AutoSamplingTheory.SALD.cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeLowerObligation

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.ProofObligation. A value of this type stores descriptions; it is not a proof of the statements in those descriptions.

def cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeLowerObligation :
    ProofObligation

Construction and field-by-field explanation

Construct a data record from explicit fields and the audited defaults shown below.

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.

id : String (explicit)

Stable obligation identifier.

sald.general_moving_target_discrete.cycle111_target_time_derivative_lower

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Cycle 111 lower compiles SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated, SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr, and SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated for appendix.tex:1358-1366. Classification: narrows-source-cited-boundary. The first theorem proves the concrete dominated target-time derivative and weighted-derivative integrability by Mathlib parametric integral. The source-ratio congruence theorem transfers the target-time integral, integrability, and HasDerivAt formula from the chosen fixed weight to the paper's density-ratio representative under a single a.e. equality. The field handoff consumes finite-KL llr regularity and feeds those concrete results into targetTimeTermIntegrable and targetTimeDerivativeFormula. Remaining exact boundary: prove that a.e. source density-ratio equality, verify target-density pointwise derivative and domination, bridge the concrete Mathlib HasDerivAt/integrability statements to the package fields, and still prove endpoint-safe first-term KL differentiation.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource — audited data reference, not expanded and not a compiled dependency edge

status : AutoSamplingTheory.ProofStatus (explicit)

Stored ProofStatus, default obligation; even an explicitly stored formalized does not independently certify a Lean theorem.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

dependsOn : List String (explicit)

List of declared dependency names as strings; may mix theorem names, obligations, source labels, or descriptions. Not the compiled dependency DAG.

Ordered data items

1. SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated
2. SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr
3. SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated
4. SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr
5. SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl
6. Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
7. MeasureTheory.integral_congr_ae
8. MeasureTheory.Integrable.congr
9. appendix.tex:1358-1366
10. sald.general_moving_target_discrete.kl_derivative
11. sald.general_moving_target_discrete.em_interpolation_fp

note : String (explicit)

Recorded evidence/caveats; may distinguish a compiled scalar helper from still-open source analysis.

Proof-producing target-time handoff only. It does not prove the source density-ratio a.e. equality, endpoint-safe KL differentiation, weak FP, hbarBCondExp, no-boundary, diffusion source-action, IBP/FI, theorem closure, SLT reuse, or any Lake dependency.

def cycle111GeneralMovingTargetDiscreteTargetTimeDerivativeLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle111_target_time_derivative_lower"
  statement := "Cycle 111 lower compiles SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated, SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr, and SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated for appendix.tex:1358-1366. Classification: narrows-source-cited-boundary. The first theorem proves the concrete dominated target-time derivative and weighted-derivative integrability by Mathlib parametric integral. The source-ratio congruence theorem transfers the target-time integral, integrability, and HasDerivAt formula from the chosen fixed weight to the paper's density-ratio representative under a single a.e. equality. The field handoff consumes finite-KL llr regularity and feeds those concrete results into targetTimeTermIntegrable and targetTimeDerivativeFormula. Remaining exact boundary: prove that a.e. source density-ratio equality, verify target-density pointwise derivative and domination, bridge the concrete Mathlib HasDerivAt/integrability statements to the package fields, and still prove endpoint-safe first-term KL differentiation."
  source := saldGeneralMovingTargetDiscreteKlWeakFpHandoffSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.generalMovingTargetDiscreteTargetTimeDerivativeOfDominated",
    "SALD.generalMovingTargetDiscreteTargetTimeDerivativeSourceRatioCongr",
    "SALD.generalMovingTargetDiscretePureRawKlTargetTimeFieldsOfDominated",
    "SALD.GeneralMovingTargetDiscretePureRawKlDerivativeNoMassAtFiniteKlLlr",
    "SALD.generalMovingTargetDiscreteKlLogRatioRegularityOfFiniteKl",
    "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le",
    "MeasureTheory.integral_congr_ae",
    "MeasureTheory.Integrable.congr",
    "appendix.tex:1358-1366",
    "sald.general_moving_target_discrete.kl_derivative",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Proof-producing target-time handoff only. It does not prove the source density-ratio a.e. equality, endpoint-safe KL differentiation, weak FP, hbarBCondExp, no-boundary, diffusion source-action, IBP/FI, theorem closure, SLT reuse, or any Lake dependency."

/-- Cycle-111 proof-DAG pane for the target-time KL derivative packet. -/