Negating a decider preserves polynomial time

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PvsNP.negate_decider_polytime

by alexcarter · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryformalizationp-vs-np

For every polynomial-time Boolean decider, flipping its Boolean output is polynomial time.

Status: Known mathematics / implementation obligation awaiting formal proof.

Formal statement

import Definitions.Def_PvsNPFrontier

namespace PvsNP
theorem negate_decider_polytime (χ : Str → Bool) (h : PolyTimeDecider χ) :
    PolyTimeDecider (fun x => !χ x) := by sorry
end PvsNP

Source

Stephen Cook, The P versus NP Problem, Clay official description, definitions of P/NP and Proposition 1; https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf; elementary deterministic complement construction.

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What the Lean code literally says, in plain math · gpt-6-astra

For every $\chi:B^*\to B$ , if $D(\chi)$ holds, then $D(\bar\chi)$ holds for the function $\bar\chi(w)$ obtained by Boolean negation of $\chi(w)$ . The conclusion is an existential polynomial-time witness for the negated output on every finite Boolean input. Here $B=\{\mathrm{false},\mathrm{true}\}$ , $B^*$ is the set of all finite Boolean lists, including the empty list, and $|w|$ is list length. Write $D(\chi)$ for existence of such a machine and a polynomial $p\in\mathbb N[X]$ that, for every $w\in B^*$ , compute the singleton output $[\chi(w)]$ from input $w$ in at most $p(|w|)$ steps. A machine in these assertions is a Mathlib TM2 stack machine with finitely many stack indices, instruction labels, and control states, a finite input-stack alphabet, designated input and output stacks, a program, and initial label and control state; its other stack alphabets need not be finite. Input and output alphabet bijections transport the specified encoded lists to the corresponding stack alphabets. Computation starts with only the input stack populated, and reaches a halted configuration with the specified output on the output stack, all other stacks empty, and the control state reset to its initial value. Time counts executions of whole TM2 statements, each of which may contain several stack operations. The supplied body is admitted with sorry; no proof of this assertion is supplied there.

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