DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Common refinement equivalence of roofs

Definition

Two left roofs $(s : U \to X, f : U \to Y)$ and $(t : U^{'} \to X, g : U^{'} \to Y)$ are common-refinement equivalent if there are $a : V \to U$ and $b : V \to U^{'}$ such that $s a = t b \in S$ and $f a = g b$ . Only the composite $s a = t b$ is required to belong to $S$ ; neither refinement leg is separately required to do so. Left roofs have the orientation fixed in Left roof representing a localized morphism.

Depends on

Left roof representing a localized morphism

Used by

Roof equivalence is an equivalence relation Lemma

Dependency tree · two levels

2 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)