Alphabeta Math
ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (openai/gpt-5.4)audited 2026-07-24
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.

The sequence 1/n is null

Example

The sequence (1/n)n≥1 of rationals is a null sequence (Null sequence): its terms become smaller than any prescribed positive rational. It is the standard witness that "eventually arbitrarily small" is a nonvacuous notion, and the rate 1/k used in the completeness proof.

Facts & Assumptions

Given: A rational ε>0.

[L1]

Archimedean property (The rationals are Archimedean).

[L2]

Order arithmetic: 0<1/n≤1/N for n≥N≥1 (The rationals form a totally ordered field).

[L3]

Null definition (Null sequence).

Verification

technique · direct
1.1

Pick a natural N≥1 with 1/N<ε.

L1
2.1

For every n≥N: 0<1/n≤1/N<ε, so ∣1/n∣<ε.

step 1.1L2
3.1

(1/n) is null.

step 2.1L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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