Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01
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.

A map vacuously transverse to a submanifold it avoids

Example

The constant map F:R→R2, F(t)=(0,1), is transverse to the x-axis Z=R×{0} because F(R)∩Z=∅.

Facts & Assumptions

Given: The constant map F(t)=(0,1) and the submanifold Z=R×{0}.

[F1]

A smooth map is transverse to a submanifold when the tangent-space condition holds at every point of the preimage, and this is vacuous when the preimage is empty (A smooth map transverse to an embedded submanifold).

Verification

technique · direct
1.1

The image of F is the single point (0,1), which does not lie on the x-axis. Hence F−1(Z)=∅.

given
2.1

By [F1], the transversality condition has no points to check when the preimage is empty. Therefore F⋔Z.

F1step 1.1
3.1

This is a concrete vacuous transversality example.

step 2.1∎

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