Samplinglib
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AutoSamplingTheory.SALD.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionDag

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 cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionDag :
    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.cycle137.middle_packet.weak_fp_density_laplacian_pointwise_ibp
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled dynamic-leaf worker theorem: SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP derives hweakFpDensityLaplacianAction from the pointwise weak Laplacian IBP equality densityLaplacianAction phi = laplacian phi. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP then feeds that smaller pointwise boundary plus hdensityLaplacianStdBasisDef into the existing cycle-136 standard-basis source formula.
    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.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionLowerObligation
    2. SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP
    3. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP
    4. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula
    5. appendix.tex:1379-1387
    6. appendix.tex:1392-1427
    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
  2. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle137.lower_1_packet.pointwise_green_ibp_scout
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled lower_1 proof-scout theorem: SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity derives the pointwise weak Laplacian IBP equality from first-Green, second-Green, and test-Laplacian pointwise identities. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity feeds those three leaves plus hdensityLaplacianStdBasisDef into the cycle-137 standard-basis consumer.
    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.cycle137GeneralMovingTargetDiscreteEmPointwiseGreenIbPScoutObligation
    2. SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity
    3. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity
    4. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP
    5. SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity
    6. appendix.tex:1379-1387
    7. appendix.tex:1392-1427
    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
  3. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle137.lower_2_packet.second_green_pointwise_box_boundary_flux
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero derives the pointwise second-Green identity negativeGradientPairAction phi = testLaplacianAction phi from residual/divergence facts, the Mathlib box-divergence package, signed-face trace identification, pointwise second-Green trace equality, and pointwise zero test trace. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux feeds that derived second-Green leaf into the lower_1 pointwise Green standard-basis consumer while leaving first-Green and test-Laplacian pointwise leaves explicit.
    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.cycle137GeneralMovingTargetDiscreteEmSecondGreenPointwiseBoxBoundaryFluxLowerObligation
    2. SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero
    3. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux
    4. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity
    5. SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox
    6. MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable
    7. Filter.Eventually.of_forall
    8. appendix.tex:1392-1427
    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
  4. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle137.remaining_density_laplacian_source_action
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact theorem boundary after cycle 137 lower_2: prove the first-Green pointwise identity densityLaplacianAction phi = negativeGradientPairAction phi and the test-Laplacian pointwise normalization testLaplacianAction phi = laplacian phi. The direct second-Green pointwise leaf is narrowed to residual/divergence, box-divergence regularity/face-trace, pointwise second-Green trace equality, and pointwise zero test trace. The already separate hdensityLaplacianActionDef and hsourceDensityLaplacianEqLaplacian source-field facts from cycle 136 also remain, and htestLaplacianStdBasisDef plus diffusion leaves stay explicit.
    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. appendix.tex:1379-1387
    2. appendix.tex:1392-1427
    3. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux
    4. SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero
    5. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity
    6. SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity
    7. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP
    8. SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP
    9. SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity
    10. SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula
    11. SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField
    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
  5. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle137.reviewer_weak_fp_density_laplacian_check
    interface : String (explicit)

    Text describing intended mathematical interface.

    Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hweakFpDensityLaplacianAction is narrowed to the pointwise weak Laplacian IBP equality through SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP, the standard-basis consumer uses SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use.
    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.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionLowerObligation
    2. SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP
    3. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP
    4. appendix.tex:1379-1427
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. cycle 137 reviewer
    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 cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle137.middle_packet.weak_fp_density_laplacian_pointwise_ibp"
      interface := "Compiled dynamic-leaf worker theorem: SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP derives hweakFpDensityLaplacianAction from the pointwise weak Laplacian IBP equality densityLaplacianAction phi = laplacian phi. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP then feeds that smaller pointwise boundary plus hdensityLaplacianStdBasisDef into the existing cycle-136 standard-basis source formula."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfSourceDensityLaplacianFormula",
        "appendix.tex:1379-1387",
        "appendix.tex:1392-1427"
      ]
      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.cycle137.lower_1_packet.pointwise_green_ibp_scout"
      interface := "Compiled lower_1 proof-scout theorem: SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity derives the pointwise weak Laplacian IBP equality from first-Green, second-Green, and test-Laplacian pointwise identities. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity feeds those three leaves plus hdensityLaplacianStdBasisDef into the cycle-137 standard-basis consumer."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle137GeneralMovingTargetDiscreteEmPointwiseGreenIbPScoutObligation",
        "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity",
        "appendix.tex:1379-1387",
        "appendix.tex:1392-1427"
      ]
      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.cycle137.lower_2_packet.second_green_pointwise_box_boundary_flux"
      interface := "Compiled lower_2 theorem: SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero derives the pointwise second-Green identity negativeGradientPairAction phi = testLaplacianAction phi from residual/divergence facts, the Mathlib box-divergence package, signed-face trace identification, pointwise second-Green trace equality, and pointwise zero test trace. SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux feeds that derived second-Green leaf into the lower_1 pointwise Green standard-basis consumer while leaving first-Green and test-Laplacian pointwise leaves explicit."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle137GeneralMovingTargetDiscreteEmSecondGreenPointwiseBoxBoundaryFluxLowerObligation",
        "SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity",
        "SALD.generalMovingTargetDiscreteBoundaryFluxIntegralOfDivergenceTheoremBox",
        "MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable",
        "Filter.Eventually.of_forall",
        "appendix.tex:1392-1427"
      ]
      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.cycle137.remaining_density_laplacian_source_action"
      interface := "Remaining exact theorem boundary after cycle 137 lower_2: prove the first-Green pointwise identity densityLaplacianAction phi = negativeGradientPairAction phi and the test-Laplacian pointwise normalization testLaplacianAction phi = laplacian phi. The direct second-Green pointwise leaf is narrowed to residual/divergence, box-divergence regularity/face-trace, pointwise second-Green trace equality, and pointwise zero test trace. The already separate hdensityLaplacianActionDef and hsourceDensityLaplacianEqLaplacian source-field facts from cycle 136 also remain, and htestLaplacianStdBasisDef plus diffusion leaves stay explicit."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "appendix.tex:1379-1387",
        "appendix.tex:1392-1427",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenSecondGreenBoxBoundaryFlux",
        "SALD.generalMovingTargetDiscreteSecondGreenPointwiseOfBoxBoundaryFluxOfPointwiseTraceEqPointwiseTestTraceZero",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseGreenIdentity",
        "SALD.generalMovingTargetDiscretePointwiseWeakLaplacianIbPOfGreenIdentity",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP",
        "SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpLaplacianIbPOfGreenIdentity",
        "SALD.generalMovingTargetDiscreteDensityLaplacianStdBasisDefOfPointwiseSourceFormula",
        "SALD.generalMovingTargetDiscreteSourceDensityLaplacianStdBasisOfLaplacianSourceField"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle137.reviewer_weak_fp_density_laplacian_check"
      interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hweakFpDensityLaplacianAction is narrowed to the pointwise weak Laplacian IBP equality through SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP, the standard-basis consumer uses SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP, and python3 tools/astis.py check passes without SLT import, non-EM fallback, wrapper churn, theorem-status promotion, fake closure, broad audit, Lake/toolchain change, or sald_version_2 use."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle137GeneralMovingTargetDiscreteEmWeakFpDensityLaplacianActionLowerObligation",
        "SALD.generalMovingTargetDiscreteWeakFpDensityLaplacianActionOfPointwiseWeakLaplacianIbP",
        "SALD.generalMovingTargetDiscreteWeakFpStdBasisDefOfPointwiseWeakLaplacianIbP",
        "appendix.tex:1379-1427"
      ]
      reusedBy := ["cycle 137 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-- Cycle-138 narrowing of the first-Green pointwise leaf to boundary-flux
cancellation facts. -/

Existing module entry · Audited data-reader index · All teaching coverage