LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.
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Roof equivalence is an equivalence relation

Statement

For any two objects $X, Y$ of a category with a two-sided multiplicative system $S$ , common refinement is an equivalence relation on left roofs from $X$ to $Y$ .

Facts & Assumptions

Given: For any two objects $X, Y$ of a category with a two-sided multiplicative system $S$ , common refinement is an equivalence relation on left roofs from $X$ to $Y$ .

[F1]

The common-refinement conditions are equality of the composite denominators in $S$ and equality of the numerators (Common refinement equivalence of roofs).

[F2]

The two Ore axioms and the two cancellation directions hold for the given multiplicative system (Multiplicative system in a category).

Proof

1.1

For a roof $(s, f)$ , take both refinement legs to be its vertex identity. This gives reflexivity. Interchanging the two refinement legs gives symmetry. These arguments also cover identity roofs and coincident vertices.

F1given

1.2

For transitivity suppose $(s, f) \sim (t, g)$ via $a, b$ and $(t, g) \sim (u, h)$ via $c, d$ . Thus $r = s a = t b \in S$ and $q = t c = u d \in S$ , while $f a = g b$ and $g c = h d$ . Apply Ore to $r$ and $q$ : obtain $v$ in $S$ and $w$ with $r v = q w \in S$ .

F1F2

2.1

Now $t b v = t c w$ . Cancellation of the postcomposed denominator $t$ supplies $e \in S$ such that $b v e = c w e$ . Therefore $s a v e = u d w e = r v e \in S$ and $f a v e = g b v e = g c w e = h d w e$ . The legs $a v e, d w e$ witness $(s, f) \sim (u, h)$ . Only $v, e$ , and the displayed composite denominator were asserted to lie in $S$ .

F2step 1.2algebra∎

Depends on

Used by

Composition of roofs is well defined Lemma

Cited to discharge well-definedness by Common refinement equivalence of roofs.

Dependency tree · two levels

5 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.

Sources

10.3.1–10.3.14, pp. 379–384 (standard reference, not scraped)