Alphabeta Math
CorollaryStatement: AI-generatedProof: AI-generatedprecheck 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.

Over an infinite integral domain, equal polynomial functions come from equal polynomials

Statement

Let D be an infinite integral domain. If f,g∈D[x] satisfy f(a)=g(a) for every a∈D, then f=g as formal polynomials.

Facts & Assumptions

Given: An infinite integral domain D and polynomials f,g∈D[x] with equal values at every element of D.

[L1]

A nonzero polynomial of degree n over an integral domain has at most n distinct roots (A nonzero polynomial of degree n over an integral domain has at most n distinct roots).

Proof

technique · contradiction
1.1

Suppose for contradiction that h=f−g is nonzero, and let n=deg⁡h; the assumed equality of values makes every element of D a root of h.

assume-contragiven
2.1

Since D is infinite it contains more than n distinct elements, contradicting [L1]; hence h=0 and f=g.

step 1.1L1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

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