Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-25
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.

1+1=2 from the recursion

Example

With 1=σ(0) and 2=σ(1)=σ(σ(0)), the recursive definition of addition gives 1+1=2. This is the archetypal proof by unfolding a recursive definition.

Facts & Assumptions

Given: addition m+0=m, m+σ(n)=σ(m+n) (Addition of natural numbers); numerals 1=σ(0), 2=σ(1) (The natural numbers N (von Neumann)).

Verification

technique · direct
1.1

By definition 1=σ(0) and 2=σ(1)=σ(σ(0)).

given
1.2

1+0=1, by m+0=m with m=1.

given
2.1

1+1=1+σ(0)=σ(1+0), by m+σ(n)=σ(m+n) with m=1, n=0.

step 1.1given
3.1

Hence 1+1=σ(1+0)=σ(1)=2.

step 2.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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