ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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 end formula checked by hand against natural transformations on the walking arrow

Example

Let $J$ be the walking arrow, with objects $0$ and $1$ and one non-identity morphism $u : 0 \to 1$ , and let $F, G : J \to S e t$ be

$F (0) = {p, q}, F (1) = {r}, F (u) (p) = F (u) (q) = r;$ $G (0) = {a, b}, G (1) = {c, d}, G (u) (a) = G (u) (b) = c .$

The end $\int_{j} S e t (F j, G j)$ is computed here twice, once from the equalizer description and once by listing the natural transformations $F \Rightarrow G$ , and the two answers are matched element for element.

Facts & Assumptions

Given: The walking arrow $J$ and the two functors $F, G$ displayed above.

[F3]

Sets as objects and functions as morphisms form a large locally small category $S e t$ (Sets and functions form the large locally small category $S e t$ ).

[F1]

An end of $T$ is a terminal object of the category of wedges over $T$ and a coend an initial object of the category of cowedges under $T$ ; in short, an end is a terminal wedge and a coend an initial cowedge (The end and the coend of a functor $C^{o p} \times C \to D$ ).

[F2]

A natural transformation $α : F \Rightarrow G$ is a family $α_{A} : F A \to G A$ such that every $f : A \to B$ satisfies the naturality equation $G f \circ α_{A} = α_{B} \circ F f$ (Natural transformation and its components).

[L2]

For a small index category and a target where the two displayed products exist, an end is the equalizer of two products, the first indexed by the objects and the second by the morphisms, the two parallel maps being built from $T (1_{c}, f)$ and $T (f, 1_{c^{'}})$ (An end is the equalizer of two products, and a coend the coequalizer of two coproducts).

[L1]

For a small source category and a locally small target, the set of natural transformations is an end of the hom-bifunctor of the values, the terminal wedge being evaluation (For a small source category, the set of natural transformations is an end of the hom-bifunctor of the values).

Verification

technique · direct

1.1

The integrand is $H (j_{1}, j_{2}) = S e t (F (j_{1}), G (j_{2}))$ , with values $H (0, 0)$ of size four, $H (1, 1)$ of size two, $H (0, 1)$ of size four and $H (1, 0)$ of size two. The object-indexed product is $S e t (F (0), G (0)) \times S e t (F (1), G (1))$ , with eight elements; the morphism-indexed product has one factor for each of $1_{0}$ , $1_{1}$ and $u$ , namely $S e t (F (0), G (0))$ , $S e t (F (1), G (1))$ and $S e t (F (0), G (1))$ .

F3given

2.1

By [L2] the two parallel maps send a pair $(ϕ_{0}, ϕ_{1})$ to the families whose $f$ -component is $G (f) \circ ϕ_{dom f}$ and $ϕ_{cod f} \circ F (f)$ respectively. At $1_{0}$ and at $1_{1}$ both components are $ϕ_{0}$ and $ϕ_{1}$ , so those factors impose nothing; at $u$ the condition is $G (u) \circ ϕ_{0} = ϕ_{1} \circ F (u)$ . Since $G (u)$ sends both $a$ and $b$ to $c$ , the left side is the constant function at $c$ , and the right side is the constant function at $ϕ_{1} (r)$ ; so the condition holds exactly when $ϕ_{1} (r) = c$ . The equalizer is therefore the subset of the eight-element product on which $ϕ_{1} (r) = c$ , which leaves $ϕ_{0}$ free: its elements are the four pairs with $ϕ_{0}$ one of $p, q \mapsto a, a$ ; $a, b$ ; $b, a$ ; $b, b$ and $ϕ_{1} (r) = c$ .

F1L2step 1.1

3.1

Listing the natural transformations independently gives the same four. By [F2] a natural transformation $α : F \Rightarrow G$ is a pair $α_{0} : {p, q} \to {a, b}$ and $α_{1} : {r} \to {c, d}$ with $G (u) α_{0} = α_{1} F (u)$ , which as in step 2.1 says exactly $α_{1} (r) = c$ ; there are four choices of $α_{0}$ and each extends in exactly one way.

F2step 1.1

4.1

The two lists agree pair for pair, and by [L1] they must: the end of $H$ is $Nat (F, G)$ with the evaluation wedge, and by [L2] it is the equalizer computed in step 2.1. So the end has four elements on this diagram, and the identification is the identity on the four pairs.

L1step 2.1step 3.1∎

Remarks

The identity morphisms of $J$ contribute factors to the morphism-indexed product and impose nothing, which is visible here rather than argued in general: their equalising condition is $ϕ_{j} = ϕ_{j}$ .

Had $G (u)$ been injective rather than constant, the condition of step 2.1 would have forced $ϕ_{0}$ to be constant as well, and the end would have had fewer elements. Nothing in the equalizer description privileges one of the two parallel maps, and both were written out.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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.