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

AutoSamplingTheory.SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderDag

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 cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderDag :
    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.cycle166.lower_packet.hfirst_from_quadratic_remainder
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder supplies hFirst from the smaller source-facing non-quotient quadratic Taylor-remainder estimate and 0 <= C1.
    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.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation
    2. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder
    3. hFirst
    4. hFirstQuadraticRemainder
    5. hC1
    6. taylorWithinEval
    7. appendix.tex:984-995
    8. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
    2. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound
    3. SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero
    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.cycle166.lower_1_packet.nonnegative_interval_taylor_remainder
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor proves the r >= 0 non-quotient quadratic remainder bound from Mathlib taylor_mean_remainder_bound, explicit ContDiffOn Real 2 on Icc 0 r, interval second-derivative domination, and interval-to-Set.univ Taylor compatibility.
    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.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation
    2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor
    3. Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound
    4. ContDiffOn Real 2
    5. iteratedDerivWithin
    6. hTaylorCompat
    7. appendix.tex:984-995
    8. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hFirstQuadraticRemainder
    2. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder
    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.cycle166.lower_2_packet.signed_interval_taylor_remainder
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor supplies the all-r non-quotient quadratic remainder from signed interval Taylor data: the nonnegative selected-line interval theorem, the reflected -e theorem for r < 0, and explicit Set.univ Taylor reflection compatibility.
    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.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation
    2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor
    3. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor
    4. hFirstQuadraticRemainder
    5. hNonnegCont
    6. hNonnegSecond
    7. hNonnegTaylorCompat
    8. hNegCont
    9. hNegSecond
    10. hNegTaylorCompat
    11. hNegTaylorReflect
    12. appendix.tex:984-995
    13. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hFirst
    2. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder
    3. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
    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.cycle166.remaining_nonquotient_quadratic_remainder
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact theorem boundary after the signed interval combination: prove the signed interval ContDiffOn/second-derivative domination, interval-to-Set.univ Taylor compatibility, and reflected Set.univ Taylor-polynomial compatibility from the paper's selected-test second-derivative/bounded-Hessian hypothesis; hSecondCoeff remains separate.
    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.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor
    2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor
    3. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder
    4. hFirstQuadraticRemainder
    5. Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound
    6. Mathlib.Analysis.Calculus.MeanValue
    7. selected-test second-derivative or bounded-Hessian source hypothesis
    8. interval-to-Set.univ taylorWithinEval compatibility
    9. reflected Set.univ taylorWithinEval compatibility
    10. appendix.tex:984-995
    11. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hFirst
    2. SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff
    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.cycle166.reviewer_hfirst_check
    interface : String (explicit)

    Text describing intended mathematical interface.

    Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hFirst is narrowed to the non-quotient quadratic remainder estimate rather than restated, hSecondCoeff and the Taylor moment/quadratic-variation/coordinate-sum leaves remain separate, and python3 tools/astis.py check passes.
    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.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation
    2. SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation
    3. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor
    4. SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder
    5. appendix.tex:984-995
    6. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. cycle 166 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 cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle166.lower_packet.hfirst_from_quadratic_remainder"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder supplies hFirst from the smaller source-facing non-quotient quadratic Taylor-remainder estimate and 0 <= C1."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation",
        "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder",
        "hFirst",
        "hFirstQuadraticRemainder",
        "hC1",
        "taylorWithinEval",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderQuadraticBoundOfTaylorQuotientBound",
        "SALD.gaussianRealSelectedTestLineSecondOrderNormalizedRemainderIntegralTendstoZero"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle166.lower_1_packet.nonnegative_interval_taylor_remainder"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor proves the r >= 0 non-quotient quadratic remainder bound from Mathlib taylor_mean_remainder_bound, explicit ContDiffOn Real 2 on Icc 0 r, interval second-derivative domination, and interval-to-Set.univ Taylor compatibility."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoIntervalTaylorLower1Obligation",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor",
        "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound",
        "ContDiffOn Real 2",
        "iteratedDerivWithin",
        "hTaylorCompat",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFirstQuadraticRemainder",
        "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle166.lower_2_packet.signed_interval_taylor_remainder"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor supplies the all-r non-quotient quadratic remainder from signed interval Taylor data: the nonnegative selected-line interval theorem, the reflected -e theorem for r < 0, and explicit Set.univ Taylor reflection compatibility."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor",
        "hFirstQuadraticRemainder",
        "hNonnegCont",
        "hNonnegSecond",
        "hNonnegTaylorCompat",
        "hNegCont",
        "hNegSecond",
        "hNegTaylorCompat",
        "hNegTaylorReflect",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFirst",
        "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle166.remaining_nonquotient_quadratic_remainder"
      interface := "Remaining exact theorem boundary after the signed interval combination: prove the signed interval ContDiffOn/second-derivative domination, interval-to-Set.univ Taylor compatibility, and reflected Set.univ Taylor-polynomial compatibility from the paper's selected-test second-derivative/bounded-Hessian hypothesis; hSecondCoeff remains separate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundNonnegOfIntervalTaylor",
        "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder",
        "hFirstQuadraticRemainder",
        "Mathlib.Analysis.Calculus.Taylor.taylor_mean_remainder_bound",
        "Mathlib.Analysis.Calculus.MeanValue",
        "selected-test second-derivative or bounded-Hessian source hypothesis",
        "interval-to-Set.univ taylorWithinEval compatibility",
        "reflected Set.univ taylorWithinEval compatibility",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle166.reviewer_hfirst_check"
      interface := "Reviewer check: accept only if the packet classification is narrows-source-cited-boundary, hFirst is narrowed to the non-quotient quadratic remainder estimate rather than restated, hSecondCoeff and the Taylor moment/quadratic-variation/coordinate-sum leaves remain separate, and python3 tools/astis.py check passes."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoFirstOrderRemainderLowerObligation",
        "SALD.cycle166GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSignedIntervalTaylorLower2Obligation",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylor",
        "SALD.gaussianRealSelectedTestLineFirstOrderTaylorQuotientBoundOfQuadraticRemainder",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := ["cycle 166 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-- Cycle-167 reflected Taylor compatibility discharge packet. -/

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