AutoSamplingTheory.SALD.discreteForwardKlFrozenDeltaObligation

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.

Lean statement of this data definition

The part after the colon is the output data type. This declaration takes no mathematical proof inputs.

def discreteForwardKlFrozenDeltaObligation : 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.discrete_forward_kl.frozen_delta_cross_lip

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 Lemma lem:frozen_delta_cross_lip_sald: under the source score space/time Lipschitz assumptions, exponential-complexity assumptions, and eta^2*L_{pi,space}^2 < 1/8, bound the frozen score-defect cross term by (1/4)*FI + 2*eta^2*alpha'^(-1)*Gamma*K + 2*eta*Delta.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldFrozenDeltaCrossLipSaldSource — 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

lem:frozen_delta_cross_lip
lem:dv_variation
eq:lip_SALD_1
eq:lip_SALD_2
eq:frozen_interp_terminal_disc_prop_additive_final

note : String (explicit)

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

The SALD-specific proof is omitted in the paper and must be obtained by specializing the later general lemma with c identically zero and sigma_eta(t)=sqrt(2); do not treat it as formalized until that specialization builds.

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 discreteForwardKlFrozenDeltaObligation : ProofObligation where
  id := "sald.discrete_forward_kl.frozen_delta_cross_lip"
  statement := "Formalize Lemma lem:frozen_delta_cross_lip_sald: under the source score space/time Lipschitz assumptions, exponential-complexity assumptions, and eta^2*L_{pi,space}^2 < 1/8, bound the frozen score-defect cross term by (1/4)*FI + 2*eta^2*alpha'^(-1)*Gamma*K + 2*eta*Delta."
  source := saldFrozenDeltaCrossLipSaldSource
  status := ProofStatus.obligation
  dependsOn := ["lem:frozen_delta_cross_lip", "lem:dv_variation", "eq:lip_SALD_1", "eq:lip_SALD_2", "eq:frozen_interp_terminal_disc_prop_additive_final"]
  note := "The SALD-specific proof is omitted in the paper and must be obtained by specializing the later general lemma with c identically zero and sigma_eta(t)=sqrt(2); do not treat it as formalized until that specialization builds."

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