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.

The kernel of an exponent-sum map in a free group

Example

Let $ϕ : F (a, b) \to Z$ send $a \mapsto 1$ and $b \mapsto 0$ , and let $H = \ker ϕ$ . Then the family

${a^{n} b a^{- n} : n \in Z}$

is a free basis of $H$ .

Facts & Assumptions

Given: The kernel $H = \ker ϕ$ of the exponent-sum map above.

[L1]

A Schreier system is a family of reduced right-coset representatives closed under initial segments (Schreier transversals and Schreier systems).

[L2]

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

Verification

technique · direct

1.1

The right cosets of $H$ are $H a^{n}$ for $n \in Z$ , so $T = {a^{n} : n \in Z}$ is a Schreier system by [L1].

L1givenconstruct

2.1

For every $n \in Z$ , one has $s (a^{n}, a) = 1$ and $s (a^{n}, b) = a^{n} b a^{- n}$ . Thus the nontrivial Schreier generators are exactly the displayed conjugates.

step 1.1algebra

3.1

By [L2], those generators form a free basis of $H$ .

L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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