Samplinglib
Lean gate passed 2026-08-19T05:09:39.794721+00:00 · 644be936998e
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.

3042 declarations
DeclarationKindLocal statusRoute statusModuleSource
AutoSamplingTheory.AutomationStage inductiveCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:5
AutoSamplingTheory.TaskKind inductiveCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:14
AutoSamplingTheory.TaskStatus inductiveCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:23
AutoSamplingTheory.AgentRole inductiveCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:32
AutoSamplingTheory.AcceptanceGate structureCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:39
AutoSamplingTheory.ArtifactSpec structureCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:46
AutoSamplingTheory.AutomationTask structureCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:53
AutoSamplingTheory.AgentContract structureCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:66
AutoSamplingTheory.PostCycleArtifactKind inductiveCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:73
AutoSamplingTheory.PostCycleArtifactSpec structureCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:81
AutoSamplingTheory.WorkflowCheckSpec structureCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:88
AutoSamplingTheory.leanBuildGate defCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:95
AutoSamplingTheory.forbiddenPatternGate defCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:101
AutoSamplingTheory.defaultGates defCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:107
AutoSamplingTheory.postCycleArtifactSpecs defCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:109
AutoSamplingTheory.workflowCheckSpecs defCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:137
AutoSamplingTheory.threeLayerAgentContracts defCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:153
AutoSamplingTheory.conversionArtifacts defCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:181
AutoSamplingTheory.seedAutomationTasks defCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:203
AutoSamplingTheory.automationTaskCount defCompiledNot mapped AutoSamplingTheory.Automation AutoSamplingTheory/Automation.lean:231
AutoSamplingTheory.ArtifactLanguage inductiveCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:14
AutoSamplingTheory.ProofStatus inductiveCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:22
AutoSamplingTheory.SourceKind inductiveCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:31
AutoSamplingTheory.SourceAnchor - Stable pointer to the source of a mathematical claim. structureCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:41
AutoSamplingTheory.ProofObligation - An honest record for content that is not yet proved in Lean. structureCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:50
AutoSamplingTheory.TheoremContract - Paper theorem or lemma translated into a Lean-facing contract. structureCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:60
AutoSamplingTheory.ProofDagBlock - A reusable proof-DAG block, usually one node in a paper proof. structureCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:72
AutoSamplingTheory.forbiddenProofPatterns - Patterns that are not allowed to close mathematical content. defCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:83
AutoSamplingTheory.sourceAnchor defCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:86
AutoSamplingTheory.localTexAnchor defCompiledNot mapped AutoSamplingTheory.Core AutoSamplingTheory/Core.lean:100
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)` theoremCompiledNot 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. structureCompiledNot 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. inductiveCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.ExampleCases.SampleWiki AutoSamplingTheory/ExampleCases/SampleWiki.lean:53
AutoSamplingTheory.ExampleCases.SampleWiki.assimilated_admissible - An assimilated SampleWiki case satisfies the graph-admission contract. theoremCompiledNot mapped AutoSamplingTheory.ExampleCases.SampleWiki AutoSamplingTheory/ExampleCases/SampleWiki.lean:59
AutoSamplingTheory.ExampleCases.SampleWiki.compiled_not_admissible - Compilation alone does not discharge the source-review boundary. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.ExampleCases.SampleWiki AutoSamplingTheory/ExampleCases/SampleWiki.lean:70
AutoSamplingTheory.ImplementationStatus inductiveCompiledNot mapped AutoSamplingTheory.Literature AutoSamplingTheory/Literature.lean:5
AutoSamplingTheory.PaperMode inductiveCompiledNot mapped AutoSamplingTheory.Literature AutoSamplingTheory/Literature.lean:13
AutoSamplingTheory.PaperEntry structureCompiledNot mapped AutoSamplingTheory.Literature AutoSamplingTheory/Literature.lean:19
AutoSamplingTheory.literature defCompiledNot mapped AutoSamplingTheory.Literature AutoSamplingTheory/Literature.lean:31
AutoSamplingTheory.literatureCount defCompiledNot mapped AutoSamplingTheory.Literature AutoSamplingTheory/Literature.lean:79
AutoSamplingTheory.OpenProblem structureCompiledNot mapped AutoSamplingTheory.OpenProblems AutoSamplingTheory/OpenProblems.lean:5
AutoSamplingTheory.openProblems defCompiledNot mapped AutoSamplingTheory.OpenProblems AutoSamplingTheory/OpenProblems.lean:14
AutoSamplingTheory.openProblemCount defCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledPartial 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 theoremCompiledNot 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 ` theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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)`. theoremCompiledNot 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 = theoremCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:466
AutoSamplingTheory.condDistribIntegralNamedLawIntegrable - Named-law variant of `condDistribIntegralMapIntegrable`. theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:505
AutoSamplingTheory.MeasureContract - A named probability measure or time-indexed law in a paper proof. structureCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:532
AutoSamplingTheory.KLContract - Forward KL divergence contract `KL(rho || pi)`. structureCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:541
AutoSamplingTheory.FIContract - Fisher information contract `FI(rho || pi)`. structureCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:550
AutoSamplingTheory.LSIContract - Log-Sobolev inequality contract. structureCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:559
AutoSamplingTheory.PIContract - Poincare inequality contract. structureCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:568
AutoSamplingTheory.TransportVelocityContract - Transport velocity field satisfying a continuity equation. structureCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:577
AutoSamplingTheory.GuidedTiltContract - Guide tilt `pi_t proportional to p_t exp(-F_t)`. structureCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:600
AutoSamplingTheory.dvVariationalObligation - Donsker--Varadhan variational formula as a cited-result contract. defCompiledNot 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. defCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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)`. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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` theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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- theoremCompiledNot 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 theoremCompiledPartial 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 theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:1079
AutoSamplingTheory.lsiToKlFiObligation - Log-Sobolev implies KL-FI comparison as a reusable proof target. defCompiledNot mapped AutoSamplingTheory.Probability AutoSamplingTheory/Probability.lean:1094
AutoSamplingTheory.RMFLD.rmfldPaperRoot defCompiledNot mapped AutoSamplingTheory.RMFLD AutoSamplingTheory/RMFLD.lean:10
AutoSamplingTheory.RMFLD.rmfldSource defCompiledNot mapped AutoSamplingTheory.RMFLD AutoSamplingTheory/RMFLD.lean:12
AutoSamplingTheory.RMFLD.exploratorySeedLabels defCompiledNot mapped AutoSamplingTheory.RMFLD AutoSamplingTheory/RMFLD.lean:16
AutoSamplingTheory.RMFLD.rmfldExploratoryContract defCompiledNot mapped AutoSamplingTheory.RMFLD AutoSamplingTheory/RMFLD.lean:25
AutoSamplingTheory.RMFLD.rmfldProofDag defCompiledNot mapped AutoSamplingTheory.RMFLD AutoSamplingTheory/RMFLD.lean:34
AutoSamplingTheory.SALD.saldPaperRoot defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27
AutoSamplingTheory.SALD.saldMainSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29
AutoSamplingTheory.SALD.saldAppendixSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33
AutoSamplingTheory.SALD.saldIterationSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:37
AutoSamplingTheory.SALD.saldGronwallSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41
AutoSamplingTheory.SALD.saldGronwallExponentRewriteSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45
AutoSamplingTheory.SALD.saldDvVariationSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:49
AutoSamplingTheory.SALD.saldPiSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:53
AutoSamplingTheory.SALD.saldPiVelocityNormSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57
AutoSamplingTheory.SALD.saldKlFiLsiSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61
AutoSamplingTheory.SALD.saldContinuousSdeSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65
AutoSamplingTheory.SALD.saldFokkerPlanckSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69
AutoSamplingTheory.SALD.saldAlphaComplexitySource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:73
AutoSamplingTheory.SALD.saldForwardKlSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:77
AutoSamplingTheory.SALD.saldForwardKlProofSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:81
AutoSamplingTheory.SALD.saldForwardKlDerivativeSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:85
AutoSamplingTheory.SALD.saldForwardKlDvEnergySource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:89
AutoSamplingTheory.SALD.saldForwardKlGronwallSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:93
AutoSamplingTheory.SALD.saldForwardKlEndpointScheduleSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:97
AutoSamplingTheory.SALD.saldForwardKlDependencyChainSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:101
AutoSamplingTheory.SALD.saldForwardKlDiscreteSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:105
AutoSamplingTheory.SALD.saldForwardKlDiscreteLipSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:109
AutoSamplingTheory.SALD.saldForwardKlDiscreteInterpolationSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:113
AutoSamplingTheory.SALD.saldFrozenDeltaCrossLipSaldSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:117
AutoSamplingTheory.SALD.saldForwardKlDiscreteProofSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:121
AutoSamplingTheory.SALD.saldForwardKlDiscreteDerivativeSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:125
AutoSamplingTheory.SALD.saldForwardKlDiscreteConditionalFpSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:129
AutoSamplingTheory.SALD.saldForwardKlDiscreteDvVelocitySource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:133
AutoSamplingTheory.SALD.saldForwardKlDiscreteGronwallSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:137
AutoSamplingTheory.SALD.saldForwardKlDiscreteAccumulatedErrorSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:141
AutoSamplingTheory.SALD.saldForwardKlDiscreteCoefficientChainSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:145
AutoSamplingTheory.SALD.saldGuidedResidualSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:149
AutoSamplingTheory.SALD.saldGuidedResidualProofSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:153
AutoSamplingTheory.SALD.saldGeneralMovingTargetSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:157
AutoSamplingTheory.SALD.saldGeneralMovingTargetDerivativeSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:161
AutoSamplingTheory.SALD.saldGeneralMovingTargetDvGronwallSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:165
AutoSamplingTheory.SALD.saldGeneralMovingTargetResidualDvSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:169
AutoSamplingTheory.SALD.saldGeneralMovingTargetPureContractionSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:173
AutoSamplingTheory.SALD.saldUnifiedForwardKlSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:177
AutoSamplingTheory.SALD.saldUnifiedForwardKlProofSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:181
AutoSamplingTheory.SALD.saldVaSaldItoSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:185
AutoSamplingTheory.SALD.saldGuidedResidualMainSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:189
AutoSamplingTheory.SALD.saldCorrectionFieldSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:193
AutoSamplingTheory.SALD.saldUnifiedTransportBridgeSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:197
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:201
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:205
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteDeltaSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:209
AutoSamplingTheory.SALD.saldFrozenDeltaCrossLipGeneralSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:213
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteDerivativeSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:217
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalDriftSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:221
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:225
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:229
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:233
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteCondDistribIntegralMathlibSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:241
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpGeneratorMathlibSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:249
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:257
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteKlLogRatioMathlibSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:265
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteYoungSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:273
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteResidualDvSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:277
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteGronwallSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:281
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteGronwallSideConditionSource defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:285
AutoSamplingTheory.SALD.firstFaithfulLabels defCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 ` theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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` theoremCompiledNot 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)* theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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- theoremCompiledNot 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, theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:1444
AutoSamplingTheory.SALD.discreteForwardKlResidualExponentBoundScalar theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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* theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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: theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 ` theoremCompiledNot 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`. theoremCompiledNot 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` theoremCompiledNot 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 theoremCompiledNot 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` theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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: theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 ` theoremCompiledNot 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_ theoremCompiledNot 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 theoremCompiledNot 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` theoremCompiledNot 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 theoremCompiledNot 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, theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 ` theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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, theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 defCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 defCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 defCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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- theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 * defCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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` theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 • theoremCompiledNot 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 ` theoremCompiledNot 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- theoremCompiledNot 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 : theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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_ theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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- theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 structureCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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- structureCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:21495
AutoSamplingTheory.SALD.generalMovingTargetDiscreteRawKlDerivativeNoMassAtFiniteKlLlrHkl - Extract the no-mass raw KL display from the finite-KL `llr` package. theoremCompiledNot 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 theoremCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:21615
AutoSamplingTheory.SALD.generalMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlrHkl - Extract the no-mass KL display from the pure finite-KL `llr` package. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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* theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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, theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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- theoremCompiledNot 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` theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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, theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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{ theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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)` theoremCompiledNot 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`: theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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. structureCompiledNot 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. structureCompiledNot 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. structureCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:24437
AutoSamplingTheory.SALD.ForwardKlDvEnergyCandidateContract - Lean-facing interface for the DV velocity-energy step in `thm:forward-KL`. structureCompiledNot 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: structureCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:24499
AutoSamplingTheory.SALD.ForwardKlGronwallInstantiationContract - Lean-facing interface for the final Gronwall instantiation in `thm:forward-KL`. structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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. structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:24743
AutoSamplingTheory.SALD.DiscreteForwardKlDerivativeCandidateContract - Lean-facing interface for the discrete forward-KL derivative block. structureCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:24781
AutoSamplingTheory.SALD.DiscreteForwardKlGronwallInstantiationContract - Lean-facing interface for the final Gronwall step in `thm:forward-KL-discrete`. structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:24919
AutoSamplingTheory.SALD.GeneralMovingTargetDerivativeCandidateContract - Lean-facing interface for the derivative block of the continuous general VA-SALD theorem. structureCompiledNot 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. structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25003
AutoSamplingTheory.SALD.GeneralMovingTargetGronwallInstantiationContract - Lean-facing interface for the Gronwall step in the continuous general VA-SALD theorem. structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25136
AutoSamplingTheory.SALD.GeneralFrozenDeltaCrossLipContract - Lean-facing interface for the general VA-SALD frozen-delta lemma. structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25155
AutoSamplingTheory.SALD.GeneralMovingTargetDiscreteDerivativeCandidateContract - Lean-facing interface for the derivative block of the discrete general VA-SALD theorem. structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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, structureCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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. structureCompiledNot 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 structureCompiledNot 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- structureCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25416
AutoSamplingTheory.SALD.saldGronwallCandidateContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25430
AutoSamplingTheory.SALD.saldGronwallEndpointCalculusContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25453
AutoSamplingTheory.SALD.saldGronwallExponentRewriteContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25500
AutoSamplingTheory.SALD.saldKLContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25526
AutoSamplingTheory.SALD.saldFIContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25533
AutoSamplingTheory.SALD.saldLSIContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25540
AutoSamplingTheory.SALD.saldLsiKlFiBridgeContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25547
AutoSamplingTheory.SALD.saldLsiKlFiDensityTestContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25568
AutoSamplingTheory.SALD.saldDvFiniteLogMgfContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25608
AutoSamplingTheory.SALD.saldPiVelocityNormDependencyContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25628
AutoSamplingTheory.SALD.cycle9FirstAppendixVocabularyPacket defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25651
AutoSamplingTheory.SALD.saldAlphaComplexityContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25683
AutoSamplingTheory.SALD.continuousForwardKlStatementContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25693
AutoSamplingTheory.SALD.forwardKlDerivativeCandidateContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25725
AutoSamplingTheory.SALD.forwardKlDerivativeSideConditionContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25757
AutoSamplingTheory.SALD.forwardKlDvEnergyCandidateContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25778
AutoSamplingTheory.SALD.forwardKlDvFiniteLogMgfWitnessContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25798
AutoSamplingTheory.SALD.forwardKlDvAlphaMonotonicityContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25834
AutoSamplingTheory.SALD.forwardKlGronwallInstantiationContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25858
AutoSamplingTheory.SALD.forwardKlMovingTargetDependencyContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25882
AutoSamplingTheory.SALD.forwardKlDependencyChainAuditContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25916
AutoSamplingTheory.SALD.cycle10ForwardKlUpperPacket defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:25973
AutoSamplingTheory.SALD.forwardKlGronwallSideConditionContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26010
AutoSamplingTheory.SALD.forwardKlEndpointScheduleContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26047
AutoSamplingTheory.SALD.cycle14ForwardKlUpperPacket defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26072
AutoSamplingTheory.SALD.cycle14ForwardKlMiddleContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26109
AutoSamplingTheory.SALD.cycle11DiscreteForwardKlUpperPacket defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26169
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlUpperPacket defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26206
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlMiddleContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26243
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlEmConditionalFpLowerContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26308
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlConditionalDriftDensityContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26347
AutoSamplingTheory.SALD.cycle19DiscreteForwardKlUpperPacket defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26649
AutoSamplingTheory.SALD.discreteSaldEulerMaruyamaContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26710
AutoSamplingTheory.SALD.discreteForwardKlStatementContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26717
AutoSamplingTheory.SALD.discreteForwardKlEmInterpolationSideConditionContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26768
AutoSamplingTheory.SALD.frozenDeltaCrossLipSaldContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26789
AutoSamplingTheory.SALD.discreteForwardKlDerivativeCandidateContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26815
AutoSamplingTheory.SALD.discreteForwardKlDvFiniteLogMgfWitnessContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26854
AutoSamplingTheory.SALD.discreteForwardKlGronwallInstantiationContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26891
AutoSamplingTheory.SALD.discreteForwardKlAccumulatedErrorBridgeContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26919
AutoSamplingTheory.SALD.cycle11DiscreteForwardKlMiddleContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:26973
AutoSamplingTheory.SALD.discreteForwardKlCoefficientChainAuditContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27032
AutoSamplingTheory.SALD.guidedResidualIdentityContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27093
AutoSamplingTheory.SALD.generalMovingTargetStatementContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27120
AutoSamplingTheory.SALD.generalMovingTargetDerivativeCandidateContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27156
AutoSamplingTheory.SALD.generalMovingTargetDvEnergyCandidateContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27199
AutoSamplingTheory.SALD.generalMovingTargetDvFiniteLogMgfWitnessContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27218
AutoSamplingTheory.SALD.generalMovingTargetDvPositiveAlphaScalingContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27255
AutoSamplingTheory.SALD.generalMovingTargetGronwallInstantiationContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27281
AutoSamplingTheory.SALD.generalMovingTargetGronwallSideConditionContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27308
AutoSamplingTheory.SALD.unifiedForwardKlSpecializationContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27349
AutoSamplingTheory.SALD.cycle12GeneralVaSaldUpperPacket defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27390
AutoSamplingTheory.SALD.cycle12GeneralVaSaldMiddleContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27426
AutoSamplingTheory.SALD.cycle16GeneralVaSaldUpperPacket defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27809
AutoSamplingTheory.SALD.cycle16UnifiedForwardKlTransportBridgeMiddleContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27862
AutoSamplingTheory.SALD.cycle16UnifiedForwardKlTransportBridgeLowerContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27907
AutoSamplingTheory.SALD.cycle13FirstAppendixVocabularyPacket defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27937
AutoSamplingTheory.SALD.cycle13FirstAppendixSourceIndexAuditContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:27971
AutoSamplingTheory.SALD.cycle13FirstAppendixMiddleAuditContract defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28309
AutoSamplingTheory.SALD.cycle25FirstAppendixPiVelocityNormMiddleObligation - Cycle-25 middle obligation for the selected PI velocity-norm sub-slice. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28364
AutoSamplingTheory.SALD.cycle25PiVelocityNormLowerObligation - Cycle-25 lower obligation after compiling the scalar PI/Cauchy--Schwarz core. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28444
AutoSamplingTheory.SALD.cycle29LsiKlFiDensityTestMiddleObligation - Cycle-29 middle obligation for the selected LSI/KL/FI density-test sub-slice. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28497
AutoSamplingTheory.SALD.cycle29LsiKlFiDensityTestLowerObligation - Cycle-29 lower obligation for the LSI/KL/FI density-test coefficient slice. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28517
AutoSamplingTheory.SALD.cycle33LsiKlFiDensityTestMiddleObligation - Cycle-33 middle obligation for the proof-producing density-test scalar slice. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28534
AutoSamplingTheory.SALD.cycle33LsiKlFiDensityTestLowerObligation - Cycle-33 lower obligation for the normalized LSI-test scalar bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28575
AutoSamplingTheory.SALD.cycle38LsiKlFiUpperObligation - Cycle-38 upper workflow obligation for the LSI/KL/FI density-test bridge. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28614
AutoSamplingTheory.SALD.cycle38LsiKlFiMiddleObligation - Cycle-38 middle obligation for the LSI/KL/FI Fisher-chain scalar slice. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28642
AutoSamplingTheory.SALD.cycle38LsiKlFiLowerObligation - Cycle-38 lower obligation after compiling a finite-coordinate Fisher-chain handoff. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28690
AutoSamplingTheory.SALD.cycle43LsiKlFiUpperObligation - Cycle-43 upper workflow obligation for the LSI/KL/FI density-test backend. defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28830
AutoSamplingTheory.SALD.cycle44MainSkeletonAnalyticInterfaceObligation - Cycle-44 upper obligation for the main skeleton analytic interface ledger. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:28906
AutoSamplingTheory.SALD.cycle44MainSkeletonAnalyticInterfaceDag - Cycle-44 proof-DAG pane for the five analytic interfaces and theorem route. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29036
AutoSamplingTheory.SALD.cycle49MainSkeletonAnalyticReadinessObligation - Cycle-49 upper obligation selecting the next theorem-level backend. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29209
AutoSamplingTheory.SALD.cycle49MainSkeletonAnalyticReadinessDag - Cycle-49 proof-DAG pane for the post-route analytic readiness check. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29356
AutoSamplingTheory.SALD.cycle45ForwardKlSkeletonDag - Cycle-45 proof-DAG pane for the continuous forward-KL theorem route. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29419
AutoSamplingTheory.SALD.cycle45ForwardKlSkeletonMiddleObligation - Cycle-45 middle obligation tying the forward-KL route audit to lower work. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29516
AutoSamplingTheory.SALD.cycle50ForwardKlSkeletonObligation - Cycle-50 obligation tying the post-readiness audit back to `thm:forward-KL`. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29600
AutoSamplingTheory.SALD.cycle50ForwardKlSkeletonMiddleObligation - Cycle-50 middle obligation selecting the continuous KL derivative backend. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29663
AutoSamplingTheory.SALD.cycle50ForwardKlDerivativeLowerObligation - Cycle-50 lower obligation for the continuous derivative/DV scalar handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29698
AutoSamplingTheory.SALD.cycle50ForwardKlSkeletonDag - Cycle-50 proof-DAG pane for the continuous forward-KL theorem route. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:29903
AutoSamplingTheory.SALD.cycle51DiscreteForwardKlSkeletonMiddleObligation - Cycle-51 middle obligation tying the discrete route audit to the derivative lower packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:30014
AutoSamplingTheory.SALD.cycle32DvVariationInterfaceObligation - Cycle-32 source-cited interface obligation for the cited DV formula. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:30074
AutoSamplingTheory.SALD.cycle32DvVariationMiddleObligation defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:30171
AutoSamplingTheory.SALD.cycle37DvVariationUpperObligation - Cycle-37 upper workflow obligation for the cited DV interface target. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:30238
AutoSamplingTheory.SALD.cycle37DvVariationMiddleObligation - Cycle-37 middle obligation tracking the new one-sided tilted backend. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:30289
AutoSamplingTheory.SALD.cycle37DvVariationLowerObligation - Cycle-37 lower obligation tracking the composed one-sided DV consequence. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:30341
AutoSamplingTheory.SALD.cycle42DvVariationMiddleObligation - Cycle-42 obligation for the selected scaled-test DV interface. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:30392
AutoSamplingTheory.SALD.cycle42DvVariationLowerObligation - Cycle-42 lower obligation tracking the post-DV scaled energy bound. defCompiledNot 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. defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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- defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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- defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31204
AutoSamplingTheory.SALD.generalVaSaldEulerMaruyamaContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31256
AutoSamplingTheory.SALD.generalFrozenDeltaCrossLipContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31263
AutoSamplingTheory.SALD.generalMovingTargetDiscreteStatementContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31297
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDerivativeCandidateContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31336
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalDriftContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31389
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31429
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31475
AutoSamplingTheory.SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31510
AutoSamplingTheory.SALD.generalMovingTargetDiscreteKlDerivativeWeakFpHandoffContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31578
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDerivativeSideConditionContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31637
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31703
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallInstantiationContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31738
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallSideConditionContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31763
AutoSamplingTheory.SALD.saldPIContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31805
AutoSamplingTheory.SALD.lsiKlFiDensityTestObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31812
AutoSamplingTheory.SALD.dvFiniteLogMgfInterfaceObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31855
AutoSamplingTheory.SALD.piVelocityNormBackendObligation defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31888
AutoSamplingTheory.SALD.cycle36GronwallUpperObligation - Cycle-36 upper workflow obligation for the Gronwall proof-closure packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31926
AutoSamplingTheory.SALD.cycle36GronwallMiddleObligation - Cycle-36 middle proof-producing Gronwall assembly record. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31949
AutoSamplingTheory.SALD.cycle41GronwallMiddleObligation - Cycle-41 middle proof-producing Gronwall derivative-source wrapper. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31973
AutoSamplingTheory.SALD.cycle41GronwallLowerObligation - Cycle-41 lower endpoint-safe Gronwall interior-derivative assembly record. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:31991
AutoSamplingTheory.SALD.gronwallAnalyticObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32010
AutoSamplingTheory.SALD.gronwallEndpointCalculusObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32039
AutoSamplingTheory.SALD.gronwallExponentRewriteObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32072
AutoSamplingTheory.SALD.firstAppendixSourceIndexAuditObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32089
AutoSamplingTheory.SALD.firstAppendixMiddleAuditObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32113
AutoSamplingTheory.SALD.forwardKlMiddleSourceToLeanMapObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32169
AutoSamplingTheory.SALD.cycle30ForwardKlDerivativeSideUpperObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32225
AutoSamplingTheory.SALD.cycle30ForwardKlDerivativeSideMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32249
AutoSamplingTheory.SALD.forwardKlDensityBoundaryObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32274
AutoSamplingTheory.SALD.cycle30ForwardKlDensityBoundaryLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32293
AutoSamplingTheory.SALD.cycle34ForwardKlDerivativeScalarObligation - Cycle-34 upper/lower scalar obligation for the derivative closure sprint. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32315
AutoSamplingTheory.SALD.cycle34ForwardKlTargetYoungLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32342
AutoSamplingTheory.SALD.cycle34ForwardKlDerivativeMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32362
AutoSamplingTheory.SALD.cycle39ForwardKlDerivativeUpperObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32383
AutoSamplingTheory.SALD.cycle39ForwardKlDerivativeMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32418
AutoSamplingTheory.SALD.forwardKlScheduleTimeChangeObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32450
AutoSamplingTheory.SALD.forwardKlDerivativeObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32478
AutoSamplingTheory.SALD.forwardKlDvEnergyObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32527
AutoSamplingTheory.SALD.forwardKlDvAlphaMonotonicityObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32541
AutoSamplingTheory.SALD.forwardKlDvFiniteLogMgfWitnessObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32553
AutoSamplingTheory.SALD.cycle26ForwardKlDvWitnessMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32573
AutoSamplingTheory.SALD.cycle26ForwardKlDvPositiveAlphaLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32595
AutoSamplingTheory.SALD.forwardKlGronwallApplicationObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32612
AutoSamplingTheory.SALD.forwardKlGronwallSideConditionObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32620
AutoSamplingTheory.SALD.forwardKlEndpointScheduleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32647
AutoSamplingTheory.SALD.forwardKlMovingTargetDependencyObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32662
AutoSamplingTheory.SALD.forwardKlCoefficientChainObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32684
AutoSamplingTheory.SALD.discreteForwardKlEmEndpointObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32702
AutoSamplingTheory.SALD.discreteForwardKlEmConditionalFpObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32710
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlConditionalDriftDensityObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32718
AutoSamplingTheory.SALD.discreteForwardKlStitchedIntervalRegularityObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32731
AutoSamplingTheory.SALD.discreteForwardKlEmInterpolationObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32739
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlMiddleEmSpineObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32747
AutoSamplingTheory.SALD.cycle15DiscreteForwardKlEmConditionalFpLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32766
AutoSamplingTheory.SALD.cycle35DiscreteForwardKlEmFpUpperObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32784
AutoSamplingTheory.SALD.cycle35DiscreteForwardKlEmFpMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32806
AutoSamplingTheory.SALD.cycle35DiscreteForwardKlEmFpLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32831
AutoSamplingTheory.SALD.cycle40DiscreteForwardKlEmFpMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32852
AutoSamplingTheory.SALD.cycle40DiscreteForwardKlEmEndpointLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32876
AutoSamplingTheory.SALD.discreteForwardKlFrozenDeltaObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32892
AutoSamplingTheory.SALD.discreteForwardKlDerivativeObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32900
AutoSamplingTheory.SALD.cycle51DiscreteForwardKlDerivativeLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32908
AutoSamplingTheory.SALD.discreteForwardKlDvFiniteLogMgfWitnessObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32927
AutoSamplingTheory.SALD.discreteForwardKlDvVelocityObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32944
AutoSamplingTheory.SALD.discreteForwardKlGronwallAccumulationObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32952
AutoSamplingTheory.SALD.discreteForwardKlLinearSlowdownObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32960
AutoSamplingTheory.SALD.discreteForwardKlResidualExponentBoundObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32968
AutoSamplingTheory.SALD.discreteForwardKlAccumulatedErrorBridgeObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:32983
AutoSamplingTheory.SALD.cycle19DiscreteForwardKlAccumulatedErrorMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33008
AutoSamplingTheory.SALD.discreteForwardKlEmDefectAccumulationMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33027
AutoSamplingTheory.SALD.cycle23DiscreteForwardKlCoefficientChainMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33048
AutoSamplingTheory.SALD.discreteForwardKlCoefficientChainObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33074
AutoSamplingTheory.SALD.cycle27DiscreteForwardKlAccumulatedCollectionUpperObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33097
AutoSamplingTheory.SALD.cycle27DiscreteForwardKlAccumulatedCollectionMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33121
AutoSamplingTheory.SALD.cycle27DiscreteForwardKlAccumulatedCollectionLowerObligation defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33228
AutoSamplingTheory.SALD.cycle46DiscreteForwardKlSkeletonDag - Cycle-46 proof-DAG pane for the discrete forward-KL theorem route. defCompiledNot 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, defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33301
AutoSamplingTheory.SALD.cycle46DiscreteForwardKlSkeletonMiddleObligation - Cycle-46 middle obligation tying the discrete route audit to lower work. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33505
AutoSamplingTheory.SALD.cycle47GuidedGeneralSkeletonMiddleObligation - Cycle-47 middle obligation tying the guided/general route audit to lower work. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33572
AutoSamplingTheory.SALD.cycle47GuidedGeneralSkeletonDag - Cycle-47 proof-DAG pane for the guided/general theorem route. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33773
AutoSamplingTheory.SALD.cycle52GuidedGeneralSkeletonMiddleObligation - Cycle-52 middle obligation tying the guided/general route audit to lower work. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33841
AutoSamplingTheory.SALD.cycle52GuidedGeneralDerivativeDvLowerObligation - Cycle-52 lower obligation for the general moving-target derivative/DV scalar handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:33895
AutoSamplingTheory.SALD.cycle52GuidedGeneralSkeletonDag - Cycle-52 proof-DAG pane for the guided/general route closure check. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:34017
AutoSamplingTheory.SALD.cycle57GuidedGeneralSkeletonObligation - Cycle-57 obligation for the upper guided/general route recheck. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:34128
AutoSamplingTheory.SALD.cycle57GuidedGeneralSkeletonMiddleObligation - Cycle-57 middle obligation tying the guided/general route audit to lower work. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:34583
AutoSamplingTheory.SALD.cycle48UnifiedDiscreteSkeletonDag - Cycle-48 proof-DAG pane for the unified and discrete general theorem route. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:34827
AutoSamplingTheory.SALD.cycle53UnifiedDiscreteGeneralMiddleObligation - Cycle-53 middle obligation tying the final route audit to lower work. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:34895
AutoSamplingTheory.SALD.cycle53UnifiedDiscreteGeneralDag - Cycle-53 proof-DAG pane for the final unified/discrete general route. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:35010
AutoSamplingTheory.SALD.cycle54MainSkeletonAnalyticInterfaceObligation - Cycle-54 upper obligation for the repeated analytic-interface ledger. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:35208
AutoSamplingTheory.SALD.cycle54MainSkeletonAnalyticInterfaceDag - Cycle-54 proof-DAG pane for the repeated analytic-interface ledger. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:35494
AutoSamplingTheory.SALD.cycle55ForwardKlDerivativeMassLowerObligation - Cycle-55 lower obligation for the first continuous derivative scalar slice. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:35535
AutoSamplingTheory.SALD.cycle55ForwardKlSkeletonDag - Cycle-55 proof-DAG pane for the focused continuous forward-KL route. defCompiledNot 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, defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:35837
AutoSamplingTheory.SALD.cycle56DiscreteForwardKlGronwallLowerObligation - Cycle-56 lower obligation for the discrete Gronwall accumulation backend. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:36032
AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralMiddleObligation - Cycle-58 middle obligation tying the route audit to lower Gronwall/display work. defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:36234
AutoSamplingTheory.SALD.cycle59MainSkeletonAnalyticInterfaceObligation - Cycle-59 upper obligation for the analytic-interface recheck. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:36485
AutoSamplingTheory.SALD.cycle59MainSkeletonAnalyticInterfaceDag - Cycle-59 proof-DAG pane for the post-cycle-58 analytic-interface check. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:36783
AutoSamplingTheory.SALD.cycle60ForwardKlDerivativeRawLowerObligation - Cycle-60 lower obligation for the raw continuous derivative scalar wrapper. defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:37181
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlAccumulatedErrorLowerObligation - Cycle-61 lower obligation for the residual integral display wrapper. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:37245
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlSkeletonDag - Cycle-61 proof-DAG pane for the recovered discrete forward-KL route. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:37601
AutoSamplingTheory.SALD.cycle62GuidedGeneralScaledResidualLowerObligation - Cycle-62 lower scalar handoff for the continuous general KL derivative. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:37655
AutoSamplingTheory.SALD.cycle62GuidedGeneralSkeletonDag - Cycle-62 proof-DAG pane for the guided/general upper route. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:37816
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralSkeletonObligation - Cycle-63 upper obligation for the unified/discrete general route refresh. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:37877
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralMeasureBackfillObligation - Cycle-63 narrow measure-theory backfill below the discrete general EM endpoint-law interface. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:38207
AutoSamplingTheory.SALD.cycle64MainSkeletonAnalyticInterfaceObligation - Cycle-64 upper obligation for the refreshed analytic-interface ledger. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:38774
AutoSamplingTheory.SALD.cycle65ForwardKlDerivativePointwiseLowerObligation - Cycle-65 lower proof-producing obligation for the pointwise continuous derivative wrapper. defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:39682
AutoSamplingTheory.SALD.cycle67GuidedGeneralResidualGronwallLowerObligation - Cycle-67 lower proof-producing obligation for the residual-to-Gronwall scalar bridge. defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:39957
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeObligation - Cycle-68 source-cited bridge for the final unified/discrete general theorem route. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40141
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralDiscreteBridgeLowerObligation - Cycle-68 lower proof-producing obligation for the final discrete general Gronwall/display bridge. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40209
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralDag - Cycle-68 proof-DAG pane for the unified and discrete general theorem route refresh. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40366
AutoSamplingTheory.SALD.cycle69MainSkeletonAnalyticInterfaceObligation - Cycle-69 upper obligation for the post-route analytic-interface ledger. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40576
AutoSamplingTheory.SALD.cycle69GeneralMovingTargetDiscreteEmFpSourceSignsLowerObligation - Cycle-69 lower obligation for the source-sign EM FP handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40616
AutoSamplingTheory.SALD.cycle70GeneralMovingTargetDiscreteConditionalLawMiddleObligation - Cycle-70 middle obligation for the conditional-law/measurability slice. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40636
AutoSamplingTheory.SALD.cycle70GeneralMovingTargetDiscreteConditionalLawLowerObligation - Cycle-70 lower obligation for the named conditional drift handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40654
AutoSamplingTheory.SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation - Cycle-71 middle obligation for endpoint-law-to-conditional-law compatibility. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40674
AutoSamplingTheory.SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalLowerObligation - Cycle-71 local wrapper obligation for the endpoint-to-conditional bridge. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40693
AutoSamplingTheory.SALD.cycle71GeneralMovingTargetDiscreteEndpointConditionalDag - Cycle-71 proof-DAG pane for endpoint-law-to-conditional-law compatibility. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40713
AutoSamplingTheory.SALD.cycle72GeneralMovingTargetDiscreteWeakFpMiddleObligation - Cycle-72 middle obligation for the weak conditional Fokker--Planck source-sign interface. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40773
AutoSamplingTheory.SALD.cycle72GeneralMovingTargetDiscreteWeakFpLowerObligation - Cycle-72 local wrapper obligation for weak-FP source signs. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40790
AutoSamplingTheory.SALD.cycle72GeneralMovingTargetDiscreteWeakFpDag - Cycle-72 proof-DAG pane for weak conditional Fokker--Planck source signs. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40873
AutoSamplingTheory.SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpUpperObligation - Cycle-73 upper obligation for the KL-derivative handoff packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40937
AutoSamplingTheory.SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpMiddleObligation - Cycle-73 middle obligation for the weak-FP-to-KL derivative source map. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40955
AutoSamplingTheory.SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpLowerObligation - Cycle-73 local wrapper obligation for weak-FP-to-KL derivative substitution. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:40974
AutoSamplingTheory.SALD.cycle73GeneralMovingTargetDiscreteKlDerivativeWeakFpDag - Cycle-73 proof-DAG pane for weak-FP-to-KL derivative handoff. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41083
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceUpperObligation - Cycle-74 upper obligation for the minimal cited measure interface. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41141
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation - Cycle-74 middle obligation for the conditional-kernel source-to-Lean map. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41158
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation - Cycle-74 lower obligation for the supplied-kernel regularity handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41182
AutoSamplingTheory.SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceDag - Cycle-74 proof-DAG pane for the conditional-kernel measure blocker. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41295
AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation - Cycle-75 upper obligation for the focused conditional-law backfill. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41386
AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation - Cycle-75 lower obligation for the swapped-orientation supplied-kernel regularity wrapper. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41417
AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillDag - Cycle-75 proof-DAG pane for the conditional-law construction backfill. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41517
AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalUpperObligation - Cycle-76 upper obligation for the endpoint-to-conditional backfill. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41579
AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation - Cycle-76 middle map for endpoint-law to conditional-law compatibility. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41595
AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalLowerObligation - Cycle-76 lower obligation for the endpoint Measure.map to swapped conditional-kernel wrapper. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41616
AutoSamplingTheory.SALD.cycle76GeneralMovingTargetDiscreteEndpointConditionalDag - Cycle-76 proof-DAG pane for endpoint-law to conditional compatibility. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41637
AutoSamplingTheory.SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorMiddleObligation - Cycle-77 middle obligation for the generator-level weak FP source-sign handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41704
AutoSamplingTheory.SALD.cycle77GeneralMovingTargetDiscreteWeakFpGeneratorLowerObligation - Cycle-77 lower obligation for the generator-level weak FP source-sign wrapper. defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41812
AutoSamplingTheory.SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorUpperObligation - Cycle-78 upper obligation for the generator-to-KL handoff packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41875
AutoSamplingTheory.SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorMiddleObligation - Cycle-78 middle obligation for the source-to-Lean KL handoff map. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:41893
AutoSamplingTheory.SALD.cycle78GeneralMovingTargetDiscreteKlDerivativeGeneratorLowerObligation - Cycle-78 local wrapper obligation for generator-piece weak-FP to KL derivative substitution. defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42009
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperPacket - Cycle-79 upper packet for the minimal cited weak-FP generator interface. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42035
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureUpperObligation - Cycle-79 upper obligation for the weak generator-to-law cited interface. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42094
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureMiddleObligation - Cycle-79 middle obligation for the weak generator-to-law source map. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42111
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureLowerObligation - Cycle-79 lower obligation for the local Measure.map weak-test handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42133
AutoSamplingTheory.SALD.cycle79GeneralMovingTargetDiscreteWeakFpGeneratorMeasureDag - Cycle-79 proof-DAG pane for the weak generator-to-law cited interface. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42240
AutoSamplingTheory.SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityUpperObligation - Cycle-80 upper obligation for the conditional-law/measurability packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42331
AutoSamplingTheory.SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityLowerObligation - Cycle-80 lower obligation for the endpoint/conditional drift-regularity wrapper. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42362
AutoSamplingTheory.SALD.cycle80GeneralMovingTargetDiscreteConditionalLawMeasurabilityDag - Cycle-80 proof-DAG pane for the conditional-law/measurability backfill. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42485
AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalUpperObligation - Cycle-81 upper obligation for the endpoint-to-conditional packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42549
AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalMiddleObligation - Cycle-81 middle obligation for the endpoint-to-conditional weak-FP readiness handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42572
AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalLowerObligation - Cycle-81 lower obligation for the endpoint-only weak-FP prerequisite handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42593
AutoSamplingTheory.SALD.cycle81GeneralMovingTargetDiscreteEndpointConditionalDag - Cycle-81 proof-DAG pane for the endpoint-to-conditional upper packet. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42784
AutoSamplingTheory.SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsMiddleObligation - Cycle-82 middle obligation for the readiness-to-source-sign bridge. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42805
AutoSamplingTheory.SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsLowerObligation - Cycle-82 lower obligation for the endpoint-readiness-to-source-sign bridge. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42826
AutoSamplingTheory.SALD.cycle82GeneralMovingTargetDiscreteWeakFpSourceSignsDag - Cycle-82 proof-DAG pane for the weak conditional FP source-sign packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:42947
AutoSamplingTheory.SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpUpperObligation - Cycle-83 upper obligation selecting the endpoint weak-FP to KL handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:43017
AutoSamplingTheory.SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpMiddleObligation - Cycle-83 middle obligation for the endpoint source-signs to KL source map. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:43034
AutoSamplingTheory.SALD.cycle83GeneralMovingTargetDiscreteKlDerivativeEndpointWeakFpLowerObligation - Cycle-83 lower obligation for the endpoint source-signs to KL wrapper. defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:43238
AutoSamplingTheory.SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendMiddleObligation - Cycle-84 middle obligation for the active EM-backend source map. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:43260
AutoSamplingTheory.SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendLowerObligation - Cycle-84 lower obligation for the endpoint log-action active-backend handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:43290
AutoSamplingTheory.SALD.cycle84GeneralMovingTargetDiscreteActiveEmBackendDag - Cycle-84 proof-DAG pane for the active EM-backend handoff/fallback decision. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:43571
AutoSamplingTheory.SALD.cycle85GeneralMovingTargetDiscreteConditionalKernelBoundaryDag - Cycle-85 proof-DAG pane for the conditional-kernel theorem boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:43752
AutoSamplingTheory.SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryUpperObligation - Cycle-86 upper obligation selecting the generator-to-law weak FP boundary. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:43875
AutoSamplingTheory.SALD.cycle86GeneralMovingTargetDiscreteWeakFpGeneratorBoundaryDag - Cycle-86 proof-DAG pane for the weak generator-to-law boundary. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44008
AutoSamplingTheory.SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryUpperObligation - Cycle-87 upper obligation selecting the KL/log-ratio analytic boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44110
AutoSamplingTheory.SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryLowerObligation - Cycle-87 lower scalar handoff for the KL/log-ratio mass-conservation drop. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44130
AutoSamplingTheory.SALD.cycle87GeneralMovingTargetDiscreteKlLogRatioBoundaryDag - Cycle-87 proof-DAG pane for the KL/log-ratio analytic boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44310
AutoSamplingTheory.SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityMiddleObligation - Cycle-88 middle source map for the log-ratio admissibility boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44388
AutoSamplingTheory.SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityLowerObligation - Cycle-88 lower handoff for log-ratio weak-test admissibility. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44411
AutoSamplingTheory.SALD.cycle88GeneralMovingTargetDiscreteKlLogRatioAdmissibilityDag - Cycle-88 proof-DAG pane for the log-ratio admissibility boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44526
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlClosurePressureUpperObligation - Cycle-89 obligation recording the pressure-test blocker. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44572
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlClosurePressureMiddleObligation - Cycle-89 middle source map for the discrete forward-KL pressure test. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44606
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlClosurePressureLowerObligation - Cycle-89 lower handoff for the discrete derivative IBP/FI split. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44642
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlClosurePressureDag - Cycle-89 proof-DAG pane for the discrete theorem closure pressure test. defCompiledNot 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: defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44819
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationUpperObligation - Cycle-90 obligation selecting the mass-conservation lower boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44864
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationMiddleObligation - Cycle-90 middle obligation for the compiled mass-derivative route. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44890
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationLowerObligation - Cycle-90 lower obligation for mapped-law constant-test mass conservation. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:44914
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationDag - Cycle-90 proof-DAG pane for the discrete KL mass-conservation blocker. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45099
AutoSamplingTheory.SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelDag - Cycle-91 proof-DAG pane for the conditional-kernel component-field backfill. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45212
AutoSamplingTheory.SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitLowerObligation - Cycle-92 lower obligation for the split-generator law-transport handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45232
AutoSamplingTheory.SALD.cycle92GeneralMovingTargetDiscreteWeakFpGeneratorSplitDag - Cycle-92 proof-DAG pane for the split-generator weak-FP boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45347
AutoSamplingTheory.SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeLowerObligation - Cycle-93 lower obligation for the compiled mapped-law mass handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45368
AutoSamplingTheory.SALD.cycle93GeneralMovingTargetDiscreteKlMassDerivativeDag - Cycle-93 proof-DAG pane for the KL/log-ratio mass derivative boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45498
AutoSamplingTheory.SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionLowerObligation - Cycle-94 lower obligation for the compiled `barB` drift-action handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45521
AutoSamplingTheory.SALD.cycle94GeneralMovingTargetDiscreteWeakFpDriftActionDag - Cycle-94 proof-DAG pane for the conditional-drift weak action boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45622
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlClosurePressureUpperObligation - Cycle-95 obligation recording the discrete theorem pressure-test blocker. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45670
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlClosurePressureMiddleObligation - Cycle-95 middle source map for the discrete forward-KL pressure test. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45735
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlClosurePressureDag - Cycle-95 proof-DAG pane for the post-cycle-94 pressure test. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45894
AutoSamplingTheory.SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation - Cycle-96 lower obligation for the compiled one-component pairing handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:45925
AutoSamplingTheory.SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingDag - Cycle-96 proof-DAG pane for the condexp generator-pairing middle packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46043
AutoSamplingTheory.SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingLowerObligation - Cycle-97 lower-ready obligation after the compiled disintegration theorem. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46069
AutoSamplingTheory.SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingDag - Cycle-97 proof-DAG pane for the canonical `condDistrib` pairing packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46175
AutoSamplingTheory.SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation - Cycle-98 lower-ready obligation for the compiled integral no-boundary handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46198
AutoSamplingTheory.SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryDag - Cycle-98 proof-DAG pane for the `barB` no-boundary integral packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46299
AutoSamplingTheory.SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeLowerObligation - Cycle-99 lower-ready obligation for the finite-KL `llr` raw-KL package. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46325
AutoSamplingTheory.SALD.cycle99GeneralMovingTargetDiscreteRawKlDerivativeDag - Cycle-99 proof-DAG pane for the raw KL finite-KL `llr` boundary. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46447
AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation - Cycle-100 lower-ready obligation for the compiled weak-pairing definition alignment handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46467
AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefDag - Cycle-100 proof-DAG pane for the `barB` weak-pairing definition alignment packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46488
AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundLowerObligation - Cycle-100 lower-ready obligation for the inner-gradient contraction handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46549
AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBInnerGradientBoundDag - Cycle-100 proof-DAG pane for the inner-gradient contraction packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46626
AutoSamplingTheory.SALD.cycle101DiscreteForwardKlNoBoundaryProductRuleLowerObligation - Cycle-101 lower product-rule handoff for the no-boundary `barB` drift boundary. defCompiledNot 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. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46784
AutoSamplingTheory.SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryDag - Cycle-102 proof-DAG pane for the trace-product zero-flux packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46879
AutoSamplingTheory.SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionLowerObligation - Cycle-103 lower obligation after the compiled `condExpKernel.map` bridge. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:46906
AutoSamplingTheory.SALD.cycle103GeneralMovingTargetDiscreteConditionalKernelVersionDag - Cycle-103 proof-DAG pane for the one-component conditional-kernel versioning packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47022
AutoSamplingTheory.SALD.cycle104GeneralMovingTargetDiscreteWeakFpNamedLawTransportLowerObligation - Cycle-104 lower obligation for the compiled named-law weak derivative transport theorem. defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47144
AutoSamplingTheory.SALD.cycle105GeneralMovingTargetDiscretePureRawKlDerivativeLowerObligation - Cycle-105 lower obligation for the compiled pure no-mass KL handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47166
AutoSamplingTheory.SALD.cycle105GeneralMovingTargetDiscretePureRawKlDerivativeDag - Cycle-105 proof-DAG pane for the pure no-mass KL/log-ratio boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47255
AutoSamplingTheory.SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftLowerObligation - Cycle-106 lower obligation for the compiled canonical `condDistrib` drift regularity theorem. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47278
AutoSamplingTheory.SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftDag - Cycle-106 proof-DAG pane for canonical conditional-integral drift regularity. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47382
AutoSamplingTheory.SALD.cycle107DiscreteForwardKlBoundaryFluxIntegralDag - Cycle-107 proof-DAG pane for the boundary-flux integral packet. defCompiledNot 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_ defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47627
AutoSamplingTheory.SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefMiddleObligation - Cycle-109 middle packet for the named `barB` source-definition bridge. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47675
AutoSamplingTheory.SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefDag - Cycle-109 proof-DAG pane for the named `barB` source-definition boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47828
AutoSamplingTheory.SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasLowerObligation - Cycle-110 lower obligation for equality-set measurability of named `barB`. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:47846
AutoSamplingTheory.SALD.cycle110GeneralMovingTargetDiscreteNamedBarBEqMeasDag - Cycle-110 proof-DAG pane for the named `barB` equality-set packet. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48035
AutoSamplingTheory.SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeLowerObligation - Cycle-111 lower obligation for the dominated target-time theorem. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48056
AutoSamplingTheory.SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeDag - Cycle-111 proof-DAG pane for the target-time KL derivative packet. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48179
AutoSamplingTheory.SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeLowerObligation - Cycle-112 lower obligation for the compiled conditional-expectation representative handoff. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48200
AutoSamplingTheory.SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeDag - Cycle-112 proof-DAG pane for the named `barB` conditional-expectation representative boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48334
AutoSamplingTheory.SALD.cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityDag - Cycle-113 proof-DAG pane for the named `barB` regularity pullback. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48427
AutoSamplingTheory.SALD.cycle114GeneralMovingTargetDiscreteCanonicalBarBStateEventSetIntegralDag - Cycle-114 proof-DAG pane for the canonical state-event set-integral narrowing. defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48522
AutoSamplingTheory.SALD.cycle115GeneralMovingTargetDiscreteNamedBarBSelectedVersionDag - Cycle-115 proof-DAG pane for the selected named `barB` version boundary. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48617
AutoSamplingTheory.SALD.cycle116GeneralMovingTargetDiscreteCanonicalBarBCondExpDag - Cycle-116 proof-DAG pane for the canonical `barB` conditional-expectation representative. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48743
AutoSamplingTheory.SALD.cycle117GeneralMovingTargetDiscreteNamedBarBVersionSelectionDag - Cycle-117 proof-DAG pane for the selected paper `barB` version boundary. defCompiledNot 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 defCompiledNot 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 ` defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48869
AutoSamplingTheory.SALD.cycle118GeneralMovingTargetDiscreteCanonicalBarBDownstreamDag - Cycle-118 proof-DAG pane for the direct canonical `barB` downstream route. defCompiledNot 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 defCompiledNot 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- defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:48974
AutoSamplingTheory.SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerDag - Cycle-119 proof-DAG pane for the canonical `barB` weak-FP consumer. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:49117
AutoSamplingTheory.SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag - Cycle-120 proof-DAG pane for the EM sample-path derivative/domination subboundary. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:50437
AutoSamplingTheory.SALD.cycle132GeneralMovingTargetDiscreteEmSecondGreenTestTraceZeroDag - Cycle-132 proof-DAG pane for the second-Green zero-test-trace narrowing. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:50593
AutoSamplingTheory.SALD.cycle133GeneralMovingTargetDiscreteEmSecondGreenPointwiseTestTraceZeroDag - Cycle-133 proof-DAG pane for the pointwise admissible-test trace boundary. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:50754
AutoSamplingTheory.SALD.cycle134GeneralMovingTargetDiscreteEmTestLaplacianSourcePullbackDag - Cycle-134 proof-DAG pane for the source-pullback test-Laplacian boundary. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:50876
AutoSamplingTheory.SALD.cycle135GeneralMovingTargetDiscreteEmSecondGreenStdBasisConsumerLowerObligation - Cycle-135 lower_2 downstream consumer for the standard-basis test-Laplacian source formulas. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:50896
AutoSamplingTheory.SALD.cycle135GeneralMovingTargetDiscreteEmTestLaplacianStdBasisDag - Cycle-135 proof-DAG pane for the test-calculus standard-basis boundary. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51023
AutoSamplingTheory.SALD.cycle136GeneralMovingTargetDiscreteEmWeakFpStdBasisSourceDensityDag - Cycle-136 proof-DAG pane for the weak-FP standard-basis source-density boundary. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51210
AutoSamplingTheory.SALD.cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxLowerObligation - Cycle-138 narrowing of the first-Green pointwise leaf to boundary-flux cancellation facts. defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51358
AutoSamplingTheory.SALD.cycle138GeneralMovingTargetDiscreteEmFirstGreenPointwiseBoundaryFluxDag - Cycle-138 proof-DAG pane for the first-Green pointwise boundary-flux sub-boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51377
AutoSamplingTheory.SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianLowerObligation - Cycle-139 weak-FP source-Laplacian field split for the pointwise test-Laplacian route. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51486
AutoSamplingTheory.SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceActionIntegralScoutObligation - Cycle-139 lower_1 state-integral scout for `hweakFpSourceActionDef`. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51505
AutoSamplingTheory.SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianStateIntegralLowerObligation - Cycle-139 lower_2 source-Laplacian state-integral narrowing. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51520
AutoSamplingTheory.SALD.cycle139GeneralMovingTargetDiscreteEmWeakFpSourceLaplacianDag - Cycle-139 proof-DAG pane for the weak-FP source-Laplacian field split. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51667
AutoSamplingTheory.SALD.cycle140GeneralMovingTargetDiscreteEmGeneratorLawIntegralScoutObligation - Cycle-140 lower_1 law-integral scout for the frozen EM generator Laplacian component. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51707
AutoSamplingTheory.SALD.cycle140GeneralMovingTargetDiscreteEmLaplacianSourceStateIntegralDag - Cycle-140 proof-DAG pane for the source-Laplacian state-integral boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51725
AutoSamplingTheory.SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceLowerObligation - Cycle-141 narrowing of the frozen EM generator source-action leaf. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51858
AutoSamplingTheory.SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceFieldSourceScoutObligation - Cycle-141 lower_1 scout narrowing below the EM generator standard-basis leaf. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51877
AutoSamplingTheory.SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorTraceLawIntegralLowerObligation - Cycle-141 lower_2 law-integral narrowing below the trace-action leaf. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51894
AutoSamplingTheory.SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralMiddleObligation - Cycle-142 state-integral narrowing below the trace-law leaf. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51911
AutoSamplingTheory.SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianStateIntegralScoutObligation - Cycle-142 lower_1 scout narrowing below the trace-state integral leaf. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51929
AutoSamplingTheory.SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceLaplacianLawIntegralLowerObligation - Cycle-142 lower_2 law-integral narrowing below the EM Laplacian state-integral leaf. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51950
AutoSamplingTheory.SALD.cycle141GeneralMovingTargetDiscreteEmGeneratorStdBasisSourceDag - Cycle-141 proof-DAG pane for the EM generator source-action split. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:51969
AutoSamplingTheory.SALD.cycle142GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralDag - Cycle-142 proof-DAG pane for the EM generator trace state-integral split. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:52118
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventLowerObligation - Cycle-143 state-event narrowing for the frozen EM generator Laplacian law integral. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:52297
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianStateEventDag - Cycle-143 proof-DAG pane for the state-event law-integral split. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:52402
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianPointwiseEventDag - Cycle-143 lower-1 proof-DAG pane for pointwise event-field narrowing. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:52505
AutoSamplingTheory.SALD.cycle143GeneralMovingTargetDiscreteEmGeneratorLaplacianActionDefDag - Cycle-143 lower-2 proof-DAG pane for the action-definition narrowing. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:52609
AutoSamplingTheory.SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisEventDag - Cycle-144 proof-DAG pane for the standard-basis frozen EM event-field narrowing. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:52719
AutoSamplingTheory.SALD.cycle144GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisActionDag - Cycle-144 lower-2 proof-DAG pane for the standard-basis action narrowing. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:52828
AutoSamplingTheory.SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianStdBasisLawDag - Cycle-145 proof-DAG pane for the law-integral to standard-basis action narrowing. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:52938
AutoSamplingTheory.SALD.cycle145GeneralMovingTargetDiscreteEmGeneratorLaplacianTraceEventDag - Cycle-145 lower_2 proof-DAG pane for the trace-field event narrowing. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:53045
AutoSamplingTheory.SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceEventTotalEventDag - Cycle-146 proof-DAG pane for the trace-event total-event narrowing. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:53136
AutoSamplingTheory.SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorTraceFieldLaplacianDag - Cycle-146 lower_1 proof-DAG pane for the trace-field Laplacian narrowing. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:53241
AutoSamplingTheory.SALD.cycle146GeneralMovingTargetDiscreteEmGeneratorPointwiseTraceEventDag - Cycle-146 lower_2 proof-DAG pane for the pointwise trace-event narrowing. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:53344
AutoSamplingTheory.SALD.cycle147GeneralMovingTargetDiscreteEmGeneratorPointwiseActionDefTraceLaplacianDag - Cycle-147 proof-DAG pane for the pointwise action-definition trace-Laplacian narrowing. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54464
AutoSamplingTheory.SALD.cycle152GeneralMovingTargetDiscreteEmGeneratorEventSourceFieldStdBasisLowerObligation - Cycle-152 direct-leaf narrowing for the EM event-field/source-field equality. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54527
AutoSamplingTheory.SALD.cycle152GeneralMovingTargetDiscreteWeakFpSourceFieldPointwiseLowerObligation - Cycle-152 lower_2 direct narrowing for the weak-FP source-field/Laplacian equality. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54545
AutoSamplingTheory.SALD.cycle152GeneralMovingTargetDiscreteEmGeneratorEventSourceFieldStdBasisDag - Cycle-152 proof-DAG pane for the source-field equality narrowing. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54561
AutoSamplingTheory.SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalLowerObligation - Cycle-153 direct-leaf narrowing for the selected-test Laplacian state integral. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54651
AutoSamplingTheory.SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralStdBasisSourceFunctionalLowerObligation - Cycle-153 lower_2 narrowing from the standard-basis source formula. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54673
AutoSamplingTheory.SALD.cycle153GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralSourceFunctionalDag - Cycle-153 proof-DAG pane for the state-integral/source-functional leaf. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54697
AutoSamplingTheory.SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalLowerObligation - Cycle-154 narrowing of the state-integral standard-basis source premise. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54793
AutoSamplingTheory.SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceLawIntegralLaplacianFieldScoutObligation - Cycle-154 lower_1 scout narrowing of the trace-field state-integral inputs. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54821
AutoSamplingTheory.SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceStateIntegralLaplacianFieldLowerObligation - Cycle-154 lower_2 narrowing of the trace-law state-integral input. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54846
AutoSamplingTheory.SALD.cycle154GeneralMovingTargetDiscreteEmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalDag - Cycle-154 proof-DAG pane for the trace-field source-functional state-integral split. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:54871
AutoSamplingTheory.SALD.cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldLowerObligation - Cycle-155 narrowing of the trace-state integral source boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55006
AutoSamplingTheory.SALD.cycle155GeneralMovingTargetDiscreteEmGeneratorTraceStateIntegralLaplacianFieldDag - Cycle-155 proof-DAG pane for the trace-state sample-integral split. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55031
AutoSamplingTheory.SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseLowerObligation - Cycle-156 narrowing of the trace-field/Laplacian source boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55097
AutoSamplingTheory.SALD.cycle156GeneralMovingTargetDiscreteEmGeneratorTraceFieldPointwiseDag - Cycle-156 proof-DAG pane for the trace-field pointwise split. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55123
AutoSamplingTheory.SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisLowerObligation - Cycle-157 narrowing of the EM Laplacian event-field standard-basis boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55240
AutoSamplingTheory.SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseLaplacianScoutObligation - Cycle-157 lower_1 scout split for the remaining pointwise event-field display. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55258
AutoSamplingTheory.SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarLowerObligation - Cycle-157 lower_2 narrowing of the pointwise event-field Laplacian leaf. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55277
AutoSamplingTheory.SALD.cycle157GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseStdBasisDag - Cycle-157 proof-DAG pane for the event-field pointwise standard-basis split. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55409
AutoSamplingTheory.SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianDefLowerObligation - Cycle-158 lower_2 source-definition boundary after the Brownian event-field split. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55429
AutoSamplingTheory.SALD.cycle158GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldPointwiseScalarAuditDag - Cycle-158 proof-DAG pane for the scalar event-field Delta blocker. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55540
AutoSamplingTheory.SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseStdBasisLowerObligation - Cycle-159 lower_2 standard-basis split of the Brownian pointwise boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55559
AutoSamplingTheory.SALD.cycle159GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldBrownianPointwiseDag - Cycle-159 proof-DAG pane for the Brownian event-field pointwise split. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55678
AutoSamplingTheory.SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseScoutObligation - Cycle-160 lower_1 pointwise scout split of the frozen scalar Ito generator boundary. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55718
AutoSamplingTheory.SALD.cycle160GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDag - Cycle-160 proof-DAG pane for the frozen scalar Brownian Ito generator split. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55866
AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorScoutObligation - Cycle-161 lower_1 scout route below the one-dimensional Brownian/Ito Taylor boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55891
AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoOneDimTaylorMomentLowerObligation - Cycle-161 lower_2 moment-algebra narrowing below the one-dimensional Taylor boundary. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:55914
AutoSamplingTheory.SALD.cycle161GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDag - Cycle-161 proof-DAG pane for the one-dimensional coordinate Taylor split. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56060
AutoSamplingTheory.SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderScoutObligation - Cycle-162 lower_1 scout route for the normalized scalar Taylor remainder. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56089
AutoSamplingTheory.SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderDctLowerObligation - Cycle-162 lower_2 compiled DCT theorem for the normalized scalar Taylor remainder. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56113
AutoSamplingTheory.SALD.cycle162GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderDag - Cycle-162 proof-DAG pane for the scalar Taylor remainder split. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56282
AutoSamplingTheory.SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderSourceEqLowerObligation - Cycle-163 lower_2 packet for the normalized-remainder source equality. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56311
AutoSamplingTheory.SALD.cycle163GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorDag - Cycle-163 proof-DAG pane for the source Taylor pointwise limit. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56513
AutoSamplingTheory.SALD.cycle164GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderMeasDag - Cycle-164 proof-DAG pane for the concrete normalized-remainder measurability leaf. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56641
AutoSamplingTheory.SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorQuotientSplitLower2Obligation - Cycle-165 lower_2 packet splitting the deterministic Taylor quotient bound. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56671
AutoSamplingTheory.SALD.cycle165GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderQuadraticBoundDag - Cycle-165 proof-DAG pane for the concrete normalized-remainder domination leaf. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56798
AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation - Cycle-166 lower_1 nonnegative interval Taylor proof-scout packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56824
AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation - Cycle-166 lower_2 signed interval Taylor combination packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56849
AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderDag - Cycle-166 proof-DAG pane for the first-order selected-line remainder split. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56877
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectMiddleObligation - Cycle-167 reflected Taylor compatibility discharge packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:56993
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectDag - Cycle-167 proof-DAG pane for the reflected Taylor compatibility discharge. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57022
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Obligation - Cycle-167 lower_1 Taylor-compatibility narrowing packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57096
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineContDiffLower2Obligation - Cycle-167 lower_2 global line regularity narrowing packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57125
AutoSamplingTheory.SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffMiddleObligation - Cycle-168 ambient selected-test regularity narrowing packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57155
AutoSamplingTheory.SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation - Cycle-168 lower_2 global line-second-derivative narrowing packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57182
AutoSamplingTheory.SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondMiddleObligation - Cycle-169 ambient directional-Hessian narrowing packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57211
AutoSamplingTheory.SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondLower2Obligation - Cycle-169 lower_2 second-Frechet-derivative operator-norm narrowing packet. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57239
AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Dag - Cycle-167 proof-DAG pane for the Taylor-data narrowing. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57259
AutoSamplingTheory.SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDag - Cycle-168 proof-DAG pane for ambient source-test regularity to selected line. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57358
AutoSamplingTheory.SALD.cycle169GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDag - Cycle-169 proof-DAG pane for the ambient directional-Hessian line-second split. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57577
AutoSamplingTheory.SALD.cycle170GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormDag - Cycle-170 lower_2 proof-DAG pane for the selected-test Hessian source interface. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57651
AutoSamplingTheory.SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag - Cycle-171 proof-DAG pane for the rejected wrapper and remaining source contract. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57675
AutoSamplingTheory.SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation - Cycle-171 lower_1 source audit for the selected-test Hessian contract. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57760
AutoSamplingTheory.SALD.cycle171GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower2Obligation - Cycle-171 lower_2 rejection of the unsourced selected-test Hessian projection route. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57812
AutoSamplingTheory.SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag - Cycle-172 proof-DAG pane for the refreshed Hessian source-contract illness area. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:57840
AutoSamplingTheory.SALD.cycle172GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceLower1Obligation - Cycle-172 lower_1 proof-scout audit for the selected-test Hessian source contract. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:58070
AutoSamplingTheory.SALD.cycle173GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDag - Cycle-173 proof-DAG pane for the middle source-contract recovery packet. defCompiledNot 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 defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:58265
AutoSamplingTheory.SALD.cycle174GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationDag - Cycle-174 proof-DAG pane for the Brownian quadratic-variation leaf. defCompiledNot 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 defCompiledNot 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 defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:58481
AutoSamplingTheory.SALD.cycle175GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDag - Cycle-175 proof-DAG pane for the selected-line Taylor-domination bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:58637
AutoSamplingTheory.SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneLower1Obligation - Cycle-176 lower_1 proof-scout route for the normalized scalar Brownian variance field. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:58707
AutoSamplingTheory.SALD.cycle176GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneDag - Cycle-176 proof-DAG pane for the normalized Brownian variance field. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:58879
AutoSamplingTheory.SALD.cycle177GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSecondTaylorCoeffDag - Cycle-177 proof-DAG pane for the scalar-line coefficient narrowing. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:59019
AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower1Obligation - Cycle-178 lower_1 proof-scout packet for the normalized coordinate law. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:59053
AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedCoordinateLawLower2Obligation - Cycle-178 lower_2 compiled bridge for the normalized Brownian coordinate law. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:59083
AutoSamplingTheory.SALD.cycle178GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDag - Cycle-178 proof-DAG pane for normalized Brownian variance law narrowing. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. defCompiledNot 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 defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:59611
AutoSamplingTheory.SALD.cycle180GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDag - Cycle-180 proof-DAG pane for the Taylor moment decomposition split. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:59820
AutoSamplingTheory.SALD.cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDag - Cycle-183 proof-DAG pane for the Brownian coordinate source-integral leaf. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:59986
AutoSamplingTheory.SALD.cycle184GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDag - Cycle-184 proof-DAG pane for the Brownian coordinate source-integral law split. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:60123
AutoSamplingTheory.SALD.cycle185GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitDag - Cycle-185 proof-DAG pane for the remainder-generator law split. defCompiledNot 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- defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:60245
AutoSamplingTheory.SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandPointwiseDag - Cycle-186 proof-DAG pane for the source Taylor integrand pointwise split. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:60355
AutoSamplingTheory.SALD.cycle186GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceLinearTermLower2Dag - Cycle-186 proof-DAG pane for the source linear term bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:60463
AutoSamplingTheory.SALD.cycle187GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceQuadraticTermDag - Cycle-187 proof-DAG pane for the source quadratic term bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:60575
AutoSamplingTheory.SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDefDag - Cycle-188 proof-DAG pane for the source Taylor integrand definition bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:60687
AutoSamplingTheory.SALD.cycle188GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorSplitLower2Dag - Cycle-188 proof-DAG pane for the selected-line Taylor split bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:60784
AutoSamplingTheory.SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsDag - Cycle-189 proof-DAG pane for the raw-term coordinate Taylor integral bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:60908
AutoSamplingTheory.SALD.cycle189GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandRawLower2Dag - Cycle-189 lower_2 proof-DAG pane for the raw source Taylor integrand bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61003
AutoSamplingTheory.SALD.cycle190GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementCoordinateLineDag - Cycle-190 proof-DAG pane for the selected-increment endpoint bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61125
AutoSamplingTheory.SALD.cycle190GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitLower2Dag - Cycle-190 lower_2 proof-DAG pane for the remainder-limit scalar-pushforward bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61268
AutoSamplingTheory.SALD.cycle191GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLower2Dag - Cycle-191 proof-DAG pane for the source-integral scalar-pushforward bridge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61411
AutoSamplingTheory.SALD.cycle192GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralLower2Dag - Cycle-192 proof-DAG pane for the Taylor-integral source-integral discharge. defCompiledNot 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 defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61631
AutoSamplingTheory.SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentRemainderLimitLower2Dag - Cycle-193 lower_2 proof-DAG pane for the one-hypothesis remainder-limit discharge. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61672
AutoSamplingTheory.SALD.cycle193GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentMiddleDag - Cycle-193 proof-DAG pane for the Taylor moment scalar-pushforward discharge. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61810
AutoSamplingTheory.SALD.cycle197GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderBoundDefLower2Dag - Cycle-197 proof-DAG pane for the normalized remainder bound-definition gap. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61884
AutoSamplingTheory.SALD.cycle198GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumLower2Dag - Cycle-198 proof-DAG pane for the frozen Brownian coordinate-sum gap. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:61952
AutoSamplingTheory.SALD.cycle201GeneralMovingTargetDiscreteEmInterpolationSelectedTestLaplacianContinuityLower2Dag - Cycle-201 proof-DAG pane for the selected-test Laplacian continuity gap. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62026
AutoSamplingTheory.SALD.cycle202GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineLower2Dag - Cycle-202 proof-DAG pane for the selected endpoint coordinate-line gap. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62113
AutoSamplingTheory.SALD.cycle203GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointLower2Dag - Cycle-203 proof-DAG pane for the selected-increment endpoint gap. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62198
AutoSamplingTheory.SALD.cycle204GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementLower2Dag - Cycle-204 proof-DAG pane for the source Taylor integrand naming gap. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62285
AutoSamplingTheory.SALD.cycle205GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitLower2Dag - Cycle-205 proof-DAG pane for the selected-line raw Taylor split gap. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62384
AutoSamplingTheory.SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackMiddleDag - Cycle-206 proof-DAG pane for the normalized-remainder pullback gap. defCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62478
AutoSamplingTheory.SALD.cycle206GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackLower2Dag - Cycle-206 proof-DAG pane for the lower_2 remainder-pullback gap. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62568
AutoSamplingTheory.SALD.cycle70GeneralMovingTargetDiscreteConditionalLawDag - Cycle-70 proof-DAG pane for the EM conditional-law/measurability backfill. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62668
AutoSamplingTheory.SALD.cycle69MainSkeletonAnalyticInterfaceDag - Cycle-69 proof-DAG pane for the post-route analytic-interface ledger. defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62728
AutoSamplingTheory.SALD.guidedResidualNormalizerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62856
AutoSamplingTheory.SALD.guidedResidualIdentityObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62864
AutoSamplingTheory.SALD.generalMovingTargetDerivativeObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62872
AutoSamplingTheory.SALD.generalMovingTargetDvEnergyObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62896
AutoSamplingTheory.SALD.generalMovingTargetDvPositiveAlphaScalingObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62910
AutoSamplingTheory.SALD.generalMovingTargetDvFiniteLogMgfWitnessObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62925
AutoSamplingTheory.SALD.generalMovingTargetGronwallApplicationObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62941
AutoSamplingTheory.SALD.cycle24GeneralVaSaldGronwallMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62949
AutoSamplingTheory.SALD.generalMovingTargetGronwallSideConditionObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62971
AutoSamplingTheory.SALD.generalMovingTargetPureContractionObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62991
AutoSamplingTheory.SALD.cycle16UnifiedForwardKlTransportBridgeMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:62999
AutoSamplingTheory.SALD.cycle16UnifiedForwardKlTransportBridgeLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63014
AutoSamplingTheory.SALD.unifiedForwardKlTransportBridgeObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63029
AutoSamplingTheory.SALD.unifiedForwardKlSpecializationObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63045
AutoSamplingTheory.SALD.generalVaSaldGuidedPathMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63063
AutoSamplingTheory.SALD.cycle64GeneralMovingTargetDiscreteConditionalDriftLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63092
AutoSamplingTheory.SALD.generalMovingTargetDiscreteEmInterpolationObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63115
AutoSamplingTheory.SALD.generalMovingTargetDiscreteConstantScheduleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63123
AutoSamplingTheory.SALD.generalMovingTargetDiscreteFrozenDeltaObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63131
AutoSamplingTheory.SALD.cycle28GeneralVaSaldDerivativeSideMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63139
AutoSamplingTheory.SALD.cycle28GeneralVaSaldDerivativeSideLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63159
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDerivativeSideConditionObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63174
AutoSamplingTheory.SALD.cycle54GeneralMovingTargetDiscreteEmFpLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63182
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDerivativeObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63197
AutoSamplingTheory.SALD.cycle53GeneralMovingTargetDiscreteDerivativeDvLowerObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63268
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDvMEnergyObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63292
AutoSamplingTheory.SALD.generalMovingTargetDiscreteDvFiniteLogMgfWitnessObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63306
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallApplicationObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63323
AutoSamplingTheory.SALD.cycle20GeneralVaSaldDiscreteGronwallMiddleObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63331
AutoSamplingTheory.SALD.generalMovingTargetDiscreteGronwallSideConditionObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63354
AutoSamplingTheory.SALD.discreteUnifiedVaSaldSpecializationObligation defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63381
AutoSamplingTheory.SALD.gronwallContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63389
AutoSamplingTheory.SALD.dvContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63408
AutoSamplingTheory.SALD.piDefinitionContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63430
AutoSamplingTheory.SALD.lsiKlFiVocabularyContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63445
AutoSamplingTheory.SALD.continuousSaldContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63469
AutoSamplingTheory.SALD.forwardKlProofDag defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:63530
AutoSamplingTheory.SALD.discreteForwardKlProofDag defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:64011
AutoSamplingTheory.SALD.discreteSaldContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:64708
AutoSamplingTheory.SALD.generalVaSaldProofDag defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:64861
AutoSamplingTheory.SALD.generalVaSaldDiscreteProofDag defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65136
AutoSamplingTheory.SALD.guidedResidualContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65432
AutoSamplingTheory.SALD.generalVaSaldContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65466
AutoSamplingTheory.SALD.unifiedForwardKlContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65517
AutoSamplingTheory.SALD.generalVaSaldDiscreteContract defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65570
AutoSamplingTheory.SALD.saldTheoremContracts defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65707
AutoSamplingTheory.SALD.saldSourceForLabel defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65721
AutoSamplingTheory.SALD.saldLeanTargetForLabel defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65734
AutoSamplingTheory.SALD.cycle49MainSkeletonDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65738
AutoSamplingTheory.SALD.cycle50ForwardKlDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65751
AutoSamplingTheory.SALD.cycle51DiscreteForwardKlDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65767
AutoSamplingTheory.SALD.cycle52GuidedGeneralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65787
AutoSamplingTheory.SALD.cycle53UnifiedDiscreteGeneralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65805
AutoSamplingTheory.SALD.cycle54MainSkeletonDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65832
AutoSamplingTheory.SALD.cycle55ForwardKlDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65859
AutoSamplingTheory.SALD.cycle56DiscreteForwardKlDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65901
AutoSamplingTheory.SALD.cycle57GuidedGeneralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65948
AutoSamplingTheory.SALD.cycle58UnifiedDiscreteGeneralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:65992
AutoSamplingTheory.SALD.cycle59MainSkeletonDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66030
AutoSamplingTheory.SALD.cycle60ForwardKlDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66059
AutoSamplingTheory.SALD.cycle65ForwardKlDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66103
AutoSamplingTheory.SALD.cycle61DiscreteForwardKlDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66157
AutoSamplingTheory.SALD.cycle66DiscreteForwardKlDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66227
AutoSamplingTheory.SALD.cycle62GuidedGeneralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66301
AutoSamplingTheory.SALD.cycle67GuidedGeneralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66361
AutoSamplingTheory.SALD.cycle68UnifiedDiscreteGeneralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66424
AutoSamplingTheory.SALD.cycle69MainSkeletonDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66486
AutoSamplingTheory.SALD.cycle70EmConditionalLawDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66532
AutoSamplingTheory.SALD.cycle71EmEndpointConditionalDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66550
AutoSamplingTheory.SALD.cycle72EmWeakFpDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66571
AutoSamplingTheory.SALD.cycle73EmKlDerivativeWeakFpDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66591
AutoSamplingTheory.SALD.cycle74EmConditionalKernelMeasureDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66614
AutoSamplingTheory.SALD.cycle75EmConditionalLawBackfillDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66640
AutoSamplingTheory.SALD.cycle76EmEndpointConditionalDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66668
AutoSamplingTheory.SALD.cycle77EmWeakFpSourceSignsDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66692
AutoSamplingTheory.SALD.cycle78EmKlDerivativeGeneratorDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66712
AutoSamplingTheory.SALD.cycle79EmWeakFpGeneratorMeasureDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66735
AutoSamplingTheory.SALD.cycle80EmConditionalLawMeasurabilityDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66764
AutoSamplingTheory.SALD.cycle81EmEndpointConditionalDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66792
AutoSamplingTheory.SALD.cycle82EmWeakFpSourceSignsDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66823
AutoSamplingTheory.SALD.cycle83EmKlDerivativeEndpointWeakFpDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66854
AutoSamplingTheory.SALD.cycle84ActiveEmBackendDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66879
AutoSamplingTheory.SALD.cycle85EmConditionalKernelBoundaryDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66907
AutoSamplingTheory.SALD.cycle86EmWeakFpGeneratorBoundaryDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66949
AutoSamplingTheory.SALD.cycle87EmKlLogRatioBoundaryDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:66977
AutoSamplingTheory.SALD.cycle88EmKlLogRatioAdmissibilityDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67013
AutoSamplingTheory.SALD.cycle89DiscreteForwardKlPressureDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67043
AutoSamplingTheory.SALD.cycle90DiscreteForwardKlMassConservationDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67085
AutoSamplingTheory.SALD.cycle91EmConditionalKernelDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67128
AutoSamplingTheory.SALD.cycle92EmWeakFpGeneratorSplitDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67154
AutoSamplingTheory.SALD.cycle93EmKlMassDerivativeDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67178
AutoSamplingTheory.SALD.cycle94EmWeakFpDriftActionDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67207
AutoSamplingTheory.SALD.cycle95DiscreteForwardKlPressureDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67232
AutoSamplingTheory.SALD.cycle96EmCondexpGeneratorPairingDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67270
AutoSamplingTheory.SALD.cycle97EmCanonicalCondDistribPairingDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67300
AutoSamplingTheory.SALD.cycle98EmBarBDivergenceNoBoundaryDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67328
AutoSamplingTheory.SALD.cycle99EmRawKlDerivativeDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67351
AutoSamplingTheory.SALD.cycle100EmBarBWeakGradDefDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67380
AutoSamplingTheory.SALD.cycle101DiscreteForwardKlPressureDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67408
AutoSamplingTheory.SALD.cycle102DiscreteForwardKlZeroFluxTraceBoundaryDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67431
AutoSamplingTheory.SALD.cycle103EmConditionalKernelComponentVersionDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67455
AutoSamplingTheory.SALD.cycle104EmWeakFpNamedLawTransportDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67483
AutoSamplingTheory.SALD.cycle105EmPureRawKlDerivativeDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67509
AutoSamplingTheory.SALD.cycle106EmCanonicalCondDistribDriftDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67537
AutoSamplingTheory.SALD.cycle107DiscreteForwardKlBoundaryFluxDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67562
AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxContinuityDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67580
AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67599
AutoSamplingTheory.SALD.cycle109EmNamedBarBSourceDefDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67620
AutoSamplingTheory.SALD.cycle110EmNamedBarBEqMeasDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67649
AutoSamplingTheory.SALD.cycle110EmWeakFpDominatedGeneratorDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67670
AutoSamplingTheory.SALD.cycle111EmKlTargetTimeDerivativeDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67688
AutoSamplingTheory.SALD.cycle112EmNamedBarBCondExpRepresentativeDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67718
AutoSamplingTheory.SALD.cycle113EmNamedBarBStateFieldRegularityDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67750
AutoSamplingTheory.SALD.cycle114EmCanonicalBarBStateEventSetIntegralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67770
AutoSamplingTheory.SALD.cycle115EmNamedBarBSelectedVersionDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67792
AutoSamplingTheory.SALD.cycle116EmCanonicalBarBCondExpDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67820
AutoSamplingTheory.SALD.cycle117EmNamedBarBVersionSelectionDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67845
AutoSamplingTheory.SALD.cycle118EmCanonicalBarBDownstreamDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67872
AutoSamplingTheory.SALD.cycle119EmCanonicalBarBWeakFpConsumerDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67897
AutoSamplingTheory.SALD.cycle120EmPathDerivativeDominationDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67925
AutoSamplingTheory.SALD.cycle121EmSampleMeasDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67950
AutoSamplingTheory.SALD.cycle122EmSampleIntDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67970
AutoSamplingTheory.SALD.cycle123EmSampleDerivMeasDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:67992
AutoSamplingTheory.SALD.cycle124EmSampleDerivBoundDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68014
AutoSamplingTheory.SALD.cycle124EmBoundIntDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68033
AutoSamplingTheory.SALD.cycle125EmPathDerivDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68053
AutoSamplingTheory.SALD.cycle126EmDerivValueDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68071
AutoSamplingTheory.SALD.cycle127EmDriftActionDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68089
AutoSamplingTheory.SALD.cycle128EmNoBoundaryTraceDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68111
AutoSamplingTheory.SALD.cycle129EmDiffusionSourceDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68137
AutoSamplingTheory.SALD.cycle130EmLaplacianIbPDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68156
AutoSamplingTheory.SALD.cycle131EmSecondGreenNoBoundaryFluxDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68182
AutoSamplingTheory.SALD.cycle132EmSecondGreenTestTraceZeroDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68208
AutoSamplingTheory.SALD.cycle133EmSecondGreenPointwiseTestTraceZeroDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68231
AutoSamplingTheory.SALD.cycle134EmTestLaplacianSourcePullbackDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68254
AutoSamplingTheory.SALD.cycle135EmTestLaplacianStdBasisDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68280
AutoSamplingTheory.SALD.cycle136EmWeakFpStdBasisSourceDensityDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68304
AutoSamplingTheory.SALD.cycle137EmWeakFpDensityLaplacianActionDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68327
AutoSamplingTheory.SALD.cycle138EmFirstGreenPointwiseBoundaryFluxDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68355
AutoSamplingTheory.SALD.cycle139EmWeakFpSourceLaplacianDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68383
AutoSamplingTheory.SALD.cycle140EmLaplacianSourceStateIntegralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68410
AutoSamplingTheory.SALD.cycle141EmGeneratorStdBasisSourceDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68437
AutoSamplingTheory.SALD.cycle142EmGeneratorTraceStateIntegralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68465
AutoSamplingTheory.SALD.cycle143EmGeneratorLaplacianStateEventDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68498
AutoSamplingTheory.SALD.cycle144EmGeneratorLaplacianStdBasisEventDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68535
AutoSamplingTheory.SALD.cycle145EmGeneratorLaplacianStdBasisLawDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68565
AutoSamplingTheory.SALD.cycle146EmGeneratorTraceEventTotalEventDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68595
AutoSamplingTheory.SALD.cycle147EmGeneratorPointwiseActionDefTraceLaplacianDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68634
AutoSamplingTheory.SALD.cycle148EmGeneratorPointwiseStateEventTraceLaplacianDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68673
AutoSamplingTheory.SALD.cycle149EmGeneratorTotalEventStdBasisSourceFunctionalDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68728
AutoSamplingTheory.SALD.cycle150EmGeneratorTotalEventTraceLawIntegralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68769
AutoSamplingTheory.SALD.cycle151EmGeneratorTotalEventTraceLaplacianStateIntegralPointwiseEventDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68814
AutoSamplingTheory.SALD.cycle152EmGeneratorEventSourceFieldStdBasisDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68845
AutoSamplingTheory.SALD.cycle153EmGeneratorLaplacianStateIntegralSourceFunctionalDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68877
AutoSamplingTheory.SALD.cycle154EmGeneratorLaplacianStateIntegralTraceFieldSourceFunctionalDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68909
AutoSamplingTheory.SALD.cycle155EmGeneratorTraceStateIntegralLaplacianFieldDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68954
AutoSamplingTheory.SALD.cycle156EmGeneratorTraceFieldPointwiseDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:68982
AutoSamplingTheory.SALD.cycle157EmGeneratorLaplacianEventFieldPointwiseStdBasisDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69016
AutoSamplingTheory.SALD.cycle158EmGeneratorLaplacianEventFieldPointwiseScalarAuditDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69051
AutoSamplingTheory.SALD.cycle159EmGeneratorLaplacianEventFieldBrownianPointwiseDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69077
AutoSamplingTheory.SALD.cycle160EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69105
AutoSamplingTheory.SALD.cycle161EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateGeneratorDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69139
AutoSamplingTheory.SALD.cycle162EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorRemainderDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69173
AutoSamplingTheory.SALD.cycle163EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoPointwiseTaylorDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69211
AutoSamplingTheory.SALD.cycle164EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoMeasDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69250
AutoSamplingTheory.SALD.cycle165EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHBoundDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69291
AutoSamplingTheory.SALD.cycle166EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHFirstDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69330
AutoSamplingTheory.SALD.cycle167EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorReflectDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69378
AutoSamplingTheory.SALD.cycle168EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69455
AutoSamplingTheory.SALD.cycle169EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoDirectionalSecondDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69499
AutoSamplingTheory.SALD.cycle170EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianOpNormDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69539
AutoSamplingTheory.SALD.cycle171EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69559
AutoSamplingTheory.SALD.cycle172EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69585
AutoSamplingTheory.SALD.cycle173EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianSourceDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69618
AutoSamplingTheory.SALD.cycle174EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoQuadraticVariationDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69655
AutoSamplingTheory.SALD.cycle175EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorDominationDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69688
AutoSamplingTheory.SALD.cycle176EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoVarianceOneDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69726
AutoSamplingTheory.SALD.cycle177EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSecondTaylorCoeffDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69763
AutoSamplingTheory.SALD.cycle178EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedVarianceDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69802
AutoSamplingTheory.SALD.cycle179EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoHessianAuditScalarLineCoeffDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69848
AutoSamplingTheory.SALD.cycle180EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69897
AutoSamplingTheory.SALD.cycle183EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:69957
AutoSamplingTheory.SALD.cycle184EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralLawDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70002
AutoSamplingTheory.SALD.cycle185EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderLimitDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70056
AutoSamplingTheory.SALD.cycle186EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70109
AutoSamplingTheory.SALD.cycle187EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceQuadraticTermDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70161
AutoSamplingTheory.SALD.cycle188EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandDefDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70205
AutoSamplingTheory.SALD.cycle189EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralRawTermsDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70256
AutoSamplingTheory.SALD.cycle190EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementCoordinateLineDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70320
AutoSamplingTheory.SALD.cycle191EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralScalarPushforwardDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70366
AutoSamplingTheory.SALD.cycle192EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorIntegralScalarPushforwardDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70413
AutoSamplingTheory.SALD.cycle193EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorMomentScalarPushforwardDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70460
AutoSamplingTheory.SALD.cycle197EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoNormalizedRemainderBoundDefDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70518
AutoSamplingTheory.SALD.cycle198EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoCoordinateSumDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70549
AutoSamplingTheory.SALD.cycle199EmInterpolationWeakFpSourceLaplacianFieldMeasDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70576
AutoSamplingTheory.SALD.cycle201EmInterpolationSelectedTestLaplacianContinuityDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70598
AutoSamplingTheory.SALD.cycle202EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedEndpointCoordinateLineDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70629
AutoSamplingTheory.SALD.cycle203EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedIncrementEndpointDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70665
AutoSamplingTheory.SALD.cycle204EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceTaylorIntegrandSelectedIncrementDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70700
AutoSamplingTheory.SALD.cycle205EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSelectedLineTaylorRawSplitDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70737
AutoSamplingTheory.SALD.cycle206EmGeneratorLaplacianEventFieldFrozenScalarBrownianItoRemainderPullbackDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70779
AutoSamplingTheory.SALD.cycle63UnifiedDiscreteGeneralDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70826
AutoSamplingTheory.SALD.cycle64MainSkeletonDependencyNames defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70875
AutoSamplingTheory.SALD.saldDependenciesForLabel defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:70921
AutoSamplingTheory.SALD.saldReusedByForLabel defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:71284
AutoSamplingTheory.SALD.saldStatusForLabel defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:71294
AutoSamplingTheory.SALD.saldFirstProofDag defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:71301
AutoSamplingTheory.SALD.saldExcludedFiles defCompiledNot mapped AutoSamplingTheory.SALD AutoSamplingTheory/SALD.lean:71312
AutoSamplingTheory.ItoDiffusionContract structureCompiledNot mapped AutoSamplingTheory.SDE AutoSamplingTheory/SDE.lean:9
AutoSamplingTheory.FokkerPlanckContract structureCompiledNot mapped AutoSamplingTheory.SDE AutoSamplingTheory/SDE.lean:19
AutoSamplingTheory.EulerMaruyamaContract structureCompiledNot mapped AutoSamplingTheory.SDE AutoSamplingTheory/SDE.lean:27
AutoSamplingTheory.DiscretizationErrorContract structureCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Algebra.ReciprocalGrowthRate AutoSamplingTheory/TechnicalLemmas/Algebra/ReciprocalGrowthRate.lean:16
AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Cutoff.smoothUnitCutoff - A smooth real cutoff equal to one on `[-1, 1]` and zero outside `(-2, 2)`. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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|`. theoremCompiledNot 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]`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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)`. defCompiledPartial 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`. theoremCompiledNot 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||`. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledPartial 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. theoremCompiledNot 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 defCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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; theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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, theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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- theoremCompiledNot 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 ` theoremCompiledCompiled 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:179
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 ∂ theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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- theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.LineDeriv AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/LineDeriv.lean:218
AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.lintegral_ofReal_ne_top_of_integrable_nonneg - A nonnegative integrable real function has finite `ℝ≥0∞` lintegral after `ENNReal.ofReal`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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]`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage AutoSamplingTheory/TechnicalLemmas/Analysis/LeftLebesgueAverage.lean:23
AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage.tendsto_sub_nhdsGT_zero_nhdsLT theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage AutoSamplingTheory/TechnicalLemmas/Analysis/LeftLebesgueAverage.lean:26
AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage.ae_tendsto_leftAverageError_real theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Analysis.LeftLebesgueAverage AutoSamplingTheory/TechnicalLemmas/Analysis/LeftLebesgueAverage.lean:104
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.prefixIntegral - Prefix integral on the finite nonnegative-time horizon. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:24
AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral.prefixIntegral_zero theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Analysis.PrefixIntegral AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:151
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.dirichletForm - The generator Dirichlet form `E(f,g) = integral f (-L)g d mu`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/Generator.lean:22
AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Generator.variance - Variance as the squared centered `L2(mu)` norm. defCompiledNot 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. defCompiledNot 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. defCompiledCompiled 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`. defCompiledNot 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. defCompiledNot 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)`. defCompiledCompiled 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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 defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledCompiled 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 structureCompiledNot 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`. theoremCompiledNot 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. theoremCompiledPartial 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- theoremCompiledPartial 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. theoremCompiledNot 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. theoremCompiledPartial 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. theoremCompiledNot 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. theoremCompiledPartial 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. theoremCompiledNot 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. theoremCompiledPartial 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. theoremCompiledNot 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. theoremCompiledPartial 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. theoremCompiledNot 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. theoremCompiledPartial 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. theoremCompiledNot 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. theoremCompiledPartial AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.SemigroupDecay AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/SemigroupDecay.lean:359
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussianPi - Product standard Gaussian measure on coordinate functions. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Gaussian AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:32
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussianPi_isProbabilityMeasure instanceCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Gaussian AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:35
AutoSamplingTheory.TechnicalLemmas.Gaussian.stdGaussianPi_isFiniteMeasure instanceCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Gaussian AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:45
AutoSamplingTheory.TechnicalLemmas.Gaussian.integral_id_gaussianReal_zero - The centered real Gaussian has zero mean. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Gaussian AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:66
AutoSamplingTheory.TechnicalLemmas.Gaussian.integrable_eval_stdGaussianPi - Coordinate projections under the ASTIS product Gaussian are integrable. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Gaussian AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:109
AutoSamplingTheory.TechnicalLemmas.Gaussian.integrable_linearForm_stdGaussianPi - Finite linear forms in product-Gaussian coordinates are integrable. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Gaussian AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:231
AutoSamplingTheory.TechnicalLemmas.Gaussian.gaussianReal_withDensity_exp_shift - One-dimensional standard Gaussian Esscher density shift. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Gaussian AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:254
AutoSamplingTheory.TechnicalLemmas.Gaussian.pi_gaussianReal_withDensity_exp_shift - Finite product standard Gaussian Esscher density shift. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Gaussian AutoSamplingTheory/TechnicalLemmas/Gaussian.lean:275
AutoSamplingTheory.TechnicalLemmas.Gaussian.pi_gaussianReal_shift_integral - Finite product Gaussian Esscher change of measure. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledCompiled 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 theoremCompiledCompiled 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 defCompiledPartial AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:29
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_iff theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:33
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_of_concave_log theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:39
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.pos theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:46
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.concaveOn_log theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:68
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.convex_domain theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:116
AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.LogConcaveOn.subset theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot 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 defCompiledCompiled 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity AutoSamplingTheory/TechnicalLemmas/Geometry/StrongConvexity.lean:57
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 theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.InformationTheory.KLDensity AutoSamplingTheory/TechnicalLemmas/InformationTheory/KLDensity.lean:33
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.renyiIntegrand - The real-valued Renyi/Hellinger-style density integrand `p^a q^(1-a)`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:27
AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi.renyiIntegrandENNReal - The `ℝ≥0∞` version used for lintegral contracts. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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]`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledPartial AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:80
AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation.IsQuadraticOptimalCoupling - A coupling is quadratic-cost optimal when it attains the Kantorovich infimum defining the squared 2-Wasserstein distance. defCompiledNot 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`. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 defCompiledCompiled 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure.DisplacementInterpolation AutoSamplingTheory/TechnicalLemmas/Measure/DisplacementInterpolation.lean:77
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.gibbsDensityENNReal - The `ℝ≥0∞` density associated with an unnormalized Gibbs potential. defCompiledPartial AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:25
AutoSamplingTheory.TechnicalLemmas.Measure.Gibbs.gibbsDensityENNReal_pos - Gibbs densities are pointwise positive. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledPartial 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 ` theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledPartial 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure.GibbsLogConcavity AutoSamplingTheory/TechnicalLemmas/Measure/GibbsLogConcavity.lean:113
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. defCompiledNot 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 defCompiledCompiled 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. theoremCompiledCompiled AutoSamplingTheory.TechnicalLemmas.Measure.KantorovichDual AutoSamplingTheory/TechnicalLemmas/Measure/KantorovichDual.lean:40
AutoSamplingTheory.TechnicalLemmas.Measure.Product.measurable_update_prod_pi - The coordinate-replacement map `(y, x) ↦ Function.update x i y` is measurable. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure.Product AutoSamplingTheory/TechnicalLemmas/Measure/Product.lean:121
AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.lintegral_fin_nat_prod_eq_prod - ENNReal Fubini for products of per-coordinate functions over `Fin n`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:209
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 defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure.Transport AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:22
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.couplingSet - The feasible set of couplings with prescribed marginals. defCompiledNot 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 defCompiledCompiled 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). theoremCompiledCompiled AutoSamplingTheory.TechnicalLemmas.Measure.Transport AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:41
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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure.Transport AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:51
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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure.Transport AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:62
AutoSamplingTheory.TechnicalLemmas.Measure.Transport.couplingSet_nonempty - The feasible set in the Kantorovich problem is nonempty for probability marginals. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure.Transport AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean:69
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.quadraticCost - Squared Euclidean transport cost as an extended nonnegative function. defCompiledNot 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. defCompiledCompiled 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. theoremCompiledCompiled AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:36
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 defCompiledCompiled AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:49
AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace.isAbsolutelyContinuousFiniteSecondMoment_iff - Expansion of the three conditions in the `P₂,ac` definition. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSpace AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSpace.lean:58
AutoSamplingTheory.TechnicalLemmas.Measure.integrable_of_measure_eq - Integrability is invariant under replacing the ambient measure by an equal measure. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Measure AutoSamplingTheory/TechnicalLemmas/Measure.lean:36
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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalKernel AutoSamplingTheory/TechnicalLemmas/Probability/ConditionalKernel.lean:30
AutoSamplingTheory.TechnicalLemmas.LemmaMemoryStatus inductiveCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:43
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. structureCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:53
AutoSamplingTheory.TechnicalLemmas.sltSourceAnchor defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:64
AutoSamplingTheory.TechnicalLemmas.analysisMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:72
AutoSamplingTheory.TechnicalLemmas.gaussianMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:295
AutoSamplingTheory.TechnicalLemmas.taylorMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:448
AutoSamplingTheory.TechnicalLemmas.calculusMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:481
AutoSamplingTheory.TechnicalLemmas.measureMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:1524
AutoSamplingTheory.TechnicalLemmas.functionalInequalityMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:2009
AutoSamplingTheory.TechnicalLemmas.stochasticProcessMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:2132
AutoSamplingTheory.TechnicalLemmas.klDensityMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:3375
AutoSamplingTheory.TechnicalLemmas.renyiDensityMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:3398
AutoSamplingTheory.TechnicalLemmas.variationalMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:3441
AutoSamplingTheory.TechnicalLemmas.geometryMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:3554
AutoSamplingTheory.TechnicalLemmas.saldExtractedMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:3967
AutoSamplingTheory.TechnicalLemmas.portQueueMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:4040
AutoSamplingTheory.TechnicalLemmas.technicalLemmaMemory defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:4063
AutoSamplingTheory.TechnicalLemmas.formalizedTechnicalLemmaCount defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Registry AutoSamplingTheory/TechnicalLemmas/Registry.lean:4068
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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/AccumulatedEnergy.lean:26
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.AccumulatedEnergy.accumulatedEnergy_zero theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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). theoremCompiledNot 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`. defCompiledNot 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 defCompiledCompiled 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 structureCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.BrownianMotion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/BrownianMotion.lean:161
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.energyLevelSet - Times in `[0,T]` at which completed energy equals a prescribed level. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:30
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.isClosed_energyLevelSet - An energy level set is closed. theoremCompiledNot 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. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalEnergyLocalizer.lean:234
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalEnergyLocalizer.canonicalLocalizingTime_mono - Canonical localizing times increase with `n`. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalLocalizationTheorem AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalLocalizationTheorem.lean:29
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalLocalizationTheorem.canonicalStoppedProgressiveL2_process theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:37
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_of_bad theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:45
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_of_good theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalRawLocalization.lean:99
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalRawLocalization.canonicalRawLocalizingTime_isStoppingTime - Mathlib-native stopping-time version. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CanonicalStoppedItoIntegral.lean:32
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral.canonicalStoppedItoProcess_stronglyAdapted - The stopped Itô process is strongly adapted. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledCompiled 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. theoremCompiledNot 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. defCompiledCompiled 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. theoremCompiledNot 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`. defCompiledCompiled 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- theoremCompiledCompiled 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 theoremCompiledCompiled 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. theoremCompiledCompiled 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. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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` theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ChewiDefinition1_1_17 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ChewiDefinition1_1_17.lean:123
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 structureCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot 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. defCompiledNot 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. theoremCompiledNot 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`. abbrevCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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]`. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:26
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.continuous_clipNat theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:28
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.stronglyMeasurable_clipNat theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:31
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clip_le_abs theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:37
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clip_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:52
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clipNat_le_abs theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:57
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clipNat_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:61
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.clipNat_eventually_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:65
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.tendsto_clipNat theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:76
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.aestronglyMeasurable_clipNat theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:80
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.clipNat_memLp theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CoefficientTruncation.lean:89
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CoefficientTruncation.abs_clipNat_sub_le theoremCompiledNot 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`. theoremCompiledNot 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. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:60
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.continuous_completedEnergy - Every completed energy path is continuous. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:77
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.monotone_completedEnergy - Every completed energy path is monotone. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:103
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.completedEnergy_zero theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedEnergy.lean:120
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedEnergy.completedEnergy_nonneg - Completed energy is nonnegative. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedIntegrand.lean:28
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand.completedIntegrand_stronglyProgressive - Completion preserves strong progressiveness. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/CompletedIntegrand.lean:83
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CompletedIntegrand.completedEnergy_stronglyProgressive - The completed energy process is strongly progressive. theoremCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:35
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.measurableSet_dyadicMaxEvent theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:38
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.measurableSet_dyadicMaxEventAll theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:45
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.monotone_dyadicMaxEvent theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:51
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.runningAbsMax_nonneg theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:116
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.activeCellIndex - The dyadic cell containing a positive time `t`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:140
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.activeCellIndex_spec theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:145
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.rightApproxTime - Right endpoint of the dyadic cell containing `t`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:154
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.rightApproxTime_eq_grid theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:158
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.rightApproxTime_mem_Icc theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:165
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.rightApproxTime_le_add_mesh theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ContinuousDoobL2.lean:183
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ContinuousDoobL2.tendsto_rightApproxTime theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot 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. theoremCompiledCompiled 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobL2.lean:21
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2.Martingale.sampled - Deterministic monotone sampling preserves the martingale property. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobL2.lean:29
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobL2.runningAbsMax - Running absolute maximum through discrete time `N`. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:67
MeasureTheory.runMax_nonneg lemmaCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:71
MeasureTheory.runMax_measurable lemmaCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:76
MeasureTheory.runMax_stronglyMeasurable lemmaCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:83
MeasureTheory.layer_meas_bound - Maximum-inequality at a fixed positive level `t`. lemmaCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:89
MeasureTheory.lintegral_runMax_rpow_eq_layer - Layer-cake step. lemmaCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:105
MeasureTheory.layer_integrand_bound - Pointwise (in `t > 0`) integrand bound. lemmaCompiledNot 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). lemmaCompiledNot 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))`. lemmaCompiledNot 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`. lemmaCompiledNot 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)`. lemmaCompiledNot 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 lemmaCompiledNot 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)`. lemmaCompiledNot 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 lemmaCompiledNot 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. lemmaCompiledNot 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)`. lemmaCompiledNot 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`. lemmaCompiledNot 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`. lemmaCompiledNot 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)`. lemmaCompiledNot 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 theoremCompiledNot 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 lemmaCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DiscreteDoobLpPort AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DiscreteDoobLpPort.lean:1114
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refinementFactor - Number of fine cells inside one coarse cell. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:29
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refinementFactor_pos theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:32
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.pow_mul_refinementFactor theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:43
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarseCell_val theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:50
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarseCell_block_left theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:55
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarseCell_block_right theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:60
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.dyadicMesh_coarse_eq_factor_mul theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:66
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarse_left_endpoint_le_fine_left theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:84
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.fine_right_endpoint_le_coarse_right theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:95
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.regularDyadic_last_time theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:110
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.DyadicElementaryProcess.horizon_pos theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:128
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_level theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:145
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_times theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:151
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_coeff theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:158
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_coeff_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:165
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_value_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:173
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_toLp_eq theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:227
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refinementEquiv_val theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:232
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.coarseCell_refinementEquiv theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:249
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.fine_block_left_endpoint theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:264
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.fine_block_right_endpoint theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:273
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_elementaryItoIntegral_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:283
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.refineDyadic_terminalToLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:326
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.terminalToLp - Terminal stochastic integral represented in `L2(mu)`. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:350
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.elementaryProcessToLp_eq_processToLp theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:365
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementLeft defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:369
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementRight defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:374
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinement_times_eq theoremCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:392
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementProcess_times_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:397
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementLeft_toLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:403
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementRight_toLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:408
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementLeft_processToLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:413
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementRight_processToLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:420
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementLeft_terminalToLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryRefinement.lean:427
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryRefinement.commonRefinementRight_terminalToLp_eq theoremCompiledNot 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. theoremCompiledCompiled 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:32
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopDyadicAtGridIndex_level theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:64
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopDyadicAtGridIndex_coeff theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:70
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.cutoffTime - The time represented by a cutoff grid index. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:81
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.cutoffTime_le_horizon theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:100
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopDyadicAtGridIndex_terminalToLp theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:230
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.level_le_stoppingLevel theoremCompiledNot 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`. defCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:247
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.cutoffTime_rightCutoffIndex theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:254
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopAtRightApprox_terminalToLp theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:262
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.stopAtRightApprox_value_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:272
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.tendsto_stoppingLevel theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:282
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.tendsto_rightApproxTime_stoppingLevel theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicElementaryStopping.lean:298
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicElementaryStopping.tendsto_stopAtRightApprox_ae theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:35
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicHorizon - The corresponding positive dyadic horizon. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:38
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalIndex_add_one theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:40
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.integerHorizon_dyadicGlobalIndex theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:46
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicHorizon_pos theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:50
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalIndex_mono - The dyadic subsequence indices are monotone. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:66
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalLocalizingTime - The global canonical localizer restricted to dyadic horizons. defCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicGlobalHorizon.lean:81
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicGlobalHorizon.dyadicGlobalLocalizingTime_mono - The dyadic global localizers are pointwise increasing. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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 defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:39
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.dyadicHorizon_mono - Dyadic horizons are monotone in their exponent. theoremCompiledNot 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)`. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:77
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.prefixIndex_val theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:104
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_level theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:151
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_times theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/DyadicHorizonExtension.lean:156
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.DyadicHorizonExtension.extendDyadicHorizon_coeff_prefix - Prefix coefficients are copied exactly. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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)`. theoremCompiledNot 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`. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:76
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.zeroLike_value theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:91
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.neg_value theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:98
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.smul_value theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:116
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.add_value theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:128
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.sub_value theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:149
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_zeroLike theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:170
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_neg theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:176
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_smul theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:183
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_add theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoAlgebra.lean:190
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoAlgebra.elementaryItoIntegral_sub theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:33
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.dyadicObservationTime_monotone theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:36
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.dyadicObservationTime_terminal theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:42
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.dyadicObservationTime_refine theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:95
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.elementaryItoProcess_commonDifference theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoDoobL2.lean:103
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoDoobL2.elementaryItoIntegral_commonDifference theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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- theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:95
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.abs_value_le_valueBound theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:125
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toProgressiveL2_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:132
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toLp_add theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:138
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toLp_sub theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoEmbedding.lean:159
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoEmbedding.toLp_smul theoremCompiledNot 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. structureCompiledNot 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. defCompiledNot 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}]`. theoremCompiledCompiled 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`. defCompiledNot 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. theoremCompiledCompiled 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). defCompiledNot 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. defCompiledNot 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]`. theoremCompiledCompiled 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. defCompiledNot 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]`. theoremCompiledCompiled 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:24
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.weightedIncrement - One summand in the elementary Ito integral. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:29
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.elementaryItoIntegral_eq_sum_weightedIncrement theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:37
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.grid_endpoint_le_of_lt theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:94
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.ordered_cross_integral_eq_zero theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledCompiled 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. theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:266
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoIsometry.interval_piece_mul_eq_zero theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledCompiled 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. theoremCompiledNot 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. theoremCompiledNot 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)`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:47
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.norm_sq_elementaryItoTerminalToLp theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoL2.lean:63
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoL2.ofReal_norm_sq_elementaryProcessToLp_eq_energy theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:28
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoProcess_zero - An elementary Ito process starts at zero. theoremCompiledNot 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. theoremCompiledNot 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). theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:85
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.stoppedWeightedIncrement ## A reusable stopped weighted Brownian increment defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:94
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.stoppedWeightedIncrement_memLp_two theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:99
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.stoppedWeightedIncrement_stronglyAdapted theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:110
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.stoppedWeightedIncrement_martingale theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoProcess.lean:235
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryItoProcess.elementaryItoSummand_martingale theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryStoppingTime.lean:48
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime.stopElementary_times theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryStoppingTime.lean:68
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime.stopElementary_coeff theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ElementaryStoppingTime AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryStoppingTime.lean:75
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.enorm_sq_eq_ofReal_sq theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyPathContinuity.lean:99
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyPathContinuity.accumulatedEnergyReal_zero theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedIntegrand.lean:31
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedIntegrand.energyStoppedIntegrand_stronglyProgressive - Energy thresholding preserves strong progressiveness. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:27
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.stoppedExtension_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:37
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.stoppedExtension_apply_of_le theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EnergyStoppedProgressiveL2.lean:151
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedProgressiveL2.stoppedProgressiveL2_process theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:35
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.coordinateDual_apply theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:40
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.norm_coordinateDual theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:46
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates.projectedIncrementVariance_coordinateDual theoremCompiledNot 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. theoremCompiledNot 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]`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EuclideanBrownianCoordinates AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/EuclideanBrownianCoordinates.lean:201
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. structureCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:42
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.integrable_boundedContinuousFunction theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:47
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.norm_kernelIntegral_le theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:65
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperatorValue_apply theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:75
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperatorValue_add theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:80
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperatorValue_smul theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:94
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerLinearMap defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:113
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.fellerOperator_apply theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledCompiled 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledPartial AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FellerSemigroup.lean:185
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FellerSemigroup.continuousLinearSemigroupOfFeller_op_apply theoremCompiledNot 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 structureCompiledNot 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 structureCompiledNot 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. defCompiledNot 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 defCompiledNot 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 theoremCompiledNot 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`. abbrevCompiledNot 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. abbrevCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FiniteDimensionalNormBridge.lean:39
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FiniteDimensionalNormBridge.matrixSquareEnergy_nonneg - The finite matrix square energy is nonnegative. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot 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. theoremCompiledNot 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. defCompiledNot 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 theoremCompiledNot 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 defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.FokkerPlanckAlgebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/FokkerPlanckAlgebra.lean:33
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Girsanov.finiteShiftedGaussianPathMeasure - The finite-dimensional shifted Gaussian cylinder measure obtained by pushing shifted product coordinates into `EuclideanSpace`. defCompiledNot 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)`. defCompiledNot 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 theoremCompiledNot 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. theoremCompiledPartial 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:35
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.integerHorizon_pos theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:37
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.integerHorizon_succ theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:40
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.integerHorizon_mono - Integer horizons are monotone. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:52
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.measure_globalBadSet_zero - The global exceptional set is null. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:102
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.globalLocalizingTime_of_bad theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalCanonicalLocalizer.lean:111
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalCanonicalLocalizer.globalLocalizingTime_of_good theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:47
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoGluingGoodSet_ae - The global gluing contract holds almost surely. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:90
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.measure_globalItoGluingBadSet_zero theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:97
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.measurableSet_globalItoGluingBadSet_at theoremCompiledNot 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. defCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot 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`. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalItoProcessGluing.lean:191
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalItoProcessGluing.globalItoProcess_stronglyAdapted - The global process is strongly adapted. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. structureCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalLocalProgressiveL2.lean:45
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalLocalProgressiveL2.GlobalLocalProgressiveL2Integrand.onHorizon_process theoremCompiledNot 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). theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:54
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale.globalStoppedItoProcess_stronglyAdapted - Every localized Itô process is strongly adapted. theoremCompiledNot 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). theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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]`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoMartingale AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoMartingale.lean:211
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess - The `k`-th globally localized Itô martingale. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:47
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess_stronglyAdapted - Each localized process is strongly adapted. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:57
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess_martingale - Each localized process is a true martingale. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedItoOverlap.lean:84
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedItoOverlap.globalStoppedItoProcess_zero - Every localized Itô process starts from zero. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. defCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:184
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedExtension_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:194
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedExtension_apply_of_le theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:302
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2.globalStoppedProgressiveL2_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.GlobalStoppedProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/GlobalStoppedProgressiveL2.lean:313
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonConsistency.extendedCanonicalApprox - Canonical small-horizon approximants, represented exactly on a larger cofinal dyadic horizon. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoHorizonProcessConsistency AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoHorizonProcessConsistency.lean:120
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess - The `n`-th canonical elementary Ito martingale. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:43
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalIncrement_eq_sub theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:51
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_martingale theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:61
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalIncrement_martingale theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:69
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_continuous_ae theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:88
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.uniformThreshold_pos theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:90
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.summable_uniformThreshold theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:93
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.fastTolerance_succ_le theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:140
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.measurableSet_uniformBadEvent theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:154
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.measure_uniformBadEvent_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:158
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.badEventMajorant_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:194
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tsum_badEventMajorant_ne_top theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:217
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tsum_measure_uniformBadEvent_ne_top theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:229
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.eventually_not_uniformBadEvent_ae theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:237
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_continuous_all_ae theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:251
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.uniformCauchyEvent_ae theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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]`. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:382
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.measure_uniformBadSet_zero theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:387
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.measurableSet_uniformBadSet_at theoremCompiledNot 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]`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:404
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendsto_canonicalItoProcess_canonicalPathLimit theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:409
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoUniformlyOn_canonicalPathLimit theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:418
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalPathLimit_continuousOn theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:440
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_continuousOn theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:449
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_continuous_ae theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:464
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalPathLimit_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:473
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_stronglyAdapted theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:482
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendsto_canonicalItoProcess_itoIntegralProcess_ae theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:515
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalRepresentative - A concrete representative of the terminal `L2` completion. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:583
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalRepresentative_memLp theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:589
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalRepresentative_integrable theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:603
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalConditionalProcess_martingale theoremCompiledNot 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`. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:616
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendsto_eLpNorm_terminalApprox_sub_representative theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:627
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendsto_eLpNorm_terminalCondApprox_sub theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:639
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_terminalCondApprox theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:669
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalCondApprox_ae_eq_canonicalItoProcess theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:685
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_canonicalItoProcess_terminalConditional theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:698
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_canonicalItoProcess_actual theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:708
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalConditionalProcess_ae_eq_actual_of_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:721
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_terminalApprox_representative theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:733
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.tendstoInMeasure_canonicalItoProcess_terminal theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:744
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalRepresentative_ae_eq_actual theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:826
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralTerminal_restrictAt_zero theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:868
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralTerminal_restrictAt_add theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:900
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralTerminal_restrictAt_smul theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:935
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_add theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:957
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_smul theoremCompiledNot 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]`. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:997
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalItoProcess_eq_terminal_of_horizon_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1052
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.canonicalPathLimit_eq_terminal_of_horizon_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1059
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_eq_terminal_of_horizon_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1068
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalConditionalProcess_ae_eq_actual_of_horizon_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1082
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.terminalConditionalProcess_ae_eq_actual theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1107
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_martingale theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoIntegralProcess.lean:1117
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess.itoIntegralProcess_integrable theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledCompiled 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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,∞)`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:29
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.terminalApprox - Canonical elementary terminal approximation. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:37
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.processApprox - Canonical elementary process approximation in product-space `L2`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:45
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.tendsto_processApprox theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:52
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.norm_terminalApprox_sub theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:61
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.terminalApprox_cauchy theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:86
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.tendsto_terminalApprox theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:101
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.norm_terminalToLp_eq_processToLp theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:151
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_norm - Completed terminal Ito isometry. theoremCompiledCompiled 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:176
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonRightProcess defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:182
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonProcess_times_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:188
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonLeftProcess_terminalToLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:193
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonRightProcess_terminalToLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:201
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonLeftProcess_processToLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:209
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.commonRightProcess_processToLp_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:218
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.addDyadic defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:227
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.smulDyadic - Scalar multiple on the same dyadic grid. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:237
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.terminalToLp_addDyadic theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:244
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.processToLp_addDyadic theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:256
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.terminalToLp_smulDyadic theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:268
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.processToLp_smulDyadic theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:287
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.integrandToLp_add theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:296
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.integrandToLp_neg theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:304
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.integrandToLp_sub theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:311
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.integrandToLp_smul theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:319
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_congr_toLp theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:326
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_add theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:338
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_smul theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:370
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_zero theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:397
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_neg theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:406
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_sub theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:433
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_elementary theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:479
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminalOnHorizon_of_pos theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:484
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion.itoIntegralTerminal_zero_horizon theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoTerminalCompletion AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:490
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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:30
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.clippedExtensionAt_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:39
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.clippedExtensionAt_apply_of_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:50
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.clippedExtensionAt_abs_le theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:82
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedCellAverage_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:89
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedCellAverage_zero theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:106
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedCellAverage_abs_le theoremCompiledNot 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`. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:155
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.dyadicMesh_le_leftTime_of_ne_zero theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:163
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.dyadicLeftTime_le_terminal theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:171
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicCoeff_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:184
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicCoeff_abs_le theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:225
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicApprox_times theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:239
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicApprox_coeff theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicApproximation.lean:246
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicApproximation.laggedDyadicApprox_last_time theoremCompiledNot 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. theoremCompiledCompiled 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:134
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.clippedHorizonFunction_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:139
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.clippedHorizonFunction_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LaggedDyadicConvergence.lean:152
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LaggedDyadicConvergence.laggedDyadicApprox_tendsto_ae_time theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot 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 ∇ theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledPartial 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledPartial 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 theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledPartial 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 theoremCompiledCompiled 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Langevin AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1253
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinCarreDuChamp.laplacian_mul - The Laplacian product rule for two globally `C²` real observables. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledCompiled 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. theoremCompiledNot 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. defCompiledNot 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`. defCompiledNot 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 structureCompiledCompiled 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 theoremCompiledNot 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 theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LangevinGenerator AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LangevinGenerator.lean:90
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. structureCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:36
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.squaredExtensionAt_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:44
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.squaredExtensionAt_apply_of_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:55
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2.squaredExtensionAt_apply_of_not_le theoremCompiledNot 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. defCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.LocalProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/LocalProgressiveL2.lean:100
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization.nnrealLebesgue abbrevCompiledNot 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. defCompiledNot 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 defCompiledCompiled 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 defCompiledCompiled AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Localization AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Localization.lean:69
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 structureCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:43
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.MeasurableENNReal.const - A constant measurable nonnegative observable. defCompiledNot 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. defCompiledCompiled AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:54
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator_const - A Markov operator preserves constant nonnegative observables. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/MarkovSemigroup.lean:80
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.MarkovSemigroup.markovOperator_comp - Chapman--Kolmogorov becomes composition of Markov operators. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledCompiled 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. defCompiledCompiled 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. theoremCompiledNot 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. structureCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot 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. defCompiledCompiled 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. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledPartial 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`. defCompiledNot 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`. theoremCompiledNot 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 theoremCompiledCompiled 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. structureCompiledNot 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. theoremCompiledPartial AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:45
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_zero - The zero vector has generator value zero. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:63
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightDifferenceQuotient_smul - Right difference quotients commute with scalar multiplication. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:72
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightDifferenceQuotient_neg - Right difference quotients commute with negation. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:89
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_add - Generator limits add. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:98
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_smul - Generator limits commute with scalar multiplication. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:107
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_neg - Generator limits commute with negation. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:115
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.hasRightGeneratorAt_sub - Generator limits subtract. theoremCompiledNot 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. defCompiledPartial AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:132
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGeneratorValue - The canonical right-generator value on its submodule domain. defCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:154
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain.rightGeneratorValue_add - The canonical generator value is additive. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledPartial 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. theoremCompiledNot 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. theoremCompiledPartial AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.OperatorGeneratorDomain AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/OperatorGeneratorDomain.lean:203
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveDriftIntegral.prefixIntegralProcess - The finite-time Bochner drift primitive. defCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. structureCompiledNot 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`. defCompiledNot 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. structureCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:42
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.toLp - The canonical product-space `Lp` representative. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:61
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictProcess_zero theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:65
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictProcess_nested theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:70
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictProcess_progressive theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:79
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.processFunction_restrictProcess theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:92
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictProcess_memLp theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:99
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt - Restriction preserves the progressive `L2` domain. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:109
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:116
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt_zero_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:121
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt_zero_toLp theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:126
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2.ProgressiveL2Integrand.restrictAt_nested_process theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2.lean:138
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.zero defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:25
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.add defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:30
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.neg defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:36
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.sub defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:42
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.smul defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:48
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.zero_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:54
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.add_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:58
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.neg_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:62
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.sub_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:66
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.smul_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:70
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_zero theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:75
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_add theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:79
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_neg theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:83
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_sub theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:87
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_smul theoremCompiledNot 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`. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:96
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_restrictAt_add theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:108
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_restrictAt_smul theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:120
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra.toLp_restrictAt_zero theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Algebra.lean:150
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.fastTolerance - Rapid geometric tolerance used for all diagonal choices. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:26
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.fastTolerance_eq theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:29
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.fastTolerance_pos theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:33
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.fastTolerance_tendsto_zero theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:36
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.summable_fastTolerance theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:47
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.summable_scaled_fastTolerance_sq theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:65
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.eventualThreshold_spec theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:69
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.strictSelection - Recursively strictify eventual thresholds without losing their bounds. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:76
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.eventualThreshold_le_strictSelection theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:81
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.strictSelection_spec theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:87
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.strictMono_strictSelection theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:92
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.TruncationGood - Truncation levels meeting the `n`th fast tolerance. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:101
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.eventually_truncationGood theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:105
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.truncationIndex - Strictly increasing clipping index selected from clipping convergence. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:114
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.truncationIndex_spec theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:118
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.truncationIndex_strictMono theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:129
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.eventually_dyadicGood theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:151
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.dyadicLevel_spec theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:155
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.dyadicLevel_strictMono theoremCompiledNot 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. structureCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:166
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.canonicalElementaryApprox - Canonical fast diagonal approximation. defCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:183
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.norm_canonicalElementaryApprox_sub_lt theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:189
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density.tendsto_canonicalElementaryApprox_toLp theoremCompiledNot 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. theoremCompiledCompiled AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Density AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Density.lean:216
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.horizonPrefix - Product-space prefix corresponding to times strictly before `T`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:30
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.measurableSet_horizonPrefix theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2HorizonExtension.lean:64
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2HorizonExtension.extendByZero_process theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Stopping.lean:102
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping.stop_process theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Stopping AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Stopping.lean:130
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipProcess - Pointwise clipping of a stochastic process. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:26
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipProcess_stronglyProgressive theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:29
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.processFunction_clipProcess theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:35
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipProcess_memLp theoremCompiledNot 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. defCompiledCompiled AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:49
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipped_process theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ProgressiveL2Truncation.lean:56
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Truncation.clipped_abs_le theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. defCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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]`. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. defCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessApprox.lean:35
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox.stopRefinedDyadic_level theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProcessApprox.lean:49
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessApprox.stopRefinedDyadic_process theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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 theoremCompiledNot 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`. theoremCompiledNot 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 theoremCompiledNot 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]`. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2 AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/RandomStoppingProgressiveL2.lean:148
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProgressiveL2.stoppedProgressiveL2_process theoremCompiledNot 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`. defCompiledCompiled 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.Reversibility AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Reversibility.lean:35
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.regularGridTimes - Equally spaced endpoints with mesh `delta`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:27
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.regularGridTimes_strictMono theoremCompiledNot 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 defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:40
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClipped_times theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:59
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClipped_coeff theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:67
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClipped_coeff_abs_le theoremCompiledNot 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]`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:85
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.dyadicMesh_pos theoremCompiledNot 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]`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:93
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation.sampledClippedDyadic_last_time theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.SampledElementaryApproximation AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/SampledElementaryApproximation.lean:100
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.StoppingGraphNull.stoppingGraph - The graph of a possibly-infinite nonnegative stopping time in `Omega × ℝ≥0`. defCompiledNot 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. theoremCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledCompiled 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot 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. defCompiledNot 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 theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasure.lean:89
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasure.upTo_singleton - Stopped Lebesgue time has no atoms. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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]`. theoremCompiledNot 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. theoremCompiledNot 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`. theoremCompiledNot 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. theoremCompiledNot 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. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:27
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realZeroExtension_measurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:30
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realZeroExtension_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:35
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realZeroExtension_coe theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:40
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realZeroExtension_eq_zero_of_neg theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:44
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.range_nnreal_coe theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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]`. defCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:135
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_coe theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:141
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_eq_zero_of_neg theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:149
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_eq_zero_of_T_lt theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:155
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_abs_le theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:172
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_stronglyMeasurable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:185
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_integrable theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/TimeMeasureRealBridge.lean:193
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.TimeMeasureRealBridge.realClippedSection_locallyIntegrable theoremCompiledNot 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. theoremCompiledNot 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. theoremCompiledNot 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 structureCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.VectorBrownianFiltration AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/VectorBrownianFiltration.lean:78
AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.WeakGenerator.IsInvariantOn - Invariance of a measure for a nonnegative-time operator family on an explicit test class. defCompiledNot 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 structureCompiledNot 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 theoremCompiledCompiled 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` theoremCompiledPartial 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 theoremCompiledNot 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. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Taylor AutoSamplingTheory/TechnicalLemmas/Taylor.lean:51
AutoSamplingTheory.TechnicalLemmas.Taylor.stdOrthonormalBasisUnit - Standard orthonormal-basis vectors are unit directions. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Taylor AutoSamplingTheory/TechnicalLemmas/Taylor.lean:68
AutoSamplingTheory.TechnicalLemmas.Taylor.quadraticVariationNormalizationOfCoeffDefAndVarianceOne - Algebraic packaging for quadratic-variation normalization. theoremCompiledNot mapped AutoSamplingTheory.TechnicalLemmas.Taylor AutoSamplingTheory/TechnicalLemmas/Taylor.lean:75
AutoSamplingTheory.Tests.CarreDuChamp.zeroGenerator defCompiledNot mapped Tests.CarreDuChamp Tests/CarreDuChamp.lean:18
AutoSamplingTheory.Tests.FellerSemigroup.identityFellerContract - The identity transition kernel is the basic Feller semigroup. theoremCompiledNot mapped Tests.FellerSemigroup Tests/FellerSemigroup.lean:17
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. defCompiledNot mapped Tests.ItoIntegralProcess Tests/ItoIntegralProcess.lean:41
AutoSamplingTheory.Tests.GeneralItoIntegral.unitDyadic defCompiledNot mapped Tests.ItoIntegralProcess Tests/ItoIntegralProcess.lean:49
AutoSamplingTheory.Tests.GeneralItoIntegral.unitDyadic_elementaryItoProcess theoremCompiledNot mapped Tests.ItoIntegralProcess Tests/ItoIntegralProcess.lean:55
AutoSamplingTheory.Tests.OperatorGenerator.identitySemigroup - The constant identity family is the simplest continuous-linear semigroup. defCompiledNot mapped Tests.OperatorGenerator Tests/OperatorGenerator.lean:18
AutoSamplingTheory.Tests.OperatorGeneratorDomain.identityStronglyContinuousSemigroup - The identity family is a strongly continuous semigroup. defCompiledNot mapped Tests.OperatorGeneratorDomain Tests/OperatorGeneratorDomain.lean:16
AutoSamplingTheory.Tests.OperatorGeneratorDomain.identity_mem_generatorDomain theoremCompiledNot mapped Tests.OperatorGeneratorDomain Tests/OperatorGeneratorDomain.lean:51
AutoSamplingTheory.Tests.Reversibility.identitySemigroup defCompiledNot 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. defCompiledNot mapped Tests.SemigroupDecay Tests/SemigroupDecay.lean:20