AutoSamplingTheory.SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation

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 cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation :
    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.cycle98_barB_divergence_no_boundary_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 98 compiles SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral, SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral, SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound, SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing, and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral. The first two remove the raw hbarBWeakDivergence premise from the cycle-94 drift-source route under explicit law-integral/no-boundary hypotheses. The lower bounded-pairing theorem discharges the paired-integrability supplied hypothesis from Integrable barB hatRhoS plus an a.e. test-gradient contraction bound. Classification: discharges-supplied-hypothesis for hpairIntegrable, while the analytic divergence/no-boundary identity itself remains source-cited.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource — 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.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation
2. SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound
3. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral
4. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing
5. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral
6. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral
7. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction
8. SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff
9. SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff
10. Mathlib.MeasureTheory.Integral.DivergenceTheorem
11. Mathlib.MeasureTheory.Integral.Bochner.Basic
12. appendix.tex:1379-1387
13. appendix.tex:1358-1387

note : String (explicit)

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

Proof-producing local handoff plus remaining source-cited boundary. The hpairIntegrable input is no longer primitive when barB is integrable and the test-gradient contraction has the recorded norm bound. Lower/reviewer should now target the concrete no-boundary IBP theorem, the weakGradPairing/driftDiv definition equalities, or the concrete test-gradient bound, not a new wrapper around hbarBWeakDivergence.

def cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle98_barB_divergence_no_boundary_lower"
  statement := "Cycle 98 compiles SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral, SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral, SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound, SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing, and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral. The first two remove the raw hbarBWeakDivergence premise from the cycle-94 drift-source route under explicit law-integral/no-boundary hypotheses. The lower bounded-pairing theorem discharges the paired-integrability supplied hypothesis from Integrable barB hatRhoS plus an a.e. test-gradient contraction bound. Classification: discharges-supplied-hypothesis for hpairIntegrable, while the analytic divergence/no-boundary identity itself remains source-cited."
  source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle98GeneralMovingTargetDiscreteBarBDivergenceNoBoundaryMiddleObligation",
    "SALD.generalMovingTargetDiscreteBarBPairIntegrableOfNormBound",
    "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryIntegral",
    "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBNoBoundaryIntegral",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpSourceSignsOfSampleSplitGeneratorBarBActionHandoff",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpLawDerivativeOfSampleSplitGeneratorBarBActionHandoff",
    "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
    "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    "appendix.tex:1379-1387",
    "appendix.tex:1358-1387"
  ]
  note := "Proof-producing local handoff plus remaining source-cited boundary. The hpairIntegrable input is no longer primitive when barB is integrable and the test-gradient contraction has the recorded norm bound. Lower/reviewer should now target the concrete no-boundary IBP theorem, the weakGradPairing/driftDiv definition equalities, or the concrete test-gradient bound, not a new wrapper around hbarBWeakDivergence."

/-- Cycle-98 proof-DAG pane for the `barB` no-boundary integral packet. -/