ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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 pentagon checked for cartesian products

Example

For sets $W, X, Y, Z$ , the pentagon for the cartesian monoidal structure compares the two canonical maps from $(((W \times X) \times Y) \times Z)$ to $W \times (X \times (Y \times Z))$ .

Facts & Assumptions

Given: The cartesian monoidal structure on a category with finite products.

[L1]

Finite products make a category monoidal, with associator given by the canonical rebracketing isomorphism (A category with finite products is monoidal).

Verification

technique · direct

1.1

Start with an element $(((w, x), y), z)$ . Along the left side of the pentagon, the first associator gives $((w, x), (y, z))$ and the second gives $(w, (x, (y, z)))$ .

1.2

Along the right side, the first associator gives $((w, (x, y)), z)$ , the second gives $(w, ((x, y), z))$ , and the third gives $(w, (x, (y, z)))$ .

2.1

Both composites send $(((w, x), y), z)$ to the same element $(w, (x, (y, z)))$ , so the pentagon commutes in this concrete cartesian example.

step 1.1step 1.2∎

Depends on

A category with finite products is monoidal

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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.