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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The empty and zero-dimensional Morse cases
Example
The Morse definitions behave as expected on the empty manifold and on -manifolds.
Facts & Assumptions
Given: A smooth function on either the empty manifold or a -manifold.
On a -manifold every point is a nondegenerate critical point of index , while the empty -manifold has no critical points (The zero-dimensional Morse convention).
Verification
If , then there are no points to test. So has no critical points, and the Morse condition is vacuous.
If is a nonempty -manifold, [F1] says that every point of is a nondegenerate critical point of index . Hence every smooth function on is Morse.
In the same -dimensional case, excellence is exactly the condition that distinct points have distinct values, again by [F1].
Therefore the empty and zero-dimensional boundary cases agree with the stated Morse conventions.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
3 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
- Ralph L. Cohen, Bundles, Manifolds, and Homotopy (standard reference, not scraped)
- Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (standard reference, not scraped)