AutoSamplingTheory.SALD.forwardKlGronwallInstantiationContract

Data definition / provenance and workflow record

Meaning and type

The result has data type AutoSamplingTheory.SALD.ForwardKlGronwallInstantiationContract. A value of this type stores descriptions; it is not a proof of the statements in those descriptions.

def forwardKlGronwallInstantiationContract : ForwardKlGronwallInstantiationContract

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.

sourceBlock : AutoSamplingTheory.SourceAnchor (explicit)

Nested provenance data.

AutoSamplingTheory.SALD.saldForwardKlGronwallSource — audited data reference, not expanded and not a compiled dependency edge

quantityK : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

K(t)=KL(rho_{s(t)} || pi_t).

gronwallA : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

a(t)=dot{s}(t)*C_LSI(t) - (1/2)*dot{s}(t)^(-1)*alpha^(-1).

gronwallB : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t).

preSplitBound : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

K(T) <= exp(-int_0^T a(t) dt)*K(0) + int_0^T exp(-int_t^T a(u) du)*b(t) dt.

splitBound : String (explicit)

Descriptive text even when field names say formula, theorem, inequality, derivative, source gap, or proof.

Separate the exponent into the negative LSI term and the positive alpha term, then drop the nonpositive LSI part in the residual exponential to match main_body.tex lines 243-246.

requiredRegularity : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. continuity or interval-integrability of a(t) and b(t) on [0,T]
2. differentiability of K(t) on the interval in the sense required by the Gronwall backend
3. positivity of dot{s}(t) so dot{s}(t)^(-1) is defined

sourceGaps : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. the source theorem does not separately state the Gronwall regularity hypotheses for K, a, and b; the proof uses them implicitly
2. the residual exponential simplification uses nonnegativity of dot{s}(u)*C_LSI(u) to discard its negative contribution

dependencies : List String (explicit)

Ordered descriptive/naming checklist, not logical conjunction or compiler dependency list.

Ordered data items

1. lem:gronwall
2. sald.forward_kl.kl_derivative
3. sald.forward_kl.dv_energy_bound
4. sald.forward_kl.gronwall_side_conditions

status : AutoSamplingTheory.ProofStatus (explicit)

Stored workflow tag; honor the exact default but do not infer mathematical certification.

AutoSamplingTheory.ProofStatus.obligation — stored label only; no proof certification

def forwardKlGronwallInstantiationContract : ForwardKlGronwallInstantiationContract where
  sourceBlock := saldForwardKlGronwallSource
  quantityK := "K(t)=KL(rho_{s(t)} || pi_t)."
  gronwallA := "a(t)=dot{s}(t)*C_LSI(t) - (1/2)*dot{s}(t)^(-1)*alpha^(-1)."
  gronwallB := "b(t)=(1/2)*dot{s}(t)^(-1)*E_alpha(pi_t,v_t)."
  preSplitBound := "K(T) <= exp(-int_0^T a(t) dt)*K(0) + int_0^T exp(-int_t^T a(u) du)*b(t) dt."
  splitBound := "Separate the exponent into the negative LSI term and the positive alpha term, then drop the nonpositive LSI part in the residual exponential to match main_body.tex lines 243-246."
  requiredRegularity := [
    "continuity or interval-integrability of a(t) and b(t) on [0,T]",
    "differentiability of K(t) on the interval in the sense required by the Gronwall backend",
    "positivity of dot{s}(t) so dot{s}(t)^(-1) is defined"
  ]
  sourceGaps := [
    "the source theorem does not separately state the Gronwall regularity hypotheses for K, a, and b; the proof uses them implicitly",
    "the residual exponential simplification uses nonnegativity of dot{s}(u)*C_LSI(u) to discard its negative contribution"
  ]
  dependencies := [
    "lem:gronwall",
    "sald.forward_kl.kl_derivative",
    "sald.forward_kl.dv_energy_bound",
    "sald.forward_kl.gronwall_side_conditions"
  ]
  status := ProofStatus.obligation