ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-05

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 Tricomi equation changes type across y = 0

Example

The Tricomi equation

$y u_{x x} + u_{y y} = 0$

is elliptic for $y > 0$ , hyperbolic for $y < 0$ , and parabolic on the line $y = 0$ .

Facts & Assumptions

Given: The principal part $y u_{x x} + u_{y y}$ .

[L1]

For a two-variable second-order principal part, the discriminant is $B^{2} - A C$ (The discriminant for a second-order equation in two variables).

[L2]

The elliptic/parabolic/hyperbolic trichotomy is only a pointwise second-order classification (Limits of the elliptic-parabolic-hyperbolic trichotomy).

Verification

technique · direct

1.1

Here $A = y$ , $B = 0$ , and $C = 1$ , so [L1] gives $Δ (y) = 0 - y \cdot 1 = - y$ .

2.1

If $y > 0$ , then $Δ < 0$ and the equation is elliptic; if $y < 0$ , then $Δ > 0$ and it is hyperbolic; and if $y = 0$ , then $Δ = 0$ and the principal part has rank one, so this pointwise change of sign is exactly the mixed-type behavior singled out in [L2].

L2step 1.1∎

Depends on

Used by

The elliptic-parabolic-hyperbolic trichotomy is not a global taxonomy Counterexample

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

Victor Ivrii, Partial Differential Equations (standard reference, not scraped)