ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

ex-inverting-a-quasi-isomorphism-by-a-reversed-roof.md

Example

Let $U = (Z \overset{2}{\to} Z)$ in degrees $- 1, 0$ and let $s : U \to Z / 2 [0]$ be reduction modulo two at degree zero. The inverse of $Q (s)$ is the roof $Z / 2 [0] \overset{s}{\leftarrow} U \overset{1}{\to} U$ .

Facts & Assumptions

Given: Let $U = (Z \overset{2}{\to} Z)$ in degrees $- 1, 0$ and let $s : U \to Z / 2 [0]$ be reduction modulo two at degree zero. The inverse of $Q (s)$ is the roof $Z / 2 [0] \overset{s}{\leftarrow} U \overset{1}{\to} U$ .

[F1]

A quasi-isomorphism is invertible in the derived category, represented by its reversed roof (The localization functor sends quasi isomorphisms to isomorphisms).

Verification

1.1givenalgebra

Multiplication by two on $Z$ is injective with cokernel $Z / 2$ . Thus $s$ is a quasi-isomorphism in all degrees, including the two boundary degrees and the zero terms elsewhere.

2.1F1step 1.1algebra∎

The inversion rule for localization identifies the displayed reversed roof with $Q (1_{U}) Q (s)^{- 1}$ . Composing on either side with $Q (s)$ gives the corresponding identity, proving the asserted inverse explicitly.

Depends on

The localization functor sends quasi isomorphisms to isomorphisms

Used by

Nothing in the library uses this result yet.

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

13.11.1–13.11.6 (standard reference, not scraped)