ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-31

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.

The five bracketings of a four-fold tensor product with no inserted unit symbol

Example

The five parenthesised tensor words on four letters with no inserted unit symbol are

$(((A \otimes B) \otimes C) \otimes D),$ $((A \otimes (B \otimes C)) \otimes D),$ $(A \otimes ((B \otimes C) \otimes D)),$ $(A \otimes (B \otimes (C \otimes D))),$ $((A \otimes B) \otimes (C \otimes D)) .$

The pentagon compares the two canonical routes from the first word to the fourth one.

Facts & Assumptions

Given: Parenthesised tensor words on four letters.

[L1]

Parenthesised tensor words are the defined expressions on this page (Parenthesised tensor words and their evaluation functors).

[L2]

The number of such words on four letters is the Catalan number $C_{3} = 5$ (Parenthesised tensor words of a fixed length are counted by the Catalan numbers).

Verification

technique · direct

1.1L1L2

Each displayed word uses the letters $A, B, C, D$ exactly once, contains no unit symbol, and differs only by the placement of parentheses, so each is a valid parenthesised tensor word of the kind counted in [L2].

2.1step 1.1L2

The list has five elements, matching [L2]. Therefore it is complete.

3.1step 2.1L1∎

The pentagon starts at $(((A \otimes B) \otimes C) \otimes D)$ and ends at $A \otimes (B \otimes (C \otimes D))$ , passing through the other three displayed bracketings exactly as the coherence diagram predicts.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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.