OAI.WitnessedSeparation.Grid.QuotedGrid.formula_value: intermediate lemma from OpenAI, Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting

Proved

OAI.WitnessedSeparation.Grid.QuotedGrid.formula_value

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.WitnessedChoice.main, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting, OpenAI Math Release, September 24, 2026, result family 243, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.WitnessedSeparation.Grid.QuotedGrid.formula_value of lean/OAI/Computability/WitnessedChoice/ChoiceSeparation.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.WitnessedSeparation.Grid.QuotedGrid, with open Classical WitnessedChoice WitnessedChoice.BGS Hereditary, open SourceProgram):

lemma formula_value (p : Polynomial ℝ) (hp : GridResources p b) (hn : 1 ≤ n) :
    program.formula.eval (atomInput b) p Fin.elim0 = some (decide (∑ v, b v = 0))

Definitions. The statement is written with OAI.WitnessedSeparation.Grid.Atom, OAI.WitnessedSeparation.Grid.GridResources, OAI.WitnessedSeparation.Grid.Scalar, OAI.WitnessedSeparation.Grid.Vertex, OAI.WitnessedSeparation.Grid.atomInput, OAI.WitnessedSeparation.Grid.instFintypeAtom, OAI.WitnessedChoice.BGS.ClosedGuardedProgram.formula and OAI.WitnessedChoice.BGS.SourceProgram.program (definitions bundle OAIQ4Wc000 of this split); OAI.WitnessedChoice.BGS.Result and OAI.WitnessedSeparation.Hereditary.HF (the published bundle Def_WitnessedChoice).

Used by. The proof of OAI.WitnessedChoice.main of this split imports it.

Preamble

import Mathlib
import Definitions.Def_WitnessedChoice
import Definitions.Def_OAIQ4Wc000

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace WitnessedSeparation.Grid

noncomputable section

open Classical WitnessedChoice WitnessedChoice.BGS Hereditary

variable {n : ℕ} {b : Vertex n → Scalar}

namespace QuotedGrid

open SourceProgram

attribute [local irreducible] SourceProgram.parameters SourceProgram.candidates
  SourceProgram.cycles SourceProgram.output

lemma formula_value (p : Polynomial ℝ) (hp : GridResources p b) (hn : 1 ≤ n) :
    program.formula.eval (atomInput b) p Fin.elim0 = some (decide (∑ v, b v = 0)) := by
  sorry

end QuotedGrid

end

end WitnessedSeparation.Grid

end OAI
end

Source

OpenAI, Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting, OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 243; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Computability/WitnessedChoice/ChoiceSeparation.lean (declaration OAI.WitnessedSeparation.Grid.QuotedGrid.formula_value)

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