Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck 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=42+2=4 from the recursion

Example

With 2=σ(σ(0))2 = \sigma(\sigma(0)) and 4=σ(σ(σ(σ(0))))4 = \sigma(\sigma(\sigma(\sigma(0)))), the recursion gives 2+2=42 + 2 = 4, by unfolding the successor twice down to 2+0=22 + 0 = 2.

Facts & Assumptions

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

Verification

technique · direct
1.1

Write 2=σ(σ(0))2 = \sigma(\sigma(0)); then 2+2=2+σ(σ(0))2 + 2 = 2 + \sigma(\sigma(0)).

given
1.2

2+0=22 + 0 = 2, by m+0=mm + 0 = m.

given
2.1

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

step 1.1given
3.1

Hence 2+2=σ(σ(2))=σ(σ(σ(σ(0))))=42 + 2 = \sigma(\sigma(2)) = \sigma(\sigma(\sigma(\sigma(0)))) = 4.

step 2.1step 1.2∎

Depends on

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: 18 results over 10 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