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.

Compressed inversion count is additive for Coxeter-reduced composites

Statement

If $x, y$ are composable ordinary root-action arrows and $ℓ (x y) = ℓ (x) + ℓ (y)$ , then $C (x y) = C (x) ⊔ x C (y)$ and $∣ C (x y) ∣ = ∣ C (x) ∣ + ∣ C (y) ∣$ .

Facts & Assumptions

[F1]

The definitions and results used in this argument are supplied, under their stated hypotheses, by Parallel classes of roots and finite inversion sets.

Proof

Given: The stated Coxeter and groupoid data, the declared prerequisite items, and the explicitly cited independent external inputs.

Goal: Establish the assertion in the Statement.

1.1given

The Coxeter inversion identity gives $N (x y) = N (x) ⊔ x N (y)$ when lengths add; the transported roots $x N (y)$ are positive in this case.

2.1givenstep 1.1∎

By the preceding parallel-class lemma, both summands are unions of parallel classes at the codomain of $x$ . Indeed $N (y)$ is such a union at its own codomain, and the bijection on all roots induced by $x$ takes an entire class to an entire class. Disjoint unions of saturated subsets induce disjoint unions of their quotient classes. This gives the asserted identity and, because the sets are finite, the cardinality equality. The inversion identity and the condition that no cancellation occurs are the usual Coxeter inversion cocycle, recorded independently in (1.4.3)–(1.4.4); they do not assume finite rank.

Depends on

Used by · 1

Every standard expression counts the same parallel classes

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, equations (1.4.3)–(1.4.4), PDF p. 4

For roots N(x)=Φ⁺∩xΦ⁻, the inversion cocycle and |N(x)|=ℓ(x) imply N(xy)=N(x) disjoint-union xN(y) when ℓ(xy)=ℓ(x)+ℓ(y).

Hypotheses: An arbitrary-rank Coxeter system with its standard based root system; a Brink–Howlett component or union of components. Finite rank is explicitly not assumed (§1.4).

Reading limits: Read extracted full statements and relevant definitions in §§1.4, 1.8–1.9, 1.19–1.27, 3.1–3.4 and Propositions 3.11–3.14; visually inspected PDF pp. 9 and 43. These are independent external inputs, not locally reconstructed proofs. Did not read the entire paper or original Brink–Howlett (1999).

Sources

Matthew Dyer and Harrison Gimenez, Root Systems, Tits Cones and Imaginary Cones of Brink-Howlett Groupoids · §1.4, equations (1.4.3)–(1.4.4), PDF p. 4