ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Laplace, heat, and wave equations have elliptic, parabolic, and hyperbolic principal symbols

Example

The model operators

$Δ = u_{x x} + u_{y y}, \partial_{t} - Δ_{x}, \partial_{t t} - \partial_{x x}$

represent the elliptic, parabolic, and hyperbolic cases respectively.

Facts & Assumptions

Given: The principal parts of Laplace, heat, and one-space-dimensional wave operators.

[L1]

Elliptic, hyperbolic, and parabolic type are read from the principal symbol definitions (Elliptic, hyperbolic, and parabolic principal symbols).

[L2]

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

Verification

technique · direct

1.1

For $u_{x x} + u_{y y}$ one has $A = C = 1$ and $B = 0$ , so [L2] gives $Δ = - 1 < 0$ and the Laplace operator is elliptic; for $u_{t t} - u_{x x}$ , the principal polynomial in $(τ, ξ)$ is $τ^{2} - ξ^{2}$ , which has two distinct real roots $τ = \pm ξ$ for $ξ \neq 0$ , so [L1] makes it hyperbolic.

L1L2

2.1

For the heat operator $\partial_{t} - Δ_{x}$ , the spatial quadratic form is $∣ ξ ∣^{2}$ , which is positive definite, while the time derivative is first order, so [L1] identifies it as parabolic.

L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)