AutoSamplingTheory.SALD.generalMovingTargetGronwallInstantiationContract

Data definition / provenance and workflow record

Meaning and type

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

def generalMovingTargetGronwallInstantiationContract :
    GeneralMovingTargetGronwallInstantiationContract

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.saldGeneralMovingTargetDvGronwallSource — 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)=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-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)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_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.

theoremBound : 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 positive sigma-weighted alpha term, then drop the nonpositive LSI part in the residual exponential to match eq:general_moving_target_KL_bound.

pureContractionSpecialization : String (explicit)

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

If c_t=v_t, then m_t=0 and E_alpha(pi_t,m_t)=alpha^(-1)*log E_{pi_t}[1]=0, leaving only the initial-error contraction factor.

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), b(t), sigma_t, dot{s}(t), C_LSI(t), and E_alpha(pi_t,m_t)
2. differentiability of K(t) in the sense required by lem:gronwall
3. positivity of sigma_t and dot{s}(t)

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 Gronwall regularity for the sigma-weighted coefficients
2. the proof drops the LSI contribution in the residual exponential using nonnegativity of C_LSI and positivity of sigma_t^2*dot{s}(t)
3. the pure-contraction clause uses the constant-zero residual energy identity, which still requires the alpha-complexity definition

dependencies : List String (explicit)

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

Ordered data items

1. lem:gronwall
2. sald.general_moving_target.kl_derivative
3. sald.general_moving_target.dv_m_energy
4. def:alpha-complexity

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 generalMovingTargetGronwallInstantiationContract :
    GeneralMovingTargetGronwallInstantiationContract where
  sourceBlock := saldGeneralMovingTargetDvGronwallSource
  quantityK := "K(t)=KL(rho_{s(t)}||pi_t)."
  gronwallA := "a(t)=(sigma_t^2/2)*dot{s}(t)*C_LSI(t)-sigma_t^(-2)*dot{s}(t)^(-1)*alpha^(-1)."
  gronwallB := "b(t)=sigma_t^(-2)*dot{s}(t)^(-1)*E_alpha(pi_t,m_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."
  theoremBound := "Separate the exponent into the negative LSI term and positive sigma-weighted alpha term, then drop the nonpositive LSI part in the residual exponential to match eq:general_moving_target_KL_bound."
  pureContractionSpecialization := "If c_t=v_t, then m_t=0 and E_alpha(pi_t,m_t)=alpha^(-1)*log E_{pi_t}[1]=0, leaving only the initial-error contraction factor."
  requiredRegularity := [
    "continuity or interval-integrability of a(t), b(t), sigma_t, dot{s}(t), C_LSI(t), and E_alpha(pi_t,m_t)",
    "differentiability of K(t) in the sense required by lem:gronwall",
    "positivity of sigma_t and dot{s}(t)"
  ]
  sourceGaps := [
    "the source theorem does not separately state Gronwall regularity for the sigma-weighted coefficients",
    "the proof drops the LSI contribution in the residual exponential using nonnegativity of C_LSI and positivity of sigma_t^2*dot{s}(t)",
    "the pure-contraction clause uses the constant-zero residual energy identity, which still requires the alpha-complexity definition"
  ]
  dependencies := [
    "lem:gronwall",
    "sald.general_moving_target.kl_derivative",
    "sald.general_moving_target.dv_m_energy",
    "def:alpha-complexity"
  ]
  status := ProofStatus.obligation