CorollaryStatement: AI-generatedProof: AI-generatedprecheck passaudited 2026-08-11

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.

The generator map $X \to W (X) / \sim$ is injective

Statement

The generator map $i_{w o r d} : X \to F_{w o r d} (X)$ of The word-quotient group $W (X) / \sim$ satisfies the universal property of the free group on $X$ , given by $i_{w o r d} (x) = [x]$ , is injective.

Facts & Assumptions

Given: Elements $x, y \in X$ with $[x] = [y]$ in $F_{w o r d} (X)$ .

[L1]

Every class in $W (X) / \sim$ contains exactly one reduced word (Every class in $W (X) / \sim$ contains exactly one reduced word).

Proof

technique · direct

1.1

The one-letter words $x$ and $y$ are reduced and lie in the same class, so uniqueness in [L1] gives $x = y$ .

L1given

2.1

Thus $i_{w o r d} (x) = i_{w o r d} (y)$ implies $x = y$ , which is injectivity; when $X$ is empty the assertion is vacuous.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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.

The generator map X→W(X)/∼X→W(X)/∼ is injective