/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn
-/
import logic ..prod algebra.relation
open prod eq.ops

namespace quotient

/- auxiliary facts about products -/

  variables {A B : Type}

  /- flip -/

  definition flip (a : A × B) : B × A := pair (pr2 a) (pr1 a)

  theorem flip_def (a : A × B) : flip a = pair (pr2 a) (pr1 a) := rfl
  theorem flip_pair (a : A) (b : B) : flip (pair a b) = pair b a := rfl
  theorem flip_pr1 (a : A × B) : pr1 (flip a) = pr2 a := rfl
  theorem flip_pr2 (a : A × B) : pr2 (flip a) = pr1 a := rfl
  theorem flip_flip : Π a : A × B, flip (flip a) = a
  | (pair x y) := rfl

  theorem P_flip {P : A → B → Prop} (a : A × B) (H : P (pr1 a) (pr2 a))
    : P (pr2 (flip a)) (pr1 (flip a)) :=
  (flip_pr1 a)⁻¹ ▸ (flip_pr2 a)⁻¹ ▸ H

  theorem flip_inj {a b : A × B} (H : flip a = flip b) : a = b :=
  have H2 : flip (flip a) = flip (flip b), from congr_arg flip H,
  show a = b, from (flip_flip a) ▸ (flip_flip b) ▸ H2

  /- coordinatewise unary maps -/

  definition map_pair (f : A → B) (a : A × A) : B × B :=
  pair (f (pr1 a)) (f (pr2 a))

  theorem map_pair_def (f : A → B) (a : A × A)
    : map_pair f a = pair (f (pr1 a)) (f (pr2 a)) :=
  rfl

  theorem map_pair_pair (f : A → B) (a a' : A)
    : map_pair f (pair a a') = pair (f a) (f a') :=
  (pr1.mk a a') ▸ (pr2.mk a a') ▸ rfl

  theorem map_pair_pr1 (f : A → B) (a : A × A) : pr1 (map_pair f a) = f (pr1 a) :=
  by esimp

  theorem map_pair_pr2 (f : A → B) (a : A × A) : pr2 (map_pair f a) = f (pr2 a) :=
  by esimp

  /- coordinatewise binary maps -/

  definition map_pair2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) : C × C :=
  pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b))

  theorem map_pair2_def {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
    map_pair2 f a b = pair (f (pr1 a) (pr1 b)) (f (pr2 a) (pr2 b)) := rfl

  theorem map_pair2_pair {A B C : Type} (f : A → B → C) (a a' : A) (b b' : B) :
    map_pair2 f (pair a a') (pair b b') = pair (f a b) (f a' b') := by esimp

  theorem map_pair2_pr1 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
  pr1 (map_pair2 f a b) = f (pr1 a) (pr1 b) := by esimp

  theorem map_pair2_pr2 {A B C : Type} (f : A → B → C) (a : A × A) (b : B × B) :
  pr2 (map_pair2 f a b) = f (pr2 a) (pr2 b) := by esimp

  theorem map_pair2_flip {A B C : Type} (f : A → B → C) : Π (a : A × A) (b : B × B),
    flip (map_pair2 f a b) = map_pair2 f (flip a) (flip b)
  | (pair a₁ a₂) (pair b₁ b₂) := rfl

-- add_rewrite flip_pr1 flip_pr2 flip_pair
-- add_rewrite map_pair_pr1 map_pair_pr2 map_pair_pair
-- add_rewrite map_pair2_pr1 map_pair2_pr2 map_pair2_pair

theorem map_pair2_comm {A B : Type} {f : A → A → B} (Hcomm : ∀a b : A, f a b = f b a)
  : Π (v w : A × A), map_pair2 f v w = map_pair2 f w v
| (pair v₁ v₂) (pair w₁ w₂) :=
  !map_pair2_pair ⬝ by rewrite [Hcomm v₁ w₁, Hcomm v₂ w₂] ⬝ (eq.symm !map_pair2_pair)

theorem map_pair2_assoc {A : Type} {f : A → A → A}
    (Hassoc : ∀a b c : A, f (f a b) c = f a (f b c)) (u v w : A × A) :
  map_pair2 f (map_pair2 f u v) w = map_pair2 f u (map_pair2 f v w) :=
show pair (f (f (pr1 u) (pr1 v)) (pr1 w)) (f (f (pr2 u) (pr2 v)) (pr2 w)) =
     pair (f (pr1 u) (f (pr1 v) (pr1 w))) (f (pr2 u) (f (pr2 v) (pr2 w))),
by rewrite [Hassoc (pr1 u) (pr1 v) (pr1 w), Hassoc (pr2 u) (pr2 v) (pr2 w)]

theorem map_pair2_id_right {A B : Type} {f : A → B → A} {e : B} (Hid : ∀a : A, f a e = a)
  : Π (v : A × A), map_pair2 f v (pair e e) = v
| (pair v₁ v₂) :=
  !map_pair2_pair ⬝ by rewrite [Hid v₁, Hid v₂]

theorem map_pair2_id_left {A B : Type} {f : B → A → A} {e : B} (Hid : ∀a : A, f e a = a)
  : Π (v : A × A), map_pair2 f (pair e e) v = v
| (pair v₁ v₂) :=
  !map_pair2_pair ⬝ by rewrite [Hid v₁, Hid v₂]

end quotient