dev.constructive.eo.laws.typeclass

Laws for dev.constructive.eo.compose.AssociativeFunctor. Operational, not category-theoretic: the "associative" part refers to packing Xo / Xi into Z, not monoidal structure.

• A1 compose-modify distributes: composed.modify(f)(s) == outer.modify(inner.modify(f))(s).
• A2 compose-modify identity: a Cogen-free specialisation of A1.

Both require ForgetfulFunctor[F] (every cats-eo AssociativeFunctor carrier also ships one).

C1 — path independence. An Iso → Affine via Tuple2 vs via Either should be modify-equivalent.

C2 — chain preserves get whenever both ends have an Accessor.

Law equations for any ForgetfulFunctor[F] instance — the usual two functor laws, stated in carrier-carrier form:

• map(id) == id
• map(g) ∘ map(f) == map(f andThen g)

Holds for every carrier EO uses: Tuple2, Either, Affine, ModifyF, Direct, Forget[F], MultiFocus[F]. The law trait is parameterised so downstream adding a new carrier can witness its ForgetfulFunctor instance here.

Equality is structural — if the carrier wraps a function (as ModifyF does), discipline-checking this law requires an extensional comparison. See dev.constructive.eo.laws.data.ModifyFLaws for that carrier-specific phrasing.

Law equations for any ForgetfulTraverse[F, Applicative] — cats' Traverse laws restated for the two-parameter carrier, at fixed effects:

• identity: traverse[Id](fa)(a => a) == fa;
• sequential composition: traversing under Nested[List, Option] equals traversing under List, then each branch under Option;
• parallel composition: traversing under Tuple2K[List, Vector] equals the pair of the two separate traversals;
• multi-shot agreement: traversing under List gives cats' cartesian product Traverse[List].traverse(foci)(f) over the carrier's foci (read off ForgetfulFold), each choice written back into fa.

The effects are deliberately MULTI-SHOT (List, Vector resume every continuation once per branch; Tuple2K once per component): a traverse that threads one mutable result through map2 passes every single-shot law (Id, Option, Const) yet hands all branches the same result. The last law is the one that pins a branch to its values; the composition laws compare the carrier only with itself.

The [_ <: Distributive] flavour (ModifyF) is not covered: it collapses to a tautology at Id.