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Rigidification

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

proven

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    Natural Language Statement

    Lemma

    For every matrix-symmetric circuit CC, there is a reduced matrix-symmetric circuit C′C' computing the same polynomial whose size, depth, and orbit size are at most those of CC.

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

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    Lean source view on GitHub

    1import Lax461425.MatrixSymmetry
    2
    3/-!
    4---
    5title: Rigidification
    6type: lemma
    7---
    8For every matrix-symmetric circuit CC, there is a reduced matrix-symmetric circuit C′C'
    9computing the same polynomial whose size, depth, and orbit size are at most those of CC.
    10-/
    11
    12namespace Lax461425.Rigidification
    13
    14open Circuits MatrixSymmetry
    15
    16/-- Every matrix-symmetric circuit can be replaced by a reduced matrix-symmetric circuit
    17computing the same polynomial, without increasing size, depth, or orbit size. -/
    18axiom exists_reduced {K : Type*} [CommSemiring K] {n : ℕ}
    19 (C : Circuit (Fin n × Fin n) K) (hsym : IsMatrixSymmetric C)
    20 {p : MvPolynomial (Fin n × Fin n) K} (hp : C.Computes p) :
    21 ∃ C' : Circuit (Fin n × Fin n) K,
    22 C'.IsReduced ∧ IsMatrixSymmetric C' ∧ C'.Computes p ∧
    23 C'.size ≤ C.size ∧ C'.depth ≤ C.depth ∧ orbitSize C' ≤ orbitSize C
    24
    25end Lax461425.Rigidification
    26
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