fl2.md

Setoid

1. S.equals(a, a) = true (reflexivity)
2. S.equals(a, b) = S.equals(b, a) (symmetry)
3. If S.equals(a, b) and S.equals(b, c), then S.equals(a, c) (transitivity)

Semigroup

1. S.concat(S.concat(a, b), c) is equivalent to S.concat(a, S.concat(b, c)) (associativity)

Monoid

1. M.concat(a, M.empty()) is equivalent to a (right identity)
2. M.concat(M.empty(), a) is equivalent to a (left identity)

Functor

1. F.map(x => x, a) is equivalent to a (identity)
2. F.map(x => f(g(x)), a) is equivalent to F.map(f, F.map(g, a)) (composition)

Apply

1. A.ap(A.ap(A.map(f => g => x => f(g(x)), a), b), c) is equivalent to A.ap(a, A.ap(b, c)) (composition)

Applicative

1. A.ap(A.of(x => x), a) is equivalent to a (identity)
2. A.ap(A.of(f), A.of(x)) is equivalent to A.of(f(x)) (homomorphism)
3. A.ap(a, A.of(x)) is equivalent to A.ap(A.of(f => f(x)), a) (interchange)

Foldable

1. F.reduce(f, seed, a) is equivalent to toArray(a).reduce(f, seed)
2. toArray derivable as a => F.reduce((r, i) => r.concat([i]), [], a)

Traversable

1. t(T.sequence(A1.of, a)) is equivalent to T.sequence(A2.of, T.map(t, a)) where t is a natural transformation from A1 to A2 (naturality)
2. T.sequence(Id.of, T.map(Id.of, a)) is equivalent to Id.of(a) (identity)
3. T.sequence(Compose.of, T.map(Compose.of, a)) is equivalent to Compose.of(A1.map(x => T.sequence(A2.of, x), T.sequence(A1.of, a))) (composition)

Chain

1. C.chain(g, C.chain(f, a)) is equivalent to C.chain(x => C.chain(g, f(x)), a) (associativity)

Monad

1. M.chain(f, M.of(x)) is equivalent to f(x) (left identity)
2. M.chain(M.of, a) is equivalent to a (right identity)

Extend

1. E.extend(f, E.extend(g, a)) is equivalent to E.extend(x => f(E.extend(g, x)), a)

Comonad

1. C.extend(C.extract, a) is equivalent to a
2. C.extract(C.extend(f, a)) is equivalent to f(a)
3. C.extend(f, a) is equivalent to C.map(f, C.extend(x => x, a))

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