Alphabeta MathHow 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.
Normalized and unnormalized bars are homotopy equivalent
Statement
The quotient map from unnormalized bars to normalized bars is a chain-homotopy equivalence.
Proof
Given: The degenerate contractible subcomplex .
The standard degeneracy splitting gives the unnormalized complex as normalized representatives plus , and the quotient is projection onto the first summand.
The inclusion of normalized representatives is a chain-map section; the contraction of supplies a homotopy from its composite with the quotient to the identity. Hence the two maps are homotopy inverses.
Depends on
Used by
Dependency tree · two levels
5 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
- Weibel, §6.5 (standard reference, not scraped)