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

Proved

OAI.WitnessedSeparation.Grid.QuotedGrid.fundamentalPool_in_cycles

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.fundamentalPool_in_cycles of lean/OAI/Computability/WitnessedChoice/TreeCorrection.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 WitnessedChoice.TreeTest Hereditary RelationPools, open QuotedTerm QuotedFormula SourceSyntax):

lemma fundamentalPool_in_cycles (env : Fin k → HF (Atom b)) (V E F R T H N : QuotedTerm k)
    (P : ParentEdges T₀) (Ns : Finset (Edge n))
    (hV : V.meaning (atomInput b) env = ofFinset (Finset.univ.image (vertexCode b)))
    (hE : E.meaning (atomInput b) env = ofFinset (Finset.univ.image (edgeCode b)))
    (hF : ∀ z, z ∈ elements (F.meaning (atomInput b) env) ↔ ∃ C : CycleFrame n, frameCode b C = z)
    (hR : R.meaning (atomInput b) env = distancesCode b (boxGraph n))
    (hT : T.meaning (atomInput b) env = ofFinset (P.treeEdges.image (edgeCode b)))
    (hH : H.meaning (atomInput b) env = connTableCode b P.treeEdges)
    (hN : N.meaning (atomInput b) env = ofFinset (Ns.image (edgeCode b)))
    (hoff : ∀ e ∈ Ns, e ∉ P.treeEdges) (hn : 1 ≤ n) :
    ∀ r ∈ P.fundamentalPool Ns,
      relationCode r ∈ elements ((cycleRelations V E F R T H N).meaning (atomInput b) env)

Definitions. The statement is written with OAI.WitnessedChoice.BGS.QuotedTerm, OAI.WitnessedChoice.BGS.relationCode, OAI.WitnessedChoice.RelationPools.Relation, OAI.WitnessedChoice.TreeTest.RootedTree, OAI.WitnessedSeparation.Grid.Atom, OAI.WitnessedSeparation.Grid.CycleFrame, OAI.WitnessedSeparation.Grid.Edge, OAI.WitnessedSeparation.Grid.EdgeX, OAI.WitnessedSeparation.Grid.EdgeY, OAI.WitnessedSeparation.Grid.EdgeZ, OAI.WitnessedSeparation.Grid.ParentEdges, OAI.WitnessedSeparation.Grid.Scalar, OAI.WitnessedSeparation.Grid.Vertex, OAI.WitnessedSeparation.Grid.atomInput, OAI.WitnessedSeparation.Grid.boxGraph, OAI.WitnessedSeparation.Grid.connTableCode, OAI.WitnessedSeparation.Grid.distancesCode, OAI.WitnessedSeparation.Grid.edgeCode, OAI.WitnessedSeparation.Grid.frameCode, OAI.WitnessedSeparation.Grid.instFintypeAtom, OAI.WitnessedSeparation.Grid.vertexCode, OAI.WitnessedChoice.BGS.SourceSyntax.cycleRelations, OAI.WitnessedSeparation.Grid.ParentEdges.fundamentalPool and OAI.WitnessedSeparation.Grid.ParentEdges.treeEdges (definitions bundle OAIQ4Wc000 of this split); OAI.WitnessedSeparation.Hereditary.HF, OAI.WitnessedSeparation.Hereditary.elements and OAI.WitnessedSeparation.Hereditary.ofFinset (the published bundle Def_WitnessedChoice).

Used by. The proof of OAI.WitnessedSeparation.Grid.QuotedGrid.selection_transitive 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 WitnessedChoice.TreeTest Hereditary RelationPools

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

namespace QuotedGrid

open QuotedTerm QuotedFormula SourceSyntax

variable {k : ℕ} {T₀ : RootedTree (Vertex n)}

attribute [local irreducible] SourceSyntax.cycleRelations SourceSyntax.cycleY

lemma fundamentalPool_in_cycles (env : Fin k → HF (Atom b)) (V E F R T H N : QuotedTerm k)
    (P : ParentEdges T₀) (Ns : Finset (Edge n))
    (hV : V.meaning (atomInput b) env = ofFinset (Finset.univ.image (vertexCode b)))
    (hE : E.meaning (atomInput b) env = ofFinset (Finset.univ.image (edgeCode b)))
    (hF : ∀ z, z ∈ elements (F.meaning (atomInput b) env) ↔ ∃ C : CycleFrame n, frameCode b C = z)
    (hR : R.meaning (atomInput b) env = distancesCode b (boxGraph n))
    (hT : T.meaning (atomInput b) env = ofFinset (P.treeEdges.image (edgeCode b)))
    (hH : H.meaning (atomInput b) env = connTableCode b P.treeEdges)
    (hN : N.meaning (atomInput b) env = ofFinset (Ns.image (edgeCode b)))
    (hoff : ∀ e ∈ Ns, e ∉ P.treeEdges) (hn : 1 ≤ n) :
    ∀ r ∈ P.fundamentalPool Ns,
      relationCode r ∈ elements ((cycleRelations V E F R T H N).meaning (atomInput b) env) := 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/TreeCorrection.lean (declaration OAI.WitnessedSeparation.Grid.QuotedGrid.fundamentalPool_in_cycles)

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