AutoSamplingTheory.SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.ProofObligation. A value of this type stores descriptions; it is not a proof of the statements in those descriptions.

def cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation :
    ProofObligation

Construction and field-by-field explanation

Construct a data record from explicit fields and the audited defaults shown below.

This Lean definition constructs provenance or workflow data. It does not prove the mathematical statements stored as text. Status labels, named dependencies and citations are data, not compilation, proof or source certificates.

id : String (explicit)

Stable obligation identifier.

sald.general_moving_target_discrete.cycle91_conditional_kernel_lower

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Cycle 91 lower compiles SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions, after SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity. Under hatRhoS = Measure.map hatXAtS P, finite-measure/standard-Borel hypotheses, AEMeasurable hatXAtS and X_k^eta, AEStronglyMeasurable and Integrable frozen guide and score integrands on the joint law Measure.map (fun omega => (hatXAtS omega, X_k^eta omega)) P, measurable equality sets, and sample-space a.e. equalities between the canonical condDistrib component integrals composed with hatXAtS and condC/condScore composed with hatXAtS, the theorem proves AEStronglyMeasurable and Integrable barB under hatRhoS. Classification: discharges-supplied-hypothesis for the law-space component-version hypotheses and component-field/barB regularity hypotheses behind SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents and SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff. Remaining source-cited boundary: prove the sample-space conditional-expectation/disintegration equalities and equality-set measurability for the SALD fields, plus conditional-kernel compatibility.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource — audited data reference, not expanded and not a compiled dependency edge

status : AutoSamplingTheory.ProofStatus (explicit)

Stored ProofStatus, default obligation; even an explicitly stored formalized does not independently certify a Lean theorem.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

dependsOn : List String (explicit)

List of declared dependency names as strings; may mix theorem names, obligations, source labels, or descriptions. Not the compiled dependency DAG.

Ordered data items

1. SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation
2. SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample
3. SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions
4. SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity
5. AutoSamplingTheory.condDistribIntegralNamedFieldRegularity
6. AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable
7. AutoSamplingTheory.condDistribIntegralNamedLawIntegrable
8. SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents
9. Mathlib.Probability.Kernel.CondDistrib
10. proof:thm:general-moving-target-SALD-discrete:conditional-drift
11. appendix.tex:1368-1377
12. sald.general_moving_target_discrete.em_interpolation_fp

note : String (explicit)

Recorded evidence/caveats; may distinguish a compiled scalar helper from still-open source analysis.

Proof-producing local theorem plus remaining obligation. It transports sample-space component version equalities through hatRhoS = Law(hatXAtS) using Mathlib ae_map_iff. It does not construct the conditional law, prove the sample-space conditional-expectation equalities, weak Fokker-Planck identity, KL derivative, LSI, DV, Gronwall, theorem closure, or any SLT result.

def cycle91GeneralMovingTargetDiscreteConditionalKernelLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle91_conditional_kernel_lower"
  statement := "Cycle 91 lower compiles SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions, after SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity. Under hatRhoS = Measure.map hatXAtS P, finite-measure/standard-Borel hypotheses, AEMeasurable hatXAtS and X_k^eta, AEStronglyMeasurable and Integrable frozen guide and score integrands on the joint law Measure.map (fun omega => (hatXAtS omega, X_k^eta omega)) P, measurable equality sets, and sample-space a.e. equalities between the canonical condDistrib component integrals composed with hatXAtS and condC/condScore composed with hatXAtS, the theorem proves AEStronglyMeasurable and Integrable barB under hatRhoS. Classification: discharges-supplied-hypothesis for the law-space component-version hypotheses and component-field/barB regularity hypotheses behind SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents and SALD.generalMovingTargetDiscreteEndpointConditionalDriftRegularityHandoff. Remaining source-cited boundary: prove the sample-space conditional-expectation/disintegration equalities and equality-set measurability for the SALD fields, plus conditional-kernel compatibility."
  source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle91GeneralMovingTargetDiscreteConditionalKernelMiddleObligation",
    "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample",
    "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfSampleVersions",
    "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularity",
    "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity",
    "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable",
    "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable",
    "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents",
    "Mathlib.Probability.Kernel.CondDistrib",
    "proof:thm:general-moving-target-SALD-discrete:conditional-drift",
    "appendix.tex:1368-1377",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Proof-producing local theorem plus remaining obligation. It transports sample-space component version equalities through hatRhoS = Law(hatXAtS) using Mathlib ae_map_iff. It does not construct the conditional law, prove the sample-space conditional-expectation equalities, weak Fokker-Planck identity, KL derivative, LSI, DV, Gronwall, theorem closure, or any SLT result."

/-- Cycle-91 proof-DAG pane for the conditional-kernel component-field
backfill. -/