AutoSamplingTheory.SALD.cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation

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 cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation :
    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.cycle109_named_barB_condExp_source_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 109 lower compiles SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The theorem uses Mathlib ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib to turn the paper's sample-space conditional-expectation definition of the frozen guide-plus-score drift into the downstream hatRhoS-a.e. canonical condDistrib equality consumed by SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq. This bypasses the middle condExpKernel.map route's measure-valued kernel-alignment hypothesis for this source-definition path; the remaining exact theorem is to identify the selected paper representative hbarBCondExp : P[dotTk • guideIntegrand(hatXAtS omega, XkEta omega) + sigmaCoeff • scoreIntegrand(hatXAtS omega, XkEta omega) | mState.comap hatXAtS] = barB (hatXAtS omega) a.e. and prove equality-set measurability for ae_map_iff under explicit finite/probability P, measurable hatXAtS, a.e.-measurable XkEta, standard-Borel state, complete vector space, and guide/score Bochner integrability hypotheses.

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. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef
2. ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib
3. SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq
4. SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity
5. SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary
6. Mathlib.Probability.Kernel.CondDistrib
7. Mathlib.MeasureTheory.Integral.Bochner.Basic
8. appendix.tex:1368-1377
9. 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 from Mathlib conditional expectation and Bochner integral linearity. It does not prove weak FP, box trace, KL/log-ratio, no-boundary, theorem status, SLT import, Lake dependency change, or sald_version_2 use.

def cycle109GeneralMovingTargetDiscreteNamedBarBCondExpSourceLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle109_named_barB_condExp_source_lower"
  statement := "Cycle 109 lower compiles SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef for appendix.tex:1368-1377. Classification: narrows-source-cited-boundary. The theorem uses Mathlib ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib to turn the paper's sample-space conditional-expectation definition of the frozen guide-plus-score drift into the downstream hatRhoS-a.e. canonical condDistrib equality consumed by SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq. This bypasses the middle condExpKernel.map route's measure-valued kernel-alignment hypothesis for this source-definition path; the remaining exact theorem is to identify the selected paper representative hbarBCondExp : P[dotTk • guideIntegrand(hatXAtS omega, XkEta omega) + sigmaCoeff • scoreIntegrand(hatXAtS omega, XkEta omega) | mState.comap hatXAtS] = barB (hatXAtS omega) a.e. and prove equality-set measurability for ae_map_iff under explicit finite/probability P, measurable hatXAtS, a.e.-measurable XkEta, standard-Borel state, complete vector space, and guide/score Bochner integrability hypotheses."
  source := saldGeneralMovingTargetDiscreteConditionalDriftSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef",
    "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib",
    "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq",
    "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity",
    "SALD.cycle109GeneralMovingTargetDiscreteNamedBarBSourceDefBoundary",
    "Mathlib.Probability.Kernel.CondDistrib",
    "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    "appendix.tex:1368-1377",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Proof-producing local theorem from Mathlib conditional expectation and Bochner integral linearity. It does not prove weak FP, box trace, KL/log-ratio, no-boundary, theorem status, SLT import, Lake dependency change, or sald_version_2 use."

/-- Cycle-109 proof-DAG pane for the named `barB` source-definition
boundary. -/