Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-06
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.

The functor of points of the affine line

Example

For a commutative ring k and a k-algebra R, the relative affine line satisfies Ak1(R)=Hom⁡k-Alg(k[t],R)≅R, naturally in R.

Facts & Assumptions

Given: A commutative ring k and a k-algebra R.

[F1]

A k-algebra homomorphism from k[t] is uniquely determined by the image of t (Universal property of R[x]: a coefficient homomorphism and the image of x determine a unique ring homomorphism).

Verification

technique · direct
1.1given

Send a k-algebra map α:k[t]→R to α(t)∈R.

1.2F1

Given r∈R, [F1] supplies the unique map with t↦r.

2.1step 1.1step 1.2∎

The assignments are inverse and commute with postcomposition, so the bijection is natural.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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