ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-09-01

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.

$1_{[0, 1]} * 1_{[0, 1]}$ is the tent function

Example

Let $f : = 1_{[0, 1]}$ on $R$ . Then

$(f * f) (x) = \int_{R} 1_{[0, 1]} (x - y) 1_{[0, 1]} (y) d y$

is the tent function

$(f * f) (x) = {\begin{cases} 0, & x \leq 0, \\ x, & 0 \leq x \leq 1, \\ 2 - x, & 1 \leq x \leq 2, \\ 0, & x \geq 2. \end{cases}$

Facts & Assumptions

Given: The indicator $f = 1_{[0, 1]}$ .

[L1]

$L^{1}$ convolution exists almost everywhere (If $f, g \in L^{1} (R^{n})$ , then $f * g$ exists almost everywhere, belongs to $L^{1}$ , and $∥ f * g ∥_{1} \leq ∥ f ∥_{1} ∥ g ∥_{1}$ ).

[L2]

The support of a convolution lies in the closure of the support sumset (The support of a convolution lies in the closure of the support sumset).

Verification

technique · direct

1.1

For fixed $x$ , the integrand is $1$ exactly when [L1, given, algebra] $y \in [0, 1] \cap [x - 1, x]$ . Therefore $(f * f) (x)$ is the length of that overlap interval.

L1givenalgebra

2.1

If $0 \leq x \leq 1$ , the overlap is $[0, x]$ , so $(f * f) (x) = x$ . If [step 1.1, algebra] $1 \leq x \leq 2$ , the overlap is $[x - 1, 1]$ , so $(f * f) (x) = 2 - x$ . For $x \leq 0$ or $x \geq 2$ , there is no overlap, so $(f * f) (x) = 0$ .

step 1.1algebra

3.1

Since $supp (f) = [0, 1]$ , [L2] predicts support inside [L2, step 2.1] $[0, 1] + [0, 1] = [0, 2]$ , exactly as the explicit computation shows.

L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)

1[0,1]∗1[0,1]1[0,1]​∗1[0,1]​ is the tent function