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-- https://github.com/BartoszMilewski/DaoFP

import Mathlib

-- universe u
def absurd' {C : Sort u} : Empty → C := Empty.rec

#check absurd'

#check id

-- Universes sad
-- #eval id id

-- Naturality condition

inductive Bool'
  | true' (a : Unit) : Bool'
  | false' (a : Unit) : Bool'

#check Bool'.true' = Bool'.false'


-- def f
--   | 0 => 1
--   | x + 1 => (x + 1) * f x

#eval f 69

def third {α β γ} (x : α × β × γ) :=
  let (_, _, c) := x
  c


universe v

-- class Natural (f : (Type u → Type v) → Type u → Type v) : Type (max (u + 1) v) where
-- Oops it's not a typeclass


def id' {α} (x : α) := x

#check id'

def yoneda {α} (m : Type u → Type v) [Functor m] (g : {β : Type u} → (α → β) → m β) : m α := g id

def yoneda' {α} (m : Type u → Type v) [Functor m] (y : m α) : {β : Type u} → (α → β) → m β := (· <$> y)


-- def map_to_T (x : String) : Type :=
-- if x = "0" then
--   Nat
-- else
--   String

-- def natOrStringThree (b : Bool) : if b then Nat else String :=
--   match b with
--   | true => (3 : Nat)
--   | false => "three"

-- abbrev map_to_T (x : String) : Type :=
--   if x = "0" then Nat else String

-- def map_to (x : String) : map_to_T x :=
--   match decide (x = "0") with
--   | true => (42 : Nat)
--   | false => x

-- def map_to (x : String) : map_to_T x :=
--   if h : x = "0" then by
--     simp [map_to_T, h]
--     exact 42
--   else by
--     simp [map_to_T, h]
--     exact x


def ap [Monad m] (fs : m (α → β)) (as : m α) : m β := do
  -- fs >>= λ f ↦ as >>= λ a ↦ pure (f a)
  -- fs >>= (· <$> as)
  return (← fs) (← as)

class Monad' (m : Type → Type) where
  fish : (β → m γ) → (α → m β) → (α → m γ)
  join : (a : m (m α)) → m α := fish id id

namespace Ch18

class Profunctor (p : Type u → Type u → Type (u + 1)) where
  dimap : (s → a) → (b → t) → (p a b → p s t)

inductive Procompose p q [Profunctor p] [Profunctor q] a b
  | mk : q a x → p x b → Procompose p q a b

def mapOut [Profunctor p] [Profunctor q] (pc : Procompose p q a b) (f : {x : Type} → q a x → p x b → c) :=
  match pc with
  | ⟨qax, pxb⟩ => f qax pxb

instance [Profunctor p] [Profunctor q] : Profunctor (Procompose p q) where
  dimap l r
  | ⟨qax, pxb⟩ => ⟨Profunctor.dimap l id qax, Profunctor.dimap id r pxb⟩

def End p [Profunctor p] := ∀ x, p x x

def Coend p [Profunctor p] := Σ x, p x x

inductive ProPair q p [Profunctor p] [Profunctor q] a b x y
  | mk : q a y → p x b → ProPair q p a b x y

instance [Profunctor p] [Profunctor q] : Profunctor (ProPair q p a b) where
  dimap l r
  | ⟨qax, pxb⟩ => ⟨Profunctor.dimap id r qax, Profunctor.dimap l id pxb⟩

inductive CoEndCompose p q [Profunctor p] [Profunctor q] a b
  | mk : Coend (ProPair q p a b) → CoEndCompose p q a b

instance [Profunctor p] [Profunctor q] : Profunctor (CoEndCompose p q) where
  dimap l r
  | ⟨x, ⟨qay, pxb⟩⟩ => ⟨x, ⟨Profunctor.dimap l id qay,Profunctor.dimap id r pxb⟩⟩

inductive Yo f [Functor f] a x y
  | mk : ((a → x) → f y) → Yo f a x y

instance [Functor f] : Profunctor (Yo f a)


def yoneda f [Functor f] : (End (Yo f a)) → f a
  | ⟨x, g⟩ => g id

inductive LensE s a
  | mk : (s → (c × a)) → (c × a → s) → LensE s a

def toGet : LensE s a → (s → a)
  | ⟨l, _⟩ => (l · |>.2)

def toSet : LensE s a → (s → a → s)
  | ⟨l, r⟩ => fun s a ↦ r ((l s).1, a)

def getResidue : LensE s a → c
  | ⟨l, _⟩ => (l _).1

end Ch18