AutoSamplingTheory.SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag

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 cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag :
    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.cycle120.middle_em_path_derivative_domination
interface : String (explicit)
Text describing intended mathematical interface.
Middle packet: split the cycle-119 remaining theorem so lower targets only the EM path derivative and local domination hypotheses behind Mathlib's dominated derivative-under-integral theorem.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — 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.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerLowerObligation
2. SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated
3. AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated
4. Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
5. appendix.tex:1379-1387
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.cycle120.lower_ready.em_sample_path_derivative_domination
interface : String (explicit)
Text describing intended mathematical interface.
Lower-ready exact theorem after the interval-neighborhood lower packet: from the frozen EM interpolation and admissible-test regularity, prove hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, and hpathDeriv for each admissible weak test. Do not include hderivValue or the drift/diffusion source-action equalities in this theorem.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — 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.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationLowerObligation
2. eq:general_moving_target_SALD_frozen_interp
3. Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le
4. appendix.tex:1379-1387
5. appendix.tex:1368-1377
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated
2. sald.general_moving_target_discrete.em_interpolation_fp
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.cycle120.lower_packet.em_interval_neighborhood
interface : String (explicit)
Text describing intended mathematical interface.
Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated specializes the canonical-barB dominated weak-FP route to the source EM interval Set.Ioo sLeft sRight and discharges hsampleNeighborhood from hs0Interval; hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, hpathDeriv, hderivValue, and source-action identities remain explicit.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — 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.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated
2. SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated
3. Mathlib.Topology.Instances.Real
4. appendix.tex:1379-1387
5. eq:general_moving_target_SALD_frozen_interp
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. sald.general_moving_target_discrete.em_interpolation_fp
2. thm:forward-KL-discrete
3. thm:general-moving-target-SALD-discrete
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.cycle120.remaining_source_actions_after_path_derivative
interface : String (explicit)
Text describing intended mathematical interface.
Remaining exact theorem after the path-derivative/domination packet: identify the sample-derivative integral as driftAction + diffusionAction, prove the canonical barB weak-action and pairing regularity inputs, prove the no-boundary divergence identity, prove the sigmaCoeff diffusion source action, and supply law-derivative/partialS uniqueness only if the normalized source-sign consumer is used.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — 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.cycle120.lower_ready.em_sample_path_derivative_domination
2. SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface
3. SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound
4. ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient
5. appendix.tex:1379-1387
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. 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.cycle120.reviewer_em_path_derivative_domination_check
interface : String (explicit)
Text describing intended mathematical interface.
Reviewer check: accept only if the packet classification remains narrows-source-cited-boundary, the target is the EM conditional-drift/weak-FP backend at appendix.tex:1368-1387, no hbarBCondExp wrapper or non-EM fallback is introduced, and python3 tools/astis.py check passes.
source : AutoSamplingTheory.SourceAnchor (explicit)
Source pointer.
AutoSamplingTheory.SALD.saldGeneralMovingTargetDiscreteWeakFpSource — 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.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationMiddleObligation
2. SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationLowerObligation
3. SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag
4. appendix.tex:1368-1387
reusedBy : List String (explicit)
Declared consumers as strings, default empty.
Ordered data items
1. cycle 120 reviewer
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 cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag :
    List ProofDagBlock :=
  [
    {
      id := "ASTIS.SALD.cycle120.middle_em_path_derivative_domination"
      interface := "Middle packet: split the cycle-119 remaining theorem so lower targets only the EM path derivative and local domination hypotheses behind Mathlib's dominated derivative-under-integral theorem."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle119GeneralMovingTargetDiscreteCanonicalBarBWeakFpConsumerLowerObligation",
        "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated",
        "AutoSamplingTheory.lawIntegralHasDerivAtOfMeasureMapEqAndDominated",
        "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le",
        "appendix.tex:1379-1387"
      ]
      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.cycle120.lower_ready.em_sample_path_derivative_domination"
      interface := "Lower-ready exact theorem after the interval-neighborhood lower packet: from the frozen EM interpolation and admissible-test regularity, prove hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, and hpathDeriv for each admissible weak test. Do not include hderivValue or the drift/diffusion source-action equalities in this theorem."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationLowerObligation",
        "eq:general_moving_target_SALD_frozen_interp",
        "Mathlib.Analysis.Calculus.ParametricIntegral.hasDerivAt_integral_of_dominated_loc_of_deriv_le",
        "appendix.tex:1379-1387",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated",
        "sald.general_moving_target_discrete.em_interpolation_fp"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle120.lower_packet.em_interval_neighborhood"
      interface := "Compiled lower theorem: SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated specializes the canonical-barB dominated weak-FP route to the source EM interval Set.Ioo sLeft sRight and discharges hsampleNeighborhood from hs0Interval; hsampleMeas, hsampleInt, hsampleDerivMeas, hsampleDerivBound, hboundInt, hpathDeriv, hderivValue, and source-action identities remain explicit."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfEmIntervalDominated",
        "SALD.generalMovingTargetDiscreteCanonicalBarBWeakConditionalFpNamedLawDerivativeOfDominated",
        "Mathlib.Topology.Instances.Real",
        "appendix.tex:1379-1387",
        "eq:general_moving_target_SALD_frozen_interp"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "thm:forward-KL-discrete",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.formalized
    },
    {
      id := "ASTIS.SALD.cycle120.remaining_source_actions_after_path_derivative"
      interface := "Remaining exact theorem after the path-derivative/domination packet: identify the sample-derivative integral as driftAction + diffusionAction, prove the canonical barB weak-action and pairing regularity inputs, prove the no-boundary divergence identity, prove the sigmaCoeff diffusion source action, and supply law-derivative/partialS uniqueness only if the normalized source-sign consumer is used."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "ASTIS.SALD.cycle120.lower_ready.em_sample_path_derivative_domination",
        "SALD.generalMovingTargetDiscreteCanonicalBarBEmStateEventInterface",
        "SALD.generalMovingTargetDiscreteWeakConditionalFpDriftSourceOfBarBInnerGradientBound",
        "ASTIS.SALD.cycle100.remaining_no_boundary_after_inner_gradient",
        "appendix.tex:1379-1387",
        "appendix.tex:1368-1377"
      ]
      reusedBy := [
        "sald.general_moving_target_discrete.em_interpolation_fp",
        "thm:general-moving-target-SALD-discrete"
      ]
      status := ProofStatus.obligation
    },
    {
      id := "ASTIS.SALD.cycle120.reviewer_em_path_derivative_domination_check"
      interface := "Reviewer check: accept only if the packet classification remains narrows-source-cited-boundary, the target is the EM conditional-drift/weak-FP backend at appendix.tex:1368-1387, no hbarBCondExp wrapper or non-EM fallback is introduced, and python3 tools/astis.py check passes."
      source := saldGeneralMovingTargetDiscreteWeakFpSource
      targetLean := "AutoSamplingTheory/SALD.lean"
      dependsOn := [
        "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationMiddleObligation",
        "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationLowerObligation",
        "SALD.cycle120GeneralMovingTargetDiscreteEmPathDerivativeDominationDag",
        "appendix.tex:1368-1387"
      ]
      reusedBy := ["cycle 120 reviewer"]
      status := ProofStatus.obligation
    }
  ]

/-! ### Cycle 121: EM sample measurability from named-law test measurability -/

/-- Cycle-121 lower obligation for discharging the sample measurability input
from the EM interval dominated packet.

This packet stays inside the cycle-120 EM path-derivative/domination boundary.
It removes only the supplied `hsampleMeas` hypothesis by using the named-law
identity `hatRhoS s = Measure.map (hatX s) P`, law-space test measurability,
and sample-path a.e. measurability.  It leaves the source-specific EM
integrability, derivative, domination, derivative-value, and source-action
facts explicit.
-/

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