Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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 unital separating real function algebra interpolates arbitrary values at two distinct points

Statement

If A⊆C(K,R) is unital and separates points, then for distinct x,y∈K and reals r,s there is g∈A with g(x)=r and g(y)=s.

Facts & Assumptions

Given: Distinct x,y∈K and r,s∈R.

[L1]

Separation gives h∈A with h(x)≠h(y); constants and affine combinations of members of A remain in A (A unital point-separating real subalgebra of C(K,R)).

Proof

technique · constructive
1.1

Choose h from [L1] and put g(z)=r+(s−r)(h(z)−h(x))/(h(y)−h(x)).

L1construct
2.1

The denominator is nonzero, g∈A, and direct substitution gives g(x)=r and g(y)=s.

step 1.1L1algebradischarge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 3 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