Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

=== Goal: combine O with monads M (State, Prob, Reader) so that O ∘ M or M ∘ O is a monad. ===

Definition 9.1 (Distributive law). A natural transformation λ : M O ⇒ O M satisfying coherence equations that make (O ∘ M, η^{OM}, μ^{OM}) a monad.

Coherence equations:
* λ ∘ η^M_O = O η^M etc — refer to Beck's distributive law conditions.

When does such λ exist? Depends on semantics:
* For M = Reader R: often we can define λ_A : (r ↦ O A) → O (r ↦ A) by pushing r inside O (if O is parametric). So Reader frequently distributes over O.
* For M = State: need to decide whether meta-levels share state or have private states. Two meaningful choices: - Shared state semantics: λ moves state inside O so that nested structures access same global state — possible if O is compatible. - Isolated state semantics (monad transformer): use StateT S O (state transformer) to manage stacking.
* For M = Dist (probability): a distributive law exists when O preserves convex combinations or is measurable; in general requires O to be measurable or a convex functor.

Practical pattern: implement StateT O or ProbT O (i.e., monad transformer that stacks O with other effects). The transformer OT may or may not exist generally; constructing according to standard monad transformer recipes is typical.