AutoSamplingTheory.SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftDag

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 cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftDag :
    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

id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle106.middle_canonical_condDistrib_drift_source_map
interface : String (explicit)
Text describing intended mathematical interface.
Middle source map: appendix.tex:1368-1377 defines bar b_{k,s} by conditioning the frozen guide and score summands on hat X_s=x. Cycle 106 chooses the canonical condDistrib representative and targets only its measurable/integrable conditional-integral regularity.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalDriftSource — 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.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftMiddleObligation
2. SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity
3. AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable
4. AutoSamplingTheory.condDistribIntegralNamedLawIntegrable
5. appendix.tex:1368-1377
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
3. thm:general-moving-target-SALD-discrete
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle106.lower_packet.canonical_condDistrib_drift_regularity
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower theorem: the canonical guide and score conditional integrals against condDistrib are each AEStronglyMeasurable and Integrable under the named law hatRhoS, and the dotTk/sigmaCoeff linear combination inherits both properties.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalDriftSource — 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.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity
2. AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable
3. AutoSamplingTheory.condDistribIntegralNamedLawIntegrable
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary
2. sald.general_moving_target_discrete.em_interpolation_fp
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle106.lower_packet.named_barB_canonical_ae_regular
interface : String (explicit)
Text describing intended mathematical interface.
Compiled versioning bridge: if a downstream named barB is hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field, SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq transfers AEStronglyMeasurable and Integrable from the canonical theorem. The equality itself remains the exact source-cited boundary.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalDriftSource — 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.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq
2. SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity
3. appendix.tex:1368-1377
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. ASTIS.SALD.cycle106.remaining_named_barB_version_boundary
2. sald.general_moving_target_discrete.em_interpolation_fp
status : AutoSamplingTheory.ProofStatus (explicit)
Recorded workflow status, default planned.
AutoSamplingTheory.ProofStatus.formalized — stored label only; no proof certification
id : String (explicit)
Stable plan-node identifier.
ASTIS.SALD.cycle106.remaining_named_barB_version_boundary
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact boundary after the compiled canonical and named-version regularity theorems: if the paper route keeps a separately named barB field, prove it is hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field. Do not replace this by another broad hcondC/hcondScore or weak-FP wrapper.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteConditionalDriftSource — 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.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq
2. SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity
3. SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample
4. ASTIS.SALD.cycle103.condC_condDistrib_condExpKernel_sample_version
5. Mathlib.Probability.Kernel.CondDistrib
6. Mathlib.Probability.Kernel.Condexp
7. appendix.tex:1368-1377
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. sald.discrete_forward_kl.em_interpolation_fp
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 cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle106.middle_canonical_condDistrib_drift_source_map"
      interface := "Middle source map: appendix.tex:1368-1377 defines bar b_{k,s} by conditioning the frozen guide and score summands on hat X_s=x.  Cycle 106 chooses the canonical condDistrib representative and targets only its measurable/integrable conditional-integral regularity."
      source := saldGeneralMovingTargetDiscreteConditionalDriftSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle106GeneralMovingTargetDiscreteCanonicalCondDistribDriftMiddleObligation",
        "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity",
        "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable",
        "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle106.lower_packet.canonical_condDistrib_drift_regularity"
      interface := "Compiled lower theorem: the canonical guide and score conditional integrals against condDistrib are each AEStronglyMeasurable and Integrable under the named law hatRhoS, and the dotTk/sigmaCoeff linear combination inherits both properties."
      source := saldGeneralMovingTargetDiscreteConditionalDriftSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity",
        "AutoSamplingTheory.condDistribIntegralNamedLawAEStronglyMeasurable",
        "AutoSamplingTheory.condDistribIntegralNamedLawIntegrable"
      ]
      reusedBy := [
        "ASTIS.SALD.cycle91.remaining_conditional_kernel_boundary",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle106.lower_packet.named_barB_canonical_ae_regular"
      interface := "Compiled versioning bridge: if a downstream named barB is hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field, SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq transfers AEStronglyMeasurable and Integrable from the canonical theorem.  The equality itself remains the exact source-cited boundary."
      source := saldGeneralMovingTargetDiscreteConditionalDriftSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq",
        "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "ASTIS.SALD.cycle106.remaining_named_barB_version_boundary",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle106.remaining_named_barB_version_boundary"
      interface := "Remaining exact boundary after the compiled canonical and named-version regularity theorems: if the paper route keeps a separately named barB field, prove it is hatRhoS-a.e. equal to the canonical condDistrib guide-plus-score field.  Do not replace this by another broad hcondC/hcondScore or weak-FP wrapper."
      source := saldGeneralMovingTargetDiscreteConditionalDriftSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq",
        "SALD.generalMovingTargetDiscreteCondDistribCanonicalDriftRegularity",
        "SALD.generalMovingTargetDiscreteCondDistribNamedFieldAeEqOfSample",
        "ASTIS.SALD.cycle103.condC_condDistrib_condExpKernel_sample_version",
        "Mathlib.Probability.Kernel.CondDistrib",
        "Mathlib.Probability.Kernel.Condexp",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 107: boundary-flux integral via Mathlib box divergence theorem -/

/-- Cycle-107 lower obligation for the Mathlib box divergence theorem
specialization behind the boundary-flux integral representation.

This packet acts on the `hboundaryFluxIntegral` premise consumed by
`generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientTraceBoundaryOfTestTraceZero`.
It keeps the product rule, `hdivergenceTheorem`, weak-test gradient bound, and
zero boundary trace as separate inputs.
-/

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