AutoSamplingTheory.SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefDag

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.

Lean statement of this data definition

The part after the colon is the output data type. This declaration takes no mathematical proof inputs.

def cycle100GeneralMovingTargetDiscreteBarBWeakGradDefDag :
    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

id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle100.middle_barB_weakGrad_def_source_map
interface : String (explicit)
Text describing intended mathematical interface.
Middle source map: appendix.tex:1379-1387 uses the drift source sign -div(hat rho_s*bar b_{k,s}). After cycle 98, align weakGradPairing with the hatRhoS law integral by taking weakGradPairing f phi to be int x, fieldPairing phi f x d hatRhoS.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — 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.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefMiddleObligation
2. ASTIS.SALD.cycle98.remaining_exact_no_boundary_theorem
3. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral
4. appendix.tex:1379-1387
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
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle100.lower_packet.barB_weakGrad_definition_alignment
interface : String (explicit)
Text describing intended mathematical interface.
Compiled local handoff: SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef remove hweakGradIntegral by definitional specialization of weakGradPairing. hpairNormBound and hdivNoBoundary remain exact analytic boundaries.
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.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation
2. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef
3. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef
4. SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing
5. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction
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
3. thm:forward-KL-discrete
4. thm:general-moving-target-SALD-discrete
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle100.remaining_no_boundary_after_weakGrad_def
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact theorem after the weakGrad definition alignment: prove the concrete a.e. norm bound for fieldPairing phi barB and prove driftDiv phi = -int x, fieldPairing phi barB x d hatRhoS by the no-boundary divergence theorem for hatRhoS * barB.
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. Mathlib.MeasureTheory.Integral.DivergenceTheorem
2. Mathlib.MeasureTheory.Integral.Bochner.Basic
3. appendix.tex:1379-1387
4. 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

Exact Lean data construction

Each field assignment stores the corresponding value shown above. Omitted fields use the explicitly identified schema defaults. Strings that name theorems remain strings; they do not call those theorems.

def cycle100GeneralMovingTargetDiscreteBarBWeakGradDefDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle100.middle_barB_weakGrad_def_source_map"
      interface := "Middle source map: appendix.tex:1379-1387 uses the drift source sign -div(hat rho_s*bar b_{k,s}).  After cycle 98, align weakGradPairing with the hatRhoS law integral by taking weakGradPairing f phi to be int x, fieldPairing phi f x d hatRhoS."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefMiddleObligation",
        "ASTIS.SALD.cycle98.remaining_exact_no_boundary_theorem",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryIntegral",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle100.lower_packet.barB_weakGrad_definition_alignment"
      interface := "Compiled local handoff: SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef and SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef remove hweakGradIntegral by definitional specialization of weakGradPairing. hpairNormBound and hdivNoBoundary remain exact analytic boundaries."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle100GeneralMovingTargetDiscreteBarBWeakGradDefLowerObligation",
        "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairingWeakGradDef",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBBoundedNoBoundaryWeakGradDef",
        "SALD.generalMovingTargetDiscreteBarBWeakDivergenceOfNoBoundaryBoundedPairing",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBWeakAction"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle100.remaining_no_boundary_after_weakGrad_def"
      interface := "Remaining exact theorem after the weakGrad definition alignment: prove the concrete a.e. norm bound for fieldPairing phi barB and prove driftDiv phi = -int x, fieldPairing phi barB x d hatRhoS by the no-boundary divergence theorem for hatRhoS * barB."
      source := saldGeneralMovingTargetDiscreteWeakFpDriftActionMathlibSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "Mathlib.MeasureTheory.Integral.DivergenceTheorem",
        "Mathlib.MeasureTheory.Integral.Bochner.Basic",
        "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-100 lower-ready obligation for the inner-gradient contraction
handoff. -/

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