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Symmetric skew circuits from symmetric ABPs

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

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    Natural Language Statement

    Theorem

    Over every commutative semiring, a family of polynomials computed by matrix-symmetric ABPs with polynomial vertex orbits is computed by matrix-symmetric skew circuits with polynomial gate orbits, that is, symVBP⊆symVSsymVBP ⊆ symVS.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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    1import Lax461425.SymmetricABPs
    2
    3/-!
    4---
    5title: Symmetric skew circuits from symmetric ABPs
    6type: theorem
    7---
    8Over every commutative semiring, a family of polynomials computed by matrix-symmetric ABPs
    9with polynomial vertex orbits is computed by matrix-symmetric skew circuits with polynomial
    10gate orbits, that is, `symVBP ⊆ symVS`.
    11-/
    12
    13namespace Lax461425.SkewFromABP
    14
    15open SymmetricABPs SymmetricFamilies SymmetricClasses
    16
    17/-- Families with symmetric polynomial-orbit ABPs have symmetric polynomial-orbit skew
    18circuits, over every commutative semiring. -/
    19axiom inSymVS_of_inSymVBP {K : Type*} [CommSemiring K] (f : MatrixFamily K)
    20 (hf : InSymVBP f) : InSymVS f
    21
    22end Lax461425.SkewFromABP
    23
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