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

AutoSamplingTheory.SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Dag

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 cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Dag :
    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.cycle167.lower_1_packet.interval_set_univ_taylor_compat_from_base_diff
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt proves first-order interval-to-Set.univ Taylor-polynomial compatibility for the selected scalar line on Icc 0 r from DifferentiableAt at 0.
    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.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt
    2. DifferentiableAt.derivWithin
    3. uniqueDiffOn_Icc
    4. derivWithin_univ
    5. taylorWithinEval_succ
    6. appendix.tex:984-995
    7. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff
    2. hNonnegTaylorCompat
    3. hNegTaylorCompat
    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.cycle167.lower_1_packet.signed_interval_remainder_from_base_diff
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff removes hNonnegTaylorCompat and hNegTaylorCompat from the signed interval first-order quadratic remainder theorem by supplying them from base differentiability.
    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.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff
    2. SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt
    3. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect
    4. hNonnegCont
    5. hNonnegSecond
    6. hNegCont
    7. hNegSecond
    8. hNonnegBaseDiff
    9. hNegBaseDiff
    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.formalized — stored label only; no proof certification
  3. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle167.lower_2_packet.signed_interval_remainder_from_global_line_contdiff
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff removes the signed interval ContDiffOn and base differentiability inputs by deriving them from global selected-line ContDiffOn Real 2 on Set.univ; the two signed interval second-derivative domination hypotheses remain 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.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineContDiffLower2Obligation
    2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff
    3. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff
    4. hLine
    5. hNonnegSecond
    6. hNegSecond
    7. ContDiffOn.mono
    8. contDiffOn_univ
    9. ContDiff.comp
    10. contDiff_neg
    11. contDiff_id
    12. ContDiffOn.contDiffAt
    13. ContDiffAt.differentiableAt
    14. appendix.tex:984-995
    15. 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.formalized — stored label only; no proof certification
  4. id : String (explicit)

    Stable plan-node identifier.

    ASTIS.SALD.cycle167.remaining_signed_interval_data_after_taylor_compat
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact theorem boundary after the lower_2 global-line narrowing: prove the global selected-line ContDiffOn Real 2 fact and the signed interval second-derivative domination from the paper's selected-test bounded-Hessian/regularity hypotheses; 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.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff
    2. hLine
    3. hNonnegSecond
    4. hNegSecond
    5. selected-test second-derivative or bounded-Hessian source hypothesis
    6. appendix.tex:984-995
    7. 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
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 cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoTaylorCompatLower1Dag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle167.lower_1_packet.interval_set_univ_taylor_compat_from_base_diff"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt proves first-order interval-to-Set.univ Taylor-polynomial compatibility for the selected scalar line on Icc 0 r from DifferentiableAt at 0."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt",
        "DifferentiableAt.derivWithin",
        "uniqueDiffOn_Icc",
        "derivWithin_univ",
        "taylorWithinEval_succ",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff",
        "hNonnegTaylorCompat",
        "hNegTaylorCompat"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle167.lower_1_packet.signed_interval_remainder_from_base_diff"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff removes hNonnegTaylorCompat and hNegTaylorCompat from the signed interval first-order quadratic remainder theorem by supplying them from base differentiability."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff",
        "SALD.gaussianRealSelectedTestLineFirstOrderTaylorCompatOfDifferentiableAt",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorNoReflect",
        "hNonnegCont",
        "hNonnegSecond",
        "hNegCont",
        "hNegSecond",
        "hNonnegBaseDiff",
        "hNegBaseDiff",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle167.lower_2_packet.signed_interval_remainder_from_global_line_contdiff"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff removes the signed interval ContDiffOn and base differentiability inputs by deriving them from global selected-line ContDiffOn Real 2 on Set.univ; the two signed interval second-derivative domination hypotheses remain explicit."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle167GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineContDiffLower2Obligation",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSignedIntervalTaylorBaseDiff",
        "hLine",
        "hNonnegSecond",
        "hNegSecond",
        "ContDiffOn.mono",
        "contDiffOn_univ",
        "ContDiff.comp",
        "contDiff_neg",
        "contDiff_id",
        "ContDiffOn.contDiffAt",
        "ContDiffAt.differentiableAt",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle167.remaining_signed_interval_data_after_taylor_compat"
      interface := "Remaining exact theorem boundary after the lower_2 global-line narrowing: prove the global selected-line ContDiffOn Real 2 fact and the signed interval second-derivative domination from the paper's selected-test bounded-Hessian/regularity hypotheses; hSecondCoeff remains separate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff",
        "hLine",
        "hNonnegSecond",
        "hNegSecond",
        "selected-test second-derivative or bounded-Hessian source hypothesis",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.obligation
    }
  ]

/-- Cycle-168 proof-DAG pane for ambient source-test regularity to selected line. -/

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