LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 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.

Elementary HNN reductions preserve the represented element

Statement

In the associated-subgroup notation for an HNN extension, replacing a pin

$t c t^{- 1} (c \in C_{-})$

by $ϕ (c)$ , or replacing a pin

$t^{- 1} d t (d \in C_{+})$

by $ϕ^{- 1} (d)$ , does not change the represented element of the HNN extension.

Facts & Assumptions

Given: The associated-subgroup presentation of an HNN extension.

[L1]

A pin is a subword $t c t^{- 1}$ with $c \in C_{-}$ or a subword $t^{- 1} d t$ with $d \in C_{+}$ , where the defining relation is $t c t^{- 1} = ϕ (c)$ . (HNN words, pins, and Britton-reduced words)

Proof

technique · direct

1.1

By [L1], every pin of the first kind is literally one side of a defining relator of the presentation, so replacing $t c t^{- 1}$ by $ϕ (c)$ preserves the represented element.

L1given

2.1

If $d \in C_{+}$ , write $d = ϕ (c)$ with $c = ϕ^{- 1} (d) \in C_{-}$ . Then step 1.1 applied to $t c t^{- 1} = ϕ (c) = d$ gives $t^{- 1} d t = c = ϕ^{- 1} (d)$ after multiplying by $t^{- 1}$ on the left and $t$ on the right. So replacing a pin of the second kind by $ϕ^{- 1} (d)$ also preserves the represented element.

L1step 1.1algebra∎

Depends on

HNN words, pins, and Britton-reduced words

Used by

Dependency tree · two levels

2 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.

Sources

Roger C. Lyndon and Paul E. Schupp, Combinatorial Group Theory (standard reference, not scraped)