AutoSamplingTheory.SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeDag

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 cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeDag :
    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.cycle112.middle_named_barB_condExp_representative
interface : String (explicit)
Text describing intended mathematical interface.
Middle source map: appendix.tex:1368-1377 names barB as the conditional expectation of the frozen guide-plus-score drift given hat X_s=x. After cycle 110 removed equality-set measurability, the remaining removable primitive is hbarBCondExp.
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.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeMiddleObligation
2. ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas
3. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef
4. 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:forward-KL-discrete
4. 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.cycle112.lower_packet.named_barB_condExp_uniqueness
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower theorem: SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq derives hbarBCondExp from Mathlib conditional-expectation uniqueness under candidate measurability, candidate integrability, guide/score integrability, and equality of Bochner set integrals on all hatXAtS-measurable finite-measure sets.
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.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq
2. MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq
3. MeasureTheory.Integrable.comp_aemeasurable
4. appendix.tex:1368-1377
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef
2. ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization
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.cycle112.lower_packet.named_barB_source_bridge_without_hbarBCondExp
interface : String (explicit)
Text describing intended mathematical interface.
Compiled downstream handoff: SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef feeds the derived hbarBCondExp into the existing product-condExp source bridge, replacing the primitive hbarBCondExp input with the set-integral characterization 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.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef
2. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef
3. SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq
4. ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq
2. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef
3. sald.general_moving_target_discrete.em_interpolation_fp
4. sald.discrete_forward_kl.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.cycle112.lower_packet.named_barB_state_event_set_integral
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower bridge: equality of Bochner set integrals on source-facing state events {omega | hatXAtS omega in t} implies the all-comap set-integral hypothesis used by conditional-expectation uniqueness, via MeasurableSpace.measurableSet_comap.
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.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents
2. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef
3. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef
4. MeasurableSpace.measurableSet_comap
5. appendix.tex:1368-1377
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization
2. sald.general_moving_target_discrete.em_interpolation_fp
3. sald.discrete_forward_kl.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.cycle112.remaining_named_barB_set_integral_characterization
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact theorem after the cycle-112 lower narrowing: prove the paper-selected representative barB(hatXAtS omega) is a.e. measurable for mState.comap hatXAtS, integrable under P, and has the same Bochner set integrals as the frozen guide-plus-score drift over source-facing events {omega | hatXAtS omega in t} for measurable state sets t.
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. ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas
2. SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq
3. SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents
4. SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef
5. MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq
6. 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 cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle112.middle_named_barB_condExp_representative"
      interface := "Middle source map: appendix.tex:1368-1377 names barB as the conditional expectation of the frozen guide-plus-score drift given hat X_s=x.  After cycle 110 removed equality-set measurability, the remaining removable primitive is hbarBCondExp."
      source := saldGeneralMovingTargetDiscreteConditionalDriftSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle112GeneralMovingTargetDiscreteNamedBarBCondExpRepresentativeMiddleObligation",
        "ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas",
        "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle112.lower_packet.named_barB_condExp_uniqueness"
      interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq derives hbarBCondExp from Mathlib conditional-expectation uniqueness under candidate measurability, candidate integrability, guide/score integrability, and equality of Bochner set integrals on all hatXAtS-measurable finite-measure sets."
      source := saldGeneralMovingTargetDiscreteConditionalDriftSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq",
        "MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq",
        "MeasureTheory.Integrable.comp_aemeasurable",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef",
        "ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle112.lower_packet.named_barB_source_bridge_without_hbarBCondExp"
      interface := "Compiled downstream handoff: SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef feeds the derived hbarBCondExp into the existing product-condExp source bridge, replacing the primitive hbarBCondExp input with the set-integral characterization boundary."
      source := saldGeneralMovingTargetDiscreteConditionalDriftSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef",
        "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSourceDef",
        "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq",
        "ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteCondDistribNamedDriftRegularityOfCanonicalAeEq",
        "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef",
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle112.lower_packet.named_barB_state_event_set_integral"
      interface := "Compiled lower bridge: equality of Bochner set integrals on source-facing state events {omega | hatXAtS omega in t} implies the all-comap set-integral hypothesis used by conditional-expectation uniqueness, via MeasurableSpace.measurableSet_comap."
      source := saldGeneralMovingTargetDiscreteConditionalDriftSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents",
        "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef",
        "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpSetIntegralDef",
        "MeasurableSpace.measurableSet_comap",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization",
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle112.remaining_named_barB_set_integral_characterization"
      interface := "Remaining exact theorem after the cycle-112 lower narrowing: prove the paper-selected representative barB(hatXAtS omega) is a.e. measurable for mState.comap hatXAtS, integrable under P, and has the same Bochner set integrals as the frozen guide-plus-score drift over source-facing events {omega | hatXAtS omega in t} for measurable state sets t."
      source := saldGeneralMovingTargetDiscreteConditionalDriftSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "ASTIS.SALD.cycle110.remaining_condExp_source_representative_after_eq_meas",
        "SALD.generalMovingTargetDiscreteNamedBarBCondExpOfSetIntegralEq",
        "SALD.generalMovingTargetDiscreteNamedBarBSetIntegralOfStateEvents",
        "SALD.generalMovingTargetDiscreteCondDistribNamedBarBAeEqOfCondExpStateSetIntegralDef",
        "MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "sald.discrete_forward_kl.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 113: named `barB` state-field regularity pullback -/

/-- Cycle-113 lower obligation for pulling the selected named `barB`
regularity back from the state marginal.

The discrete theorem pressure test reaches the cycle-112 named `barB`
representative boundary.  This lower packet discharges the sample-space
candidate-regularity inputs of that boundary from source-facing state-field
regularity, leaving the state-event set-integral characterization as the next
non-wrapper blocker.
-/

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