ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

Currying for sets of three variables

Example

Let $A = B = C = {0, 1}$ and $D = {0, 1, 2, 3}$ . Define

$f : A \times B \times C \to D, f (a, b, c) = a + b + c,$

where the sum is taken in the integers.

Facts & Assumptions

Given: The concrete map $f (a, b, c) = a + b + c$ on ${0, 1}^{3}$ .

[L1]

Currying and uncurrying are mutually inverse, and repeated currying is associative up to reassociation (Currying and uncurrying are mutually inverse).

Verification

technique · direct

1.1

Currying in the $C$ -variable sends $(a, b)$ to the function $c \mapsto a + b + c$ . Currying again in the $B$ -variable sends $a$ to the function $b \mapsto (c \mapsto a + b + c)$ .

givenalgebra

2.1

Uncurrying the result reverses those two assignments: from $a$ recover the map $b \mapsto (c \mapsto a + b + c)$ , then evaluate at $(b, c)$ to recover $a + b + c = f (a, b, c)$ .

step 1.1algebra

3.1

So the twice-curried and twice-uncurried maps return the original $f$ , exactly as Currying and uncurrying are mutually inverse predicts.

step 2.1L1∎

Depends on

Currying and uncurrying are mutually inverse

Used by

Nothing in the library uses this result yet.

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

S. Mac Lane, Categories for the Working Mathematician, 2nd ed., IV.6 (standard reference, not scraped)