coordinate other than the selected one is retained
almost surely. Singleton measurability is needed only at the retained coordinate
`j`: equality of a marginal law with a Dirac measure on a coa |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/CoordinateHeatBath.lean:53 |
AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBath.heatBath_eq_cond
- On a positive retained-coordinate fiber, the actual coordinate heat-bath
kernel equals the normalized restriction of the finite target to that fiber.
Only the selected coordinate is required to be nonempty and Standard |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.CoordinateHeatBathConditional |
AutoSamplingTheory/TechnicalLemmas/Probability/CoordinateHeatBathConditional.lean:21 |
AutoSamplingTheory.TechnicalLemmas.Probability.FiniteProductPairMarginal.map_pair_eval_eq_prod
- Distinct coordinate projections of a finite product probability measure
have joint law equal to the product of their coordinate laws. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.FiniteProductPairMarginal |
AutoSamplingTheory/TechnicalLemmas/Probability/FiniteProductPairMarginal.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Probability.FiniteProductPairMarginal.integral_comp_pair_eval
- Integral form: an integrand depending on two distinct coordinates can be
integrated against the corresponding two-coordinate product law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.FiniteProductPairMarginal |
AutoSamplingTheory/TechnicalLemmas/Probability/FiniteProductPairMarginal.lean:49 |
AutoSamplingTheory.TechnicalLemmas.Probability.FiniteProductSupport.pi_box_apply_eq_one
- Coordinate probability-one sets form a probability-one box. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.FiniteProductSupport |
AutoSamplingTheory/TechnicalLemmas/Probability/FiniteProductSupport.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Probability.FiniteProductSupport.ae_mem_pi_box
- Under coordinate measurability, the product tuple belongs to the
probability-one box almost surely. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.FiniteProductSupport |
AutoSamplingTheory/TechnicalLemmas/Probability/FiniteProductSupport.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Probability.GaussianConditionalKernel.exists_tilted_isCondKernel
- The normalized quadratic tilt is a measurable Markov kernel and a backward
conditional law of the actual Gaussian augmentation, for every positive noise
variance. The input law may be singular and need not have any fin |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.GaussianConditionalKernel |
AutoSamplingTheory/TechnicalLemmas/Probability/GaussianConditionalKernel.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath.heatBathSnd
- Keep the first coordinate and draw the second from the selected regular
conditional distribution under the finite measure `μ`.
The chosen conditional version is characterized only almost everywhere for
`μ.map Prod.fst |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/HeatBath.lean:29 |
AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath.heatBathSnd_isMarkovKernel
- The selected conditional distribution gives a Markov update at every state,
including states over first-marginal null fibers. This is a mass-one assertion,
not a conditional-support assertion on such fibers. |
instance | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/HeatBath.lean:37 |
AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath.heatBathSnd_apply
- Pointwise product-law form of the update, using the selected conditional
version. The Dirac factor retains the first coordinate. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/HeatBath.lean:44 |
AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath.heatBathSnd_invariant
- A second-coordinate heat-bath update leaves its finite joint target invariant.
The zero measure is allowed. This theorem does not imply irreducibility,
reversibility, convergence from another initial law, or any mixin |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/HeatBath.lean:54 |
AutoSamplingTheory.TechnicalLemmas.Probability.KernelInvariance.invariant_pow
- If `μ` is invariant for the one-step kernel `κ`, then it is invariant for
its `n`-step kernel `κ ^ n` for every `n : ℕ`.
This is an invariance statement only. It does not assert irreducibility,
aperiodicity, converge |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.KernelInvariance |
AutoSamplingTheory/TechnicalLemmas/Probability/KernelInvariance.lean:31 |
AutoSamplingTheory.TechnicalLemmas.Probability.KernelInvariance.bind_pow_eq
- Measure-level form of `invariant_pow`: starting an invariant law and taking
`n` transitions leaves the law unchanged. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.KernelInvariance |
AutoSamplingTheory/TechnicalLemmas/Probability/KernelInvariance.lean:46 |
AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture
- A finite fixed-weight mixture, formed through constant-density kernels.
Normalization is required by the correctness theorems, not the definition. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture |
AutoSamplingTheory/TechnicalLemmas/Probability/KernelMixture.lean:27 |
AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture_apply
- Pointwise measure law of the finite mixture. Constant uncurry densities are
measurable, so the totalized `withDensity` zero fallback is never used. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture |
AutoSamplingTheory/TechnicalLemmas/Probability/KernelMixture.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture_isMarkovKernel
- Fixed normalized nonnegative weights mix Markov kernels into a Markov kernel.
Zero weights are allowed. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture |
AutoSamplingTheory/TechnicalLemmas/Probability/KernelMixture.lean:44 |
AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture.finiteMixture_invariant
- A fixed normalized finite mixture preserves any common invariant measure.
There is no finite or s-finite assumption on `μ`. The component s-finiteness is
only the `withDensity` construction contract; intended Markov c |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.KernelMixture |
AutoSamplingTheory/TechnicalLemmas/Probability/KernelMixture.lean:60 |
AutoSamplingTheory.TechnicalLemmas.Probability.KernelReversibility.isReversible_of_singleton_balance
- Atomic detailed balance implies equality of the two set flux integrals.
The given measure and kernel need not have finite total mass. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.KernelReversibility |
AutoSamplingTheory/TechnicalLemmas/Probability/KernelReversibility.lean:19 |
AutoSamplingTheory.TechnicalLemmas.Probability.KernelTotalVariation.abs_real_comp_sub_le
- An eventwise probability discrepancy bound is preserved by a common Markov
kernel. The hypothesis on the empty event already implies `0 ≤ δ`. All helper
functions and integrability proofs are local; the conclusion conc |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.KernelTotalVariation |
AutoSamplingTheory/TechnicalLemmas/Probability/KernelTotalVariation.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Probability.KernelTransport.invariant_map_comap
- Conjugating a kernel by a measurable equivalence preserves invariance of
the corresponding pushforward measure. The input is pulled back with `e.symm`
and the output is pushed forward with `e`.
This is exact invarianc |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.KernelTransport |
AutoSamplingTheory/TechnicalLemmas/Probability/KernelTransport.lean:21 |
AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasure.compl_null_of_toMeasure_le_restrict
- Domination by a restriction forces the dominated measure to give zero mass
to the complement. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasure |
AutoSamplingTheory/TechnicalLemmas/Probability/NormalizedFiniteMeasure.lean:27 |
AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasure.normalize_apply_eq_one_of_toMeasure_le_restrict
- A positive finite measure dominated by an ambient restriction gives
probability one to that restriction set after normalization. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasure |
AutoSamplingTheory/TechnicalLemmas/Probability/NormalizedFiniteMeasure.lean:39 |
AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasure.ae_mem_of_toMeasure_le_restrict
- Almost-sure form consumed directly by product-probability arguments. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasure |
AutoSamplingTheory/TechnicalLemmas/Probability/NormalizedFiniteMeasure.lean:54 |
AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasureIntegral.integral_eq_mass_mul_integral_normalize
- Integrating a real observable against a finite measure is its total mass
times the expectation under the normalized probability measure. The Bochner
integral is totalized, so no separate integrability hypothesis is nee |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasureIntegral |
AutoSamplingTheory/TechnicalLemmas/Probability/NormalizedFiniteMeasureIntegral.lean:33 |
AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasureIntegral.integral_lt_integral_iff_normalize_of_mass_pos
- Positive-mass specialization, packaged with the scalar positivity needed
to transport strict inequalities between normalized expectations and finite
measure integrals. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.NormalizedFiniteMeasureIntegral |
AutoSamplingTheory/TechnicalLemmas/Probability/NormalizedFiniteMeasureIntegral.lean:50 |
AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan
- One uniform random-site update of a finite Boolean probability target.
The conditional versions at null fibers are inherited from `heatBath`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/RandomScanHeatBath.lean:21 |
AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_isMarkovKernel
- The uniform weights and the existing Markov components give mass one at
every input, independently of the positive-start condition in the law below. |
instance | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/RandomScanHeatBath.lean:28 |
AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_apply_singleton
- Exact probability of a singleton after one uniformly selected coordinate
update. Every positive fiber is derived from the positive starting atom.
The retained-coordinate test is literal equality at every unselected sit |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath |
AutoSamplingTheory/TechnicalLemmas/Probability/RandomScanHeatBath.lean:38 |
AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.singleton_balance
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBathReversibility |
AutoSamplingTheory/TechnicalLemmas/Probability/RandomScanHeatBathReversibility.lean:20 |
AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_isReversible
- The actual uniform random-site heat-bath kernel is reversible for every
Boolean probability target, without a full-support or starting-atom premise.
Conditional versions at target-null inputs are immaterial to the flux |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBathReversibility |
AutoSamplingTheory/TechnicalLemmas/Probability/RandomScanHeatBathReversibility.lean:54 |
AutoSamplingTheory.TechnicalLemmas.Probability.UniformExpectationGap.integral_lt_integral_of_ae_add_le
- If `f` dominates `g` by one uniform positive margin almost everywhere,
then the expectation of `f` is strictly larger than that of `g`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.UniformExpectationGap |
AutoSamplingTheory/TechnicalLemmas/Probability/UniformExpectationGap.lean:28 |
AutoSamplingTheory.TechnicalLemmas.Probability.UniformExpectationGap.integral_lt_integral_of_ae_gap
- Difference form, convenient when a pointwise cost identity is already
available. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Probability.UniformExpectationGap |
AutoSamplingTheory/TechnicalLemmas/Probability/UniformExpectationGap.lean:46 |
AutoSamplingTheory.TechnicalLemmas.LemmaMemoryStatus
|
inductive | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:45 |
AutoSamplingTheory.TechnicalLemmas.LemmaMemoryEntry
- Metadata for a lemma-memory entry. The executable proof is the declaration
named in `localDecl`; this structure is only the retrieval record used by
agents and documentation exports. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:55 |
AutoSamplingTheory.TechnicalLemmas.sltSourceAnchor
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:66 |
AutoSamplingTheory.TechnicalLemmas.analysisMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:74 |
AutoSamplingTheory.TechnicalLemmas.gaussianMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:727 |
AutoSamplingTheory.TechnicalLemmas.taylorMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:880 |
AutoSamplingTheory.TechnicalLemmas.calculusMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:913 |
AutoSamplingTheory.TechnicalLemmas.measureMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:1956 |
AutoSamplingTheory.TechnicalLemmas.functionalInequalityMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:2451 |
AutoSamplingTheory.TechnicalLemmas.stochasticProcessMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:2584 |
AutoSamplingTheory.TechnicalLemmas.klDensityMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:3827 |
AutoSamplingTheory.TechnicalLemmas.renyiDensityMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:3850 |
AutoSamplingTheory.TechnicalLemmas.variationalMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:3893 |
AutoSamplingTheory.TechnicalLemmas.geometryMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:4006 |
AutoSamplingTheory.TechnicalLemmas.saldExtractedMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:4419 |
AutoSamplingTheory.TechnicalLemmas.portQueueMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:4492 |
AutoSamplingTheory.TechnicalLemmas.technicalLemmaMemory
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:4515 |
AutoSamplingTheory.TechnicalLemmas.formalizedTechnicalLemmaCount
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Registry |
AutoSamplingTheory/TechnicalLemmas/Registry.lean:4520 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy.accumulatedEnergy
- Squared energy accumulated strictly before `min t T`. The strict endpoint
choice differs from the closed-interval convention only on a time-null
singleton and makes monotonicity pointwise. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/AccumulatedEnergy.lean:26 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy.accumulatedEnergy_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/AccumulatedEnergy.lean:32 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy.accumulatedEnergy_mono
- Accumulated energy is monotone in the observation time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/AccumulatedEnergy.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy.accumulatedEnergy_eq_terminal_of_le
- Once the observation time is beyond the terminal horizon, the accumulated
energy no longer changes. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/AccumulatedEnergy.lean:56 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy.accumulatedEnergy_le_terminal
- Energy at any time before the horizon is bounded by terminal energy. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/AccumulatedEnergy.lean:64 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy.accumulatedEnergy_nonneg
- Accumulated energy is nonnegative (recorded as an explicit reusable leaf
for order-theoretic stopping-time arguments). |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/AccumulatedEnergy.lean:73 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.projectedIncrementVariance
- The variance of a standard Brownian increment after applying a continuous
linear functional `ell`: `(t-s) * ||ell||^2`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:32 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsStandardBrownianMotion
- Chewi Definition 1.1.1: a standard Brownian motion in a finite-dimensional
real Hilbert space.
Independent increments are stated for every finite family of pairwise
disjoint half-open time intervals. The Gaussian law |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:43 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration
- A real Brownian motion relative to a specified filtration.
The last field is the condition needed for stochastic integration: the
increment after `s` is independent of the whole past sigma-algebra `F_s`.
Bare independ |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:64 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.isProbabilityMeasure
- A Brownian-filtration contract carries a probability measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.increment_stronglyMeasurable
- Every Brownian increment is strongly measurable in the ambient
sigma-algebra. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:86 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.indepFun_increment_of_stronglyMeasurable
- Any real random variable measurable at time `s` is independent of a
future Brownian increment. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:94 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.integral_increment_eq_zero
- A Brownian increment has mean zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:104 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.condExp_increment_eq_zero
- The conditional mean of a future Brownian increment given the past is
zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:114 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.integral_increment_sq
- The second moment of a Brownian increment is its elapsed time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:130 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsBrownianMotionWithFiltration.condExp_increment_sq
- The conditional second moment of a future Brownian increment is its
elapsed time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:161 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.increment_memLp_two
- A Brownian increment is square-integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:31 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.increment_sq_integrable
- The square of a Brownian increment is integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:40 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.centered_increment_sq_integrable
- The compensated square of one future Brownian increment is integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.integral_centered_increment_sq_eq_zero
- The basic compensated-increment identity behind Brownian quadratic
variation: `E[(B_t-B_s)^2-(t-s)] = 0`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:63 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.centeredSquaredIncrement
- One compensated cell of a deterministic finite time grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:74 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.quadraticVariationError
- Finite-grid error in the Brownian quadratic-variation rule. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:81 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.quadraticVariationSum
- The uncompensated finite-grid quadratic-variation sum. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:87 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.grid_cell_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:93 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.centeredSquaredIncrement_integrable
- Every compensated grid cell is integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:100 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.integral_quadraticVariationError_eq_zero
- Every deterministic finite partition has zero-mean compensated quadratic
variation error. This is the finite-sum identity that precedes the mesh-limit
argument in Chewi's quadratic-variation calculation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:114 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation.integral_quadraticVariationSum_eq_sum_cellLengths
- Expected quadratic variation on a finite deterministic grid is exactly the
sum of the cell lengths. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianQuadraticVariation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianQuadraticVariation.lean:140 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.energyLevelSet
- Times in `[0,T]` at which completed energy equals a prescribed level. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:30 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.isClosed_energyLevelSet
- An energy level set is closed. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalEnergyLocalizer
- First equality-level time, or `T` if the level is not reached. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:50 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalEnergyLocalizer_mem
- If the level set is nonempty, the canonical localizer belongs to it. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:60 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalEnergyLocalizer_le_of_mem
- The canonical localizer is no later than any member of its level set. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:73 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalEnergyLocalizer_le_terminal
- Every canonical localizer is capped by the terminal horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:85 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.exists_level_time_of_le
- If completed energy at time `t` dominates a nonnegative level, continuity
produces an equality-level time no later than `t`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:97 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalEnergyLocalizer_le_iff
- Fixed-time characterization of the canonical localizer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:117 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.measurableSet_canonicalEnergyLocalizer_le
- Canonical-localizer events are measurable at the observation time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:152 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.completedEnergy_at_canonical_eq_of_nonempty
- On a reached level, completed energy at the canonical localizer equals the
level exactly. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:178 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.completedEnergy_at_canonical_le
- Whether or not the level is reached before `T`, stopped completed energy is
bounded by the requested nonnegative level. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:189 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalEnergyLocalizer_mono_level
- Higher energy levels are reached no earlier than lower levels. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:210 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalLocalizingTime
- Chewi's canonical localizer uses the positive integer level `n+1`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:234 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalLocalizingTime_mono
- Canonical localizing times increase with `n`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:241 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalLocalizingTime_eventually_eq_terminal
- Every path is eventually left unstopped: once the integer level exceeds
terminal energy, the localizer equals `T`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:252 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.tendsto_canonicalLocalizingTime
- Canonical localizing times converge pointwise to the terminal horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:280 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.completedEnergy_at_canonicalLocalizingTime_le
- The stopped completed energy at the `n`-th localizer is bounded by `n+1`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:291 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyStoppingTime.canonicalEnergyLocalizer_isChewiStoppingTime
- The canonical hitting time of any nonnegative energy level is a Chewi
stopping time. Keeping this theorem level-generic lets later nested-stopping
arguments use arbitrary `c ≤ d`, not only integer thresholds. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyStoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyStoppingTime.lean:24 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyStoppingTime.canonicalLocalizingTime_isChewiStoppingTime
- Each canonical integer energy localizer is a stopping time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyStoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyStoppingTime.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyStoppingTime.canonicalLocalizingTime_isStoppingTime
- The same statement at Mathlib's native stopping-time interface. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyStoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyStoppingTime.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalLocalizationTheorem.canonicalStoppedProgressiveL2
- The globally square-integrable stopped integrand at the `n`-th canonical
energy level. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalLocalizationTheorem |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalLocalizationTheorem.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalLocalizationTheorem.canonicalStoppedProgressiveL2_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalLocalizationTheorem |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalLocalizationTheorem.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalLocalizationTheorem.chewi_proposition_1_1_13
- Chewi, Proposition 1.1.13: the energy-level first-hitting times form a
canonical localizing sequence, and every stopped integrand is globally square
integrable with its exact pathwise energy bound. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalLocalizationTheorem |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalLocalizationTheorem.lean:48 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime
- Source-facing canonical time: use the completed-energy localizer off the
null bad-energy set and stop immediately on that exceptional set. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_of_bad
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_of_good
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_mono
- The raw localizers increase with the energy level. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.measurableSet_canonicalRawLocalizingTime_le
- Fixed-time occurrence events for the raw localizer are measurable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:73 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_isChewiStoppingTime
- Each raw canonical localizer is a stopping time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:99 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_isStoppingTime
- Mathlib-native stopping-time version. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:111 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_eventually_eq_terminal_of_good
- Outside the null bad-energy set, the raw localizer is eventually the
terminal horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:122 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_tendsto_terminal_ae
- The source-facing raw localizing times converge to `T` almost surely,
with the codomain exactly matching Definition 1.1.12. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:134 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.stoppedIntegrand_ae_eq_energyStoppedIntegrand
- For each sample path, the source-facing stopped raw integrand and the
completed energy-stopped representative agree almost everywhere in time. The
only possible pointwise discrepancy on a good path is the single hittin |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:159 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalizationL2.rawStoppedTimeLintegral_le
- On each sample path, the literal source stopped integrand has nonnegative
Lebesgue energy at most the canonical level `n+1`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalizationL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalizationL2.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalizationL2.rawStoppedProductEnergy_lt_top
- Integrating the pathwise `n+1` bound over a probability space gives the
finite expected stopped energy required by Definition 1.1.12. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalizationL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalizationL2.lean:71 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalizationL2.canonicalRaw_isLocalizingSequence
- Chewi Proposition 1.1.13 in the exact repository-native source contract:
the canonical raw times form a `Localization.IsLocalizingSequence` for the
original progressive locally square-integrable integrand. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalizationL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalizationL2.lean:101 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral.canonicalStoppedItoProcess
- The `n`-th globally square-integrable stopped Itô martingale. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalStoppedItoIntegral.lean:32 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral.canonicalStoppedItoProcess_stronglyAdapted
- The stopped Itô process is strongly adapted. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalStoppedItoIntegral.lean:42 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral.canonicalStoppedItoProcess_martingale
- The stopped Itô process is a genuine martingale. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalStoppedItoIntegral.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral.canonicalStoppedItoProcess_continuousOn
- The process has continuous paths on the construction horizon, including
on the completed exceptional set where it is patched by zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalStoppedItoIntegral.lean:67 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral.canonicalStoppedItoProcess_at_eq_terminal
- At every deterministic time, the stopped process represents the terminal
`L2` Itô integral of the correspondingly restricted stopped integrand. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalStoppedItoIntegral.lean:81 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral.chewi_display_1_1_14
- Chewi display (1.1.14): every canonical energy truncation is fed into the
already-constructed global Itô map and yields an adapted continuous
martingale, with the exact deterministic-time restriction compatibility. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalStoppedItoIntegral.lean:98 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.carreDuChamp
- Chewi Definition 1.2.12: the carre du champ of a linear generator. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.carreDuChamp_comm
- The carre du champ is symmetric in its observable arguments. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:42 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.iteratedCarreDuChamp
- Chewi Definition 1.2.28: the iterated carre du champ. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.iteratedCarreDuChamp_comm
- The iterated carre du champ inherits symmetry from the first one. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:63 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.SatisfiesBakryEmery
- Chewi Definition 1.2.29: the curvature-dimension condition
`CD(alpha, infinity)`, including the source requirement `alpha > 0`. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:74 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.carreDuChamp_nonneg_of_markov_jensen_rightGenerator
- Chewi Lemma 1.2.13: the Markov-semigroup Jensen inequality implies
nonnegativity of the carre du champ after taking the right-generator limit.
The theorem is pointwise. `hf` and `hf2` are the actual right difference- |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:87 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.fundamental_integration_by_parts
- Chewi Theorem 1.2.14: stationarity and generator symmetry imply the
fundamental integration-by-parts identity between the Dirichlet form and the
integrated carre du champ.
The three integrability hypotheses are the ex |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:153 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp.negativeGenerator_quadratic_nonneg
- Chewi Corollary 1.2.15: the negative reversible generator has a
nonnegative quadratic form once Gamma is pointwise nonnegative. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CarreDuChamp.lean:192 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17.stochasticIntegral
- The vector stochastic term in Chewi Definition 1.1.17.
Each state coordinate is the finite sum of scalar Itô integrals against the
coordinates of one and the same Euclidean Brownian motion. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiDefinition1_1_17.lean:41 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17.stochasticIntegral_coordinate
- The `i`-th coordinate of the vector stochastic integral is exactly the
finite coordinate sum used by the scalar construction. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiDefinition1_1_17.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17.drift_integral_coordinate
- A Bochner integral of the Euclidean drift evaluates coordinatewise.
This is the finite-dimensional bridge needed to turn the already-compiled
coordinate identities into the literal vector equation in the textbook. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiDefinition1_1_17.lean:76 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17.definition_1_1_17_vector_display
- Literal vector form of Chewi Definition 1.1.17.
For every deterministic time `t`, the source-facing process satisfies
`X_t = X_0 + ∫_0^t b_s ds + ∫_0^t σ_s dB_s`
almost surely. The last term is `stochasticIntegral` |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiDefinition1_1_17.lean:95 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17.chewi_definition_1_1_17
- Source-complete formalization of Chewi Definition 1.1.17.
The theorem records both facts stated around the definition in the textbook:
the vector process is progressive, and it satisfies the finite-dimensional Itô
int |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiDefinition1_1_17.lean:123 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDisplay1_1_18.chewi_display_1_1_18_integral_meaning
- Source-faithful meaning of Chewi display (1.1.18).
The notation `dX_t = b_t dt + sigma_t dB_t` means exactly that the source
finite-dimensional Itô process satisfies the vector integral equation from
Definition 1.1.17 |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDisplay1_1_18 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiDisplay1_1_18.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess.SourceData
- Source-shaped coefficient data for Chewi Definition 1.1.17.
`driftIntegrable` is the Bochner formulation of the textbook's condition
`∫ ‖b_s‖ ds < ∞`: for a progressive finite-dimensional process it records
strong mea |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcess.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess.SourceData.initial_coordinate_stronglyMeasurable
- Initial-value measurability descends from the Euclidean vector to each
coordinate by the norm-one continuous coordinate functional. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcess.lean:69 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess.SourceData.drift_coordinate_progressive
- Progressive measurability of the vector drift descends to each scalar
coordinate. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcess.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess.SourceData.drift_coordinate_integrable
- Local Bochner integrability of the vector drift implies local integrability
of every scalar coordinate. This is a continuous-linear-map consequence, not
an additional coordinatewise assumption. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcess.lean:93 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess.SourceData.diffusion_entry_progressive
- Progressive measurability of the flattened matrix process descends to each
matrix entry. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcess.lean:105 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess.SourceData.toCoordinateItoData
- Compile Chewi's vector/matrix source coefficients into the scalar
coordinate ABI used by the Chapter 1 Itô integral. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcess.lean:119 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess.process
- The source-facing Itô process associated with `SourceData` and one
Euclidean Brownian driver. Internally this is assembled coordinatewise from
the scalar global Itô integral, then repackaged as one Euclidean vector. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcess.lean:136 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess.definition_1_1_17_coordinate_display
- Chewi Definition 1.1.17, displayed at an arbitrary state coordinate.
The stochastic term is a finite sum over coordinates of the same
`R^N`-valued Brownian motion. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcess.lean:152 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess.ChewiSourceData
- Literal textbook dimensions: state space `R^d` and Brownian space `R^N`. |
abbrev | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcess.lean:170 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcessProgressive.process_stronglyProgressive
- The `R^d`-valued process constructed from Chewi Definition 1.1.17 source
data is strongly progressive. This is the missing process-level regularity
claim in the textbook definition, not merely a coordinate display. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiItoProcessProgressive |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiItoProcessProgressive.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiProposition1_1_16.chewi_proposition_1_1_16
- Chewi, Proposition 1.1.16.
Assume the filtration satisfies the usual conditions, `B` is Brownian motion
with respect to that filtration, and `eta` is strongly progressive with finite
pathwise square energy on ever |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiProposition1_1_16 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiProposition1_1_16.lean:51 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiProposition1_1_16.chewi_proposition_1_1_16_stopped_integral_representation
- Localized Itô representation for Proposition 1.1.16.
For every canonical dyadic localizer `tau_k`, stopping the globally glued local
Itô process at `tau_k` recovers, almost surely and at every deterministic time
i |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiProposition1_1_16 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiProposition1_1_16.lean:72 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiProposition1_1_16.chewi_proposition_1_1_16_localizers
- Source-facing localization certificate accompanying Proposition 1.1.16:
the cofinal dyadic energy localizers are the concrete witness used by the
local-martingale proof. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiProposition1_1_16 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiProposition1_1_16.lean:107 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.clip
- Projection of a real value onto `[-M, M]`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:23 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.clipNat
- Natural truncation levels used by the canonical coefficient sequence. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:26 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.continuous_clipNat
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:28 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.stronglyMeasurable_clipNat
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:31 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clip_le_abs
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clip_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clipNat_le_abs
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clipNat_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.clipNat_eventually_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:65 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.tendsto_clipNat
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:76 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.aestronglyMeasurable_clipNat
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.clipNat_memLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:89 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clipNat_sub_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:97 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.tendsto_clipNat_toLp
- Clipping converges to the original coefficient in `L2`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:105 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.badEnergySet
- Sample points whose squared integrand is not time-integrable. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:27 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.measure_badEnergySet_zero
- The bad path set is null by the source local-square-integrability
assumption. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:34 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.measurableSet_badEnergySet
- The bad set belongs to every time sigma-algebra under the usual
completeness condition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:43 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.completedEnergy
- Accumulated energy with all bad paths replaced by the zero path. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:50 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.completedEnergy_stronglyMeasurable
- Fixed-time completed energy is strongly measurable at the observation
time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:60 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.continuous_completedEnergy
- Every completed energy path is continuous. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:77 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.monotone_completedEnergy
- Every completed energy path is monotone. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:103 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.completedEnergy_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:120 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.completedEnergy_nonneg
- Completed energy is nonnegative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:131 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.completedEnergy_eq_terminal_of_le
- Completed energy stabilizes after the terminal horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:140 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.measurableSet_completedEnergy_ge
- Fixed-time threshold events for completed energy are measurable at the
observation time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:154 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand.completedIntegrand
- The original progressive integrand with every nonintegrable sample path
replaced by zero. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedIntegrand.lean:28 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand.completedIntegrand_stronglyProgressive
- Completion preserves strong progressiveness. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedIntegrand.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand.sectionSquare_integrable
- Every completed sample path has an integrable square on the finite time
horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedIntegrand.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand.completedEnergy_eq_prefixIntegral
- Prefix energy of the completed integrand is exactly the completed energy
process. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedIntegrand.lean:83 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand.completedEnergy_stronglyProgressive
- The completed energy process is strongly progressive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedIntegrand.lean:112 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.dyadicMaxEvent
- The path exceeds `a` on the finite level-`level` dyadic grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.dyadicMaxEventAll
- The path exceeds `a` on at least one finite dyadic grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.measurableSet_dyadicMaxEvent
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.measurableSet_dyadicMaxEventAll
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.monotone_dyadicMaxEvent
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:51 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.runningAbsMax_nonneg
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.pow_mul_measure_dyadicMaxEvent_le
- Chebyshev combined with finite-grid Doob, in a form stable under taking
the increasing union of dyadic grids. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.pow_mul_measure_dyadicMaxEventAll_le
- The same probability bound for exceedance on the union of all dyadic
observation grids. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:105 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.measure_dyadicMaxEventAll_le
- Divided form of the all-dyadic-grid estimate for a positive threshold. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:116 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.activeCellIndex
- The dyadic cell containing a positive time `t`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:140 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.activeCellIndex_spec
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:145 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.rightApproxTime
- Right endpoint of the dyadic cell containing `t`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:154 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.rightApproxTime_eq_grid
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:158 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.rightApproxTime_mem_Icc
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:165 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.rightApproxTime_le_add_mesh
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:183 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.tendsto_rightApproxTime
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:191 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.continuousOn_mem_dyadicMaxEventAll
- On a continuous path, exceeding a threshold anywhere on `[0,T]` is
detected on one of the finite dyadic observation grids. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:214 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.continuousExceedEvent
- Exceedance somewhere on the whole compact interval. It need not be
declared measurable: the following theorem controls its outer measure through
the source-derived countable dyadic event. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:262 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.doobL2_continuous
- Continuous-time Doob `L2` maximal inequality in threshold/outer-measure
form. The right side is the usual constant-four terminal second-moment
bound. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:268 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2.sampledFiltration
- Pull a filtration back along a monotone deterministic time map. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobL2.lean:21 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2.Martingale.sampled
- Deterministic monotone sampling preserves the martingale property. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobL2.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2.runningAbsMax
- Running absolute maximum through discrete time `N`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobL2.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2.doobL2_finite
- Doob's finite discrete `L2` inequality in its canonical `eLpNorm` form. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobL2.lean:43 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2.doobL2_finite_sq
- Squared form of the finite discrete Doob estimate. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobL2.lean:55 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2.doobL2_sampled
- Doob's `L2` inequality along any deterministic monotone observation grid. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobL2.lean:67 |
MeasureTheory.runMax
- Internal abbreviation for the running maximum
`max {f 0 ω, f 1 ω, …, f n ω}`. Kept `private`: the public theorem
states the bound in terms of the explicit `Finset.sup'` form to match
`MeasureTheory.maximal_ineq`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:67 |
MeasureTheory.runMax_nonneg
|
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:71 |
MeasureTheory.runMax_measurable
|
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:76 |
MeasureTheory.runMax_stronglyMeasurable
|
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:83 |
MeasureTheory.layer_meas_bound
- Maximum-inequality at a fixed positive level `t`. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:89 |
MeasureTheory.lintegral_runMax_rpow_eq_layer
- Layer-cake step. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:105 |
MeasureTheory.layer_integrand_bound
- Pointwise (in `t > 0`) integrand bound. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:118 |
MeasureTheory.A_le_layer_integral
- Combining steps: A ≤ ofReal p · ∫⁻ t in Ioi 0, ofReal(t^(p-2)) · ofReal(∫_{fstar ≥ t} f_n). |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:136 |
MeasureTheory.lintegral_rpow_Ioc
- Inner integral evaluation: `∫⁻ t in Ioc 0 f, ofReal(t^(p-2)) = ofReal(f^(p-1)/(p-1))`. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:156 |
MeasureTheory.ofReal_setIntegral_eq_setLIntegral_ofReal
- Convert `ofReal` of Bochner set integral to `setLIntegral` of `ofReal`. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:175 |
MeasureTheory.inner_t_integral
- Pointwise inner integral: for `fstar ≥ 0`, integrating `t^(p-2)` against
the indicator `𝟙{0 < t ≤ fstar}` evaluates to `fstar^(p-1)/(p-1)`. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:187 |
MeasureTheory.fubini_swap
- Fubini swap stage (Tier A.2 Stage 1).
For `p > 1`, a non-negative submartingale `f`, and a time `n`, the
iterated integral
`∫⁻ t in Ioi 0, ofReal(t^(p-2)) ⋅ ∫⁻_{fstar ≥ t} ofReal(f_n) dμ`
equals
|
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:246 |
MeasureTheory.holder_apply
- Stage 2a: apply Hölder to the post-Fubini integral.
For non-negative f, g and Hölder conjugates p, q (so 1/p + 1/q = 1):
`∫⁻ ω, ofReal(f_n) ⋅ ofReal(fstar^(p-1)) ≤ (∫⁻ f_n^p)^(1/p) ⋅ (∫⁻ fstar^p)^(1/q)`.
|
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:393 |
MeasureTheory.inner_t_integral_truncated
- Truncated inner t-integral: for `fstar ≥ 0` and `K > 0`,
`∫⁻ t in Ioi 0, t^(p-2) ⋅ 𝟙{0 < t ≤ K ∧ t ≤ fstar}
= ofReal(min fstar K^(p-1) / (p-1))`.
Identical to `inner_t_integral` but with an extra `t |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:448 |
MeasureTheory.fubini_swap_truncated
- Truncated Fubini swap. Analog of `fubini_swap` but with the outer
`t`-integral restricted to `Ioc 0 K`, producing
`min (runMax f n) K` in the post-swap formula. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:522 |
MeasureTheory.A_K_le_layer_integral
- Truncated layer-cake bound: for `Z_K = min (runMax f n) K`,
`∫⁻ Z_K^p ≤ ofReal(p) * ∫⁻ t in Ioc 0 K, ofReal(t^(p-2)) * ofReal(∫_{fstar ≥ t} f_n)`. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:658 |
MeasureTheory.holder_step_truncated
- Truncated holder_step: master bound for `A_K = ∫⁻ (min fstar K)^p`. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:743 |
MeasureTheory.holder_step
- Stage 2 (Hölder + algebra): combining Fubini's output with Hölder
yields the master bound on `∫⁻ fstar^p`. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:840 |
MeasureTheory.eLpNorm_eq_lintegral_ofReal_pow
- Conversion lemma: for a non-negative `f : Ω → ℝ` and `1 < p`,
`eLpNorm f (ofReal p) μ = (∫⁻ ω, ofReal(f ω ^ p) ∂μ)^(1/p)`. |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:902 |
MeasureTheory.maximal_ineq_Lp
- Doob's L^p maximal inequality for discrete-time non-negative submartingales.
For a non-negative submartingale `f : ℕ → Ω → ℝ` and `1 < p`, the L^p norm
of the running maximum `f*_n(ω) = max_{k ≤ n} f_k(ω)` is boun |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:931 |
MeasureTheory.martingale_norm_submartingale
- The norm process of a Banach-valued martingale is a non-negative submartingale.
Internal lemma used to derive the Banach version of Doob's L^p inequality from the
real-valued case via Jensen (`MeasureTheory.norm_condEx |
lemma | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:1094 |
MeasureTheory.Martingale.eLpNorm_norm_runMax_le
- Doob's L^p maximal inequality, Banach-valued martingale form.
For a Banach-valued martingale `f : ℕ → Ω → E` and `1 < p`,
`‖max_{k ≤ n} ‖f_k‖‖_{L^p} ≤ (p / (p - 1)) · ‖f_n‖_{L^p}`.
Derived from `MeasureTheory.max |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:1114 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refinementFactor
- Number of fine cells inside one coarse cell. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refinementFactor_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:32 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.pow_mul_refinementFactor
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarseCell
- Fine-cell index viewed in its containing coarse cell. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:43 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarseCell_val
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:50 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarseCell_block_left
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:55 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarseCell_block_right
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:60 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.dyadicMesh_coarse_eq_factor_mul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:66 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarse_left_endpoint_le_fine_left
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:84 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.fine_right_endpoint_le_coarse_right
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:95 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.regularDyadic_last_time
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:110 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.DyadicElementaryProcess.horizon_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:116 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic
- Refine a dyadic process to a finer dyadic level by repeating each coarse
coefficient across the fine cells in its block. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:128 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_level
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:145 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_times
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:151 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_coeff
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:158 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_coeff_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:165 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_value_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:173 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_toLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:212 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refinementEquiv
- Product indexing of a fine grid by coarse cell and within-cell offset. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:227 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refinementEquiv_val
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:232 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarseCell_refinementEquiv
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:239 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.sum_brownianIncrements_block
- A finite block of consecutive increments telescopes. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:249 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.fine_block_left_endpoint
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:264 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.fine_block_right_endpoint
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:273 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_elementaryItoIntegral_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:283 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_terminalToLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:326 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.terminalToLp
- Terminal stochastic integral represented in `L2(mu)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:343 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.processToLp
- Product-space representative, with finiteness supplied by the Brownian
probability contract rather than exposed as an extra theorem parameter. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:350 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.elementaryProcessToLp_eq_processToLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:357 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonDyadicLevel
- Least dyadic level containing the grids of both processes. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:365 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementLeft
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:369 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementRight
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:374 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinement_times_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:379 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementLeftProcess
- Left common refinement with the shared cell count exposed in its type. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:386 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementRightProcess
- Right common refinement with the shared cell count exposed in its type. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:392 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementProcess_times_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:397 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementLeft_toLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:403 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementRight_toLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:408 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementLeft_processToLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:413 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementRight_processToLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:420 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementLeft_terminalToLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:427 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementRight_terminalToLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:434 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.norm_terminal_sub_eq_process_sub
- The elementary Ito terminal map is an exact distance isometry even when
the two processes are initially represented on different dyadic grids. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:443 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopDyadicAtGridIndex
- Refine `eta` and retain exactly the cells strictly before the grid point
`cutoff`. Values at the cutoff itself are immaterial in product `L2`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:32 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopDyadicAtGridIndex_level
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:64 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopDyadicAtGridIndex_coeff
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.cutoffTime
- The time represented by a cutoff grid index. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:81 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.cutoffTime_le_horizon
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:85 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopDyadicAtGridIndex_elementaryItoIntegral
- Stopping coefficients at a grid index is exactly the same finite Ito sum
as integrating the refined process up to that grid time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:100 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopDyadicAtGridIndex_terminalToLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:157 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopDyadicAtGridIndex_value_eq
- Pointwise description of a grid-stopped process. The closed endpoint is
kept here; it differs from `restrictProcess` only on one null time slice. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:178 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stoppingLevel
- Target level used by the right-endpoint stopping approximation. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:230 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.level_le_stoppingLevel
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:233 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.rightCutoffIndex
- Grid index of the right endpoint of the cell containing `t`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:239 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopAtRightApprox
- Dyadic elementary process stopped at right grid endpoints decreasing to
the deterministic time `t`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:247 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.cutoffTime_rightCutoffIndex
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:254 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopAtRightApprox_terminalToLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:262 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopAtRightApprox_value_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:272 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.tendsto_stoppingLevel
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:282 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.tendsto_rightApproxTime_stoppingLevel
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:288 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.tendsto_stopAtRightApprox_value_of_ne
- Away from the single cutoff time, the stopped dyadic representatives
converge pointwise to the strict time restriction. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:298 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.tendsto_stopAtRightApprox_ae
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:323 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.abs_stopAtRightApprox_error_le
- A stopped representative and the strict restriction are uniformly
dominated by twice the deterministic elementary-process bound. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:361 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.tendsto_stopAtRightApprox_toLp
- Right-grid stopping converges in the actual product-space `L2` object to
strict restriction at `t`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:396 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalIndex
- Index in the original integer-horizon ladder corresponding to horizon
`2^k`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicHorizon
- The corresponding positive dyadic horizon. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalIndex_add_one
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:40 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.integerHorizon_dyadicGlobalIndex
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:46 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicHorizon_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:50 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalIndex_mono
- The dyadic subsequence indices are monotone. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalIndex_ge_self
- The dyadic subsequence is cofinal in the natural-number index set. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:60 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.tendsto_dyadicGlobalIndex_atTop
- Topological cofinality of `k ↦ 2^k - 1`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:66 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalLocalizingTime
- The global canonical localizer restricted to dyadic horizons. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:74 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalLocalizingTime_isChewiStoppingTime
- Every member of the dyadic subsequence remains a Chewi stopping time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:81 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalLocalizingTime_mono
- The dyadic global localizers are pointwise increasing. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:90 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalLocalizingTime_le_horizon
- The `k`-th dyadic global localizer is bounded by the matching horizon
`2^k`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:99 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalLocalizingTime_tendsto_top_ae
- The cofinal dyadic subsequence still tends to infinity almost surely. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:108 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGridStoppingIto.IsGridValuedFor
- A stopping time is grid-valued relative to a dyadic elementary process if
we have chosen, for every sample point, the grid endpoint that represents its
value. Keeping the witness explicit is useful in the finite-sum p |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGridStoppingIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGridStoppingIto.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGridStoppingIto.stopElementary_coeff_eq_gridCutoff
- On one sample point, the coefficient retained by random stopping agrees
with the deterministic coefficient cutoff at the selected grid index. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGridStoppingIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGridStoppingIto.lean:46 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGridStoppingIto.elementaryItoIntegral_stop_gridValued
- Exact finite-sum stopped-Itô identity for a dyadic grid-valued stopping
time. This is pointwise in `omega`: no expectation, completion, or limiting
argument is hidden in the statement. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGridStoppingIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGridStoppingIto.lean:71 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extensionLevel
- New dyadic level after enlarging the horizon from `2^a` to `2^b` while
keeping the physical mesh fixed. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.dyadicHorizon_mono
- Dyadic horizons are monotone in their exponent. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:44 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.dyadicMesh_dyadicHorizon_align
- Exact mesh alignment under the level shift `L ↦ L + (b-a)`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.oldCellCount_le_extension
- The old cell count embeds into the enlarged dyadic cell count. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:69 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.prefixIndex
- Old cell index regarded as a prefix index of the enlarged grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:77 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.prefixIndex_val
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:82 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.prefix_time_eq
- Prefix grid times are exactly preserved by the horizon extension. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:89 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon
- Dyadic zero extension from `H_a` to `H_b`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:104 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_level
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:151 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_times
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:156 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_coeff_prefix
- Prefix coefficients are copied exactly. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:165 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_coeff_tail
- Every new tail coefficient is exactly zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:174 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_value_eq_of_le
- The enlarged process agrees with the old elementary process on the whole
old closed horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:183 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_value_eq_zero_of_old_lt
- The enlarged process is zero strictly after the old horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:233 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_value_eq_restrictProcess_of_ne_terminal
- Away from the old terminal slice, the enlarged elementary process is
pointwise the strict zero extension used by `ProgressiveL2Integrand.restrictProcess`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:290 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.processFunction_extendDyadicHorizon_ae_eq_restrictProcess
- The enlarged elementary process and the strict zero extension agree almost
everywhere for the larger product process-time measure. The only possible
disagreement is the old deterministic terminal slice. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:308 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_toLp_eq_extendByZero
- At the `L²` level, dyadic horizon extension is exactly the general
zero-extension isometry. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:333 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto.fin_sum_eq_sum_prefix_of_tail_zero
- A finite sum over a larger `Fin M` reduces to a prefix `Fin N` if every
new tail term is zero and the prefix terms agree. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonIto.lean:32 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto.old_time_le_horizon
- Every old grid endpoint lies below the old terminal horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonIto.lean:62 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto.extend_time_castSucc_eq
- Left endpoint of an old cell is unchanged in the enlarged grid. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonIto.lean:75 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto.extend_time_succ_eq
- Right endpoint of an old cell is unchanged in the enlarged grid. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonIto.lean:91 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto.extendDyadicHorizon_elementaryItoIntegral_eq
- Exact finite-sum cross-horizon identity. Extending a dyadic elementary
integrand from `2^a` to `2^b` by zero leaves its terminal Itô integral
unchanged for every sample point. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonIto.lean:109 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto.extendDyadicHorizon_terminalToLp_eq
- The same finite-sum identity in terminal `L²(mu)`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonIto.lean:164 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.zeroLike
- The zero process carried by the strict grid of `eta`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:26 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.neg
- Pointwise negation preserves elementary adaptedness and the time grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.smul
- Scalar multiplication preserves elementary adaptedness and the time grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:46 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.add
- Addition of elementary processes represented on the same strict grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:60 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.sub
- Subtraction of elementary processes represented on the same strict grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:76 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.zeroLike_value
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:91 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.neg_value
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:98 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.smul_value
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:116 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.add_value
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:128 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.sub_value
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:149 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_zeroLike
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:170 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_neg
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:176 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:183 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_add
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:190 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_sub
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:207 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.dyadicObservationTime
- The `k`-th point of the level-`level` dyadic observation grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.dyadicObservationTime_monotone
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.dyadicObservationTime_terminal
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:42 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.dyadicObservationTime_refine
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:48 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.measurable_runningAbsMax_dyadic
- Finite dyadic running maxima are measurable random variables. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.runningAbsMax_dyadic_mono_level
- Refining a dyadic observation grid can only increase its running maximum. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:75 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.commonDifference
- Put two dyadic elementary processes on their common grid and subtract. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:95 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.elementaryItoProcess_commonDifference
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:103 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.elementaryItoIntegral_commonDifference
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:128 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.eLpNorm_commonDifference_terminal
- The terminal `eLpNorm` of the common-grid difference is exactly the
product-space `L2` distance of the two integrands. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:139 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.doobL2_elementaryItoProcess
- Finite-grid Doob control for an elementary Ito martingale. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:169 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.doobL2_elementaryItoProcess_sub
- Finite-grid Doob control for the difference of two heterogeneous dyadic
elementary Ito processes. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:185 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.processFunction_stronglyMeasurable
- The probability-time representative of an elementary process is jointly
strongly measurable in the repository's sample-first product orientation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:27 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.value_stronglyProgressive
- Elementary left-endpoint processes are strongly progressive. Cells whose
left endpoint lies after the inspected horizon vanish on that restricted
product; all earlier coefficients are measurable in the terminal sigma- |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.valueBound
- A deterministic bound obtained from the finitely many coefficient bounds. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:95 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.abs_value_le_valueBound
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:98 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.value_memLp_two
- Bounded elementary processes belong to product `L2` on every finite
probability-time horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:113 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toProgressiveL2
- Canonical inclusion of elementary adapted processes into the progressive
`L2` domain used for general Ito integration. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:125 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toProgressiveL2_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:132 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toLp_add
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:138 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toLp_sub
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:159 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toLp_smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:180 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.ElementaryAdaptedProcess
- The data and regularity conditions of the elementary adapted process in
Chewi display (1.1.2). There are `n` half-open time intervals and `n + 1`
strictly increasing endpoints. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:25 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.ElementaryAdaptedProcess.value
- Value of an elementary adapted process at a time and sample point. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.chewi_display_1_1_2
- Chewi display (1.1.2): an elementary adapted process is the finite sum of
its left-endpoint measurable coefficients on `(t_i, t_{i+1}]`. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.elementaryItoIntegral
- The finite Brownian-increment sum used to define the Ito integral of an
elementary process at terminal time `T`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:56 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.chewi_display_1_1_3
- Chewi display (1.1.3): the elementary Ito integral is exactly the finite
sum of adapted coefficients times stopped Brownian increments. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:67 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.processTimeMeasure
- Product measure `P tensor m|[0,T]` used for the square-integrability
condition in Chewi display (1.1.7). |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.processL2Energy
- Squared `L2(P tensor m|[0,T])` energy of a real process, in `ENNReal` so
finiteness is not hidden by totalized real integration. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:87 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.chewi_display_1_1_7
- Chewi display (1.1.7): Tonelli identifies the product-space squared `L2`
energy with the expected time integral over `[0,T]`. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:94 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.IsLocallySquareIntegrableOn
- Almost-sure local square integrability on `[0,T]`, the weaker condition
used when Chewi extends stochastic integration by localization. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:107 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral.chewi_display_1_1_10
- Chewi display (1.1.10): local square integrability is precisely almost-
sure finiteness of the accumulated squared integrand on `[0,T]`. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIntegral.lean:115 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.brownianIncrement
- Brownian increment over `(a, b]`, clipped at terminal time `T`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:24 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.weightedIncrement
- One summand in the elementary Ito integral. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.elementaryItoIntegral_eq_sum_weightedIncrement
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.grid_endpoint_le_of_lt
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:46 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.coeff_memLp
- A bounded elementary coefficient belongs to every finite `Lp` space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:55 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.brownianIncrement_memLp_two
- A clipped Brownian increment is square integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:67 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.weightedIncrement_memLp_two
- Every weighted elementary Brownian increment is square integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:78 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.integral_weightedIncrement_sq
- Diagonal term: an adapted coefficient factors from the squared future
increment, whose second moment is the clipped interval length. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:94 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.ordered_cross_integral_eq_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:142 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.integral_weightedIncrement_mul_eq_zero
- Distinct adapted weighted Brownian increments are orthogonal in `L2`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:197 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.chewi_display_1_1_5
- Chewi display (1.1.5): expanding the finite square leaves only diagonal
terms because distinct adapted weighted increments are orthogonal. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:211 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.elementaryItoIntegral_sq_eq_sum
- The probabilistic part of Chewi display (1.1.6): each diagonal term is
the coefficient's second moment times the clipped time-step length. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:253 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.sq_sum_eq_sum_sq_of_pairwise_mul_eq_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:266 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.interval_piece_mul_eq_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:279 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.ofReal_value_sq_eq_sum
- Pointwise square of an elementary process: disjoint time cells remove all
cross terms before time integration. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:298 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.lintegral_value_sq
- Time `L2` energy of one elementary sample path, evaluated exactly on the
clipped grid cells. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:326 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.value_sq_aemeasurable
- The elementary process square is measurable on sample-path/time product
space, so Tonelli applies without an extra supplied hypothesis. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:366 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.processL2Energy_value
- Exact `ENNReal` expansion of the product-space energy of an elementary
adapted process. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:393 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.chewi_display_1_1_6
- Chewi display (1.1.6), in the repository's nonnegative product-space
energy representation. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:426 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.norm_sq_toLp_eq_integral_sq
- The square of the `L2` norm of a real representative is its second
moment. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:30 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.elementaryItoIntegral_memLp_two
- The finite elementary stochastic sum is square integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.elementaryItoTerminalToLp
- The elementary terminal Ito integral as an actual element of `L2(mu)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.norm_sq_elementaryItoTerminalToLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.elementaryProcessToLp
- The elementary integrand as a product-space `L2` element in the Brownian
probability environment. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:63 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.ofReal_norm_sq_elementaryProcessToLp_eq_energy
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.norm_elementaryItoTerminalToLp
- The elementary terminal map is an isometry between the product-space
integrand `L2` norm and the terminal random-variable `L2` norm. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:91 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.elementaryItoTerminalToLp_add
- On a fixed grid, the terminal `L2` representative respects addition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:109 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.elementaryProcessToLp_add
- The Brownian-environment product-space representatives respect same-grid
addition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:133 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.inner_elementaryItoTerminalToLp
- On a common grid, the elementary terminal map preserves the real Hilbert
inner product, not only norms. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:146 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.elementaryItoTerminalToLp_sub
- On a fixed grid, the terminal `L2` representative respects subtraction. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:163 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.elementaryProcessToLp_sub
- The Brownian-environment product-space representatives respect same-grid
subtraction. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:187 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.norm_elementaryItoTerminalToLp_sub
- Same-grid differences satisfy the exact distance form of the elementary
Ito isometry. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:200 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoProcess
- The elementary Ito integral accumulated up to `t` and stopped at `T`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:28 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoProcess_zero
- An elementary Ito process starts at zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:34 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoProcess_stronglyAdapted
- Every elementary Ito value is measurable with respect to the information
available at that time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:43 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoProcess_integrable
- Every elementary Ito value is integrable (in fact square integrable). |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:85 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.stoppedWeightedIncrement
## A reusable stopped weighted Brownian increment |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:94 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.stoppedWeightedIncrement_memLp_two
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:99 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.stoppedWeightedIncrement_stronglyAdapted
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:110 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.stoppedWeightedIncrement_martingale
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:128 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoSummand
- One grid-cell contribution to the elementary Ito process. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:235 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoSummand_martingale
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:243 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoProcess_martingale
- The elementary Ito integral process is a genuine martingale, obtained as
a finite sum of stopped weighted Brownian-increment martingales. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:277 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoProcess_terminal
- At the stopping horizon, the process agrees definitionally with the
terminal elementary Ito integral. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:298 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoProcess_continuous_ae
- Elementary Ito paths are continuous outside the Brownian null set. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:306 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime.activeBefore
- The event on which the coefficient attached to the cell beginning at `t`
remains active after stopping at `tau`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryStoppingTime.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime.measurableSet_activeBefore
- A stopping time makes every left-endpoint activity event measurable at
that left endpoint. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryStoppingTime.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime.stopElementary
- Stop an elementary adapted integrand by a stopping time. On cell `i`, its
left-endpoint coefficient is retained exactly when the stopping time is still
strictly after that left endpoint. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryStoppingTime.lean:48 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime.stopElementary_times
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryStoppingTime.lean:68 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime.stopElementary_coeff
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryStoppingTime.lean:75 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.enorm_sq_eq_ofReal_sq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:28 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.accumulatedEnergyReal_eq_prefixIntegral
- The measurable fixed-time representative agrees with the ordinary prefix
integral; the only pointwise discrepancy is the null upper endpoint. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:34 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.sectionSquare_aestronglyMeasurable
- A time section of the squared process is strongly measurable under the
finite time measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.sectionSquare_integrable_ae
- The source local-square-integrability assumption is exactly almost-sure
Bochner integrability of the squared time section. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:73 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.continuous_accumulatedEnergyReal_of_integrable
- Every finite-energy sample path has continuous accumulated energy. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:89 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.continuous_accumulatedEnergyReal_ae
- Accumulated energy is continuous for almost every sample point. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:99 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.accumulatedEnergyReal_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:106 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.accumulatedEnergyReal_mono_of_integrable
- On every finite-energy path, accumulated energy is monotone. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:113 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.accumulatedEnergyReal_eq_terminal_of_le
- Accumulated energy stabilizes at the terminal horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:123 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.measurableSet_accumulatedEnergyReal_ge
- Threshold events for fixed-time accumulated energy are measurable at that
time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:132 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand.energyStoppedIntegrand
- Completed integrand stopped immediately when completed energy reaches the
specified level. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedIntegrand.lean:31 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand.energyStoppedIntegrand_stronglyProgressive
- Energy thresholding preserves strong progressiveness. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedIntegrand.lean:41 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand.completedEnergy_lt_iff_lt_canonicalEnergyLocalizer
- Before terminal time, being below the energy level is equivalent to being
strictly before the canonical equality-level localizer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedIntegrand.lean:59 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand.sectionSquare_integrable
- On every sample path, the stopped square is time-integrable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedIntegrand.lean:97 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand.integral_energyStoppedIntegrand_sq
- The real time integral of the stopped square is the completed energy at
its canonical localizer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedIntegrand.lean:131 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand.integral_energyStoppedIntegrand_sq_le
- Stopping at a nonnegative energy level bounds the pathwise square energy
by that level. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedIntegrand.lean:165 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedItoOverlap.energyStoppedItoProcess_overlap_ae
- A lower energy-stopped Itô process is the stopped version of every higher
energy-stopped Itô process, at every deterministic time in the common finite
horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedItoOverlap.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedItoOverlap.canonicalStoppedItoProcess_overlap_ae
- Natural-number form matching Chewi's canonical levels `n+1`. For `n ≤ m`,
the `n`-th canonical stopped Itô martingale agrees with the `m`-th one stopped
at the `n`-th canonical energy localizer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedItoOverlap.lean:89 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.stoppedExtension
- Zero extension of an energy-stopped process from `[0,T] × Ω`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:27 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.stoppedExtension_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.stoppedExtension_apply_of_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.processFunction_aestronglyMeasurable
- Product-space representative of the stopped process is strongly
measurable almost everywhere. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:71 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.processFunction_sq_integrable
- The square of the stopped product-space process is integrable whenever the
level is nonnegative and the sample measure is a probability measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:98 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.stoppedProgressiveL2
- Energy stopping upgrades a local progressive integrand to the global
progressive `L2` domain. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:151 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.stoppedProgressiveL2_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:163 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingBoundaryBridge.energyStoppedIntegrand_eq_closedStop_larger_of_ne_boundary
- Away from the smaller hitting-time boundary and before the terminal
horizon, stopping a larger energy truncation at the smaller canonical hitting
time is exactly the smaller energy truncation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingBoundaryBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppingBoundaryBridge.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingBoundaryBridge.energyStoppedIntegrand_ae_eq_closedStop_larger
- For every fixed sample path, the strict canonical truncation and the closed
stopping of any larger truncation agree for almost every time in `[0,T]`.
Only the hitting-time singleton and the terminal singleton are disca |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingBoundaryBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppingBoundaryBridge.lean:84 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingL2Bridge.ae_time_le_terminal
- Under the process-time measure, the time coordinate lies in `[0,T]` almost
everywhere. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingL2Bridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppingL2Bridge.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingL2Bridge.ae_time_ne_terminal
- The terminal time slice is product-null. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingL2Bridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppingL2Bridge.lean:51 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingL2Bridge.processFunction_energyStopped_ae_eq_closedStop_larger
- The smaller strict energy truncation agrees product-a.e. with the larger
energy truncation stopped in Chewi's closed convention at the smaller hitting
time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingL2Bridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppingL2Bridge.lean:64 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingL2Bridge.stoppedProgressiveL2_toLp_eq_stop_larger
- In completed progressive `L²`, strict truncation at level `c` is exactly
closed stopping at `τ_c` of any larger truncation level `d ≥ c`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingL2Bridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppingL2Bridge.lean:110 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.coordinateDual
- The continuous linear functional selecting the `j`-th Euclidean
coordinate. It is represented through the standard orthonormal basis so that
its norm and its action are inherited from the inner-product-space API. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.coordinateDual_apply
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:40 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.norm_coordinateDual
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:46 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.projectedIncrementVariance_coordinateDual
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:51 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.coordinate_increment_hasLaw
- A coordinate increment has exactly the one-dimensional Gaussian law with
variance equal to elapsed time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.coordinate_eval_hasLaw
- Each coordinate has the correct Brownian one-time law. The `t = 0`
case is discharged from the source's pointwise `B₀ = 0` clause; positive times
come from the increment law over `[0,t]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:71 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.IsStandardBrownianMotion.coordinate_isBrownianReal
- The `j`-th coordinate of a Chewi-standard Euclidean Brownian motion is a
Mathlib real Brownian motion. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:96 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.IsStandardBrownianMotionWithFiltration.coordinate_stronglyAdapted
- Strong adaptedness passes from the vector process to each coordinate by
composition with the continuous coordinate functional. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:125 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.IsStandardBrownianMotionWithFiltration.coordinate_incrementIndependent
- Independence of a future vector increment from the whole past filtration
passes to every coordinate increment by shrinking the second sigma-algebra
along the measurable coordinate projection. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:137 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.IsStandardBrownianMotionWithFiltration.coordinate
- The exact scalar Brownian-filtration contract consumed by the Chapter 1
Itô integral, derived from one source Euclidean Brownian motion. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:187 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.coordinateFamily
- Package all coordinates as the integration-facing family used by the
finite-dimensional Itô-process ABI. Every member comes from the same vector
Brownian motion and common filtration. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:201 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerGeneratorBridge.fellerOperator_const
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerGeneratorBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerGeneratorBridge.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerGeneratorBridge.continuousLinearSemigroupOfFeller_op_const
- The continuous-linear semigroup induced by a Feller kernel fixes every
constant observable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerGeneratorBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerGeneratorBridge.lean:46 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerGeneratorBridge.hasRightGeneratorAt_const
- Every constant bounded continuous observable has right-generator value
zero for the Feller continuous-linear semigroup. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerGeneratorBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerGeneratorBridge.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerGeneratorBridge.const_mem_generatorDomainSubmodule
- Constants belong to the canonical Feller generator domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerGeneratorBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerGeneratorBridge.lean:68 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.FellerTransitionKernelContract
- A transition-kernel Markov semigroup which maps bounded continuous real
observables to continuous observables. Boundedness of the image is derived
from the Markov property rather than included as a field. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:42 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.integrable_boundedContinuousFunction
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.norm_kernelIntegral_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperatorValue
- The bounded continuous observable obtained by integrating against the
transition kernel at time `t`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:65 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperatorValue_apply
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:75 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperatorValue_add
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperatorValue_smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:94 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerLinearMap
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:104 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperator
- The Feller Markov operator as a continuous linear map on bounded
continuous real observables. Its operator norm is at most one. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:113 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperator_apply
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:125 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.norm_fellerOperator_apply_le
- The pointwise contraction estimate inherited from integration against a
probability kernel. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:132 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.sq_fellerOperator_apply_le
- Jensen's inequality for the square under a Feller Markov operator:
`(P_t f x)^2 ≤ P_t(f^2)(x)`. This is equation (1.2.11) in Chewi's
2026-08-09 edition. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:143 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperator_zero
- The zero-time Feller operator is the identity continuous linear map. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:161 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperator_add
- Chapman--Kolmogorov yields the continuous-linear operator semigroup law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:170 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.continuousLinearSemigroupOfFeller
- A Feller transition-kernel contract therefore supplies the exact
continuous-linear semigroup consumed by the right-generator development. |
def | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:185 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.continuousLinearSemigroupOfFeller_op_apply
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:193 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess.CoordinateBrownianFamilyWithFiltration
- Scalar Brownian coordinates equipped with the filtration contract required
by the Chapter 1 stochastic-integral construction.
This is an integration-facing interface. It intentionally does not claim that
coordinatewi |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcess.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess.CoordinateItoData
- Coordinate data behind a finite-dimensional Itô process.
`iota` indexes state coordinates and `kappa` indexes Brownian coordinates.
The diffusion field stores one already-audited globally locally square
integrable pro |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcess.lean:64 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess.coordinateStochasticTerm
- The finite-coordinate stochastic integral
`sum_j integral sigma^{i,j} dB^j` built exclusively from the scalar global
local Itô integral already proved in Chewi Proposition 1.1.16. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcess.lean:78 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess.coordinateItoProcess
- The coordinatewise finite-dimensional Itô process associated with the
source data. Lebesgue time integration uses exactly the same `TimeMeasure.upTo`
measure as the stochastic-integration foundation, so endpoint conve |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcess.lean:93 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess.chewi_definition_1_1_17_coordinate_display
- Coordinate display behind Chewi Definition 1.1.17.
This theorem is intentionally named `coordinate_display`: it certifies the
finite-sum assembly but does not by itself close the source item. Source
completion additi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcess.lean:112 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess.ChewiItoData
- Chewi's literal finite dimensions are obtained by taking state coordinates
`Fin d` and Brownian coordinates `Fin N`. |
abbrev | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcess.lean:130 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess.ChewiBrownianCoordinates
- Integration-facing Brownian-coordinate contract for the literal `N`
coordinates in Chewi Definition 1.1.17. |
abbrev | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcess.lean:137 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcessProgressive.initialCoordinateProcess_stronglyProgressive
- A time-constant initial coordinate is progressive once its `F_0`
measurability is known. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcessProgressive |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcessProgressive.lean:32 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcessProgressive.coordinateStochasticTerm_stronglyProgressive
- The finite stochastic sum over Brownian coordinates is progressive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcessProgressive |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcessProgressive.lean:44 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcessProgressive.coordinateItoProcess_coordinate_stronglyProgressive
- Every scalar coordinate of the finite-dimensional Itô process is strongly
progressive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalItoProcessProgressive |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalItoProcessProgressive.lean:63 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.matrixSquareEnergy
- Sum of squares of all entries of a finite real matrix, written as a
curried function so it can be used without committing the stochastic layer to
a particular `Matrix` wrapper. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.matrixSquareEnergy_nonneg
- The finite matrix square energy is nonnegative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.entry_sq_le_matrixSquareEnergy
- Any individual matrix-entry square is bounded by the total finite matrix
square energy. This is the algebraic core of the Hilbert--Schmidt-to-entrywise
`L²` bridge. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:56 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.matrixAsEuclidean
- Flatten a finite matrix into one Euclidean vector indexed by coordinate
pairs. No information is lost; this is only a norm/notation bridge. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:72 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.norm_sq_matrixAsEuclidean
- The squared Euclidean norm of the flattened matrix is exactly the sum of
squares of its entries. In finite-dimensional Euclidean spaces this is the
Frobenius/Hilbert--Schmidt norm squared. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.MatrixLocallySquareIntegrableOn
- Pathwise local finiteness of the finite matrix square energy on `[0,T]`.
This is the matrix analogue of Chewi's scalar condition (1.1.10), before the
separate progressive-measurability contract is attached. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:92 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.MatrixLocallySquareIntegrableOn.entry
- Finite matrix energy implies Chewi's scalar local-square-integrability
condition for every matrix entry. No expectation over sample paths is added:
the implication remains pathwise almost surely, exactly as in display
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:105 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.MatrixLocallySquareIntegrableNormOn
- Chewi's finite-dimensional diffusion condition written literally with the
squared Euclidean/Frobenius norm of the matrix coefficient. This source-facing
predicate keeps the public statement free of the implementation-o |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:122 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.MatrixLocallySquareIntegrableNormOn.toEnergy
- The literal Frobenius-norm formulation implies the finite-sum energy
formulation used by the scalar integration layer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:133 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.MatrixLocallySquareIntegrableNormOn.entry
- A source-level Frobenius local-`L²` hypothesis yields local square
integrability for every scalar matrix entry. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:145 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.entryGlobalLocalProgressiveL2
- Package one diffusion entry into the global scalar progressive-`L²` ABI
used by the Itô integral, from finite-dimensional source assumptions.
Progressive measurability is supplied componentwise here; the coordinate
mea |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:159 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.strictGrid_existsUnique_cell
- A point strictly after the first endpoint and at most the final endpoint
belongs to a unique cell of a strictly increasing finite grid. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:28 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.ElementaryAdaptedProcess.value_eq_coeff_of_mem_cell
- Inside a cell, the elementary process is exactly that cell's coefficient. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:73 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.ElementaryAdaptedProcess.value_eq_zero_of_le_first
- An elementary process vanishes at and before its first grid endpoint. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:101 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.ElementaryAdaptedProcess.value_eq_zero_of_last_lt
- An elementary process vanishes strictly after its final grid endpoint. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:116 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.dyadicMesh_tendsto_zero
- The real-valued mesh of the dyadic partition tends to zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:131 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.eventually_two_mul_dyadicMesh_lt
- Eventually twice the dyadic mesh is below every positive real tolerance. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:144 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.eventually_dyadicMesh_lt
- Eventually the dyadic mesh is below every positive real tolerance. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:152 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.dyadic_activeCell
- Every positive time up to `T` has a unique active dyadic cell. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:159 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.dyadic_activeCell_left_le
- The left endpoint of an active dyadic cell is at most the point. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:176 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.dyadic_activeCell_right
- The right endpoint of a regular dyadic cell is one mesh after its left endpoint. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:184 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid.dyadic_previousCell_subset_leftNeighborhood
- For every nonfirst active cell, its preceding cell lies in the left
neighborhood of radius twice the mesh. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteTimeGrid |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteTimeGrid.lean:194 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FokkerPlanckAlgebra.fpRewriteScalarAlgebra
- Scalar algebra behind the rewrite
`-div(q b) + a lap q = a div(q A) + div(q V)` once the analytic identities
for `lap q` and `V` have been supplied. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FokkerPlanckAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FokkerPlanckAlgebra.lean:19 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FokkerPlanckAlgebra.fisherIbpAlgebra
- Scalar algebra behind the Fisher/IBP conclusion once the two integration
by parts identities are supplied. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FokkerPlanckAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FokkerPlanckAlgebra.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GaussianFourthMoment.integral_pow_four_gaussianReal_zero
- Fourth moment of a centered real Gaussian with variance `v`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GaussianFourthMoment |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GaussianFourthMoment.lean:25 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GeneratorStationarity.hasRightGeneratorAt_zero_of_fixed
- A vector fixed by the entire semigroup has right-generator value zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GeneratorStationarity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GeneratorStationarity.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GeneratorStationarity.mem_generatorDomainSubmodule_of_fixed
- Consequently a fixed vector belongs to the canonical generator domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GeneratorStationarity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GeneratorStationarity.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GeneratorStationarity.invariantFunctional_generator_eq_zero
- An invariant continuous linear functional annihilates every right-generator
value.
This is the abstract infinitesimal-stationarity argument: apply the functional
to the semigroup difference quotient. Invariance makes |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GeneratorStationarity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GeneratorStationarity.lean:60 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GeneratorStationarity.invariantFunctional_rightGenerator_eq_zero
- Bundled generator-domain form of infinitesimal stationarity. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GeneratorStationarity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GeneratorStationarity.lean:89 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov.finiteShiftedGaussianPathMeasure
- The finite-dimensional shifted Gaussian cylinder measure obtained by
pushing shifted product coordinates into `EuclideanSpace`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Girsanov.lean:27 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov.finiteGaussianGirsanovWeight
- The finite-dimensional Girsanov/Esscher likelihood ratio against
`stdGaussian (EuclideanSpace ℝ ι)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Girsanov.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov.finiteGaussianGirsanovCylinderIntegral
- Finite-dimensional cylindrical Girsanov change of measure.
This is the PATH-facing wrapper around the Gaussian
`stdGaussian_shift_integral_map_toLp` leaf. It is the right finite-dimensional
base case for later Browni |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Girsanov.lean:44 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov.finiteGaussianGirsanovCylinderMeasure_eq_withDensity
- Measure-level finite-dimensional cylindrical Girsanov density identity. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Girsanov.lean:56 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov.integral_finiteGaussianGirsanovWeight_eq_one
- The finite-dimensional Girsanov weight has unit mass under the centered
`stdGaussian` cylinder. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Girsanov.lean:109 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.integerHorizon
- Positive integer horizon used by the global localization ladder. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.integerHorizon_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.integerHorizon_succ
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:40 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.integerHorizon_mono
- Integer horizons are monotone. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.globalBadSet
- The countable exceptional set where local square integrability fails on at
least one positive integer horizon. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.measure_globalBadSet_zero
- The global exceptional set is null. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.measurableSet_globalBadSet
- Completeness puts the shared exceptional set in every filtration sigma
algebra. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:66 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.not_bad_on_integerHorizon
- A globally good path is good on every positive integer horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:73 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.completedEnergy_eq_of_le_horizons
- On a globally good path, completed accumulated energy before the smaller
horizon is independent of which larger integer horizon is used. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:83 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.globalLocalizingTime
- Global canonical localizer: zero on the shared null set; otherwise use the
usual finite-horizon energy hitting time at matching level and horizon. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:102 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.globalLocalizingTime_of_bad
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:111 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.globalLocalizingTime_of_good
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:118 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.globalLocalizingTime_le_horizon
- Every global localizer is capped by its matching integer horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:128 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.globalLocalizingTime_isChewiStoppingTime
- Each global localizer is a Chewi stopping time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:140 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.globalLocalizingTime_mono
- The global localizing times are pointwise increasing. The proof uses both
increasing energy thresholds and the fact that accumulated energy before an
earlier time is independent of the larger ambient horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:198 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizerLimit.eventually_lt_globalLocalizingTime_of_good
- On a globally good path, every deterministic time lies strictly below all
sufficiently late canonical localizers. The proof freezes the energy at that
time on one integer horizon and then lets both the energy threshol |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizerLimit |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizerLimit.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizerLimit.tendsto_globalLocalizingTime_top_of_good
- Pointwise divergence to the top element of `WithTop ℝ≥0` on every good
path. This is the topological notion of tending to infinity used in Chewi's
local-martingale definition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizerLimit |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizerLimit.lean:77 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizerLimit.globalLocalizingTime_tendsto_top_ae
- The global canonical localizing sequence tends to infinity almost surely.
This is the exact limiting clause required by `Localization.IsLocalMartingale`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizerLimit |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizerLimit.lean:92 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoGluingGoodSet
- The full-measure pathwise contract used for global gluing: the cofinal
localizers diverge and all countably many localized martingale pairs agree
before the smaller localizer fires. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoGluingGoodSet_ae
- The global gluing contract holds almost surely. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:63 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoGluingBadSet
- The single exceptional set patched by zero in the global continuous
version. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:90 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.measure_globalItoGluingBadSet_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:97 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.measurableSet_globalItoGluingBadSet_at
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:105 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.rawGlobalItoProcess
- Pointwise candidate obtained from the coherent localized martingale family. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:115 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.rawGlobalItoProcess_stronglyMeasurable
- The raw pointwise limit is measurable at every deterministic time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:124 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.tendsto_globalStoppedItoProcess_of_good_of_le_localizer
- On a good path, before `tau_k`, the localized martingale sequence is
literally eventually constant at `M_k`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:138 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.rawGlobalItoProcess_eq_globalStopped_of_good_of_le_localizer
- Identification of the raw `limUnder` with any localized martingale before
its localizer fires. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:156 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoProcess
- Everywhere-defined continuous version: keep the coherent limit on the good
set and patch the single null exceptional set by zero. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:178 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoProcess_eq_globalStopped_of_good_of_le_localizer
- On every good path, the patched global process agrees with `M_k` before
`tau_k`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:191 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoProcess_stronglyAdapted
- The global process is strongly adapted. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:208 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoProcess_continuousAt_of_good
- On a good path the global process is continuous at an arbitrary time: choose
a localizer strictly beyond that time, then the global process equals the
corresponding continuous localized martingale throughout a neighbor |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:224 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoProcess_continuous
- The patched global Itô process has continuous paths for every sample point,
including the exceptional null set where it is identically zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:255 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoProcess_zero
- The global Itô process starts at zero exactly, not merely almost surely. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:272 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.stopped_globalItoProcess_eq_stopped_globalStopped_ae
- Every globally stopped version agrees almost surely, at every deterministic
time, with the corresponding stopped localized martingale. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:291 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.stopped_globalItoProcess_martingale
- Stopping the glued global process at any member of the dyadic localizing
sequence gives a genuine martingale. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:312 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoProcess_isLocalMartingale
- The glued process satisfies Chewi's exact local-martingale definition with
the cofinal dyadic global localizers. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:337 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessProgressive.globalItoProcess_stronglyProgressive
- The everywhere-continuous, strongly adapted global Itô integral built for
Chewi Proposition 1.1.16 is strongly progressive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessProgressive |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessProgressive.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2.GlobalLocalProgressiveL2Integrand
- A progressive process satisfying Chewi's local square-integrability
condition (1.1.10) on every finite horizon. This is the rigorous global domain
needed by Proposition 1.1.16. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalLocalProgressiveL2.lean:31 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2.GlobalLocalProgressiveL2Integrand.onHorizon
- Restrict the global source domain to a finite horizon, recovering the
existing localization input type exactly. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalLocalProgressiveL2.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2.GlobalLocalProgressiveL2Integrand.onHorizon_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalLocalProgressiveL2.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2.GlobalLocalProgressiveL2Integrand.chewi_global_condition_1_1_10
- Source-facing restatement of the implicit global form of (1.1.10). |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalLocalProgressiveL2.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.globalStoppedItoProcess
- The `k`-th genuine martingale obtained by integrating the literal globally
stopped source integrand on its matching dyadic horizon. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.globalStoppedItoProcess_stronglyAdapted
- Every localized Itô process is strongly adapted. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:65 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.globalStoppedItoProcess_martingale
- Every localized Itô process is a genuine martingale on the whole
nonnegative time axis (constant after its construction horizon). |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:79 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.globalStoppedItoProcess_continuous
- Every localized Itô process has an everywhere-continuous path on all
nonnegative times. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:93 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.globalStoppedItoProcess_zero
- Every localized Itô process starts at zero exactly. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:106 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.globalStoppedItoProcess_eq_horizon_of_le
- The `k`-th process is exactly constant after `H_k`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:119 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.globalStoppedItoProcess_overlap_pathwise_ae
- Compact-path coherence of the global localized martingales.
For `k <= ell`, on one full-measure event the lower process is the larger
process stopped at `tau_k`, simultaneously at every time of `[0,H_k]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:136 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.globalStoppedItoProcess_eq_of_le_localizer_ae
- Before the lower localizer has fired, all later localized martingales agree
with the lower one. This is the eventual pathwise stability used in global
gluing. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:188 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.stopped_globalStoppedItoProcess_martingale
- Stopping one localized martingale at its own global localizer is again a
martingale, proved by identifying it with the Itô process of the correspondingly
stopped progressive-`L²` integrand. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:211 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess
- The `k`-th globally localized Itô martingale. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess_stronglyAdapted
- Each localized process is strongly adapted. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess_martingale
- Each localized process is a true martingale. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess_continuous
- Every localized process has a continuous path on the whole nonnegative
axis, not only on its construction horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:84 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess_zero
- Every localized Itô process starts from zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:97 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.stoppedProcess_coe_eq_min
- For a finite-valued stopping time, Mathlib's `WithTop` stopped-process
notation is the ordinary `NNReal` minimum. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:111 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.stopped_globalStoppedItoProcess_continuous
- The stopped larger martingale has continuous paths. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:129 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.stopped_large_eq_ito_stop_pathwise_ae
- On the larger finite horizon, completed random stopping identifies the
stopped larger path with the Itô process of the twice-stopped integrand,
simultaneously for every time on one full-measure event. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:157 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess_overlap_pathwise_ae
- Global pairwise coherence. If `k <= ell`, then on one full-measure
event the `k`-th localized martingale is exactly the `ell`-th martingale
stopped at `tau_k`, simultaneously for every nonnegative time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:220 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedL2Overlap.dyadicGlobalLocalizingTime_coe_le
- Pointwise monotonicity of the dyadic global localizers, coerced to
`WithTop NNReal`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedL2Overlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedL2Overlap.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedL2Overlap.stopped_globalStoppedIntegrand_eq
- Re-stopping the larger raw stopped integrand at the smaller localizer gives
exactly the smaller raw stopped integrand, pointwise in time and sample path. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedL2Overlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedL2Overlap.lean:49 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedL2Overlap.stop_globalStoppedProgressiveL2_process
- The direct progressive-`L²` stop of the larger finite-horizon package has
exactly the smaller raw stopped process as its process field. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedL2Overlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedL2Overlap.lean:69 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedL2Overlap.stop_globalStopped_toLp_eq_extendByZero
- Cross-horizon nested `L²` identity.
Stopping the `ell`-th package at `tau_k` is the same element of
`L²(P ⊗ dt|[0,H_ell])` as zero-extending the `k`-th package from `H_k` to
`H_ell`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedL2Overlap |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedL2Overlap.lean:87 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedIntegrand
- The literal globally stopped source integrand at the `k`-th dyadic
localizing time. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedIntegrand_stronglyProgressive
- Closed stopping preserves strong progressiveness for the global source
process. This is the same measurable-event argument used by
`ProgressiveL2Stopping`, but it only needs progressiveness of the source, not a
pre-ex |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.dyadicGlobalLocalizingTime_eq_canonicalRaw_of_good
- On a globally good sample path, the dyadic global time is literally the
finite-horizon canonical raw localizer at the matching index. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:110 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedTimeLintegral_le
- Pathwise stopped energy is bounded by the matching finite canonical level.
The statement uses exactly the stopped time measure `upTo H_k` used by the
completed Itô domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:136 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedExtension
- A strongly measurable ambient extension of the stopped source process from
`[0,H_k] × Omega`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:184 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedExtension_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:194 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedExtension_apply_of_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:216 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedProcessFunction_aestronglyMeasurable
- Product-space strong measurability of the literal globally stopped source
integrand on its matching finite horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:230 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedProcessFunction_sq_integrable
- The squared globally stopped process is integrable on its matching finite
horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:260 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedProgressiveL2
- The literal source process `eta * 1_{s <= tau_k}` packaged in the finite
progressive `L²` domain on `H_k`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:302 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedProgressiveL2_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:313 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.diffusionColumn
- The `j`-th state-space column of a finite diffusion matrix. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:31 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.driftGradientTerm
- First-order deterministic drift contribution in Itô's formula. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.diffusionGradientCoefficient
- Brownian-coordinate coefficient of the stochastic first-order term. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:44 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.secondDerivativeQuadraticForm
- Second Fréchet derivative evaluated twice in the same direction. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.diffusionHessianContraction
- The second-order diffusion/Hessian contraction in column form.
For a `d x N` coefficient matrix this is
`sum_j D^2 f(x)[sigma_j, sigma_j]`, hence the finite-dimensional form of
`<nabla^2 f(x), sigma sigma^T>`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:62 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.itoDriftCorrection
- The deterministic finite-variation coefficient in the source Itô formula. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.driftGradientTerm_eq_fderiv
- The drift-gradient contraction is the Fréchet derivative applied to the
drift whenever `f` is differentiable at the point. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.diffusionGradientCoefficient_eq_fderiv
- Each stochastic first-order coefficient is the Fréchet derivative in the
corresponding diffusion-column direction. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:92 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.driftGradientTerm_eq_sum
- Coordinate expansion of the first-order drift contraction. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:105 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.diffusionGradientCoefficient_eq_sum
- Coordinate expansion of the Brownian-coordinate stochastic coefficient. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:116 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.secondDerivativeQuadraticForm_eq_of_hasFDerivAt
- If a supplied Hessian representative is the derivative of `fderiv f` at
`x`, the quadratic form uses that representative literally. This is the local
bridge used later to turn a `C^2` source function into the Hessian |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:131 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra.diffusionHessianContraction_eq_of_hasFDerivAt
- Rewrite the full diffusion/Hessian contraction using a supplied Hessian
representative at the point. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoFormulaAlgebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoFormulaAlgebra.lean:142 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency.extendedCanonicalApprox
- Canonical small-horizon approximants, represented exactly on a larger
cofinal dyadic horizon. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonConsistency.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency.tendsto_extendedCanonicalApprox_toLp
- The extended canonical approximants converge to analytic zero extension in
product-space `L²` on the larger horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonConsistency.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency.tendsto_extendedCanonicalApprox_processToLp
- The large-horizon process-space representatives of the extended canonical
approximants converge to the completed zero extension. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonConsistency.lean:82 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency.extendedCanonicalApprox_terminal_eq
- The terminal sequence obtained after exact horizon extension is literally
the canonical small-horizon terminal approximation sequence. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonConsistency.lean:99 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency.itoIntegralTerminal_extendByZero_eq
- Completed terminal cross-horizon identity. Integrating an `L²`
integrand on `H_a` gives exactly the same terminal `L²(mu)` element as first
zero-extending it to any larger dyadic horizon `H_b` and integrating ther |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonConsistency.lean:114 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonProcessConsistency.extendByZero_restrictAt_toLp_eq
- Deterministic time restriction commutes with zero extension in product
`L²`, provided the restriction time lies in the smaller horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonProcessConsistency.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonProcessConsistency.itoIntegralTerminal_restrict_cross_horizon_eq
- Restricted terminal completions agree across dyadic horizons. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonProcessConsistency.lean:56 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonProcessConsistency.itoIntegralProcess_extendByZero_ae
- Fixed deterministic times agree almost surely across horizons. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonProcessConsistency.lean:93 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonProcessConsistency.itoIntegralProcess_extendByZero_pathwise_ae
- Pathwise compact-interval cross-horizon consistency.
There is one full-measure event on which the larger zero-extended Itô process
and the smaller Itô process agree simultaneously for every time in the smaller
clo |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonProcessConsistency.lean:120 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess
- The `n`-th canonical elementary Ito martingale. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalIncrement
- The common-grid elementary martingale representing the difference of two
successive canonical approximants. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:43 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalIncrement_eq_sub
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:51 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_martingale
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalIncrement_martingale
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:69 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_continuous_ae
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:79 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.uniformThreshold
- Geometric uniform threshold used in the Borel--Cantelli argument. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:88 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.uniformThreshold_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:90 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.summable_uniformThreshold
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:93 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.fastTolerance_succ_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:98 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.norm_canonical_process_consecutive_lt
- Explicit `L2` estimate for successive canonical integrands. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:107 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.uniformBadEvent
- Measurable event on which the `n`-th process increment exceeds its
uniform threshold on some dyadic observation grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:140 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.measurableSet_uniformBadEvent
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:145 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.badEventMajorant
- Explicit probability majorant supplied by Doob and the fast diagonal
approximation rate. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:154 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.measure_uniformBadEvent_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:158 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.badEventMajorant_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:194 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tsum_badEventMajorant_ne_top
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:217 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tsum_measure_uniformBadEvent_ne_top
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:229 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.eventually_not_uniformBadEvent_ae
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:237 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_continuous_all_ae
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:243 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.uniformCauchyEvent
- Full-measure event on which all elementary paths are continuous and only
finitely many maximal increment events occur. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:251 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.uniformCauchyEvent_ae
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:257 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalIncrement_abs_le_of_not_mem_bad
- Outside the `n`-th bad event, continuity upgrades the dyadic maximal
bound to the whole compact time interval. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:269 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.sum_canonicalIncrement_Ico
- Successive canonical increments telescope between any two approximation
levels. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:287 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_sub_abs_le_sum_threshold
- If all bad events after `N` are absent, differences between canonical
processes are bounded by the corresponding geometric tail. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:311 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_uniformCauchyOn
- On the full-measure good event, the canonical elementary Ito processes
are uniformly Cauchy on `[0,T]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:333 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.uniformBadSet
- Null exceptional set used to define an everywhere continuous patched
version of the limit process. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:382 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.measure_uniformBadSet_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:387 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.measurableSet_uniformBadSet_at
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:393 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalPathLimit
- Pointwise complete-space limit of the canonical elementary Ito processes.
On the good event the convergence is uniform on `[0,T]`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:404 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendsto_canonicalItoProcess_canonicalPathLimit
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:409 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoUniformlyOn_canonicalPathLimit
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:418 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalPathLimit_continuousOn
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:429 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess
- The actual process-level Ito integral: use the uniform path limit off the
completed null exceptional set and patch by zero on that set. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:440 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_continuousOn
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:449 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_continuous_ae
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:464 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalPathLimit_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:473 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_stronglyAdapted
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:482 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendsto_canonicalItoProcess_itoIntegralProcess_ae
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:494 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralTerminal_restrictAt_elementary_ae
- For a dyadic elementary integrand, the completed integral of its strict
restriction at `t` is represented by the elementary Ito process at `t`.
Right dyadic stopping supplies the common approximation sequence; converge |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:515 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalRepresentative
- A concrete representative of the terminal `L2` completion. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:583 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalRepresentative_memLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:589 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalRepresentative_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:595 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalConditionalProcess
- The canonical martingale obtained by conditioning the completed terminal
integral on each filtration level. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:603 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalConditionalProcess_martingale
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:608 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalApprox_ae_eq_canonicalItoProcess_terminal
- The raw terminal value of a canonical elementary martingale represents
the corresponding `terminalApprox` element of `L2`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:616 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendsto_eLpNorm_terminalApprox_sub_representative
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:627 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendsto_eLpNorm_terminalCondApprox_sub
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:639 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_terminalCondApprox
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:669 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalCondApprox_ae_eq_canonicalItoProcess
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:685 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_canonicalItoProcess_terminalConditional
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:698 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_canonicalItoProcess_actual
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:708 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalConditionalProcess_ae_eq_actual_of_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:721 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_terminalApprox_representative
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:733 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_canonicalItoProcess_terminal
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:744 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalRepresentative_ae_eq_actual
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:753 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_at_eq_terminal_of_pos
- At every positive time before the horizon, the actual continuous process
represents the completed terminal integral of the restricted integrand. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:765 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_at_zero
- The constructed process starts at zero, including on the patched null
set. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:826 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralTerminal_restrictAt_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:841 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_at_eq_terminal
- Fixed-time compatibility for every time in the construction horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:851 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_isometry_restrictAt
- Fixed-time Ito isometry, first in the exact product-space restriction
form used by the Lean construction. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:868 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralTerminal_restrictAt_add
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:900 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralTerminal_restrictAt_smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:918 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_zero
- The process construction respects the zero integrand at every time in
the horizon, up to the unavoidable representative equality. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:935 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_add
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:957 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:977 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_unique
- Any other continuous adapted version representing the same restricted
terminal integrals at every deterministic time is indistinguishable from the
constructed process on `[0,T]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:997 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_eq_terminal_of_horizon_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1052 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalPathLimit_eq_terminal_of_horizon_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1059 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_eq_terminal_of_horizon_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1068 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalConditionalProcess_ae_eq_actual_of_horizon_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1082 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalConditionalProcess_ae_eq_actual
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1107 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_martingale
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1117 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1126 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_terminal_eq
- The continuous process agrees at the horizon with the `L2` terminal
completion used to construct it. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1137 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.chewi_display_1_1_9_terminal
- Chewi's Ito isometry for a progressive globally square-integrable
integrand, stated at the fixed horizon used by the construction. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1147 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.chewi_display_1_1_9
- Chewi display (1.1.9) at every deterministic time. The right side uses
the strict restriction representative on the fixed product horizon; the
single omitted endpoint is null, so this is the formal `integral_0^t`
stat |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1183 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.chewi_theorem_1_1_8
- Process-level existence theorem behind Chewi Theorem 1.1.8. It packages
the constructed adapted continuous martingale, its terminal completion, and
the terminal Ito isometry; no stochastic-integral contract is assumed |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1196 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon.canonicalItoProcess_eq_terminal_of_le
- Every canonical elementary Itô approximant is exactly constant after the
construction horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcessAfterHorizon.lean:32 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon.canonicalPathLimit_eq_terminal_of_le
- The pointwise complete-space path limit is constant after the horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcessAfterHorizon.lean:43 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon.itoIntegralProcess_eq_terminal_of_le
- The patched continuous Itô process is exactly constant after its finite
construction horizon, on every sample path including the null-set patch. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcessAfterHorizon.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon.itoIntegralProcess_continuous
- A finite-horizon completed Itô version is in fact continuous on the whole
nonnegative time axis: it is continuous on `[0,T]` and exactly constant on
`[T,∞)`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcessAfterHorizon.lean:76 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon.stoppedProcess_eq_terminal_of_le
- If a stopping time is pointwise bounded by `T`, its stopped process is
exactly constant after `T`, independently of any stochastic assumptions. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessAfterHorizon |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcessAfterHorizon.lean:108 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessCongruence.itoIntegralProcess_congr_toLp_ae
- Equal product-space `L²` integrands have almost-surely equal completed Itô
process values at every deterministic time in the construction horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessCongruence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcessCongruence.lean:30 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessCongruence.itoIntegralProcess_congr_toLp_pathwise_ae
- Equal product-space `L²` integrands determine the same continuous Itô
version simultaneously at every time of the finite construction horizon, on
one full-measure event.
This avoids intersecting an uncountable family |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessCongruence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcessCongruence.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessGlobalContinuity.itoIntegralProcess_continuous
- The completed finite-horizon Itô process is globally continuous because it
is constant after the construction horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessGlobalContinuity |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcessGlobalContinuity.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.integrandToLp
- Product-space representative of a progressive integrand, with finiteness
obtained from the Brownian probability contract. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.terminalApprox
- Canonical elementary terminal approximation. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.processApprox
- Canonical elementary process approximation in product-space `L2`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.tendsto_processApprox
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:52 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.norm_terminalApprox_sub
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.terminalApprox_cauchy
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:72 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal
- Terminal Ito integral as the complete-space limit of elementary terminal
integrals. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:86 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.tendsto_terminalApprox
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:91 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.tendsto_terminal_of_tendsto_elementary
- Every elementary approximation converging to the integrand in product
`L2` has terminal integrals converging to the completed terminal integral. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:101 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.norm_terminalToLp_eq_processToLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:151 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_norm
- Completed terminal Ito isometry. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:159 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonLeftProcess
- Same-grid sum after refining both operands to their least common dyadic
level. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:176 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonRightProcess
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:182 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonProcess_times_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:188 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonLeftProcess_terminalToLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:193 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonRightProcess_terminalToLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:201 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonLeftProcess_processToLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:209 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonRightProcess_processToLp_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:218 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.addDyadic
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:227 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.smulDyadic
- Scalar multiple on the same dyadic grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:237 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.terminalToLp_addDyadic
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:244 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.processToLp_addDyadic
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:256 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.terminalToLp_smulDyadic
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:268 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.processToLp_smulDyadic
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:287 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.integrandToLp_add
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:296 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.integrandToLp_neg
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:304 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.integrandToLp_sub
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:311 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.integrandToLp_smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:319 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_congr_toLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:326 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_add
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:338 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:370 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:397 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_neg
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:406 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_sub
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:418 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.elementaryIntegrand
- Progressive integrand induced by an elementary process in the Brownian
probability environment. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:433 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_elementary
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:440 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_isometry_sub
- Distance form of the completed Ito isometry. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:459 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_inner
- The completed terminal map preserves the real Hilbert inner product. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:467 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminalOnHorizon
- Total horizon interface: the positive-horizon completion and the unique
zero integral on a degenerate horizon. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:479 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminalOnHorizon_of_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:484 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_zero_horizon
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:490 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.L2GeneratorIdentities.integral_rightGenerator_eq_zero_of_expectation_invariant
- If expectation under `pi` is invariant under an `L²(pi)` semigroup, then
its generator integrates to zero on the generator domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.L2GeneratorIdentities |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/L2GeneratorIdentities.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.L2GeneratorIdentities.integral_rightGenerator_mul_eq_integral_mul_rightGenerator
- Reversibility of an `L²(pi)` semigroup gives the generator pair symmetry in
source-facing integral form:
`integral (Lf) g dpi = integral f (Lg) dpi`.
The actual construction of the Markov/Langevin semigroup on `L²(pi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.L2GeneratorIdentities |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/L2GeneratorIdentities.lean:59 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.clippedExtensionAt
- Zero extension of a clipped progressive process from `[0,b] x Omega`.
The target measurable space on `Omega` is the filtration at time `b`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:30 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.clippedExtensionAt_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.clippedExtensionAt_apply_of_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:50 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.clippedExtensionAt_abs_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedCellAverage
- Average of the clipped process over `(a,b]`, normalized by `delta`.
The extension makes the joint measurability used by parameterized Bochner
integration explicit. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:82 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedCellAverage_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:89 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedCellAverage_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:106 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedCellAverage_abs_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:113 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.dyadicLeftTime
- The left endpoint of the dyadic cell indexed by `i`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:150 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicCoeff
- The lagged coefficient used on dyadic cell `i`. Cell zero has coefficient
zero; every later cell uses the average over the immediately preceding cell. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:155 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.dyadicMesh_le_leftTime_of_ne_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:163 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.dyadicLeftTime_le_terminal
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:171 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicCoeff_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:184 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicCoeff_abs_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:204 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicApprox
- The bounded elementary adapted process obtained by lagging dyadic cell
averages by one cell. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:225 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicApprox_times
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:239 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicApprox_coeff
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:246 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicApprox_last_time
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:253 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicApprox_isElementaryAdapted
- The two obligations that matter downstream: each coefficient is known at
its cell's left endpoint and remains bounded by the clipping level. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:264 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.abs_laggedDyadicApprox_sub_le
- Away from the initial cell, the lagged dyadic error is controlled by twice
the mean pointwise error on a left neighborhood of radius two mesh widths. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.clippedHorizonFunction
- Measurable sample-first extension of the clipped process, equal to it on
the stopped horizon and zero beyond the horizon. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:134 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.clippedHorizonFunction_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:139 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.clippedHorizonFunction_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:152 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.laggedDyadicApprox_tendsto_ae_time
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:158 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.laggedDyadicApprox_tendsto_ae
- At every fixed clipping level, lagged dyadic approximations converge
pointwise almost everywhere on the repository's sample-first product space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:219 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.laggedDyadicApprox_abs_le
- The value of a lagged dyadic approximation inherits the coefficient bound;
there is no factor equal to the number of cells because active cells are
unique. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:263 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.abs_laggedDyadic_error_le_two_mul
- Uniform pointwise error bound at a fixed clipping level. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:291 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.tendsto_integral_sq_laggedDyadicApprox_sub
- Dominated convergence for the squared fixed-clipping error. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:307 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.tendsto_laggedDyadicApprox_toLp_clipped
- Fixed-clipping convergence in the actual product-space `L2` object. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:358 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.hasDerivAt_gibbsWeight_mul_testDeriv_eq_langevinGenerator_1d
- Pointwise product-rule identity for the one-dimensional Gibbs weight
`exp (-V)`. In source notation this is the local calculation
`(exp (-V) f')' = exp (-V) * (f'' - V' * f')`.
This is only an ordinary derivative sta |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:43 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.deriv_gibbsWeight_mul_testDeriv_eq_langevinGenerator_1d
- Derivative-form version of
`hasDerivAt_gibbsWeight_mul_testDeriv_eq_langevinGenerator_1d`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.weightedDivergence_gibbsWeight_langevinGenerator_algebra
- Algebraic multidimensional handoff behind the weighted-divergence form of
the overdamped Langevin generator.
The hypotheses are intentionally supplied product-rule and chain-rule outputs:
`hdiv` stands for
`div (rho ∇ |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:78 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.expNeg_weightedDivergence_langevinGenerator_algebra
- Source-facing specialization of
`weightedDivergence_gibbsWeight_langevinGenerator_algebra` with the Gibbs weight
`rho = exp (-Vx)`. This is still only algebra after the product-rule and
chain-rule facts have been supp |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:95 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteCoord_weightedDivergence_langevinGenerator_algebra
- Finite-coordinate aggregation of supplied product-rule and chain-rule
identities for the weighted-divergence form of the Langevin generator.
Here `divCoord i` represents the already-supplied coordinate derivative
`∂ᵢ |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:114 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteCoord_named_weightedDivergence_langevinGenerator_algebra
- Named finite-coordinate wrapper for
`finiteCoord_weightedDivergence_langevinGenerator_algebra`.
The hypotheses `hlap` and `hinner` are supplied identifications of the
coordinate sums with a named Laplacian scalar and |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:147 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteCoord_toLpInner_weightedDivergence_langevinGenerator_algebra
- Finite-coordinate Langevin divergence-form handoff using the Mathlib
`EuclideanSpace` inner-product notation for the coordinate gradients.
The coordinate product rule, Gibbs-weight chain rule, and Laplacian-coordinate |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:170 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteCoord_euclideanInner_weightedDivergence_langevinGenerator_algebra
- Finite-coordinate Langevin divergence-form handoff using direct
`EuclideanSpace` inner-product notation for supplied coordinate gradients.
The coordinate product rule, Gibbs-weight chain rule, and Laplacian-coordinate |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:201 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_langevinGenerator_basisDisplay
- Pointwise finite-dimensional Euclidean display of the formal Langevin
differential expression.
This rewrites Mathlib's `Laplacian.laplacian f x - inner ℝ (gradient V x)
(gradient f x)` into a finite coordinate-basis s |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:227 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_langevinGenerator_coordinateDisplay
- Explicit coordinate-unit version of
`finiteEuclidean_langevinGenerator_basisDisplay`.
The additional `[DecidableEq ι]` instance is only used to unfold Mathlib's
`EuclideanSpace.basisFun` into `EuclideanSpace.single i |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:257 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_weightedDivergence_langevinGenerator_basisHandoff
- Supplied-hypothesis finite-coordinate handoff from weighted-divergence
algebra to the Mathlib pointwise expression `Δ f - <∇V, ∇f>`.
The hypotheses still provide the coordinate product-rule output, the
Gibbs-weight ch |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:274 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_weightedDivergence_langevinGenerator_coordinateHandoff
- Explicit coordinate-unit version of
`finiteEuclidean_weightedDivergence_langevinGenerator_basisHandoff`.
This is still a supplied-hypothesis algebra/display handoff. The theorem does
not prove the coordinate product |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:321 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_expNeg_weightedDivergence_langevinGenerator_basisHandoff
- Basis-coordinate handoff with the Gibbs-weight chain rule discharged by
Mathlib's gradient API.
The coordinate product-rule output and divergence-sum identity remain supplied
as hypotheses. The only removed hypothesi |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:374 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_expNeg_weightedDivergence_langevinGenerator_coordinateHandoff
- Coordinate-unit version of
`finiteEuclidean_expNeg_weightedDivergence_langevinGenerator_basisHandoff`.
This removes only the Gibbs-weight chain-rule hypothesis from the explicit
coordinate-unit display. Divergence, c |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:409 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_expNeg_lineDeriv_fderiv_coordinateSum_langevinGenerator_display
- Coordinate-line-derivative sum display for the explicit Gibbs-weighted
first-derivative field.
This theorem aggregates the compiled pointwise leaf
`lineDeriv_expNegPotential_mul_fderiv_coordinate_eq` across all finite |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:454 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.finiteEuclidean_expNeg_lineDeriv_fderiv_coordinateSum_langevinGenerator_display_of_differentiableAt
- Coordinate-line-derivative sum display with the local gradient-coordinate
bridge discharged.
Compared with
`finiteEuclidean_expNeg_lineDeriv_fderiv_coordinateSum_langevinGenerator_display`,
this theorem removes the su |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:521 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.coordinateDivergence_expNeg_fderivCoordinateField_langevinGenerator_display_of_differentiableAt
- Named coordinate-divergence version of the finite Euclidean Gibbs-weighted
first-derivative display.
The vector field is the coordinate representative
`y ↦ exp (-V y) * fderiv ℝ f y eᵢ`. The theorem only rewrites the |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:549 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.trace_expNeg_fderivCoordinateField_langevinGenerator_display_of_hasFDerivAt
- Trace-summand display for the explicit Pi-space vector field
`x ↦ exp (-V x) * fderiv f x eᵢ`.
This is the pointwise bridge from Mathlib's finite-box divergence-theorem trace
integrand to the Langevin display
`exp (-V |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:582 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.continuousOn_expNeg_langevinGenerator_rhs_of_components
- Continuity of the scalar Langevin display on a finite Pi-box from
component continuity.
The hypotheses keep the analytic regularity inputs explicit: continuity of the
potential, the Mathlib Laplacian display, and the |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:629 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.continuousOn_expNeg_langevinGenerator_rhs_of_contDiff
- Continuity of the scalar Langevin display on a finite Pi-box from global
`C¹/C²` test-function regularity.
The assumptions are deliberately global total-derivative hypotheses:
`V` is `C¹` and `f` is `C²` on the finite |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:674 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.hasFDerivAt_expNeg_fderivCoordinateField_of_differentiableAt
- The explicit Pi-space vector field
`z ↦ (i ↦ exp (-V (toLp z)) * fderiv f (toLp z) eᵢ)` is differentiable when
the potential is differentiable and the total first-derivative map of `f` is
differentiable at the transpor |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:725 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.hasFDerivAt_expNeg_fderivCoordinateField_of_contDiff
- Global `C¹/C²` version of
`hasFDerivAt_expNeg_fderivCoordinateField_of_differentiableAt`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:770 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrableOn_trace_expNeg_fderivCoordinateField_of_continuousOn
- Finite-box trace integrability for the explicit Gibbs-weighted Langevin
trace display, assuming the displayed scalar RHS is continuous on the box.
This closes the integrability handoff only under explicit regularity d |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:804 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrableOn_trace_expNeg_fderivCoordinateField_of_component_continuousOn
- Finite-box trace integrability for the explicit Gibbs-weighted Langevin
trace display from component continuity.
This is a convenience wrapper around
`integrableOn_trace_expNeg_fderivCoordinateField_of_continuousOn`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:868 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrableOn_trace_expNeg_fderivCoordinateField_of_contDiff
- Finite-box trace integrability for the explicit Gibbs-weighted Langevin
trace display from global `C¹/C²` regularity, still assuming the explicit
Pi-space trace field has the supplied Frechet derivative on the box.
Co |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:927 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrableOn_trace_expNeg_fderivCoordinateField_of_contDiff_fderiv
- Finite-box trace integrability for the explicit Gibbs-weighted Langevin
field under global `C¹/C²` regularity, with the field derivative chosen as
Mathlib's `fderiv`.
This removes the remaining supplied `hF` input fro |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:973 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrable_expNeg_langevinGenerator_rhs_of_contDiff_of_hasCompactSupport
- Whole-space integrability of the concrete Gibbs-weighted Langevin
generator display for a compactly supported `C²` test function.
The compact support belongs to the test function, not to the Gibbs weight.
Outside `tsu |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1017 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrable_expNeg_langevinGenerator_rhs_comp_toLp_of_contDiff_of_hasCompactSupport
- Raw finite-coordinate form of
`integrable_expNeg_langevinGenerator_rhs_of_contDiff_of_hasCompactSupport`.
This transports the Euclidean-space result through Mathlib's
volume-preserving `WithLp.toLp 2` equivalence. It |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1074 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrable_expNeg_comp_toLp_of_lintegral_expNeg_ne_top
- Whole-space integrability of the unnormalized Gibbs weight in raw
finite-Pi coordinates.
The hypothesis is the finite `ℝ≥0∞` Gibbs mass on Euclidean space. Continuity
supplies measurability, and Mathlib's volume-pres |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1104 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.tendsto_setIntegral_expNeg_norm_ge_comp_toLp_of_lintegral_expNeg_ne_top
- The unnormalized Gibbs mass outside expanding Euclidean balls tends to
zero, in the raw finite-Pi coordinates used by the radial cutoff route.
This combines finite Gibbs mass with the generic `L¹` tail theorem. It is |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1132 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integral_expNeg_langevinGenerator_rhs_eq_zero_of_contDiff_of_hasCompactSupport
- Whole-space Gibbs-weighted Langevin integration by parts for a compactly
supported `C²` test function:
`∫ exp (-V) * (Δ f - ⟪∇V, ∇f⟫) = 0`.
The proof builds the raw finite-Pi vector field `exp (-V) * Df`, proves that
|
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1164 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin.integrable_expNeg_fderivCoordinateField_of_lintegral_expNeg_ne_top_of_fderiv_norm_le
- Whole-space integrability of the Gibbs-weighted coordinate derivative
field from finiteness of the unnormalized Gibbs mass and a uniform operator
norm bound on the test-function derivative.
This theorem proves only so |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1253 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCanonicalFisherGamma.SmoothCanonicalPairDomain
- Local generator-domain agreement needed for the canonical density/log-ratio
pair. No assertion is made about arbitrary observables or about a closed
operator domain. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCanonicalFisherGamma |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinCanonicalFisherGamma.lean:48 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCanonicalFisherGamma.carreDuChamp_density_logRatio_eq_inner_ae
- Abstract carré du champ of the actual generator agrees almost everywhere
with the concrete Langevin gradient inner product on the canonical pair. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCanonicalFisherGamma |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinCanonicalFisherGamma.lean:84 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCanonicalFisherGamma.hasCanonicalFisherGamma
- The local operator agreement plus the score chain rule discharges the
canonical Fisher-Gamma contract used by the abstract Dirichlet layer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCanonicalFisherGamma |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinCanonicalFisherGamma.lean:112 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCanonicalFisherGamma.dirichletForm_density_logRatio_eq_information
- Concrete Langevin specialization of the canonical Dirichlet--Fisher edge.
All remaining obligations are now visibly split between the smooth local
operator-domain contract and the abstract pairwise stationarity/symmetr |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCanonicalFisherGamma |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinCanonicalFisherGamma.lean:133 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp.laplacian_mul
- The Laplacian product rule for two globally `C²` real observables. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinCarreDuChamp.lean:25 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp.gradient_mul
- The gradient product rule used by the Langevin drift cancellation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinCarreDuChamp.lean:104 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp.langevinCarreDuChamp_eq_inner
- Chewi Example 1.2.17: the carre-du-champ expression of the formal
Langevin differential operator equals the gradient inner product. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinCarreDuChamp.lean:114 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp.langevinCarreDuChamp_self_eq_norm_sq
- Diagonal form of Chewi Example 1.2.17. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinCarreDuChamp.lean:132 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.CompactlySupportedC2
- The compactly supported twice continuously differentiable test core used
for the finite-dimensional Langevin generator. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinGenerator.lean:28 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.operator
- The displayed overdamped Langevin differential operator associated with
the potential `V`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinGenerator.lean:34 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.CoreContract
- An explicit domain contract for a candidate Langevin generator.
`domain` is kept separate from the operator action: the first field requires
the whole `C_c²` test core to belong to the candidate domain, while the seco |
structure | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinGenerator.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.integral_operator_normalizedGibbs_eq_zero_on_compactlySupportedC2
- The normalized Gibbs measure annihilates the displayed Langevin operator
on the compactly supported `C²` core.
This is a normalized-measure corollary of the whole-space weighted-IBP theorem.
It is a core-level infinit |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinGenerator.lean:65 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator.isInvariantOn_normalizedGibbs_on_compactlySupportedC2
- A semigroup satisfying the integrated-generator contract on the
compactly supported `C²` core preserves normalized Gibbs expectations on that
core.
This theorem composes the concrete Gibbs integration-by-parts identit |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinGenerator.lean:90 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinKLDissipation.kl_hasDerivAt_eq_neg_information
- Source-facing smooth finite-domain Langevin entropy dissipation:
`d/dt KL(mu_t || pi) = - FI(mu_t || pi)`.
Every non-algebraic obligation remains explicit in the three domain contracts.
This is the Chapter 1.2 join n |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinKLDissipation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinKLDissipation.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.LocalProgressiveL2Integrand
- A progressive process whose squared time integral on `[0,T]` is finite
almost surely. No finite expected energy is assumed. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:28 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.squaredExtensionAt
- Zero extension of the squared process from `[0,b] × Ω`. The sample-space
measurable structure is the filtration at `b`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.squaredExtensionAt_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:44 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.squaredExtensionAt_apply_of_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:55 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.squaredExtensionAt_apply_of_not_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:64 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.accumulatedEnergyReal
- Real-valued accumulated energy. On the almost-sure finite-energy set it
agrees with the exact `ENNReal` accumulated energy and is continuous in time;
those comparison and continuity statements are proved downstream. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.accumulatedEnergyReal_stronglyMeasurable
- At each fixed time the real energy is measurable with respect to the
filtration at the stopped time `min t T`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:88 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.accumulatedEnergyReal_stronglyMeasurable_ambient
- Fixed-time energy is measurable in the ambient sample sigma-algebra. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:100 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization.nnrealLebesgue
|
abbrev | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Localization.lean:30 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization.stoppedIntegrand
- The stopped real integrand used in the local square-integrability
condition. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Localization.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization.IsLocalizingSequence
- Chewi Definition 1.1.12: an increasing stopping-time sequence which makes
the stopped integrand square-integrable on `[0,T]` and converges almost surely
to `T`.
The iterated `lintegral` is the literal nonnegative form |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Localization.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization.IsLocalMartingale
- Chewi Definition 1.1.15: an adapted process is a local martingale when a
monotone sequence of stopping times tends to infinity almost surely and every
stopped, initially centered process is a martingale.
The limit is |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Localization.lean:69 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.evolveMeasure
- Evolve a measure by the transition kernel at elapsed time `t`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:31 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.evolveMeasure_add
- Measure-level Chapman--Kolmogorov: evolving for `s+t` is the same as
first evolving for `s` and then for `t`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.isProbabilityMeasure_evolveMeasure
- A Markov kernel sends a probability law to a probability law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:49 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.lintegral_markovOperator_eq_lintegral_evolveMeasure
- Integrating a Markov observable against the input law is the same as
integrating the original observable against the evolved law.
This is the measure/operator compatibility identity behind the usual formula
`∫ P_t f d |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:62 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.IsStationary
- A measure is stationary for a transition-kernel semigroup when it is fixed
by every nonnegative-time law evolution.
This is the direct measure-level notion used in Chewi Proposition 1.2.7. The
generator characterizati |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:77 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.isStationary_iff
- Unfold the stationary-measure predicate. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:83 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.isStationary_iff_kernel_invariant
- ASTIS semigroup stationarity is exactly Mathlib kernel invariance at every
time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:91 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.isStationary_of_kernel_reversible
- Detailed balance for every transition kernel implies stationarity of the
whole semigroup. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:105 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.IsStationary.lintegral_markovOperator_eq
- Stationarity implies invariance of every measurable nonnegative expectation
under the Markov operator. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:117 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.isStationary_of_lintegral_markovOperator_eq
- Conversely, invariance of all measurable nonnegative expectations under
every Markov operator determines the stationary measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:126 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.isStationary_iff_lintegral_markovOperator_eq
- Measure stationarity is equivalent to expectation invariance for all
measurable nonnegative observables. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:142 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.IsStationary.evolveMeasure_eq
- A stationary law remains unchanged at every named time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:154 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.IsStationary.after
- Stationarity is preserved after any elapsed time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovMeasureEvolution |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovMeasureEvolution.lean:161 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.TransitionKernelContract
- A time-homogeneous Markov transition-kernel contract at nonnegative times.
`chapmanKolmogorov` is oriented so that first evolving for time `s` and then
for time `t` is the kernel composition `K t ∘ₖ K s`. The contrac |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.MeasurableENNReal
- The measurable nonnegative observables on a measurable state space. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:43 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.MeasurableENNReal.const
- A constant measurable nonnegative observable. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:50 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator
- The Markov operator induced by a transition-kernel contract. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator_const
- A Markov operator preserves constant nonnegative observables. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator_apply_mono
- Markov integration is monotone in the observable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:72 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator_zero
- At time zero, the transition-kernel Markov operator is the identity. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator_comp
- Chapman--Kolmogorov becomes composition of Markov operators. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:91 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator_comm
- Time-homogeneous Markov operators commute because nonnegative-time
addition is commutative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:106 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.chewi_lemma_1_2_2
- Chewi, Lemma 1.2.2: the zero-time and two-time Markov-operator laws,
under the explicit transition-kernel contract. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:119 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Martingale.IsChewiMartingale
- Chewi Definition 1.1.4 for a real process indexed by nonnegative time.
Mathlib's predicate includes strong adaptedness and the conditional-
expectation identity; integrability follows from that identity. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Martingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Martingale.lean:21 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Martingale.isChewiMartingale_const
- A constant real process is a martingale under a finite measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Martingale |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Martingale.lean:28 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.ContinuousLinearSemigroup
- A nonnegative-time semigroup of continuous linear operators on a real
normed space. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.ContinuousLinearSemigroup.op_add_apply
- Application form of the semigroup law. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:40 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.ContinuousLinearSemigroup.op_comm_apply
- Operators in a one-parameter semigroup commute. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.rightDifferenceQuotient
- The right difference quotient used to define the infinitesimal generator
in the chosen norm topology. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.HasRightGeneratorAt
- `g` is the right-generator value of `f` when the semigroup difference
quotient converges to `g` through strictly positive times. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:64 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.generatorDomain
- The domain of the right generator in the chosen norm topology. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.rightDifferenceQuotient_map
- The right difference quotient commutes with every semigroup operator. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:74 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.HasRightGeneratorAt.map
- The generator graph is invariant under the semigroup, and the generator
commutes with the semigroup on its domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:85 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.generatorDomain_map
- In particular, the right-generator domain is preserved by the semigroup. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:100 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.rightOrbitDifferenceQuotient
- The forward right difference quotient of the semigroup orbit at time `t`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:108 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.rightOrbitDifferenceQuotient_eq
- Semigroup algebra rewrites the orbit quotient at time `t` as the generator
quotient applied to `S.op t f`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:114 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator.kolmogorov_backward_right
- Chewi's backward-equation calculation, in a precise right-difference
quotient form: if `g` is the generator value of `f`, then the orbit derivative
at time `t` converges to `S.op t g`, and this is also the generator va |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGenerator.lean:126 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.StronglyContinuousSemigroup
- A continuous-linear semigroup whose orbit is strongly continuous at time
zero for every vector in the ambient normed space. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.StronglyContinuousSemigroup.tendsto_op_add
- Strong continuity at zero propagates to right continuity of every orbit at
an arbitrary nonnegative starting time. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_zero
- The zero vector has generator value zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:57 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightDifferenceQuotient_add
- Right difference quotients are additive in the observable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:63 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightDifferenceQuotient_smul
- Right difference quotients commute with scalar multiplication. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:72 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightDifferenceQuotient_neg
- Right difference quotients commute with negation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:81 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_unique
- The right-generator value is unique. The relevant one-sided filter is
nontrivial because positive nonnegative reals accumulate at zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:89 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_add
- Generator limits add. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:98 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_smul
- Generator limits commute with scalar multiplication. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:107 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_neg
- Generator limits commute with negation. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:115 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_sub
- Generator limits subtract. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:122 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.generatorDomainSubmodule
- The right-generator domain is a real submodule of the ambient normed
space. |
def | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:132 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGeneratorValue
- The canonical right-generator value on its submodule domain. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:147 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGeneratorValue_spec
- The canonical value really is the right-generator limit. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:154 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGeneratorValue_add
- The canonical generator value is additive. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:162 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGeneratorValue_smul
- The canonical generator value commutes with real scalar multiplication. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:173 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGenerator
- The infinitesimal right generator as a genuine linear map on its domain. |
def | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:182 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGenerator_map
- The canonical generator commutes with the semigroup on its invariant
domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:190 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.kolmogorov_backward_right_generator
- Chewi's right Kolmogorov backward equation using the canonical bundled
generator rather than an existential generator witness. |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:203 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveDriftIntegral.prefixIntegralProcess
- The finite-time Bochner drift primitive. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveDriftIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveDriftIntegral.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveDriftIntegral.prefixIntegralProcess_eq_fixedHorizon
- On a larger fixed horizon, the moving-prefix integral can be represented
by an indicator integrand against one fixed finite time measure. Clipping the
time fed to `b` by `T` makes the integrand globally well-typed for |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveDriftIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveDriftIntegral.lean:42 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveDriftIntegral.prefixIntegralProcess_stronglyProgressive
- Progressive measurability is preserved by deterministic prefix Bochner
integration. No pathwise integrability hypothesis is needed for this
measurability theorem. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveDriftIntegral |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveDriftIntegral.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.SatisfiesUsualConditions
- The usual conditions needed in Chewi's stochastic-calculus setup.
`completeAt` says that every ambient `mu`-null set belongs to every time
sigma-algebra. Right continuity uses Mathlib's right-continuation interface. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:28 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.processFunction
- The product-space representative, with sample point first and time
second, matching `processTimeMeasure`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand
- A progressively measurable process with finite global `L2` energy on
`[0,T]`. Keeping `process` as data preserves filtration information that an
abstract `Lp` element alone would erase. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:42 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.toLp
- The canonical product-space `Lp` representative. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:55 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictProcess
- Restrict a process to times strictly before `t`. This representative
differs from the closed interval convention only at one Lebesgue-null time. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictProcess_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:65 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictProcess_nested
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictProcess_progressive
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:79 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.processFunction_restrictProcess
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:92 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictProcess_memLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:99 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt
- Restriction preserves the progressive `L2` domain. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:109 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:116 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt_zero_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:121 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt_zero_toLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:126 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt_nested_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:132 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.norm_restrictAt_le
- Time restriction cannot increase the product-space `L2` norm. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:138 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.zero
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:25 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.add
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:30 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.neg
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.sub
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:42 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.smul
|
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:48 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.zero_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:54 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.add_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.neg_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:62 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.sub_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:66 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.smul_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:75 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_add
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:79 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_neg
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:83 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_sub
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:87 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:91 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_restrictAt_sub
- Restriction commutes with subtraction in product-space `L2`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:96 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_restrictAt_add
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:108 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_restrictAt_smul
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:120 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_restrictAt_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:131 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.norm_restrictAt_sub_le
- Restricting both integrands cannot increase their product-space `L2`
distance. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:140 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.restrictAt_toLp_eq_of_toLp_eq
- Equality in product-space `L2` is preserved by every deterministic time
restriction. This is the congruence principle used to turn completed
integrand identities into process-level Itô identities. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:150 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.fastTolerance
- Rapid geometric tolerance used for all diagonal choices. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:26 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.fastTolerance_eq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.fastTolerance_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.fastTolerance_tendsto_zero
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.summable_fastTolerance
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:47 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.summable_scaled_fastTolerance_sq
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:53 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.eventualThreshold
- A threshold beyond which an eventual predicate always holds. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:65 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.eventualThreshold_spec
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:69 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.strictSelection
- Recursively strictify eventual thresholds without losing their bounds. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:76 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.eventualThreshold_le_strictSelection
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:81 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.strictSelection_spec
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:87 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.strictMono_strictSelection
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:92 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.TruncationGood
- Truncation levels meeting the `n`th fast tolerance. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:101 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.eventually_truncationGood
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:105 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.truncationIndex
- Strictly increasing clipping index selected from clipping convergence. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:114 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.truncationIndex_spec
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:118 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.truncationIndex_strictMono
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:123 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.DyadicGood
- Dyadic levels meeting the discretization half of the `n`th tolerance. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:129 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.eventually_dyadicGood
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:136 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.dyadicLevel
- Strictly increasing dyadic level selected after clipping has been fixed. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:151 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.dyadicLevel_spec
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:155 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.dyadicLevel_strictMono
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:160 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.DyadicElementaryProcess
- A dyadic elementary process with its level recorded in the type. |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:166 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.canonicalElementaryApprox
- Canonical fast diagonal approximation. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:174 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.DyadicElementaryProcess.toLp
- Product-space `L2` embedding of a heterogeneous dyadic process. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:183 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.norm_canonicalElementaryApprox_sub_lt
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:189 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.tendsto_canonicalElementaryApprox_toLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:205 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.progressiveL2_elementary_dense
- Genuine density of bounded dyadic elementary adapted processes in the
progressive product-space `L2` domain. |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:216 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.horizonPrefix
- Product-space prefix corresponding to times strictly before `T`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:30 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.measurableSet_horizonPrefix
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.restrict_processTimeMeasure_horizonPrefix
- Restricting the larger process-time measure to the strict smaller prefix
recovers the smaller process-time measure exactly. The only omitted point is
the terminal slice, which is time-null. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:40 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.processFunction_restrictProcess_eq_indicator
- The product representative of deterministic zero extension is exactly the
indicator of the strict time prefix. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:51 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.extendByZero
- Extend a progressive `L²` integrand from `T₁` to `T₂ ≥ T₁` by zero after
`T₁`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:64 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.extendByZero_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:76 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.norm_extendByZero_eq
- Zero extension preserves the product-space `L²` norm exactly. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:83 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.extendByZero_sub_toLp
- Zero extension commutes with subtraction in `L²`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:95 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.norm_extendByZero_sub_extendByZero_eq
- Zero extension is an isometry for the product-space `L²` distance. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:109 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping.measurableSet_stoppingSet
- The product-space closed stopping event is measurable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Stopping.lean:34 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping.stoppedIntegrand_stronglyProgressive
- Chewi's closed stopping convention preserves strong progressiveness. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Stopping.lean:46 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping.stoppedIntegrand_memLp
- The closed stopped integrand remains in product-space `L²`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Stopping.lean:92 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping.stop
- Generic closed stopping operator on completed progressive `L²` integrands. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Stopping.lean:102 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping.stop_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Stopping.lean:112 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping.norm_stop_sub_stop_le
- Closed stopping is a contraction on the completed progressive `L²` space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Stopping.lean:119 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping.tendsto_stop_toLp_of_tendsto
- Stopping preserves convergence in the completed progressive `L²` space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Stopping.lean:130 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipProcess
- Pointwise clipping of a stochastic process. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:26 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipProcess_stronglyProgressive
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.processFunction_clipProcess
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipProcess_memLp
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:41 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipped
- Clipping as an endomorphism of the progressive `L2` integrand domain. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:49 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipped_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:56 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipped_abs_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.tendsto_clipped_toLp
- Bounded progressive truncations converge to the original integrand in
the actual product-space `Lp` object. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:69 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingBoundary.stopRefined_coeff_eq_zero_of_stoppingValue_eq_zero
- At a sample point where the stopping value is zero, every coefficient of
any refined elementary process stopped by that random time is exactly zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingBoundary |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingBoundary.lean:34 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingBoundary.stopRefined_elementaryItoIntegral_eq_zero_of_stoppingValue_eq_zero
- If the stopping value at the chosen sample point is zero, every refined
stopped elementary Ito sum is exactly zero. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingBoundary |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingBoundary.lean:60 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingBoundary.tendsto_stopRefined_elementaryItoIntegral
- Pointwise stopped-Ito convergence for an elementary integrand and an
arbitrary bounded nonnegative stopping value. At positive stopping values the
finite stopped sum is exactly evaluation at the dyadic right endpoint |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingBoundary |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingBoundary.lean:83 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox.IsBoundedNNRealStoppingTime
- A finite nonnegative stopping time bounded by the construction horizon. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingDyadicApprox.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox.stopRefined_coeff_eq_rightCutoff
- At a sample point where `tau` is positive, stopping the refined elementary
integrand by the original stopping time retains exactly the same coefficients
as the deterministic right-grid cutoff of the cell containing `ta |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingDyadicApprox.lean:45 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox.stopRefined_elementaryItoIntegral_eq_rightApprox
- At a positive sample value of the bounded stopping time, the *whole*
finite Itô sum of the refined process stopped by the original random time is
exactly the original elementary Itô sum evaluated at the deterministic r |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingDyadicApprox.lean:106 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox.tendsto_rightApproxTime_stoppingValue
- For every positive sample value of the bounded stopping time, the right
endpoints chosen on successively finer dyadic refinements converge to that
sample value. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingDyadicApprox.lean:159 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox.tendsto_rightApproxTime_stoppingValue_nhdsWithin
- The same convergence, recorded in the subspace topology of the construction
interval. This is the exact interface needed to compose with a path that is
known to be continuous only on `[0,T]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingDyadicApprox.lean:173 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox.tendsto_continuousOn_rightApproxTime_stoppingValue
- Continuous paths may be evaluated along the dyadic right approximations:
if the path is continuous on the construction interval, its values at the
selected right endpoints converge to its value at the original stopping |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingDyadicApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingDyadicApprox.lean:192 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingGeneralIto.tendsto_stoppedCanonical_toLp
- The stopped canonical elementary approximants converge to the generic
closed stop of the completed integrand. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingGeneralIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingGeneralIto.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingGeneralIto.tendsto_stoppedCanonical_terminal
- Ito isometry transfers stopped-integrand convergence to terminal `L²`
convergence. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingGeneralIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingGeneralIto.lean:55 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingGeneralIto.itoIntegralTerminal_stop_ae
- General bounded random-stopping identity.
For `tau ≤ T`, the completed Ito integral of `eta_s 1_{s ≤ tau}` equals the
continuous Ito process of `eta` evaluated at `tau`, almost surely. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingGeneralIto |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingGeneralIto.lean:82 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingIntegrandLimit.tendsto_stopRefinedDyadic_value_stoppedIntegrand
- The randomly stopped dyadic refinements converge at every time/sample pair
to the closed stopped integrand used in Chewi's localization definition.
The endpoint `s = tau omega` is included: every dyadic right endpoint |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingIntegrandLimit |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingIntegrandLimit.lean:42 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingItoTerminal.tendsto_stopRefinedDyadic_terminalToLp
- Product-space convergence of the stopped dyadic refinements transfers
through the completed Ito isometry to terminal `L2(mu)` convergence. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingItoTerminal |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingItoTerminal.lean:41 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingItoTerminal.itoIntegralTerminal_stopped_elementary_ae
- Elementary random-stopping consistency. The completed Ito integral of
Chewi's closed stopped elementary integrand is represented almost everywhere
by evaluating the original elementary Ito path at the bounded stop |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingItoTerminal |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingItoTerminal.lean:75 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction.stoppingSet
- Product-space event on which a closed stopped integrand is active. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingL2Contraction.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction.processFunction_stoppedIntegrand_eq_indicator
- Chewi's closed stopped integrand is literally multiplication by the
product-space stopping indicator. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingL2Contraction.lean:40 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction.norm_stopped_sub_le
- Stopping contraction. Any two already-constructed closed stopped
representatives are no farther apart in product-space `L²` than their original
integrands. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingL2Contraction.lean:60 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction.norm_stoppedElementary_sub_target_le
- Specialization of the contraction to a dyadic elementary approximant and
its legal closed random stop. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingL2Contraction.lean:103 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction.tendsto_stoppedCanonicalApprox_toLp
- Closed stopping preserves convergence of the canonical elementary density
sequence. This is the analytic extension step needed before stochastic
integration can be commuted with a bounded random stopping time for an a |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Contraction |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingL2Contraction.lean:133 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Convergence.valueBound_nonneg
- The deterministic coefficient-sum bound used below is nonnegative. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Convergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingL2Convergence.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Convergence.abs_stopRefinedDyadic_sub_stoppedIntegrand_le
- The pointwise error between a stopped refinement and its stopped target is
bounded by twice the original elementary-process bound. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Convergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingL2Convergence.lean:44 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Convergence.tendsto_integral_sq_stopRefinedDyadic_sub
- Dominated convergence for the squared product-space error. This is the
measure-theoretic core of random-stopping convergence. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Convergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingL2Convergence.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Convergence.tendsto_stopRefinedDyadic_toLp
- The stopped dyadic refinements converge to the stopped elementary
integrand in the actual product-space `L2` object. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingL2Convergence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingL2Convergence.lean:143 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox.stopRefinedDyadic
- The refined elementary process stopped by the original random time,
repackaged with its regular dyadic grid. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessApprox.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox.stopRefinedDyadic_level
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessApprox.lean:49 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox.stopRefinedDyadic_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessApprox.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox.stopRefinedDyadic_value_eq_rightApprox
- At a positive stopping value, not only the coefficients and terminal Ito
sum but the whole stopped refined time process agrees exactly with the
process cut off at the deterministic right endpoint of the fine cell
conta |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessApprox.lean:75 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox.stopRefinedDyadic_value_eq_zero_of_stoppingValue_eq_zero
- At a zero stopping value, the whole stopped refined time process vanishes,
not merely its terminal finite Ito sum. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessApprox.lean:101 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency.ae_time_ne
- Every deterministic time slice is null under the product process-time
measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessConsistency.lean:44 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency.minStoppingValue_isChewiStoppingTime
- The pointwise minimum of a finite-valued stopping time and a deterministic
time is again a finite-valued stopping time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessConsistency.lean:56 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency.restrictAt_stop_toLp_eq_stop_min
- Restricting a closed stopped integrand at deterministic time `t` is the
same element of product-space `L²` as stopping the original integrand at
`min tau t`. The representatives differ at most on the deterministic sli |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessConsistency.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency.itoIntegralProcess_stop_ae
- Process-level bounded random-stopping identity.
For every deterministic `t ≤ T`, the completed Itô process of the closed
stopped integrand agrees almost surely with the original continuous Itô
process evaluated at |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessConsistency.lean:132 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency.itoIntegralProcess_stop_eq_stoppedProcess_ae
- Stopped-process form of `itoIntegralProcess_stop_ae`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessConsistency.lean:178 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency.stoppedProcess_coe_apply
- Finite-valued stopped processes are ordinary composition with
`t ↦ min t (tau omega)`. We unfold Mathlib's definition so the finite
`WithTop` value reduces definitionally, avoiding any theorem that expects the
stoppin |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessConsistency.lean:214 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency.itoIntegralProcess_stop_eq_stoppedProcess_pathwise_ae
- Pathwise bounded random-stopping identity.
On one full-measure event, the completed Itô process of the stopped integrand
and the stopped continuous Itô process agree simultaneously at every time in
`[0,T]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessConsistency.lean:236 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2.stoppedIntegrand_stronglyProgressive
- Chewi's closed stopped elementary integrand is strongly progressive. The
proof is by pointwise limit of the already legal elementary stopped
refinements, so the stopping-time measurability is inherited rather than
rec |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProgressiveL2.lean:38 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2.stoppedProcessFunction_stronglyMeasurable
- The sample-first product representative of the stopped integrand is
strongly measurable. This is the product-space analogue of the progressive
limit theorem above. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProgressiveL2.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2.abs_stoppedIntegrand_le_valueBound
- Stopping never increases the deterministic elementary-process value bound. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProgressiveL2.lean:80 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2.abs_stopRefinedDyadic_value_le_valueBound
- Every legal stopped dyadic refinement is dominated by the same bound as
the original elementary process. This uniform bound is the domination used
by the next product-space `L2` convergence layer. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProgressiveL2.lean:97 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2.stoppedIntegrand_memLp_two
- The stopped elementary integrand belongs to the product-space `L2` domain
on every finite horizon and finite sample measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProgressiveL2.lean:125 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2.stoppedProgressiveL2
- The actual progressive `L2` object represented by Chewi's stopped
integrand. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProgressiveL2.lean:148 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2.stoppedProgressiveL2_process
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2 |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProgressiveL2.lean:160 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Reversibility.IsReversible
- Chewi Definition 1.2.10: every time operator is symmetric in the ambient
real Hilbert-space inner product. Taking `H = L²(pi)` gives the source
definition of reversibility with respect to `pi`. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Reversibility |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Reversibility.lean:29 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Reversibility.isReversible_identity
- The constant identity semigroup is reversible on every real inner-product
space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Reversibility |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Reversibility.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ReversibleGenerator.inner_rightDifferenceQuotient_eq
- Reversibility already makes every finite positive-time generator difference
quotient symmetric. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ReversibleGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ReversibleGenerator.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ReversibleGenerator.inner_rightGenerator_eq
- The canonical right generator of a reversible semigroup is symmetric on
its generator domain.
No closedness, self-adjointness, or concrete `L²(pi)` realization is claimed;
this is exactly the pairwise identity inherit |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ReversibleGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ReversibleGenerator.lean:55 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.regularGridTimes
- Equally spaced endpoints with mesh `delta`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:27 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.regularGridTimes_strictMono
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:30 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClipped
- Sample at each deterministic left endpoint and clip at a natural level.
The result inhabits the actual elementary-process structure used by the Ito
isometry, including its strict grid, filtration measurability, and bou |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:40 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClipped_times
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:59 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClipped_coeff
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:67 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClipped_coeff_abs_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:76 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.dyadicMesh
- Mesh for the level-`level` dyadic partition of `[0,T]`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:85 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.dyadicMesh_pos
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:88 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClippedDyadic
- Canonical clipped left-step process on the dyadic partition of `[0,T]`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:93 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClippedDyadic_last_time
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:100 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StationarityEquivalence.chewi_proposition_1_2_7_invariant_implies_generator_zero
- Chewi Proposition 1.2.7, invariant-to-infinitesimal direction:
`ell(P_t f) = ell(f)` for all `t >= 0` implies `ell(Af) = 0` on the
right-generator domain. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StationarityEquivalence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StationarityEquivalence.lean:34 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StationarityEquivalence.chewi_proposition_1_2_7_generator_zero_implies_invariant
- Chewi Proposition 1.2.7, infinitesimal-to-invariant direction on an
explicit generator domain.
The integrated-generator contract is precisely the analytic input needed to
turn `∫ Lf dμ = 0` into `∫ P_t f dμ = ∫ f dμ`; |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StationarityEquivalence |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StationarityEquivalence.lean:49 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull.stoppingGraph
- The graph of a possibly-infinite nonnegative stopping time in
`Omega × ℝ≥0`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingGraphNull.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull.measurableSet_stoppingGraph
- A Chewi stopping time has a measurable graph in product space. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingGraphNull.lean:37 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull.timeSection_stoppingGraph_zero
- Every fixed-sample-path section of the stopping graph has zero stopped
Lebesgue-time measure. The proof avoids choosing an `untop`: if a finite
section point exists, injectivity of the `WithTop` coercion makes the who |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingGraphNull.lean:65 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull.processTimeMeasure_stoppingGraph_zero
- The stopping-time graph is null under Chewi's product process-time
measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingGraphNull.lean:93 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull.ae_notMem_stoppingGraph
- Almost every product-space point avoids the stopping-time graph. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingGraphNull.lean:103 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull.ae_time_ne_stoppingTime
- Pointwise form of `ae_notMem_stoppingGraph`: almost every product-space
point has time coordinate different from the stopping time. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingGraphNull.lean:112 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime.IsChewiStoppingTime
- Chewi Definition 1.1.11: at time `t`, the information in the filtration
decides whether the extended nonnegative stopping time has occurred. |
def | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingTime.lean:27 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime.isChewiStoppingTime_const
- Constant nonnegative times satisfy the source stopping-time definition. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingTime.lean:33 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime.IsChewiStoppingTime.min
- The pointwise minimum of two Chewi stopping times is again a stopping
time. This is the stopping-time algebra needed for repeated stopping. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingTime.lean:41 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime.IsChewiStoppingTime.min_const
- Truncating a stopping time by a deterministic nonnegative horizon preserves
the stopping-time property. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingTime.lean:51 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime.stoppedProcess_stoppedProcess_inf
- Repeated stopping is exactly stopping at the pointwise infimum. This is a
pure process identity; no stopping-time or martingale hypotheses are needed. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingTime.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime.stoppedProcess_stoppedProcess_of_le
- If the second stopping time occurs no later than the first, stopping twice
reduces to the earlier stop. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingTime |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/StoppingTime.lean:70 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.nnrealLebesgue
- Lebesgue measure on nonnegative real time, pulled back along the canonical
embedding into the real line. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:21 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.upTo
- Finite Lebesgue measure on nonnegative time up to `T`. Defining the
restriction before pulling back supplies Mathlib's finite-measure instance. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:27 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.upTo_eq_restrict_nnrealLebesgue
- The stopped time measure is literally nonnegative Lebesgue measure
restricted to `[0,T]`. This bridge lets source statements written with a
restricted Lebesgue integral reuse the finite `upTo T` measure used by the
It |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:39 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.restrict_upTo_Iio_eq_of_le
- Restricting two larger finite horizons to the same earlier prefix gives
exactly the same time measure. This is the cross-horizon consistency used by
global localization. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:71 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.upTo_univ
- The total mass of nonnegative time stopped at `T` is `T`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:89 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.upTo_singleton
- Stopped Lebesgue time has no atoms. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:113 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.upTo_Ioc
- The mass of `(a, b]` under time measure stopped at `T` is the length of
the clipped interval. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:127 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.ae_mem_Ioc_zero_upTo
- Stopped nonnegative Lebesgue time lies in `(0,T]` almost everywhere;
the omitted initial endpoint is null. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:157 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.restrict_upTo_Ioc_zero
- Restricting stopped time to `(0,T]` leaves the measure unchanged. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:164 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.upTo_Ioi_terminal
- The finite time measure is supported on `[0,T]`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:169 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.ae_le_terminal
- Almost every time under `upTo T` lies below the terminal horizon. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:195 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.ae_lt_terminal
- The terminal endpoint itself is null, so almost every stopped time is
strictly before `T`. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:201 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.restrict_upTo_Iio_terminal
- Restricting `[0,T]` to the open terminal prefix `[0,T)` changes nothing,
because the omitted endpoint has zero Lebesgue mass. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:209 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realZeroExtension
- Extend a function on nonnegative time by zero to the negative real axis. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:27 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realZeroExtension_measurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:30 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realZeroExtension_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realZeroExtension_coe
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:40 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realZeroExtension_eq_zero_of_neg
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:44 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.range_nnreal_coe
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:49 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.map_restrict_upTo_Ioc
- On an interval contained in `[0,T]`, pushing `upTo T` forward along the
canonical embedding gives ordinary Lebesgue measure restricted to that real
interval. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:61 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.integral_upTo_restrict_Ioc_eq_real
- Exact Bochner-integral bridge on `(a,b]`. No endpoint regularity is
assumed; the interval convention agrees on both sides. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:87 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.ae_restrict_upTo_Ioc_iff_real
- An a.e. statement on a nonnegative interval is equivalent to its real
zero-coordinate form under ordinary restricted Lebesgue measure. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:110 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.ae_prod_restrict_upTo_of_forall_ae
- Upgrade pointwise-in-`omega` time-a.e. facts to product-a.e. facts once
the target event is known measurable. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:125 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection
- Real-time section of the clipped process, zero outside `[0,T]`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:135 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_coe
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:141 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_eq_zero_of_neg
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:149 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_eq_zero_of_T_lt
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:155 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_abs_le
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:172 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_stronglyMeasurable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:185 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_integrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:193 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_locallyIntegrable
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:219 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsStandardBrownianMotion.hasIndepIncrements
- Chewi's arbitrary finite-family independent-increment clause implies
Mathlib's consecutive-grid `HasIndepIncrements` predicate. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.VectorBrownianFiltration |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/VectorBrownianFiltration.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsStandardBrownianMotion.projected_hasIndepIncrements
- Every continuous linear projection of a Chewi-standard vector Brownian
motion has Mathlib independent increments. All projections still come from
the same vector process. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.VectorBrownianFiltration |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/VectorBrownianFiltration.lean:66 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion.IsStandardBrownianMotionWithFiltration
- Source-level Brownian-filtration contract for a vector process.
The process itself is Chewi-standard. Adaptedness and independence of each
future *vector* increment from the whole past filtration are added explicitly |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.VectorBrownianFiltration |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/VectorBrownianFiltration.lean:78 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation.dualAction
- The dual action of a semigroup on a continuous linear functional:
`Pₜ* ℓ = ℓ ∘ Pₜ`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakForwardEquation.lean:36 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation.dualAction_apply
|
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakForwardEquation.lean:42 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation.dualAction_zero
- The dual action starts at the identity. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakForwardEquation.lean:49 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation.dualAction_add
- Dual Chapman--Kolmogorov law. The order is reversed by composition, as
expected for the adjoint action. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakForwardEquation.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation.rightDualPairingDifferenceQuotient
- Right difference quotient of the weak pairing
`⟨f, Pₜ* ell⟩ = ell (Pₜ f)`. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakForwardEquation.lean:68 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation.rightDualPairingDifferenceQuotient_eq
- The weak dual quotient is obtained by applying the functional to the
ordinary semigroup orbit quotient. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakForwardEquation.lean:76 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation.kolmogorov_forward_weak_right
- Weak right-hand Kolmogorov forward equation.
For an observable in the canonical generator domain, differentiating the dual
pairing gives the evolved functional applied to the generator:
`d⁺/dt (Pₜ* ell)(f) = (Pₜ* ell |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakForwardEquation |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakForwardEquation.lean:95 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator.IsInvariantOn
- Invariance of a measure for a nonnegative-time operator family on an
explicit test class. |
def | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakGenerator.lean:25 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator.IntegratedSemigroupGeneratorContract
- Semigroup and integrated-generator data sufficient for the standard
generator-to-invariance argument on an explicit domain.
The time parameter is represented by `ℝ`, but every law is required only for
nonnegative time |
structure | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakGenerator.lean:35 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator.isInvariantOn_of_integral_generator_eq_zero
- A semigroup is invariant on its declared generator domain when integrated
generator action vanishes throughout that domain.
This theorem is the operator-domain-to-invariance bridge. All analytic
content is visible in |
theorem | Partial | Compiled | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakGenerator.lean:58 |
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator.weakGeneratorFromSampleDerivative
- Move a supplied sample-space generator derivative to a named law path.
In SDE applications, `hderiv` is usually the Ito-generator derivative for a
test function composed with a process, while `hDrift` and `hDiffusion` |
theorem | Partial | Partial | AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator |
AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/WeakGenerator.lean:84 |
AutoSamplingTheory.TechnicalLemmas.Taylor.hessianOpNormOfSourceHessianField
- A source-backed Hessian representative supplies the operator-norm bound on
`fderiv (fderiv f)`.
This is the reusable version of a SALD Brownian/Ito bridge: once source
correspondence gives a Hessian field and a unifor |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Taylor |
AutoSamplingTheory/TechnicalLemmas/Taylor.lean:35 |
AutoSamplingTheory.TechnicalLemmas.Taylor.iteratedFDerivTwoOpNormOfFDerivFDerivOpNorm
- Convert an operator-norm bound on `fderiv (fderiv f)` to the corresponding
Mathlib `iteratedFDeriv` bound of order two. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Taylor |
AutoSamplingTheory/TechnicalLemmas/Taylor.lean:51 |
AutoSamplingTheory.TechnicalLemmas.Taylor.stdOrthonormalBasisUnit
- Standard orthonormal-basis vectors are unit directions. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Taylor |
AutoSamplingTheory/TechnicalLemmas/Taylor.lean:68 |
AutoSamplingTheory.TechnicalLemmas.Taylor.quadraticVariationNormalizationOfCoeffDefAndVarianceOne
- Algebraic packaging for quadratic-variation normalization. |
theorem | Partial | Not mapped | AutoSamplingTheory.TechnicalLemmas.Taylor |
AutoSamplingTheory/TechnicalLemmas/Taylor.lean:75 |
AutoSamplingTheory.Tests.CarreDuChamp.zeroGenerator
|
def | Partial | Not mapped | Tests.CarreDuChamp |
Tests/CarreDuChamp.lean:18 |
AutoSamplingTheory.Tests.CoordinateHeatBath.retained_marginal
|
theorem | Partial | Not mapped | Tests.CoordinateHeatBath |
Tests/CoordinateHeatBath.lean:25 |
AutoSamplingTheory.Tests.CoordinateHeatBath.coarseCoordinates
|
def | Partial | Not mapped | Tests.CoordinateHeatBath |
Tests/CoordinateHeatBath.lean:77 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.fiber_ne_zero_of_atom
|
theorem | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:13 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.singleton_transition
|
theorem | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:31 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.State
|
abbrev | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:51 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.a
|
def | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:52 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.b
|
def | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:53 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.c
|
def | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:54 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.off
|
def | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:55 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.joint
|
def | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:58 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.fiber_a
|
theorem | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:72 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.fiber_a_mass
|
theorem | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:78 |
AutoSamplingTheory.Tests.CoordinateHeatBathConditional.null_fiber
|
theorem | Partial | Not mapped | Tests.CoordinateHeatBathConditional |
Tests/CoordinateHeatBathConditional.lean:125 |
AutoSamplingTheory.Tests.FellerSemigroup.identityFellerContract
- The identity transition kernel is the basic Feller semigroup. |
theorem | Partial | Not mapped | Tests.FellerSemigroup |
Tests/FellerSemigroup.lean:17 |
Tests.IsotropicGaussianDensity.real_scale_density
|
theorem | Partial | Not mapped | Tests.IsotropicGaussianDensity |
Tests/IsotropicGaussianDensity.lean:8 |
Tests.IsotropicGaussianDensity.two_dimensional_density
|
theorem | Partial | Not mapped | Tests.IsotropicGaussianDensity |
Tests/IsotropicGaussianDensity.lean:16 |
Tests.IsotropicGaussianDensity.zero_dimensional_density
|
theorem | Partial | Not mapped | Tests.IsotropicGaussianDensity |
Tests/IsotropicGaussianDensity.lean:25 |
AutoSamplingTheory.Tests.GeneralItoIntegral.unitElementary
- A one-cell deterministic unit integrand used to exercise the complete
general-Ito construction rather than only checking its declaration names. |
def | Partial | Not mapped | Tests.ItoIntegralProcess |
Tests/ItoIntegralProcess.lean:41 |
AutoSamplingTheory.Tests.GeneralItoIntegral.unitDyadic
|
def | Partial | Not mapped | Tests.ItoIntegralProcess |
Tests/ItoIntegralProcess.lean:49 |
AutoSamplingTheory.Tests.GeneralItoIntegral.unitDyadic_elementaryItoProcess
|
theorem | Partial | Not mapped | Tests.ItoIntegralProcess |
Tests/ItoIntegralProcess.lean:55 |
AutoSamplingTheory.Tests.KernelMixture.lazyHeatBathComponents
|
def | Partial | Not mapped | Tests.KernelMixture |
Tests/KernelMixture.lean:26 |
AutoSamplingTheory.Tests.KernelTransport.coordinateSplit
|
def | Partial | Not mapped | Tests.KernelTransport |
Tests/KernelTransport.lean:23 |
AutoSamplingTheory.Tests.KernelTransport.coordinateUpdate
|
def | Partial | Not mapped | Tests.KernelTransport |
Tests/KernelTransport.lean:34 |
AutoSamplingTheory.Tests.KernelTransport.coordinateUpdate_invariant
|
theorem | Partial | Not mapped | Tests.KernelTransport |
Tests/KernelTransport.lean:42 |
AutoSamplingTheory.Tests.OperatorGenerator.identitySemigroup
- The constant identity family is the simplest continuous-linear semigroup. |
def | Partial | Not mapped | Tests.OperatorGenerator |
Tests/OperatorGenerator.lean:18 |
AutoSamplingTheory.Tests.OperatorGeneratorDomain.identityStronglyContinuousSemigroup
- The identity family is a strongly continuous semigroup. |
def | Partial | Not mapped | Tests.OperatorGeneratorDomain |
Tests/OperatorGeneratorDomain.lean:16 |
AutoSamplingTheory.Tests.OperatorGeneratorDomain.identity_mem_generatorDomain
|
theorem | Partial | Not mapped | Tests.OperatorGeneratorDomain |
Tests/OperatorGeneratorDomain.lean:51 |
Tests.ProximalBPSGaussianAugmentation.dirac_real_input
|
theorem | Partial | Not mapped | Tests.ProximalBPSGaussianAugmentation |
Tests/ProximalBPSGaussianAugmentation.lean:8 |
Tests.ProximalBPSGaussianAugmentation.gaussian_two_dimensional_input
|
theorem | Partial | Not mapped | Tests.ProximalBPSGaussianAugmentation |
Tests/ProximalBPSGaussianAugmentation.lean:17 |
AutoSamplingTheory.Tests.RandomScanHeatBath.State
|
abbrev | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:11 |
AutoSamplingTheory.Tests.RandomScanHeatBath.a
|
def | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:12 |
AutoSamplingTheory.Tests.RandomScanHeatBath.b
|
def | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:13 |
AutoSamplingTheory.Tests.RandomScanHeatBath.c
|
def | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:14 |
AutoSamplingTheory.Tests.RandomScanHeatBath.d
|
def | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:15 |
AutoSamplingTheory.Tests.RandomScanHeatBath.uniform
|
def | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:17 |
AutoSamplingTheory.Tests.RandomScanHeatBath.uniform_a_pos
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:26 |
AutoSamplingTheory.Tests.RandomScanHeatBath.diagonal
|
def | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:72 |
AutoSamplingTheory.Tests.RandomScanHeatBath.diagonal_a_pos
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:81 |
AutoSamplingTheory.Tests.RandomScanHeatBath.null_fibers
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBath |
Tests/RandomScanHeatBath.lean:122 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.State
|
abbrev | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:38 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.a
|
def | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:39 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.b
|
def | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:40 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.d
|
def | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:41 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.nonuniform
|
def | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:43 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.nonuniform_a_pos
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:52 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.nonuniform_b_pos
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:56 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.forward_move
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:60 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.reverse_move
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:71 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.forbidden_mass
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:97 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.null_fibers
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:109 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.diagonal
|
def | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:128 |
AutoSamplingTheory.Tests.RandomScanHeatBathReversibility.diagonal_a_pos
|
theorem | Partial | Not mapped | Tests.RandomScanHeatBathReversibility |
Tests/RandomScanHeatBathReversibility.lean:137 |
AutoSamplingTheory.Tests.Reversibility.identitySemigroup
|
def | Partial | Not mapped | Tests.Reversibility |
Tests/Reversibility.lean:14 |
AutoSamplingTheory.Tests.SemigroupDecay.zeroDissipationCurve
- The identically zero curve exercises every interface without adding an
analytic assumption hidden inside the tests. |
def | Partial | Not mapped | Tests.SemigroupDecay |
Tests/SemigroupDecay.lean:20 |
diagonalH
|
def | Partial | Not mapped | Tests.Shared.QuadraticGradientDescent |
Tests/Shared/QuadraticGradientDescent.lean:8 |
diagonalH_symmetric
|
theorem | Partial | Not mapped | Tests.Shared.QuadraticGradientDescent |
Tests/Shared/QuadraticGradientDescent.lean:11 |
diagonalH_mode
|
theorem | Partial | Not mapped | Tests.Shared.QuadraticGradientDescent |
Tests/Shared/QuadraticGradientDescent.lean:14 |
quadratic_strong
|
theorem | Partial | Not mapped | Tests.Shared.StrongConvexPLPullback |
Tests/Shared/StrongConvexPLPullback.lean:10 |
RecursiveDepthTest.condition
|
def | Partial | Not mapped | Tests.SmoothedPicardRecursiveDepth |
Tests/SmoothedPicardRecursiveDepth.lean:12 |
RecursiveDepthTest.heat
|
def | Partial | Not mapped | Tests.SmoothedPicardRecursiveDepth |
Tests/SmoothedPicardRecursiveDepth.lean:14 |
RecursiveDepthTest.stepVariance
|
def | Partial | Not mapped | Tests.SmoothedPicardRecursiveDepth |
Tests/SmoothedPicardRecursiveDepth.lean:17 |
RecursiveDepthTest.nextPrecision
|
def | Partial | Not mapped | Tests.SmoothedPicardRecursiveDepth |
Tests/SmoothedPicardRecursiveDepth.lean:20 |
RecursiveDepthTest.precision
|
def | Partial | Not mapped | Tests.SmoothedPicardRecursiveDepth |
Tests/SmoothedPicardRecursiveDepth.lean:23 |