AutoSamplingTheory.SALD.cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingMiddleObligation

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 cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingMiddleObligation :
    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.cycle97_canonical_condDistrib_pairing_middle

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 97 middle keeps the active backend on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex lines 1358-1387, narrowed to appendix.tex lines 1368-1377. It compiles AutoSamplingTheory.condDistribIntegralMapIntegral and AutoSamplingTheory.condDistribIntegralNamedLawIntegral: for X=hatX_s, Y=X_k^eta, finite P, hatRhoS=P.map hatX_s, and an integrable paired component test f on the joint law of (hatX_s,X_k^eta), the law-space integral of the canonical condDistrib conditional integral equals the sample-space paired component integral. This discharges the Mathlib disintegration/pullout sub-hypothesis behind the canonical condDistrib component generator action, but it does not prove weak FP, divergence/no-boundary, log-ratio admissibility, or theorem closure. Classification: discharges-supplied-hypothesis for the map-law disintegration part of the cycle-96 hcanonical boundary.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalDriftSource — 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. AutoSamplingTheory.condDistribIntegralMapIntegral
2. AutoSamplingTheory.condDistribIntegralNamedLawIntegral
3. SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion
4. SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation
5. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings
6. AutoSamplingTheory.condDistribIntegralNamedFieldRegularity
7. ProbabilityTheory.compProd_map_condDistrib
8. MeasureTheory.Measure.integral_compProd
9. MeasureTheory.integral_map
10. ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib
11. Mathlib.Probability.Kernel.CondDistrib
12. Mathlib.Probability.Kernel.Condexp
13. appendix.tex:1368-1377
14. sald.general_moving_target_discrete.em_interpolation_fp
15. sald.discrete_forward_kl.em_interpolation_fp

note : String (explicit)

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

Exact lower theorem boundary after this proof: expand componentAction and weakGradPairing to the paired Bochner integrals used by AutoSamplingTheory.condDistribIntegralNamedLawIntegral, prove the required integrability for that paired test-gradient integrand, and then feed hcanonical in SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion. The divergence/no-boundary theorem for hatRhoS * barB remains separate.

def cycle97GeneralMovingTargetDiscreteCanonicalCondDistribPairingMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle97_canonical_condDistrib_pairing_middle"
  statement := "Cycle 97 middle keeps the active backend on sald.general_moving_target_discrete.em_interpolation_fp over appendix.tex lines 1358-1387, narrowed to appendix.tex lines 1368-1377. It compiles AutoSamplingTheory.condDistribIntegralMapIntegral and AutoSamplingTheory.condDistribIntegralNamedLawIntegral: for X=hatX_s, Y=X_k^eta, finite P, hatRhoS=P.map hatX_s, and an integrable paired component test f on the joint law of (hatX_s,X_k^eta), the law-space integral of the canonical condDistrib conditional integral equals the sample-space paired component integral. This discharges the Mathlib disintegration/pullout sub-hypothesis behind the canonical condDistrib component generator action, but it does not prove weak FP, divergence/no-boundary, log-ratio admissibility, or theorem closure. Classification: discharges-supplied-hypothesis for the map-law disintegration part of the cycle-96 hcanonical boundary."
  source := saldGeneralMovingTargetDiscreteConditionalDriftSource
  status := ProofStatus.obligation
  dependsOn := [
    "AutoSamplingTheory.condDistribIntegralMapIntegral",
    "AutoSamplingTheory.condDistribIntegralNamedLawIntegral",
    "SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion",
    "SALD.cycle96GeneralMovingTargetDiscreteCondexpGeneratorPairingLowerObligation",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftActionOfBarBComponentPairings",
    "AutoSamplingTheory.condDistribIntegralNamedFieldRegularity",
    "ProbabilityTheory.compProd_map_condDistrib",
    "MeasureTheory.Measure.integral_compProd",
    "MeasureTheory.integral_map",
    "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib",
    "Mathlib.Probability.Kernel.CondDistrib",
    "Mathlib.Probability.Kernel.Condexp",
    "appendix.tex:1368-1377",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.discrete_forward_kl.em_interpolation_fp"
  ]
  note := "Exact lower theorem boundary after this proof: expand componentAction and weakGradPairing to the paired Bochner integrals used by AutoSamplingTheory.condDistribIntegralNamedLawIntegral, prove the required integrability for that paired test-gradient integrand, and then feed hcanonical in SALD.generalMovingTargetDiscreteCondDistribComponentWeakPairingOfAeVersion. The divergence/no-boundary theorem for hatRhoS * barB remains separate."

/-- Cycle-97 lower-ready obligation after the compiled disintegration theorem. -/