LemmaStatement: AI-generatedProof: AI-generated

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.

Every finite generator path reduces to a standard expression

Statement

Modulo atomic braid relations and the quadratic relations $t t^{- 1} = 1$ for elementary inverse pairs, every finite composable elementary-generator path is congruent to a standard expression of its product.

Statement references

Standard expressions and their lengths

Atomic braid relations

Facts & Assumptions

[F1]

Use the arbitrary-rank Coxeter and left-to-right groupoid conventions of Standard expressions and their lengths. All paths are finite.

[F2]

Atomic braid moves substitute the two words in a relation of Atomic braid relations. Here congruence means the equivalence generated by these substitutions in contiguous subwords and by insertion or deletion of $s s^{- 1}$ for every elementary arrow $s$ , at its source object. Replacements may be made within any composable prefix and suffix.

[F3]

Every article morphism has a standard expression, by Every standard expression counts the same parallel classes, first assertion.

[F4]

Appending an elementary factor satisfies the two-case length dichotomy of Appending an elementary factor is additive or removes its inverse.

[F5]

Standard expressions for the same morphism are connected by finitely many endpoint-preserving atomic braid moves, by Standard expressions are connected by atomic braid moves.

[F6]

For a component $D$ with $W_{D}$ finite and $b = ι_{D} (a)$ , the identities $w_{D}^{2} = 1$ , $ι_{D}^{2} = 1$ and $w_{D} w_{D ∖ {a}} w_{D} = w_{D ∖ {b}}$ are proved in Transfer to the root-action Brink–Howlett groupoid, Proof, steps 1.2 and 2.1. They require no finite-rank hypothesis on the ambient Coxeter system.

Proof

Given: The arbitrary-rank Coxeter and left-to-right groupoid conventions in [F1], with the congruence of [F2]. All paths are finite.

Goal: Establish the assertion in the Statement.

Proof technique: direct.

1.1F1F2given

Induct on the number of factors of the given path. The empty path is standard. For a nonempty path $P t$ , normalise its shorter prefix $P$ by induction, obtaining $P \equiv U$ with $U$ a standard word for $w : I \to K$ . Write the last elementary factor as $t = v_{a, K} : K \to L$ . Congruence gives $P t \equiv U t$ . If $ℓ (w t) = ℓ (w) + ℓ (t)$ , then $U t$ is standard by length additivity, completing this case.

2.1F1F6step 1.1

Put $D = [K, a]$ and $b = ι_{D} (a)$ . By [F1], $W_{D}$ is finite and $L = (K \cup {a}) ∖ {b}$ . Hence $L \cup {b} = K \cup {a}$ and $[L, b] = D$ . By [F6], $ι_{D} (b) = a$ and $v_{b, L} = w_{D ∖ {b}} w_{D} = w_{D} w_{D ∖ {a}} = t^{- 1}$ . Thus $t^{- 1} : L \to K$ is elementary. Reversing any simple-generator word represents its inverse with the same number of letters; applying this in both directions to shortest words gives $ℓ (t^{- 1}) = ℓ (t)$ .

3.1F1F3F4step 1.1step 2.1

In the nonadditive case of step 1.1, [F4] supplies a morphism $u : I \to L$ with $w = u t^{- 1}$ and $ℓ (w) = ℓ (u) + ℓ (t)$ . Choose a standard word $V$ for $u$ by [F3]. Together with step 2.1, this length equality makes $V t^{- 1}$ a standard word for $w$ .

4.1F2F5step 1.1step 2.1step 3.1∎

By [F5], $U$ is connected to $V t^{- 1}$ by finitely many atomic braid moves. Append $t$ throughout and delete $t^{- 1} t$ using [F2] with the elementary arrow $s = t^{- 1}$ from step 2.1. This gives $P t \equiv U t \equiv V t^{- 1} t \equiv V$ . The product of $V$ is $u = w t$ , so $V$ is the required standard expression of the original product. Together with the additive case, this completes the induction. All words and move sequences used are finite, and only $W_{D}$ was required to be finite.

Depends on

Used by · 1

The atomic-quadratic path quotient is faithful

Dependency tree · 2 levels

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