DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

Distributive law between two monads

Definition

Let $(S, η^{S}, μ^{S})$ and $(T, η^{T}, μ^{T})$ be monads on a category $C$ . A distributive law of $S$ over $T$ is a natural transformation $λ : S T \Rightarrow T S$ satisfying

$λ \circ η^{S} T = T η^{S}, λ \circ S η^{T} = η^{T} S,$

and

$λ \circ μ^{S} T = T μ^{S} \circ λ S \circ S λ,$

$λ \circ S μ^{T} = μ^{T} S \circ T λ \circ λ T .$

Depends on

Monad on a category

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Cheng, Distributive laws for Lawvere theories, Definition 2.1 (standard reference, not scraped)