LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-02

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.

Polynomial-time many-one reductions compose

Statement

If $A \leq_{p} B$ and $B \leq_{p} C$ , then $A \leq_{p} C$ .

Facts & Assumptions

Given: Languages $A, B, C$ with $A \leq_{p} B$ witnessed by a total map $f$ and $B \leq_{p} C$ witnessed by a total map $g$ , in the sense of Polynomial-time many-one reductions.

[L1]

A polynomial-time many-one reduction is a total polynomial-time function preserving membership in both directions, by Polynomial-time many-one reductions.

Proof

technique · direct

1.1

Define $h : = g \circ f$ . Because both $f$ and $g$ are total by [L1], the composite $h$ is also total.

L1givenconstruct

2.1

For every source word $x$ , [L1] gives $x \in A ⟺ f (x) \in B ⟺ g (f (x)) \in C$ . So $x \in A ⟺ h (x) \in C$ .

L1step 1.1

3.1

The function $f$ runs in polynomial time and can output only polynomially many symbols, so the length of $f (x)$ is polynomially bounded in $∣ x ∣$ . Running $g$ on that output therefore still takes polynomial time in $∣ x ∣$ . Hence $h$ is a polynomial-time many-one reduction from $A$ to $C$ .

L1step 1.1step 2.1∎

Depends on

Polynomial-time many-one reductions

Used by

Polynomial-time many-one reductions transfer P-, NP-, and coNP-membership Theorem

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

Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)