ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-31‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The power-set functor is lax monoidal but not strong

Example

Let $P : S e t \to S e t$ be the covariant power-set functor: on a function $f : X \to Y$ , it sends $A \subseteq X$ to its direct image $f (A) \subseteq Y$ . For sets $X, Y$ , define

$P_{2} : P (X) \times P (Y) ⟶ P (X \times Y), (A, B) ⟼ A \times B,$

and let

$P_{0} : 1 ⟶ P (1)$

pick the unique subset ${*}$ of the singleton set.

Facts & Assumptions

Given: The power-set construction and the cartesian monoidal structure on $S e t$ .

[L1]

$P (X)$ is the set of all subsets of $X$ (The power set $P (x) = {z : z \subseteq x}$ ).

[L2]

Lax, strong, and strict monoidal functors are distinguished by their structure maps and whether those maps are isomorphisms (Lax, strong, and strict monoidal functors).

[L3]

$S e t$ is cartesian monoidal (Set, Cat, and every complete category are cartesian monoidal).

Verification

technique · direct

1.1

Direct images preserve identities and composition, so the displayed action on functions makes $P$ a functor. By [L1] and [L3], the map $P_{2}$ is well typed: if $A \subseteq X$ and $B \subseteq Y$ , then $A \times B \subseteq X \times Y$ . For functions $f : X \to X^{'}$ and $g : Y \to Y^{'}$ , one has $(f \times g) (A \times B) = f (A) \times g (B)$ , so $P_{2}$ is natural. The unit map $P_{0}$ is also well typed because a singleton set has exactly one subset equal to itself.

L1L3algebra

2.1

For subsets $A \subseteq X$ , $B \subseteq Y$ , and $C \subseteq Z$ , both sides of the lax associativity axiom send $((A, B), C)$ to the subset of $X \times (Y \times Z)$ consisting of all $(x, (y, z))$ with $x \in A$ , $y \in B$ , and $z \in C$ . The left and right unit axioms likewise send $(*, A)$ and $(A, *)$ back to $A$ . Hence $P$ is lax monoidal.

step 1.1L2L3

2.2

Take $X = Y = {0, 1}$ . The diagonal subset $Δ = {(0, 0), (1, 1)} \subseteq X \times Y$ is not of the form $A \times B$ for subsets $A \subseteq X$ and $B \subseteq Y$ . So $P_{2}$ is not surjective, hence not an isomorphism.

L1step 1.1

3.1

Therefore the power-set functor is lax monoidal but not strong.

step 2.1step 2.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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.