AutoSamplingTheory.SALD.cycle34ForwardKlTargetYoungLowerObligation

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 cycle34ForwardKlTargetYoungLowerObligation : 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.cycle34_target_young_lower

statement : String (explicit)

Desired mathematical or workflow content as String; it is not a proposition in Prop and the def does not prove it.

Maintain the lower scalar target-side Young slice for appendix.tex lines 199-208: once target transport, integration by parts, Cauchy--Schwarz, and L2/FI identifications supply targetTerm <= sqrt(FI)*sqrt(||tilde v_s||^2) with nonnegative FI and velocity square, SALD.forwardKlTargetTransportYoungBoundScalar proves targetTerm <= (1/2)*FI+(1/2)*||tilde v_s||^2, and SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar feeds that bound into the derivative bookkeeping.

source : AutoSamplingTheory.SourceAnchor (explicit)

SourceAnchor supporting the intended requirement.

AutoSamplingTheory.SALD.saldForwardKlDerivativeSource — 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.cycle34ForwardKlDerivativeUpperPacket
SALD.cycle34ForwardKlDerivativeMiddleContract
SALD.forwardKlTargetTransportYoungBoundScalar
SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar
SALD.forwardKlPostYoungDerivativeBoundScalar
SALD.forwardKlFirstTermFisherSubstitutionScalar
SALD.forwardKlDerivativeSideConditionContract
SALD.forwardKlDensityBoundaryObligation
sald.forward_kl.cycle34_derivative_scalar
sald.forward_kl.density_boundary_regular
sald.forward_kl.kl_derivative

note : String (explicit)

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

Lower cycle 34 formalizes only the Real Young inequality and its composition with existing scalar derivative bookkeeping. The target-side transport identity, integration by parts, Cauchy--Schwarz estimate, Fisher/L2 square identifications, density regularity, LSI, schedule time change, DV, and Gronwall remain explicit obligations.

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 cycle34ForwardKlTargetYoungLowerObligation : ProofObligation where
  id := "sald.forward_kl.cycle34_target_young_lower"
  statement := "Maintain the lower scalar target-side Young slice for appendix.tex lines 199-208: once target transport, integration by parts, Cauchy--Schwarz, and L2/FI identifications supply targetTerm <= sqrt(FI)*sqrt(||tilde v_s||^2) with nonnegative FI and velocity square, SALD.forwardKlTargetTransportYoungBoundScalar proves targetTerm <= (1/2)*FI+(1/2)*||tilde v_s||^2, and SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar feeds that bound into the derivative bookkeeping."
  source := saldForwardKlDerivativeSource
  status := ProofStatus.obligation
  dependsOn := [
    "SALD.cycle34ForwardKlDerivativeUpperPacket",
    "SALD.cycle34ForwardKlDerivativeMiddleContract",
    "SALD.forwardKlTargetTransportYoungBoundScalar",
    "SALD.forwardKlPostYoungDerivativeBoundOfCauchyScalar",
    "SALD.forwardKlPostYoungDerivativeBoundScalar",
    "SALD.forwardKlFirstTermFisherSubstitutionScalar",
    "SALD.forwardKlDerivativeSideConditionContract",
    "SALD.forwardKlDensityBoundaryObligation",
    "sald.forward_kl.cycle34_derivative_scalar",
    "sald.forward_kl.density_boundary_regular",
    "sald.forward_kl.kl_derivative"
  ]
  note := "Lower cycle 34 formalizes only the Real Young inequality and its composition with existing scalar derivative bookkeeping. The target-side transport identity, integration by parts, Cauchy--Schwarz estimate, Fisher/L2 square identifications, density regularity, LSI, schedule time change, DV, and Gronwall remain explicit obligations."

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