ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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.

Special values of the real Gamma and Beta functions

Example

The real Gamma and Beta functions have the values

$Γ (1) = 1, Γ (2) = 1, Γ (3) = 2, Γ (1 / 2) = \sqrt{π}, Γ (3 / 2) = \frac{\sqrt{π}}{2}, B (1, 1) = 1, B (2, 3) = \frac{1}{12} .$

Facts & Assumptions

Given: The displayed positive arguments.

[F1]

For every $s > 0$ , $Γ (s + 1) = s Γ (s)$ , and $Γ (1) = 1$ (The real Gamma functional equation $Γ (s + 1) = s Γ (s)$ ).

[F2]

$Γ (1 / 2) = \sqrt{π}$ ( $Γ (1 / 2) = \sqrt{π}$ from the Gaussian integral).

[F3]

For $p, q > 0$ , $B (p, q) = Γ (p) Γ (q) / Γ (p + q)$ (The real Beta--Gamma identity).

Verification

technique · direct

1.1

Repeated use of [F1] gives $Γ (1) = 1$ , $Γ (2) = 1$ , and $Γ (3) = 2$ .

F1algebra

1.2

Fact [F2] and [F1] give $Γ (1 / 2) = \sqrt{π}$ and $Γ (3 / 2) = (1 / 2) \sqrt{π}$ .

F1F2algebra

2.1

Substituting the integer values into [F3] gives $B (1, 1) = 1$ and $B (2, 3) = 1 \cdot 2 / 24 = 1 / 12$ .

step 1.1F3algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §§1.2, 1.4, 2.2 (standard reference, not scraped)