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

AutoSamplingTheory.SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDag

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 cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDag :
    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.cycle168.middle_packet.ambient_source_contdiff_to_selected_line
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn proves global selected-line ContDiffOn Real 2 from ambient sourceTest ContDiffOn Real 2 by composing with q |-> x + q • e.
    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.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn
    2. hSource
    3. contDiffOn_univ
    4. ContDiff.comp
    5. ContDiff.add
    6. ContDiff.smul_const
    7. contDiff_const
    8. contDiff_id
    9. appendix.tex:984-995
    10. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. hLine
    2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff
    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.cycle168.middle_packet.hline_consumer_source_contdiff
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn removes hLine from the cycle-167 global-line bridge, keeping hNonnegSecond and hNegSecond 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.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn
    2. SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn
    3. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff
    4. hSource
    5. hNonnegSecond
    6. hNegSecond
    7. appendix.tex:984-995
    8. 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.cycle168.lower_2_packet.signed_interval_second_bounds_from_global_line_second
    interface : String (explicit)

    Text describing intended mathematical interface.

    Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds removes hNonnegSecond and hNegSecond as primitive signed interval iteratedDerivWithin 2 bounds by deriving them from global selected/reflected scalar-line iteratedDeriv 2 bounds.
    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.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation
    2. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds
    3. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn
    4. hSource
    5. hC1
    6. hLineSecond
    7. hNegLineSecond
    8. hNonnegSecond
    9. hNegSecond
    10. accPt_iff_nhds
    11. derivWithin_zero_of_not_accPt
    12. uniqueDiffOn_Icc
    13. iteratedDerivWithin_eq_iteratedDeriv
    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.cycle168.remaining_global_line_second_derivative_bounds
    interface : String (explicit)

    Text describing intended mathematical interface.

    Remaining exact theorem boundary after the lower_2 interval-second-derivative narrowing: prove hLineSecond and hNegLineSecond from the paper's selected-test bounded-Hessian regularity, with ambient source ContDiffOn Real 2 and nonnegative Taylor constant hC1 still explicit; 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. hSource
    2. hC1
    3. hLineSecond
    4. hNegLineSecond
    5. selected-test bounded-Hessian/regularity source hypothesis
    6. iteratedDeriv_vcomp_two
    7. affine-line derivative algebra
    8. appendix.tex:984-995
    9. appendix.tex:1379-1387
    reusedBy : List String (explicit)

    Declared consumers as strings, default empty.

    Ordered data items

    1. SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds
    2. hFirst
    3. 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 cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoSourceContDiffDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle168.middle_packet.ambient_source_contdiff_to_selected_line"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn proves global selected-line ContDiffOn Real 2 from ambient sourceTest ContDiffOn Real 2 by composing with q |-> x + q • e."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn",
        "hSource",
        "contDiffOn_univ",
        "ContDiff.comp",
        "ContDiff.add",
        "ContDiff.smul_const",
        "contDiff_const",
        "contDiff_id",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hLine",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle168.middle_packet.hline_consumer_source_contdiff"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn removes hLine from the cycle-167 global-line bridge, keeping hNonnegSecond and hNegSecond explicit."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn",
        "SALD.gaussianRealSelectedTestLineContDiffOnOfSourceContDiffOn",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfGlobalLineContDiff",
        "hSource",
        "hNonnegSecond",
        "hNegSecond",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle168.lower_2_packet.signed_interval_second_bounds_from_global_line_second"
      interface := "Compiled theorem: SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds removes hNonnegSecond and hNegSecond as primitive signed interval iteratedDerivWithin 2 bounds by deriving them from global selected/reflected scalar-line iteratedDeriv 2 bounds."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle168GeneralMovingTargetDiscreteEmGeneratorLaplacianEventFieldFrozenScalarBrownianItoLineSecondLower2Obligation",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds",
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOn",
        "hSource",
        "hC1",
        "hLineSecond",
        "hNegLineSecond",
        "hNonnegSecond",
        "hNegSecond",
        "accPt_iff_nhds",
        "derivWithin_zero_of_not_accPt",
        "uniqueDiffOn_Icc",
        "iteratedDerivWithin_eq_iteratedDeriv",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle168.remaining_global_line_second_derivative_bounds"
      interface := "Remaining exact theorem boundary after the lower_2 interval-second-derivative narrowing: prove hLineSecond and hNegLineSecond from the paper's selected-test bounded-Hessian regularity, with ambient source ContDiffOn Real 2 and nonnegative Taylor constant hC1 still explicit; hSecondCoeff remains separate."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "hSource",
        "hC1",
        "hLineSecond",
        "hNegLineSecond",
        "selected-test bounded-Hessian/regularity source hypothesis",
        "iteratedDeriv_vcomp_two",
        "affine-line derivative algebra",
        "appendix.tex:984-995",
        "appendix.tex:1379-1387"
      ]
      reusedBy := [
        "SALD.gaussianRealSelectedTestLineFirstOrderQuadraticRemainderBoundOfSourceContDiffOnAndLineSecondBounds",
        "hFirst",
        "SALD.gaussianRealSelectedTestLineSecondOrderTaylorQuotientBoundOfFirstOrderAndSecondCoeff"
      ]
      status := ProofStatus.obligation
    }
  ]

/-- Cycle-169 proof-DAG pane for the ambient directional-Hessian line-second split. -/

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