AutoSamplingTheory.SALD.forwardKlGronwallSideConditionObligation

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 forwardKlGronwallSideConditionObligation : 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.forward_kl.gronwall_side_conditions

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 endpoint and exponent side conditions in appendix lines 244-252 and main_body.tex lines 243-246: identify K(T)=KL(rho_S||pi_T) and K(0)=KL(rho_0||pi_0), provide the Gronwall coefficient regularity for a(t), b(t), split exp(-int a), and justify dropping the nonpositive LSI contribution from the residual exponent.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlGronwallSource — 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.cycle22ForwardKlUpperPacket
2. SALD.cycle22ForwardKlMiddleContract
3. sald.forward_kl.moving_target_dependency_chain
4. sald.forward_kl.endpoint_schedule_identities
5. sald.forward_kl.schedule_time_change
6. sald.forward_kl.kl_derivative
7. sald.forward_kl.dv_energy_bound
8. sald.gronwall.integrating_factor
9. SALD.gronwallExpProductRewriteIntervalIntegral
10. SALD.gronwallExpProductRewriteIntegralCongr
11. SALD.forwardKlGronwallCoeffIntervalIntegrable
12. SALD.forwardKlGronwallCoeffAdjacentIntervalIntegrable
13. SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces
14. SALD.forwardKlGronwallCoeffIntegralSub
15. SALD.forwardKlGronwallInitialExponentSplitScalar
16. SALD.forwardKlGronwallInitialExponentSplitOfPieces
17. SALD.forwardKlGronwallResidualExponentDropScalar
18. SALD.forwardKlGronwallResidualExponentDropIntegral

note : String (explicit)

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

forwardKlGronwallSideConditionContract records the endpoint schedule identities, coefficient regularity, exponent split, and sign facts as obligations rather than extra assumptions on thm:forward-KL. The adjacent-interval Gronwall exponent bridge, outer-integral congruence, forward-KL coefficient-piece assembly, initial exponent split, and residual exponent drop now build as local algebra under explicit interval-integrability/nonnegativity hypotheses. Theorem-specific regularity of the LSI/alpha pieces, endpoint rewrites, b(t) regularity/nonnegativity, nonnegativity of the LSI integral, and full Gronwall application remain obligations.

def forwardKlGronwallSideConditionObligation : ProofObligation where
  id := "sald.forward_kl.gronwall_side_conditions"
  statement := "Formalize the endpoint and exponent side conditions in appendix lines 244-252 and main_body.tex lines 243-246: identify K(T)=KL(rho_S||pi_T) and K(0)=KL(rho_0||pi_0), provide the Gronwall coefficient regularity for a(t), b(t), split exp(-int a), and justify dropping the nonpositive LSI contribution from the residual exponent."
  source := saldForwardKlGronwallSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle22ForwardKlUpperPacket",
    "SALD.cycle22ForwardKlMiddleContract",
    "sald.forward_kl.moving_target_dependency_chain",
    "sald.forward_kl.endpoint_schedule_identities",
    "sald.forward_kl.schedule_time_change",
    "sald.forward_kl.kl_derivative",
    "sald.forward_kl.dv_energy_bound",
    "sald.gronwall.integrating_factor",
    "SALD.gronwallExpProductRewriteIntervalIntegral",
    "SALD.gronwallExpProductRewriteIntegralCongr",
    "SALD.forwardKlGronwallCoeffIntervalIntegrable",
    "SALD.forwardKlGronwallCoeffAdjacentIntervalIntegrable",
    "SALD.forwardKlGronwallExpProductRewriteIntegralCongrOfPieces",
    "SALD.forwardKlGronwallCoeffIntegralSub",
    "SALD.forwardKlGronwallInitialExponentSplitScalar",
    "SALD.forwardKlGronwallInitialExponentSplitOfPieces",
    "SALD.forwardKlGronwallResidualExponentDropScalar",
    "SALD.forwardKlGronwallResidualExponentDropIntegral"
  ]
  note := "forwardKlGronwallSideConditionContract records the endpoint schedule identities, coefficient regularity, exponent split, and sign facts as obligations rather than extra assumptions on thm:forward-KL. The adjacent-interval Gronwall exponent bridge, outer-integral congruence, forward-KL coefficient-piece assembly, initial exponent split, and residual exponent drop now build as local algebra under explicit interval-integrability/nonnegativity hypotheses. Theorem-specific regularity of the LSI/alpha pieces, endpoint rewrites, b(t) regularity/nonnegativity, nonnegativity of the LSI integral, and full Gronwall application remain obligations."