AutoSamplingTheory.SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDag

Data definition / provenance and workflow record

Meaning and type

The result has data type List AutoSamplingTheory.ProofDagBlock. A value of this type stores descriptions; it is not a proof of the statements in those descriptions.

def cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDag :
    List ProofDagBlock

Construction and field-by-field explanation

Construct an ordered list of the following data items. It is not a logical conjunction or proof DAG.

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.

Ordered data items

1. id : String (explicit)

   Stable plan-node identifier.

   ASTIS.SALD.forward_KL_discrete.cycle108.lower_packet.hatRho_barB_flux_product_derivative

   interface : String (explicit)

   Text describing intended mathematical interface.

   Compiled lower theorem: SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt proves the pointwise product derivative for x |-> hatRhoDensity x • barB x; SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion lifts it off the union of separate density and drift exception sets; SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv feeds that derivative into the concrete box theorem.

   source : AutoSamplingTheory.SourceAnchor (explicit)

   Source pointer.

   AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource — audited data reference, not expanded and not a compiled dependency edge

   targetLean : String (explicit)

   Textual planned implementation destination.

   AutoSamplingTheory/SALD.lean

   dependsOn : List String (explicit)

   Declared input names as strings, default empty.

   Ordered data items

   1. SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeLowerObligation
   2. SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt
   3. SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion
   4. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv
   5. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox
   6. Mathlib.Analysis.Calculus.FDeriv.Mul
   7. appendix.tex:1379-1387
   8. appendix.tex:1368-1377

   reusedBy : List String (explicit)

   Declared consumers as strings, default empty.

   Ordered data items

   1. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero
   2. sald.general_moving_target_discrete.em_interpolation_fp
   3. sald.discrete_forward_kl.em_interpolation_fp
   4. thm:forward-KL-discrete

   status : AutoSamplingTheory.ProofStatus (explicit)

   Recorded workflow status, default planned.

   AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification

2. id : String (explicit)

   Stable plan-node identifier.

   ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_after_product_derivative

   interface : String (explicit)

   Text describing intended mathematical interface.

   Remaining exact boundary after the compiled product-flux derivative handoff: prove divergence integrability for the product derivative, boundaryFlux equals the interior divergence integral, and signed Mathlib faces equal the existing testTrace/normalFluxTrace boundary integral. Keep hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero separate.

   source : AutoSamplingTheory.SourceAnchor (explicit)

   Source pointer.

   AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource — audited data reference, not expanded and not a compiled dependency edge

   targetLean : String (explicit)

   Textual planned implementation destination.

   AutoSamplingTheory/SALD.lean

   dependsOn : List String (explicit)

   Declared input names as strings, default empty.

   Ordered data items

   1. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv
   2. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox
   3. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox
   4. ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_boundary
   5. Mathlib.MeasureTheory.Integral.DivergenceTheorem
   6. appendix.tex:1379-1387
   7. appendix.tex:1368-1377

   reusedBy : List String (explicit)

   Declared consumers as strings, default empty.

   Ordered data items

   1. sald.general_moving_target_discrete.em_interpolation_fp
   2. sald.discrete_forward_kl.em_interpolation_fp

   status : AutoSamplingTheory.ProofStatus (explicit)

   Recorded workflow status, default planned.

   AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

def cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.forward_KL_discrete.cycle108.lower_packet.hatRho_barB_flux_product_derivative"
      interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt proves the pointwise product derivative for x |-> hatRhoDensity x • barB x; SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion lifts it off the union of separate density and drift exception sets; SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv feeds that derivative into the concrete box theorem."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle108DiscreteForwardKlHatRhoBarBFluxDerivativeLowerObligation",
        "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAt",
        "SALD.generalMovingTargetDiscreteHatRhoBarBWeightedFieldHasFDerivAtOffUnion",
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv",
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox",
        "Mathlib.Analysis.Calculus.FDeriv.Mul",
        "appendix.tex:1379-1387",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero",
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_after_product_derivative"
      interface := "Remaining exact boundary after the compiled product-flux derivative handoff: prove divergence integrability for the product derivative, boundaryFlux equals the interior divergence integral, and signed Mathlib faces equal the existing testTrace/normalFluxTrace boundary integral. Keep hproductRule, hdivergenceTheorem, hgradNormBound, and htestTraceZero separate."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBoxProductDeriv",
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfHatRhoBarBBox",
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox",
        "ASTIS.SALD.forward_KL_discrete.cycle108.remaining_hatRho_barB_box_trace_boundary",
        "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
        "appendix.tex:1379-1387",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 109: named `barB` source-definition boundary -/

/-- Cycle-109 exact missing theorem for the named `barB` source definition.

This is the lower-ready boundary behind `appendix.tex:1368-1377` after the
canonical `condDistrib` regularity theorem from cycle 106.  The target is not
another direct a.e. equality hypothesis for `barB`; it is the source theorem
that selects the paper's conditional-expectation representative and aligns it
with Mathlib's conditional-distribution kernel.
-/