LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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.

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C prime(lambda) implies C(lambda)

Statement

Let $0 < λ \leq 1$ . If a symmetrised presentation has only nonempty relators and satisfies $C^{'} (λ)$ , then it satisfies $C (λ)$ .

Facts & Assumptions

Given: A symmetrised relator set of nonempty words satisfying $C^{'} (λ)$ .

[L1]

Under $C^{'} (λ)$ , every piece $p$ lying in a relator $r$ satisfies $∣ p ∣ < λ ∣ r ∣$ , while $C (λ)$ asks that a factorisation of $r$ into pieces use more than $1 / λ$ pieces (The small-cancellation conditions C(lambda) and C prime(lambda)).

Proof

technique · direct

1.1

Let $r = p_{1} \dots p_{n}$ be a factorisation of a relator into pieces. Applying [L1] to each $p_{i}$ gives $∣ p_{i} ∣ < λ ∣ r ∣$ for every $i$ . Summing these inequalities yields $∣ r ∣ = \sum_{i = 1}^{n} ∣ p_{i} ∣ < n λ ∣ r ∣ .$

L1givenalgebra

2.1

Because every relator is nonempty, $∣ r ∣ > 0$ . Thus step 1.1 implies $1 < n λ$ , hence $n > 1 / λ$ . This is exactly the $C (λ)$ condition from [L1].

step 1.1L1givenalgebra∎

Depends on

The small-cancellation conditions C(lambda) and C prime(lambda)

Used by

Nothing in the library uses this result yet.

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

GAP SmallCancellation manual, Chapter 1: Small Cancellation Theory — the classical conditions (standard reference, not scraped)
Jay Williams, Universal Countable Borel Quasi-Orders (standard reference, not scraped)
Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, Section 3.5 (standard reference, not scraped)
Clara Löh, Geometric Group Theory: An Introduction, Section 7.4.1 (standard reference, not scraped)