Samplinglib
Lean gate not recorded for this source state main · 0e31a3cda412

AutoSamplingTheory.SALD.cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDag

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 cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDag :
    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.cycle183.middle_packet.source_integral_ae_bridge
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled bridge: derive hBrownianCoordinateGeneratorTaylorIntegralDef from the source integral definition hBrownianCoordinateGeneratorSourceIntegralDef and the a.e. integrand equality hBrownianCoordinateGeneratorTaylorIntegrandAE by MeasureTheory.integral_congr_ae.
    source : AutoSamplingTheory.SourceAnchor (explicit)

    Source pointer.

    AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE
    2. MeasureTheory.integral_congr_ae
    3. hBrownianCoordinateGeneratorSourceIntegralDef
    4. hBrownianCoordinateGeneratorTaylorIntegrandAE
    5. hBrownianCoordinateGeneratorTaylorIntegralDef
    6. appendix.tex:984-995
    7. appendix.tex:1170-1176
    8. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hBrownianCoordinateGeneratorTaylorIntegralDef
    2. SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder
    3. sald.general_moving_target_discrete.em_interpolation_fp
    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.cycle183.lower_1_packet.taylor_integrand_ae_route
    interface : String (explicit)

    Text describing intended mathematical interface.

    Lower_1 proof route: prove hBrownianCoordinateGeneratorTaylorIntegrandAE by identifying the paper's sourceTaylorIntegrand for the frozen scalar Brownian coordinate with linearCoeff*z + quadraticCoeff*z^2 + normalizedRemainder phi x i z a.e. under gaussianReal 0 (variance phi x i). Keep sigma_eta^2/2 outside the Brownian event field.
    source : AutoSamplingTheory.SourceAnchor (explicit)

    Source pointer.

    AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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. hBrownianCoordinateGeneratorTaylorIntegrandAE
    2. sourceTaylorIntegrand
    3. hScalarLineTaylorCoeffDef
    4. hNormalizedVectorLaw
    5. hCoordinateLawDef
    6. hVarianceDef
    7. hRemainderGeneratorLimitDef
    8. appendix.tex:984-995
    9. appendix.tex:1170-1176
    10. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. future lower_2 source Taylor identity packet
    status : AutoSamplingTheory.ProofStatus (explicit)

    Recorded workflow status, default planned.

    AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification
  3. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle183.lower_2_packet.pointwise_source_taylor_identity
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled lower_2 bridge: SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise reduces hBrownianCoordinateGeneratorTaylorIntegrandAE to the pointwise source Taylor identity hSourceTaylorIntegrandPointwise by Filter.Eventually.of_forall.
    source : AutoSamplingTheory.SourceAnchor (explicit)

    Source pointer.

    AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise
    2. Filter.Eventually.of_forall
    3. hBrownianCoordinateGeneratorTaylorIntegrandAE
    4. hSourceTaylorIntegrandPointwise
    5. hScalarLineTaylorCoeffDef
    6. normalizedRemainder
    7. appendix.tex:984-995
    8. appendix.tex:1170-1176
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hBrownianCoordinateGeneratorTaylorIntegrandAE
    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.cycle183.remaining_brownian_taylor_backend
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact Brownian/Ito frozen backend after the lower_2 pointwise-to-a.e. bridge: prove hBrownianCoordinateGeneratorSourceIntegralDef, hSourceTaylorIntegrandPointwise, hRemainderGeneratorLimitDef, and the concrete normalized-remainder measurability/domination package hRemainderMeas/hRemainderBound/hRemainderBoundInt. hSourceHasHessian and hSourceHessianBound stay source-contract gaps.
    source : AutoSamplingTheory.SourceAnchor (explicit)

    Source pointer.

    AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteEmSource — 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. hBrownianCoordinateGeneratorSourceIntegralDef
    2. hSourceTaylorIntegrandPointwise
    3. hRemainderGeneratorLimitDef
    4. hRemainderMeas
    5. hRemainderBound
    6. hRemainderBoundInt
    7. hSourceHasHessian
    8. hSourceHessianBound
    9. source-contract-gap for selected weak-test Hessian fields
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. sald.general_moving_target_discrete.em_interpolation_fp
    2. sald.general_moving_target_discrete.kl_derivative
    3. divergence/FI/IBP handoff
    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 cycle183GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceIntegralDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle183.middle_packet.source_integral_ae_bridge"
      interface := "Compiled bridge: derive hBrownianCoordinateGeneratorTaylorIntegralDef from the source integral definition hBrownianCoordinateGeneratorSourceIntegralDef and the a.e. integrand equality hBrownianCoordinateGeneratorTaylorIntegrandAE by MeasureTheory.integral_congr_ae."
      source := saldGeneralMovingTargetDiscreteEmSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegralDefOfSourceIntegralAndAE",
        "MeasureTheory.integral_congr_ae",
        "hBrownianCoordinateGeneratorSourceIntegralDef",
        "hBrownianCoordinateGeneratorTaylorIntegrandAE",
        "hBrownianCoordinateGeneratorTaylorIntegralDef",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hBrownianCoordinateGeneratorTaylorIntegralDef",
        "SALD.selectedWeakTestFrozenScalarBrownianItoTaylorMomentDecompositionOfIntegralDefsAndDominatedRemainder",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle183.lower_1_packet.taylor_integrand_ae_route"
      interface := "Lower_1 proof route: prove hBrownianCoordinateGeneratorTaylorIntegrandAE by identifying the paper's sourceTaylorIntegrand for the frozen scalar Brownian coordinate with linearCoeff*z + quadraticCoeff*z^2 + normalizedRemainder phi x i z a.e. under gaussianReal 0 (variance phi x i). Keep sigma_eta^2/2 outside the Brownian event field."
      source := saldGeneralMovingTargetDiscreteEmSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "hBrownianCoordinateGeneratorTaylorIntegrandAE",
        "sourceTaylorIntegrand",
        "hScalarLineTaylorCoeffDef",
        "hNormalizedVectorLaw",
        "hCoordinateLawDef",
        "hVarianceDef",
        "hRemainderGeneratorLimitDef",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176",
        "appendix.tex:1379-1387"
      ]
      reusedBy := ["future lower_2 source Taylor identity packet"]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle183.lower_2_packet.pointwise_source_taylor_identity"
      interface := "Compiled lower_2 bridge: SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise reduces hBrownianCoordinateGeneratorTaylorIntegrandAE to the pointwise source Taylor identity hSourceTaylorIntegrandPointwise by Filter.Eventually.of_forall."
      source := saldGeneralMovingTargetDiscreteEmSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.selectedWeakTestBrownianCoordinateGeneratorTaylorIntegrandAEOfPointwise",
        "Filter.Eventually.of_forall",
        "hBrownianCoordinateGeneratorTaylorIntegrandAE",
        "hSourceTaylorIntegrandPointwise",
        "hScalarLineTaylorCoeffDef",
        "normalizedRemainder",
        "appendix.tex:984-995",
        "appendix.tex:1170-1176"
      ]
      reusedBy := ["hBrownianCoordinateGeneratorTaylorIntegrandAE"]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle183.remaining_brownian_taylor_backend"
      interface := "Remaining exact Brownian/Ito frozen backend after the lower_2 pointwise-to-a.e. bridge: prove hBrownianCoordinateGeneratorSourceIntegralDef, hSourceTaylorIntegrandPointwise, hRemainderGeneratorLimitDef, and the concrete normalized-remainder measurability/domination package hRemainderMeas/hRemainderBound/hRemainderBoundInt. hSourceHasHessian and hSourceHessianBound stay source-contract gaps."
      source := saldGeneralMovingTargetDiscreteEmSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "hBrownianCoordinateGeneratorSourceIntegralDef",
        "hSourceTaylorIntegrandPointwise",
        "hRemainderGeneratorLimitDef",
        "hRemainderMeas",
        "hRemainderBound",
        "hRemainderBoundInt",
        "hSourceHasHessian",
        "hSourceHessianBound",
        "source-contract-gap for selected weak-test Hessian fields"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.general_moving_target_discrete.kl_derivative",
        "divergence/FI/IBP handoff"
      ]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 184 middle: Brownian coordinate source-integral law split -/

/-- Cycle-184 middle packet for the Brownian coordinate source-integral leaf.

The compiled bridge narrows `hBrownianCoordinateGeneratorSourceIntegralDef` to
the actual normalized scalar-coordinate law definition plus the source-backed
standard-Gaussian vector law, coordinate-law definition, and variance-field
definition already tracked in the Brownian/Ito backend.
-/

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