CorollaryStatement: AI-generatedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $i_{\mathrm{word}}:X\to F_{\mathrm{word}}(X)$ of The word-quotient group $W(X)/{\sim}$ satisfies the universal property of the free group on $X$ , given by $i_{\mathrm{word}}(x)=[x]$ , is injective.

Facts & Assumptions

Given: Elements $x,y\in X$ with $[x]=[y]$ in $F_{\mathrm{word}}(X)$ .

[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 $i_{\mathrm{word}}(x)=i_{\mathrm{word}}(y)$ implies $x=y$ , which is injectivity; when $X$ is empty the assertion is vacuous.

step 1.1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

The generator map X→W(X)/∼X\to W(X)/{\sim}X→W(X)/∼ is injective