Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passaudited 2026-08-11
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 polynomial x2+1 is irreducible over R

Example

The polynomial x2+1 is irreducible in R[x].

Facts & Assumptions

Given: The polynomial f=x2+1∈R[x].

[L1]

A quadratic over a field is irreducible exactly when it has no root in that field (A polynomial of degree two or three over a field is irreducible exactly when it has no root in the field).

[L2]

The real numbers form an ordered field, so every square is nonnegative and 1>0 (The reals form a totally ordered field).

Verification

technique · direct
1.1

For every real a, [L2] gives a2≥0 and hence a2+1>0.

givenL2algebra
2.1

Thus f has no real root, and [L1] makes it irreducible in R[x].

step 1.1L1∎

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