AutoSamplingTheory.SALD.guidedResidualNormalizerObligation

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 guidedResidualNormalizerObligation : 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.guided_path_residual.normalizer_derivative

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Formalize appendix lines 630-656: for Z_t=int p_t exp(-f_t), use partial_t p_t=-div(p_t*u_t), differentiation under the integral, and integration by parts to prove dot Z_t/Z_t=-E_{pi_t}[partial_t f_t+nabla f_t^T u_t].

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGuidedResidualProofSource — 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. TransportVelocityContract
2. GuidedTiltContract

note : String (explicit)

Recorded evidence/caveats; may distinguish a compiled scalar helper from still-open source analysis.

The paper performs this calculation directly; Lean must expose positive finite Z_t, dominated convergence, and boundary decay as obligations.

def guidedResidualNormalizerObligation : ProofObligation where
  id := "sald.guided_path_residual.normalizer_derivative"
  statement := "Formalize appendix lines 630-656: for Z_t=int p_t exp(-f_t), use partial_t p_t=-div(p_t*u_t), differentiation under the integral, and integration by parts to prove dot Z_t/Z_t=-E_{pi_t}[partial_t f_t+nabla f_t^T u_t]."
  source := saldGuidedResidualProofSource
  status := ProofStatus.obligation
  dependsOn := ["TransportVelocityContract", "GuidedTiltContract"]
  note := "The paper performs this calculation directly; Lean must expose positive finite Z_t, dominated convergence, and boundary decay as obligations."