DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-07-31

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.

Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise

A continuous map $r : X \to A$ is a retraction of $X$ onto $A$ if

$r \circ i = {id}_{A},$

equivalently, if $r (a) = a$ for every $a \in A$ . When such an $r$ exists, $A$ is a retract of $X$ .

The subspace $A$ is a deformation retract of $X$ if there are a retraction $r : X \to A$ and a homotopy

$H : {id}_{X} ≃_{A} i \circ r .$

Thus $H (x, 0) = x$ , $H (x, 1) = i (r (x))$ , and $H (a, t) = a$ for all $a \in A$ and $t \in I$ (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints). The pair $(r, H)$ is a deformation retraction.

Some sources call the pointwise-fixed condition a strong deformation retract. In this library the word deformation retract always includes it.

Depends on