Pipeline-generated

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 result · all verified · 1 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs; all 1 also cleared it.

Pointwise Extended-Real Function Products: Examples

1 · Prerequisites

2 · Summary

The worked example checks all four zero-infinity pairs, the four sign outcomes for a nonzero infinite factor paired with a nonzero finite value, and ordinary finite multiplication. See Zero, sign, and finite cases for the pointwise function product.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

Zero, sign, and finite cases for the pointwise function product

Example

Let $X = {x_{1}, \dots, x_{10}}$ and define $f, g : X \to \overline{R}$ by the following values. The last column is the value of the function-level product from A total pointwise product for extended-real-valued functions.

$x$	$f (x)$	$g (x)$	$(f ⊙_{X} g) (x)$
$x_{1}$	$0$	$+ \infty$	$0$
$x_{2}$	$0$	$- \infty$	$0$
$x_{3}$	$+ \infty$	$0$	$0$
$x_{4}$	$- \infty$	$0$	$0$
$x_{5}$	$+ \infty$	$2$	$+ \infty$
$x_{6}$	$+ \infty$	$- 2$	$- \infty$
$x_{7}$	$- \infty$	$2$	$- \infty$
$x_{8}$	$- \infty$	$- 2$	$+ \infty$
$x_{9}$	$2$	$- 3$	$- 6$
$x_{10}$	$0$	$5$	$0$

Facts & Assumptions

Given: The set $X$ and the functions $f, g$ with values in the table above. Verify the product at all ten points, including the four zero-infinity pairs and the sign cases.

[F1]

A total pointwise product for extended-real-valued functions: The value is zero if either input is zero; it is the real product when both inputs are finite and nonzero; otherwise both inputs are nonzero and at least one is infinite, and the output infinity is determined by whether the signs agree.

Verification

Proof technique: apply the definition separately to the zero, signed-infinite, and finite rows.

1.1givenF1cases

At $x_{1}, x_{2}, x_{3}, x_{4}$ , one factor is zero, so [F1] gives respectively $0, 0, 0, 0$ . These are the four ordered pairs $(0, + \infty)$ , $(0, - \infty)$ , $(+ \infty, 0)$ , and $(- \infty, 0)$ .

1.2givenF1cases

At $x_{5}, x_{6}, x_{7}, x_{8}$ , both factors are nonzero and one is infinite. At $x_{5}$ and $x_{8}$ the signs agree, giving $+ \infty$ ; at $x_{6}$ and $x_{7}$ the signs differ, giving $- \infty$ . This checks all four sign combinations of a nonzero infinite value with a nonzero finite value.

1.3givenF1algebra

At $x_{9}$ both values are finite and nonzero, so the result is the real product $2 (- 3) = - 6$ . At $x_{10}$ one finite factor is zero, so the zero clause gives $0$ , also the real product $0 \cdot 5$ .

2.1step 1.1step 1.2step 1.3∎

The values from steps 1.1, 1.2, and 1.3 cover every point of $X$ and give the output tuple $(0, 0, 0, 0, + \infty, - \infty, - \infty, + \infty, - 6, 0)$ in the listed order.