AutoSamplingTheory.SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation

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 cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation :
    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_middle

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 middle sharpens the appendix.tex:1368-1377 lower packet after checking the local SLT Measure.map style and Mathlib Probability.Kernel.CondDistrib/Condexp APIs. To condition X_k^eta on hat X_s=x, lower should instantiate condDistrib with X=hat X_s and Y=X_k^eta, so compProd_map_condDistrib produces the joint law in the order (hat X_s,X_k^eta). The existing SALD cycle-71 compatibility names the joint law as (X_k^eta,hat X_s) with hat rho_s as the second marginal, so lower must either work with the swapped joint law or use AutoSamplingTheory.lawMapProdSwap to bridge orientations before applying component conditional-integral measurability and integrability facts for the two frozen drift summands.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteDerivativeSource — 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.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation
2. SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface
3. SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation
4. SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation
5. SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract
6. SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract
7. SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents
8. SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents
9. SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents
10. AutoSamplingTheory.lawMapProdSwap
11. ProbabilityTheory.compProd_map_condDistrib
12. ProbabilityTheory.condDistrib_ae_eq_iff_measure_eq_compProd
13. MeasureTheory.StronglyMeasurable.integral_condDistrib
14. MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map
15. MeasureTheory.Integrable.integral_condDistrib_map
16. MeasureTheory.Integrable.norm_integral_condDistrib_map
17. ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib
18. ProbabilityTheory.condExp_ae_eq_integral_condExpKernel
19. 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.

Middle source map only. It records the condDistrib orientation, named-marginal compatibility, and vector-valued component-integral measurability/integrability targets; it imports no Mathlib kernel module, constructs no conditional law, proves no weak FP/KL derivative theorem, and keeps standard-Borel, finite/probability, AEMeasurable, Integrable, density/AC, and admissible-test hypotheses explicit.

def cycle75GeneralMovingTargetDiscreteConditionalLawBackfillMiddleObligation :
    ProofObligation where
  id := "sald.general_moving_target_discrete.cycle75_conditional_law_middle"
  statement := "Cycle 75 middle sharpens the appendix.tex:1368-1377 lower packet after checking the local SLT Measure.map style and Mathlib Probability.Kernel.CondDistrib/Condexp APIs. To condition X_k^eta on hat X_s=x, lower should instantiate condDistrib with X=hat X_s and Y=X_k^eta, so compProd_map_condDistrib produces the joint law in the order (hat X_s,X_k^eta). The existing SALD cycle-71 compatibility names the joint law as (X_k^eta,hat X_s) with hat rho_s as the second marginal, so lower must either work with the swapped joint law or use AutoSamplingTheory.lawMapProdSwap to bridge orientations before applying component conditional-integral measurability and integrability facts for the two frozen drift summands."
  source := saldGeneralMovingTargetDiscreteDerivativeSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle75GeneralMovingTargetDiscreteConditionalLawBackfillUpperObligation",
    "SALD.cycle74GeneralMovingTargetDiscreteConditionalKernelMeasureInterface",
    "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceMiddleObligation",
    "SALD.cycle74GeneralMovingTargetDiscreteMeasureInterfaceLowerObligation",
    "SALD.generalMovingTargetDiscreteConditionalLawMeasurabilityContract",
    "SALD.generalMovingTargetDiscreteEndpointConditionalCompatibilityContract",
    "SALD.generalMovingTargetDiscreteConditionalKernelRegularityOfComponents",
    "SALD.generalMovingTargetDiscreteNamedConditionalDriftComponents",
    "SALD.generalMovingTargetDiscreteNamedConditionalDriftRegularityOfComponents",
    "AutoSamplingTheory.lawMapProdSwap",
    "ProbabilityTheory.compProd_map_condDistrib",
    "ProbabilityTheory.condDistrib_ae_eq_iff_measure_eq_compProd",
    "MeasureTheory.StronglyMeasurable.integral_condDistrib",
    "MeasureTheory.AEStronglyMeasurable.integral_condDistrib_map",
    "MeasureTheory.Integrable.integral_condDistrib_map",
    "MeasureTheory.Integrable.norm_integral_condDistrib_map",
    "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib",
    "ProbabilityTheory.condExp_ae_eq_integral_condExpKernel",
    "sald.general_moving_target_discrete.em_interpolation_fp"
  ]
  note := "Middle source map only. It records the condDistrib orientation, named-marginal compatibility, and vector-valued component-integral measurability/integrability targets; it imports no Mathlib kernel module, constructs no conditional law, proves no weak FP/KL derivative theorem, and keeps standard-Borel, finite/probability, AEMeasurable, Integrable, density/AC, and admissible-test hypotheses explicit."

/-- Cycle-75 lower obligation for the swapped-orientation supplied-kernel
regularity wrapper. -/