ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

The state monad threads a fixed state set through a computation

Example

Fix a set $S$ . The product–exponential adjunction induces the state monad

$T (X) = (X \times S)^{S} .$

An element of $T (X)$ takes an initial state and returns a value together with a new state.

Facts & Assumptions

Given: A fixed set $S$ .

[L1]

The functor $- \times S$ is left adjoint to $(-)^{S}$ on $S e t$ (Currying gives the adjunction $- \times A ⊣ (-)^{A}$ in $S e t$ ).

[L2]

Every adjunction induces a monad on the domain of its left adjoint (Every adjunction induces a monad on the domain of its left adjoint).

Verification

technique · direct

1.1L1L2

Applying [L2] to [L1] gives $T (X) = (X \times S)^{S}$ .

2.1L1L2step 1.1

The unit is $η_{X} (x) (s) = (x, s)$ . For $k \in T (T X)$ , write $k (s) = (c, s^{'})$ with $c : S \to X \times S$ and define $μ_{X} (k) (s) = c (s^{'})$ : the outer computation selects the inner computation and intermediate state, then the inner computation runs at that state.

3.1L2step 2.1∎

Evaluating at any initial state, either unit inserts a computation that returns its input without changing state, and either associativity bracketing runs the same outer, selected middle, and selected inner computations in order. If $S = \emptyset$ , every relevant function has empty domain and the same equations hold vacuously.

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

E. Riehl, Category Theory in Context, 2nd ed., Example 5.2.11(iii) (standard reference, not scraped)