AutoSamplingTheory.SALD.cycle107DiscreteForwardKlBoundaryFluxIntegralLowerObligation

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 cycle107DiscreteForwardKlBoundaryFluxIntegralLowerObligation :
    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.discrete_forward_kl.cycle107_boundary_flux_integral_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 107 lower compiles SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox. Classification: narrows-source-cited-boundary. The old hboundaryFluxIntegral premise in SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero is narrowed to a Mathlib box divergence theorem package: continuity of the weighted field on the box, Frechet differentiability off a countable interior set, integrability of the divergence, identification of boundaryFlux with the box divergence integral, and identification of the signed face sum with int_y testTrace phi y * normalFluxTrace y d boundaryMeasure. The theorem invokes MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable directly and leaves hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero explicit.

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.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox
2. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero
3. SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero
4. SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero
5. MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable
6. Mathlib.MeasureTheory.Integral.DivergenceTheorem
7. Mathlib.MeasureTheory.Integral.Bochner.Basic
8. appendix.tex:1379-1387
9. appendix.tex:1368-1377
10. sald.general_moving_target_discrete.em_interpolation_fp
11. sald.discrete_forward_kl.em_interpolation_fp

note : String (explicit)

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

Proof-producing Mathlib-backed theorem only. It does not prove the Euclidean weighted-field product rule, the paper-specific boundaryFlux/divergence identification, the face-to-trace parametrization, compact-support/zero-trace theorem, hgradNormBound, weak FP/KL closure, theorem-status promotion, SLT import, Lake dependency change, or sald_version_2 use.

def cycle107DiscreteForwardKlBoundaryFluxIntegralLowerObligation :
    ProofObligation where
  id := "sald.discrete_forward_kl.cycle107_boundary_flux_integral_lower"
  statement := "Cycle 107 lower compiles SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox. Classification: narrows-source-cited-boundary. The old hboundaryFluxIntegral premise in SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero is narrowed to a Mathlib box divergence theorem package: continuity of the weighted field on the box, Frechet differentiability off a countable interior set, integrability of the divergence, identification of boundaryFlux with the box divergence integral, and identification of the signed face sum with int_y testTrace phi y * normalFluxTrace y d boundaryMeasure. The theorem invokes MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable directly and leaves hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero explicit."
  source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox",
    "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero",
    "SALD.generalMovingTargetDiscreteTraceProductZeroOfTestTraceZero",
    "SALD.generalMovingTargetDiscreteZeroBoundaryFluxOfTraceProductZero",
    "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable",
    "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
    "Mathlib.MeasureTheory.Integral.Bochner.Basic",
    "appendix.tex:1379-1387",
    "appendix.tex:1368-1377",
    "sald.general_moving_target_discrete.em_interpolation_fp",
    "sald.discrete_forward_kl.em_interpolation_fp"
  ]
  note := "Proof-producing Mathlib-backed theorem only. It does not prove the Euclidean weighted-field product rule, the paper-specific boundaryFlux/divergence identification, the face-to-trace parametrization, compact-support/zero-trace theorem, hgradNormBound, weak FP/KL closure, theorem-status promotion, SLT import, Lake dependency change, or sald_version_2 use."

/-- Cycle-107 proof-DAG pane for the boundary-flux integral packet. -/