AutoSamplingTheory.SALD.cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityLowerObligation

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 cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityLowerObligation :
    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.cycle113_named_barB_state_field_regularity_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 113 lower compiles SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField and SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef for appendix.tex:1368-1377. Classification: discharges-supplied-hypothesis. The exact supplied hypotheses discharged from SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef are hbarBMeas and hbarBInt: if the selected named state-field barB is StronglyMeasurable on State and Integrable under hatRhoS, with hatRhoS = Measure.map hatXAtS P, then barB(hatXAtS omega) is AEStronglyMeasurable for mState.comap hatXAtS and Integrable under P. The downstream theorem feeds those derived regularity facts into the existing cycle-112 state-event set-integral bridge. Remaining exact non-wrapper blocker after the forward-KL-discrete pressure test: prove the source Bochner set-integral characterization over events {omega | hatXAtS omega in t} for measurable state sets t, plus any source proof of state-field Integrable barB hatRhoS not already supplied by the selected representative.

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.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField
2. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef
3. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef
4. SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents
5. MeasureTheory.Integrable.comp_measurable
6. Measurable.of_comap_le
7. appendix.tex:1368-1377
8. sald.general_moving_target_discrete.em_interpolation_fp
9. thm:forward-KL-discrete

note : String (explicit)

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

Proof-producing lower theorem only. It does not prove the state-event set-integral characterization, conditional law construction, weak Fokker-Planck equation, KL derivative, LSI, DV, Gronwall, theorem closure, SLT reuse, Lake changes, source-index rebaseline, or sald_version_2 content.

def cycle113GeneralMovingTargetDiscreteNamedBarBStateFieldRegularityLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle113_named_barB_state_field_regularity_lower"
  statement := "Cycle 113 lower compiles SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField and SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef for appendix.tex:1368-1377. Classification: discharges-supplied-hypothesis. The exact supplied hypotheses discharged from SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef are hbarBMeas and hbarBInt: if the selected named state-field barB is StronglyMeasurable on State and Integrable under hatRhoS, with hatRhoS = Measure.map hatXAtS P, then barB(hatXAtS omega) is AEStronglyMeasurable for mState.comap hatXAtS and Integrable under P. The downstream theorem feeds those derived regularity facts into the existing cycle-112 state-event set-integral bridge. Remaining exact non-wrapper blocker after the forward-KL-discrete pressure test: prove the source Bochner set-integral characterization over events {omega | hatXAtS omega in t} for measurable state sets t, plus any source proof of state-field Integrable barB hatRhoS not already supplied by the selected representative."
  source := saldGeneralMovingTargetDiscreteConditionalDriftSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.generalMovingTargetDiscreteNamedBarBComapRegularityOfStateField",
    "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateFieldSetIntegralDef",
    "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef",
    "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents",
    "MeasureTheory.Integrable.comp_measurable",
    "Measurable.of_comap_le",
    "appendix.tex:1368-1377",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "thm:forward-KL-discrete"
  ]
  note := "Proof-producing lower theorem only. It does not prove the state-event set-integral characterization, conditional law construction, weak Fokker-Planck equation, KL derivative, LSI, DV, Gronwall, theorem closure, SLT reuse, Lake changes, source-index rebaseline, or sald_version_2 content."

/-- Cycle-113 proof-DAG pane for the named `barB` regularity pullback. -/