ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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.

Displacement data versus velocity data in one dimension

Example

Let $c > 0$ , $a > 0$ , $t \geq 0$ , and let $u_{0} \in C_{c}^{2} (R)$ and $u_{1} \in C_{c}^{1} (R)$ , with $supp u_{0}, supp u_{1} \subseteq [- a, a]$ . The two terms of the classical d'Alembert formula of d'Alembert's formula and uniqueness in one dimension behave differently: (i) pure displacement, $u_{1} = 0$ : $u (x, t) = \frac{1}{2} (u_{0} (x - c t) + u_{0} (x + c t))$ is the sum of two half-amplitude copies of the profile translating rigidly at speed $c$ ; each component preserves its own values and the support lies in $[- a - c t, a + c t]$ . (ii) pure velocity, $u_{0} = 0$ : $u (x, t) = \frac{1}{2 c} \int_{x - c t}^{x + c t} u_{1}$ is $t$ times the interval average of $u_{1}$ for $t > 0$ , and hence is an integral over a growing interval: where the whole support of $u_{1}$ lies inside the interval the value is the constant $\frac{1}{2 c} \int u_{1}$ , and the profile is smoothed by integration rather than generally undergoing a rigid translation. For the bounded indicator extension $u_{1} = 1_{[- a, a]}$ , this integral is the expanding plateau of A compactly supported velocity datum produces an expanding interval; that indicator example describes the formula extension, while the present Cauchy-problem data are classical. Both classical solutions are supported in $[- a - c t, a + c t]$ , consistent with The one-dimensional value depends on the characteristic interval.

Facts & Assumptions

Given: a speed $c > 0$ , $a > 0$ , compactly supported data $u_{0} \in C_{c}^{2} (R)$ , $u_{1} \in C_{c}^{1} (R)$ with supports in $[- a, a]$ , and the classical d'Alembert solution $u$ of d'Alembert's formula and uniqueness in one dimension.

[F1]

For admissible data the d'Alembert solution is $u (x, t) = \frac{1}{2} (u_{0} (x - c t) + u_{0} (x + c t)) + \frac{1}{2 c} \int_{x - c t}^{x + c t} u_{1} (y) d y$ (d'Alembert's formula and uniqueness in one dimension, The d'Alembert expression attains both initial data).

[F2]

The value at $(x, t)$ depends on the data only through their restrictions to $[x - c t, x + c t]$ (The one-dimensional value depends on the characteristic interval).

[F3]

For $u_{1} = 1_{[- a, a]}$ the integral extension is the expanding plateau computed in A compactly supported velocity datum produces an expanding interval.

Verification

1.1F1algebra

Pure displacement. Setting $u_{1} = 0$ in [F1] leaves $u (x, t) = \frac{1}{2} u_{0} (x - c t) + \frac{1}{2} u_{0} (x + c t)$ . Each summand is a rigid translate of the half-amplitude profile $u_{0} / 2$ . Their supports lie in the translates $[- a + c t, a + c t]$ and $[- a - c t, a - c t]$ , so the total support lies in $[- a - c t, a + c t]$ . Where the two profiles overlap they add, and can reinforce or cancel; preservation of amplitude is a claim about the individual translating summands.

1.2F1F3algebra

Pure velocity. Setting $u_{0} = 0$ leaves $u (x, t) = \frac{1}{2 c} \int_{x - c t}^{x + c t} u_{1}$ , the overlap integral of the datum with the interval $[x - c t, x + c t]$ , which equals the constant $\frac{1}{2 c} \int u_{1}$ whenever $supp u_{1} \subseteq [x - c t, x + c t]$ ; for $u_{1} \in C_{c}^{1}$ the profile gains one derivative and generally changes shape through integration rather than translating a fixed profile. The bounded indicator extension in [F3] has the exact plateau computed there, though the present classical solution claim uses the stated $C_{c}^{1}$ data.

2.1F1F2algebra∎

Support. In both cases the data vanish outside $[- a, a]$ , so by [F1] the value is zero unless $[x - c t, x + c t] \cap [- a, a] \neq \emptyset$ , that is unless $∣ x ∣ \leq a + c t$ ; continuity of the compactly supported data makes the value zero also at $∣ x ∣ = a + c t$ ; this is the one-dimensional instance of the domain of dependence [F2].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

John K. Hunter, Notes on Partial Differential Equations (revised 18 June 2014, UC Davis) (standard reference, not scraped)
Jared Speck, MIT 18.152 Introduction to Partial Differential Equations, Class Meeting #10: Introduction to the Wave Equation (Fall 2011) (standard reference, not scraped)