From 448b32385bd0a1ba7310b8c83fae72963bbc3295 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Tom=C3=A1s=20Ruiz-L=C3=B3pez?= <truizlop@gmail.com>
Date: Tue, 4 Dec 2018 10:48:58 +0100
Subject: [PATCH] Translate chapter 3.9 into Swift

---
 3.9-algebras-for-monads.md | 149 +++++++++++++++++++++++++++++++++++++
 1 file changed, 149 insertions(+)
 create mode 100644 3.9-algebras-for-monads.md

diff --git a/3.9-algebras-for-monads.md b/3.9-algebras-for-monads.md
new file mode 100644
index 0000000..c69c850
--- /dev/null
+++ b/3.9-algebras-for-monads.md
@@ -0,0 +1,149 @@
+```Haskell
+alg . return = id
+alg . join = alg . fmap alg
+```
+```swift
+compose(alg, monad.pure) = id
+compose(alg, monad.flatten) = compose(alg, monad.lift(alg))
+```
+................
+```Haskell
+foldr :: (a -> a -> a) -> a -> [a] -> a
+```
+```swift
+func foldr<A>(_ f : @escaping (A, A) -> A, _ z : A, _ l : [A]) -> A
+```
+................
+```Haskell
+foldr f z [x] = x `f` z
+```
+```swift
+func foldr<A>(_ f : @escaping (A, A) -> A, _ z : A, _ l : [A]) -> A {
+  return f(l[0], z)
+}
+```
+................
+```Haskell
+x `f` z = x
+```
+```swift
+f(x, z) == x
+```
+................
+```Haskell
+h = join . fmap g . f
+```
+```swift
+h == compose(monad.flatten, compose(monad.lift(g), f))
+```
+................
+```Haskell
+return . f
+```
+```swift
+compose(pure, f)
+```
+................
+```Haskell
+data Store s a = Store (s -> a) s
+```
+```swift
+class Store<S, A> : Kind2<ForStore, S, A> {
+  let run : (S) -> A
+  let s : S
+
+  init(_ run : @escaping (S) -> A, _ s : S) {
+    self.run = run
+    self.s = s
+  }
+}
+```
+................
+```Haskell
+set :: a -> s -> a
+get :: a -> s
+```
+```swift
+func set<A, S>(_ a : A, _ s : S) -> A
+func get<A, S>(_ a : A) -> S
+```
+................
+```Haskell
+coalgs a = Store (set a) (get a)
+```
+```swift
+func coalg<A, S>(_ a : A) -> Store<S, A> {
+  return Store<S, A>({ s in set(a, s) }, get(a))
+}
+```
+................
+```Haskell
+instance Comonad (Store s) where
+  extract (Store f s) = f s
+  duplicate (Store f s) = Store (Store f) s
+```
+```swift
+class StoreComonad<S> : Comonad {
+  typealias F = Kind<ForStore, S>
+
+  func extract<A>(_ wa : Kind2<ForStore, S, A>) -> A {
+    let store = Store<S, A>.fix(wa)
+    return store.run(store.s)
+  }
+
+  func duplicate<A>(_ wa : Kind2<ForStore, S, A>) -> Kind2<ForStore, S, Kind2<ForStore, S, A>> {
+    let store = Store<S, A>.fix(wa)
+    return Store<S, Store<S, A>>({ s in Store(store.run, s) }, store.s)
+  }
+}
+```
+................
+```Haskell
+fmap g (Store f s) = Store (g . f) s
+```
+```swift
+functor.map(Store(f, s), g) == Store(compose(g, f), s)
+```
+................
+```Haskell
+Store (coalg . set a) (get a)
+```
+```swift
+Store(compose(coalg, { s in set(a, s) }), get(a))
+```
+................
+```Haskell
+Store (Store (set a)) (get a)
+```
+```swift
+Store(Store({ s in set(a, s)), get(a))
+```
+................
+```Haskell
+coalg (set a s) = Store (set a) s
+```
+```swift
+coalg(set(a, s)) == Store({ s in set(a, s) }, s)
+```
+................
+```Haskell
+Store (set (set a s)) (get (set a s)) = Store (set a) s
+```
+```scala
+Store({ s1 in set(set(a, s), s1) }, get(set(a, s))) ==
+  Store({ s in set(a, s) }, s)
+```
+................
+```Haskell
+set (set a s) = set a
+```
+```swift
+{ s1 in set(set(a, s), s1)} == { s1 in set(a, s1) }
+```
+................
+```Haskell
+get (set a s) = s
+```
+```swift
+get(set(a, s)) == s
+```