Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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.

Relative CW inclusions are cofibrations

Statement

If (X,A) is a relative CW complex, then A↪X has the homotopy extension property; in particular it is a cofibration.

Facts & Assumptions

Given: A map g:X→Y and a homotopy H:A×I→Y with H(a,0)=g(a).

Proof

technique · direct
1.1

On each attached disk, the already-defined map on Sn−1×I∪Dn×{0} extends over Dn×I because this subspace is a deformation retract of a collar-containing neighborhood in the disk cylinder.

givenconstruct
2.1

Induct over attachment stages, choosing these extensions; their restrictions agree on prior stages. The weak topology of the relative CW construction makes the assembled map X×I→Y continuous and it extends H.

step 1.1discharge-construct∎

Depends on

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