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ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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/n1/n is null

Example

The sequence (1/n)n≥1(1/n)_{n \ge 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/k1/k used in the completeness proof.

Facts & Assumptions

Given: A rational ε>0\varepsilon > 0.

[L1]

Archimedean property (The rationals are Archimedean).

[L2]

Order arithmetic: 0<1/n≤1/N0 < 1/n \le 1/N for n≥N≥1n \ge N \ge 1 (The rationals form a totally ordered field).

[L3]

Null definition (Null sequence).

Verification

technique · direct
1.1

Pick a natural N≥1N \ge 1 with 1/N<ε1/N < \varepsilon.

L1
2.1

For every n≥Nn \ge N: 0<1/n≤1/N<ε0 < 1/n \le 1/N < \varepsilon, so ∣1/n∣<ε|1/n| < \varepsilon.

step 1.1L2
3.1

(1/n)(1/n) is null.

step 2.1L3∎

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: 29 results over 13 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