AutoSamplingTheory.SALD.discreteForwardKlLinearSlowdownObligation

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 discreteForwardKlLinearSlowdownObligation : 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.linear_slowdown_specialization

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 the algebra from the appendix general-schedule Gronwall bound to main_body.tex lines 309-323 under t(s)=s/r: dot{s}(t)=r, dot{s}(t)^(-1)=1/r, barGamma=int_0^T Gamma(t)dt, barDelta_{alpha'}=int_0^T Delta(t)dt, and the residual exponential is bounded by exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha') after dropping the nonpositive LSI contribution.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlDiscreteSource — 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. sald.discrete_forward_kl.gronwall_accumulation
2. def:alpha-complexity

note : String (explicit)

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

The paper states the linear slowdown before the theorem, but the appendix proof ends at the general-schedule Gronwall display; this obligation records the exact specialization without changing constants.

def discreteForwardKlLinearSlowdownObligation : ProofObligation where
  id := "sald.discrete_forward_kl.linear_slowdown_specialization"
  statement := "Formalize the algebra from the appendix general-schedule Gronwall bound to main_body.tex lines 309-323 under t(s)=s/r: dot{s}(t)=r, dot{s}(t)^(-1)=1/r, barGamma=int_0^T Gamma(t)dt, barDelta_{alpha'}=int_0^T Delta(t)dt, and the residual exponential is bounded by exp(T/(r*alpha)+2*r*eta^2*barGamma/alpha') after dropping the nonpositive LSI contribution."
  source := saldForwardKlDiscreteSource
  status := ProofStatus.obligation
  dependsOn := ["sald.discrete_forward_kl.gronwall_accumulation", "def:alpha-complexity"]
  note := "The paper states the linear slowdown before the theorem, but the appendix proof ends at the general-schedule Gronwall display; this obligation records the exact specialization without changing constants."