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Formula-witness equivalence

Lax461425.FormulaWitnessEquivalence · concepts/Lax461425/FormulaWitnessEquivalence.lean · lax-461425

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    Natural Language Statement

    Theorem

    Over a field of characteristic zero, symmetric VNP is unchanged when the witnessing circuits are required to be formulas.

    Concept map
    11 concepts
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax461425.BooleanSumSymmetry
    2import Lax461425.SymmetricFormulaWitness
    3
    4/-!
    5---
    6title: Formula-witness equivalence
    7type: theorem
    8---
    9Over a field of characteristic zero, symmetric VNP is unchanged when the witnessing
    10circuits are required to be formulas.
    11-/
    12
    13namespace Lax461425.FormulaWitnessEquivalence
    14
    15open SymmetricFamilies SymmetricClasses SymmetricVNP
    16
    17/-- Formula witnesses and circuit witnesses define the same Boolean-sum class. -/
    18axiom formula_witnesses_iff {K : Type*} [Field K] [CharZero K] (f : MatrixFamily K) :
    19 InSymVNP f ↔ InSymVNPFormula f
    20
    21end Lax461425.FormulaWitnessEquivalence
    22
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