Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Symmetric membership and hardness of the permanent

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    The permanent family lies in symmetric VNP over every commutative ring. Over a field of characteristic zero, every polynomial of a family in symmetric VNP is a symmetric affine projection of a permanent of some order.

    Concept map
    9 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 hardness proven

    2 membership proven

    Lean source view on GitHub

    1import Lax461425.Permanent
    2import Lax461425.BooleanSumSymmetry
    3
    4/-!
    5---
    6title: Symmetric membership and hardness of the permanent
    7type: theorem
    8---
    9The permanent family lies in symmetric VNP over every commutative ring. Over a field of
    10characteristic zero, every polynomial of a family in symmetric VNP is a symmetric affine
    11projection of a permanent of some order.
    12-/
    13
    14namespace Lax461425.PermanentCompleteness
    15
    16open Permanent SymmetricFamilies SymmetricClasses SymmetricVNP
    17
    18/-- The permanent has a jointly symmetric Boolean witness over every commutative ring. -/
    19axiom membership {K : Type*} [CommRing K] : InSymVNP (permanent K)
    20
    21/-- Every symmetric-VNP family projects symmetrically to the permanent, allowing
    22unrestricted source order independently for every matrix size. -/
    23axiom hardness {K : Type*} [Field K] [CharZero K] :
    24 ∀ f : MatrixFamily K, InSymVNP f →
    25 ∀ n, IsSymmetricProjection (matrixAction n) (f n)
    26
    27end Lax461425.PermanentCompleteness
    28
    Show ProofShow Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…