AutoSamplingTheory.SALD.gronwallExponentRewriteObligation

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 gronwallExponentRewriteObligation : 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.gronwall.exponent_rewrite

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 final exponent algebra in appendix lines 63-69: after multiplying by exp(-int_0^t1 a), rewrite exp(-int_0^t1 a)*exp(int_0^t a) as exp(-int_t^t1 a) inside the b_t integral using interval-integral additivity on 0<=t<=t1, the compiled scalar Real.exp product lemma, and interval-integral congruence.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldGronwallExponentRewriteSource — 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

SALD.saldGronwallEndpointCalculusContract
SALD.saldGronwallExponentRewriteContract
SALD.gronwallNegIntegralRewriteScalar
SALD.gronwallExpProductRewriteScalar
SALD.gronwallIntervalIntegralAdditivityScalar
SALD.gronwallExpProductRewriteIntervalIntegral
SALD.gronwallExpProductRewriteIntegralCongr
SALD.gronwallEndpointIntegralRewrite

note : String (explicit)

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

This is a lower sub-obligation of sald.gronwall.endpoint_calculus. Cycle 17 lower formalized scalar negation/Real.exp algebra; cycle 18 lower added the adjacent-interval bridge from intervalIntegral.integral_add_adjacent_intervals to that scalar helper; cycle 21 lower added the interval congruence wrapper SALD.gronwallExpProductRewriteIntegralCongr; cycle 36 middle added SALD.gronwallEndpointIntegralRewrite to package the final integral term. Proving the adjacent interval-integrability from source regularity remains an obligation. No sign assumptions on a or b were added.

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 gronwallExponentRewriteObligation : ProofObligation where
  id := "sald.gronwall.exponent_rewrite"
  statement := "Formalize the final exponent algebra in appendix lines 63-69: after multiplying by exp(-int_0^t1 a), rewrite exp(-int_0^t1 a)*exp(int_0^t a) as exp(-int_t^t1 a) inside the b_t integral using interval-integral additivity on 0<=t<=t1, the compiled scalar Real.exp product lemma, and interval-integral congruence."
  source := saldGronwallExponentRewriteSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.saldGronwallEndpointCalculusContract",
    "SALD.saldGronwallExponentRewriteContract",
    "SALD.gronwallNegIntegralRewriteScalar",
    "SALD.gronwallExpProductRewriteScalar",
    "SALD.gronwallIntervalIntegralAdditivityScalar",
    "SALD.gronwallExpProductRewriteIntervalIntegral",
    "SALD.gronwallExpProductRewriteIntegralCongr",
    "SALD.gronwallEndpointIntegralRewrite"
  ]
  note := "This is a lower sub-obligation of sald.gronwall.endpoint_calculus. Cycle 17 lower formalized scalar negation/Real.exp algebra; cycle 18 lower added the adjacent-interval bridge from intervalIntegral.integral_add_adjacent_intervals to that scalar helper; cycle 21 lower added the interval congruence wrapper SALD.gronwallExpProductRewriteIntegralCongr; cycle 36 middle added SALD.gronwallEndpointIntegralRewrite to package the final integral term. Proving the adjacent interval-integrability from source regularity remains an obligation. No sign assumptions on a or b were added."

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