LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Arithmetization preserves Boolean values

Statement

For every field $F$ , Boolean formula $φ$ on $n \geq 0$ variables and assignment $a \in {0, 1}^{n}$ , $P_{φ} (a)$ equals the Boolean value of $φ (a)$ , embedded as $0$ or $1$ in $F$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

Arithmetization uses the stated constant, variable, NOT, AND and OR gates (Arithmetization of Boolean formulas).

Proof

1.1

Constants and variable leaves have their prescribed Boolean values by the recursive construction. This includes a constant formula on the singleton cube ${0, 1}^{0}$ .

F1base

1.2

For inputs $u = 0, 1$ , $1 - u$ gives respectively $1, 0$ . For input pairs $(0, 0), (0, 1), (1, 0), (1, 1)$ , the product $u v$ gives $0, 0, 0, 1$ , and $u + v - u v$ gives $0, 1, 1, 1$ . These computations hold also in characteristic two.

F1algebra

2.1

Assume each immediate subformula has the correct value. The gate calculations then give the correct value for its parent. This proves the parent implication, including when its value is zero or one.

step 1.1step 1.2ih

3.1

Starting at the leaves and applying the parent implication up the finite formula tree proves agreement for the whole formula, including formulas that are identically false or true.

step 1.1step 2.1discharge-induction∎

Depends on

Arithmetization of Boolean formulas

Used by

Verifying formula counts over a supplied field Proposition

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Arora and Barak, Computational Complexity, January 2007 web draft, §8.5.1, p.158 (standard reference, not scraped)