AutoSamplingTheory.SALD.discreteForwardKlDvVelocityObligation

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 discreteForwardKlDvVelocityObligation : 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.dv_velocity_bound

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 appendix lines 493-523: apply Donsker--Varadhan with Z=alpha*||v_{t(s)}||^2 to bound ||tilde v_s||_{L2(hat rho_s)}^2 by dot{t}(s)^2*(alpha^(-1)*KL(hat rho_s||tilde pi_s)+E_alpha(pi_{t(s)},v_{t(s)})).

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlDiscreteDvVelocitySource — 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.forward_kl.dv_energy_bound
sald.discrete_forward_kl.dv_finite_log_mgf_witness
probability.dv_variational_formula
def:alpha-complexity

note : String (explicit)

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

The theorem reuses the continuous alpha-complexity DV pattern but the measure nu is the discrete EM interpolation law hat rho_s; the finite-log-mgf and common-space witness is tracked separately.

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 discreteForwardKlDvVelocityObligation : ProofObligation where
  id := "sald.discrete_forward_kl.dv_velocity_bound"
  statement := "Formalize appendix lines 493-523: apply Donsker--Varadhan with Z=alpha*||v_{t(s)}||^2 to bound ||tilde v_s||_{L2(hat rho_s)}^2 by dot{t}(s)^2*(alpha^(-1)*KL(hat rho_s||tilde pi_s)+E_alpha(pi_{t(s)},v_{t(s)}))."
  source := saldForwardKlDiscreteDvVelocitySource
  status := ProofStatus.obligation
  dependsOn := ["sald.forward_kl.dv_energy_bound", "sald.discrete_forward_kl.dv_finite_log_mgf_witness", "probability.dv_variational_formula", "def:alpha-complexity"]
  note := "The theorem reuses the continuous alpha-complexity DV pattern but the measure nu is the discrete EM interpolation law hat rho_s; the finite-log-mgf and common-space witness is tracked separately."

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