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.

A rank-two free group contains an infinite-rank subgroup

Example

The rank-two free group $F (a, b)$ contains a subgroup of infinite rank, namely the kernel of the exponent-sum map $ϕ (a) = 1$ , $ϕ (b) = 0$ .

Facts & Assumptions

Given: The kernel $H = \ker ϕ \leq F (a, b)$ of the exponent-sum map.

[L1]

Nielsen-Schreier makes the nontrivial Schreier generators of a Schreier system into 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. Its nontrivial Schreier generators are the infinitely many elements $a^{n} b a^{- n}$ for $n \in Z$ .

givenconstruct

2.1

By [L1], the family in step 1.1 is a free basis of $H$ . Since that basis is infinite, $H$ has infinite rank.

L1step 1.1∎

Depends on

Under the stated choice boundary, every subgroup of a free group is free with its nontrivial Schreier generators as a basis

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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.