Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-12
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.

Factorial is both lambda-definable and partial recursive

Example

Let fact⁡(0)=1 and fact⁡(n+1)=(n+1)fact⁡(n). This is the standard primitive-recursive presentation of factorial. The lambda-definability theorem therefore supplies a closed lambda term F such that F n‾ beta-reduces to fact⁡(n)‾ for each n.

Facts & Assumptions

Given: The factorial function displayed above.

[L1]

A partial function is lambda-definable if and only if it is partial recursive, by A partial function is lambda-definable if and only if it is partial recursive.

Verification

technique · direct
1.1

The displayed recursion for factorial uses the initial value 1 and the recursive update (n,u)↦(n+1)u, so factorial is a primitive-recursive total function.

given
2.1

Every primitive-recursive function is partial recursive. Applying [L1] to step 1.1 therefore gives a closed lambda term F representing factorial on Church numerals.

L1step 1.1
3.1

Step 1.1 gives the partial-recursive side and step 2.1 gives the lambda-definable side, so factorial is a concrete example of the equivalence asserted in [L1].

L1step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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