Lax Lean Archive

P is closed under complement

Lax434930.ComplementClosure · concepts/Lax434930/ComplementClosure.lean · lax-434930

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    Natural Language Statement

    Lemma

    If a language of binary strings belongs to P\mathrm{P}, then its complement also belongs to P\mathrm{P}. The complement is taken in the set of all finite binary strings.

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

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    In the paper

    • page 4 of this submission's paper

    Lean source view on GitHub

    1import Lax434930.PolynomialTime
    2
    3/-!
    4---
    5title: P is closed under complement
    6type: lemma
    7---
    8If a language of binary strings belongs to P\mathrm{P}, then its complement
    9also belongs to P\mathrm{P}. The complement is taken in the set of all finite
    10binary strings.
    11-/
    12
    13namespace Lax434930.ComplementClosure
    14
    15open Lax434930.PolynomialTime
    16
    17/-- The complement of a polynomial-time decidable language is polynomial-time decidable. -/
    18axiom closed_under_complement (L : Language) : L ∈ P → Lᶜ ∈ P
    19
    20end Lax434930.ComplementClosure
    21
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