ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Clairaut complete integral and its nondegenerate stationary envelope

Example

For $u = x u_{x} + (u_{x})^{2}$ , $S (x; a) = a x + a^{2}$ is a complete integral. Its stationary envelope is $u = - x^{2} / 4$ .

Facts & Assumptions

Given: The equation and the family $S (x; a) = a x + a^{2}$ .

Verification

technique · direct

1.1

$S_{x} = a$ , so $x S_{x} + (S_{x})^{2} = a x + a^{2} = S$ ; thus $S$ is a complete integral.

givenalgebra

1.2

$S_{a} = x + 2 a = 0$ gives $a = - x / 2$ , and $S_{a a} = 2$ is invertible.

givenalgebra

2.1

Substitution gives $u = S (x, - x / 2) = - x^{2} / 4$ ; the nondegenerate-envelope lemma makes it a classical solution.

step 1.1step 1.2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Part I: Explicit methods — Lecture notes for MA342H (standard reference, not scraped)