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.

Appending an elementary factor is additive or removes its inverse

Statement

Let $w : I \to K$ and let $t = v_{a, K} : K \to L$ be elementary. Exactly one of the following holds: $ℓ (w t) = ℓ (w) + ℓ (t)$ ; or $w = u t^{- 1}$ for an article morphism $u : I \to L$ with $ℓ (w) = ℓ (u) + ℓ (t)$ . Write $N (x) = Φ^{+} \cap x (Φ^{-})$ for the root inversion set (not the reflection inversion notation of the cited source).

External input

The independent external inputs are the inversion and weak-order identities in Dyer–Gimenez §§1.4–1.5, and the preprincipal property and identification of atoms in Proposition 3.14(a),(b), with preprincipal defined in §1.19. In the notation $N (x)$ used here, the latter property says $N (t) \subseteq N (w^{- 1})$ or $N (t) \cap N (w^{- 1}) = \emptyset$ . The source proves this for arbitrary rank. No presentation theorem is imported.

Facts & Assumptions

[F99]

Proof

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

Goal: Establish the assertion in the Statement.

Proof technique: direct.

1.1givenF99

Use the arrow reversal in Transfer to the root-action Brink–Howlett groupoid. The article arrows $w^{- 1} : K \to I$ and $t : K \to L$ correspond to root-action arrows with common target $K$ , carrying the same labels $w^{- 1}$ and $t$ . The elementary arrow is atomic in that root-action groupoid. Proposition 3.14 and the definition of preprincipal therefore give either $N (t) \cap N (w^{- 1}) = \emptyset$ or $N (t) \subseteq N (w^{- 1})$ .

2.1step 1.1givenF99

In the disjoint case, §1.5 gives the asserted additivity. In the inclusion case it gives $ℓ (w^{- 1}) = ℓ (t) + ℓ (t^{- 1} w^{- 1})$ . Inverting the equality $w^{- 1} = t (t^{- 1} w^{- 1})$ and setting $u = w t$ gives $w = u t^{- 1}$ and $ℓ (w) = ℓ (u) + ℓ (t)$ , using invariance of Coxeter length under inversion. The groupoid laws give the asserted endpoints.

3.1step 1.1step 2.1∎

An elementary generator has positive Coxeter length. The two displayed length equalities are consequently incompatible. The preprincipal alternatives exhaust the possibilities, proving the claim.

Depends on

Used by · 1

Every finite generator path reduces to a standard expression

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · §§1.4–1.5, PDF p.4; §1.19, PDF p.8; Proposition 3.14(a),(b) and proof, PDF pp.43–44

For root-action Brink–Howlett arrows with common target, an elementary arrow s is atomic; its inversion set is either contained in or disjoint from that of any arrow g. Inversion inclusion is length-additive prefix order, and disjointness of N(x inverse) and N(y) is equivalent to length additivity of xy.

Hypotheses: Arbitrary-rank Coxeter system with standard based root system, and a component or union of components of the root-action Brink–Howlett groupoid. No spherical or finite-rank assumption. The article convention reverses root-action arrows with unchanged labels.

Reading limits: For this adjudication, read extracted PDF text in §§1.1–1.5, 1.17–1.19 and 1.21, and Proposition 3.14(a),(b) with its proof (PDF pp.43–44). Checked the inversion convention, arbitrary-rank scope, common-target condition and elementary-generator atomicity at their stated locators. This is not a full-paper reading or a reading of Brink–Howlett (1999); no presentation theorem is used.

Sources

Matthew Dyer and Harrison Gimenez, Root systems, Tits cones and imaginary cones of Brink-Howlett groupoids · §§1.4–1.5, PDF p.4; §1.19, PDF p.8; Proposition 3.14(a),(b) and proof, PDF pp.43–44