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

1.1

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.

givenalgebra

2.1

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.

F1step 1.1algebra∎

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)