LemmaStatement: AI-adaptedProof: 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.

Formula degree and point-evaluation cost

Statement

Let $φ$ be a formula with $s \geq 1$ syntax nodes on $n \geq 0$ variables, and let $t_{i}$ count the leaves labelled $x_{i}$ . Then $P_{φ}$ has individual degree at most $t_{i}$ in $X_{i}$ and total degree at most $\sum_{i} t_{i}$ . The bounds include the zero polynomial (which satisfies every nonnegative degree upper bound). At any supplied point, $P_{φ}$ can be evaluated with $O (s)$ field operations.

Facts & Assumptions

Given: The objects and hypotheses in the statement above.

[F1]

The polynomial is represented by a formula tree with the specified arithmetic gates (Arithmetization of Boolean formulas).

Proof

1.1

For a constant leaf all degree bounds are zero; for a variable leaf the individual bound is one in that variable and zero otherwise, and the total bound is one. Each leaf needs no arithmetic operation.

F1base

1.2

Assume the bounds for the children. Subtracting a polynomial from $1$ does not raise any nonnegative degree bound. At a binary gate with child bounds $a, b$ (individual or total), the product has bound $a + b$ and the sum has bound $\max (a, b) \leq a + b$ . Thus AND and OR obey the sum of the child occurrence bounds, even if coefficients cancel or a child is zero.

F1ihalgebra

2.1

Occurrence counts add over child trees, so the preceding leaf and parent argument proves all degree bounds by structural induction. A bottom-up evaluator visits each syntax node once and applies at most a fixed number of operations to its stored child values; the OR gate reuses those values. Hence its field-operation count is $O (s)$ . This includes variable-free formulas and unused variables.

step 1.1step 1.2discharge-inductionalgebra∎

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, 3CNF construction generalized by the same gate induction (standard reference, not scraped)