LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-12

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.

Computable many-one reducibility is reflexive and transitive

Statement

Computable many-one reducibility is reflexive and transitive: every language $A$ satisfies $A \leq_{m} A$ , and whenever $A \leq_{m} B$ and $B \leq_{m} C$ , one also has $A \leq_{m} C$ .

Facts & Assumptions

Given: Languages $A, B, C$ over finite alphabets.

[L1]

A computable many-one reduction is a total computable function preserving membership in both directions, by Computable many-one reductions between languages.

Proof

technique · direct

1.1L1givenconstruct

For reflexivity, use the identity map $id (x) = x$ on $A$ 's alphabet. It is total and computable, and $x \in A ⟺ id (x) \in A$ for every $x$ , so [L1] gives $A \leq_{m} A$ .

1.2L1givenconstruct

For transitivity, let $f$ witness $A \leq_{m} B$ and let $g$ witness $B \leq_{m} C$ . By [L1], both maps are total and computable, so $h : = g \circ f$ is total and computable as well.

2.1L1step 1.2∎

For every $x$ , [L1] gives $x \in A ⟺ f (x) \in B ⟺ g (f (x)) \in C$ . Thus $x \in A ⟺ h (x) \in C$ , so [L1] yields $A \leq_{m} C$ .

Depends on

Computable many-one reductions between languages

Used by

Nothing in the library uses this result yet.

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

MIT 6.045J / 18.400J, Lecture 9: Mapping Reducibility and Rice's Theorem (standard reference, not scraped)
Kevin Kelly, Many-one Reduction (standard reference, not scraped)