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.

2+2=4 from the recursion

Example

With 2=σ(σ(0)) and 4=σ(σ(σ(σ(0)))), the recursion gives 2+2=4, by unfolding the successor twice down to 2+0=2.

Facts & Assumptions

Given: addition m+0=m, m+σ(n)=σ(m+n) (Addition of natural numbers); 2=σ(σ(0)), 4=σ(σ(σ(σ(0)))) (The natural numbers N (von Neumann)).

Verification

technique · direct
1.1

Write 2=σ(σ(0)); then 2+2=2+σ(σ(0)).

given
1.2

2+0=2, by m+0=m.

given
2.1

2+σ(σ(0))=σ(2+σ(0))=σ(σ(2+0)), applying m+σ(n)=σ(m+n) twice.

step 1.1given
3.1

Hence 2+2=σ(σ(2))=σ(σ(σ(σ(0))))=4.

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