AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation

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 cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation :
    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.cycle75_conditional_law_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 75 lower compiles SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap and SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents for appendix.tex lines 1368-1377. If the Mathlib condDistrib/condExpKernel backend supplies kernel compatibility for the swapped joint law (hat X_s,X_k^eta), component integral fields for the two frozen drift summands, and their measurability/integrability consequences, the wrapper identifies the named hat rho_s first marginal, bridges back to the existing SALD (X_k^eta,hat X_s) joint-law orientation with AutoSamplingTheory.lawMapProdSwap, and derives measurability/integrability of bar b_{k,s}. It does not construct the conditional kernel or prove weak FP.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource — 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.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation
2. SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface
3. SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation
4. SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap
5. SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents
6. SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents
7. SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents
8. AutoSamplingTheory.lawMapProdSwap
9. ProbabilityTheory.compProd_map_condDistrib
10. ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib
11. ProbabilityTheory.condExp_ae_eq_integral_condExpKernel
12. 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 local wrapper under supplied hypotheses only. The regular conditional kernel, Mathlib conditional-expectation instantiation, component integral theorem, density/AC, weak conditional Fokker-Planck identity, log-ratio admissibility, KL differentiation, integration by parts, and theorem closure remain obligations.

def cycle75GeneralMovingTargetDiscreteConditionalLawBackfillLowerObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle75_conditional_law_lower"
  statement := "Cycle 75 lower compiles SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap and SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents for appendix.tex lines 1368-1377. If the Mathlib condDistrib/condExpKernel backend supplies kernel compatibility for the swapped joint law (hat X_s,X_k^eta), component integral fields for the two frozen drift summands, and their measurability/integrability consequences, the wrapper identifies the named hat rho_s first marginal, bridges back to the existing SALD (X_k^eta,hat X_s) joint-law orientation with AutoSamplingTheory.lawMapProdSwap, and derives measurability/integrability of bar b_{k,s}. It does not construct the conditional kernel or prove weak FP."
  source := saldGeneralMovingTargetDiscreteConditionalKernelMathlibSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation",
    "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface",
    "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation",
    "SALD.generalMovingTargetDiscreteHatRhoFirstMarginalOfSwappedJointMap",
    "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfSwappedComponents",
    "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents",
    "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents",
    "AutoSamplingTheory.lawMapProdSwap",
    "ProbabilityTheory.compProd_map_condDistrib",
    "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib",
    "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Proof-producing local wrapper under supplied hypotheses only. The regular conditional kernel, Mathlib conditional-expectation instantiation, component integral theorem, density/AC, weak conditional Fokker-Planck identity, log-ratio admissibility, KL differentiation, integration by parts, and theorem closure remain obligations."

/-- Cycle-75 proof-DAG pane for the conditional-law construction backfill. -/