CorollaryStatement: AI-generatedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $f,g\in D[x]$ satisfy $f(a)=g(a)$ for every $a\in D$ , then $f=g$ as formal polynomials.

Facts & Assumptions

Given: An infinite integral domain $D$ and polynomials $f,g\in D[x]$ 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

A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Neil Donaldson, Math 120B Notes, Sections 22-23 (standard reference, not scraped)