ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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.

An index-two subgroup of a rank-two free group has rank three

Example

Let $H \leq F (a, b)$ be the subgroup of reduced words with even exponent sum in $a$ . Then $H$ has index $2$ and free basis

${b, a^{2}, a b a^{- 1}} .$

Consequently $rank (H) = 3$ .

Facts & Assumptions

Given: The subgroup $H \leq F (a, b)$ of words with even exponent sum in $a$ .

[L1]

Nielsen-Schreier identifies a free basis from the nontrivial Schreier generators of a Schreier system (Under the stated choice boundary, every subgroup of a free group is free with its nontrivial Schreier generators as a basis).

[L2]

The index-rank formula gives $rank (H) = 1 + [F (a, b) : H]$ (The Schreier index-rank formula).

Verification

technique · direct

1.1

The two right cosets are $H$ and $H a$ , so $T = {1, a}$ is a Schreier system. Its nontrivial Schreier generators are $s (1, b) = b$ , $s (a, a) = a^{2}$ , and $s (a, b) = a b a^{- 1}$ , while $s (1, a) = 1$ .

givenconstruct

2.1

By [L1], the three nontrivial generators from step 1.1 form a free basis of $H$ . Since $[F (a, b) : H] = 2$ , [L2] also gives $rank (H) = 1 + 2 (2 - 1) = 3$ , agreeing with the computed basis.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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