-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura

abbreviation id {A : Type} (a : A) := a
abbreviation compose {A : Type} {B : Type} {C : Type} (g : B → C) (f : A → B) := λ x, g (f x)
infixl `∘`:60 := compose

inductive path {A : Type} (a : A) : A → Type :=
| refl : path a a

infix `=`:50 := path

definition transport {A : Type} {a b : A} {P : A → Type} (H1 : a = b) (H2 : P a) : P b
:= path_rec H2 H1

namespace logic
  notation p `*(`:75 u `)` := transport p u
end
using logic

definition symm {A : Type} {a b : A} (p : a = b) : b = a
:= p*(refl a)

definition trans {A : Type} {a b c : A} (p1 : a = b) (p2 : b = c) : a = c
:= p2*(p1)

calc_subst transport
calc_refl  refl
calc_trans trans

namespace logic
  postfix `⁻¹`:100 := symm
  infixr `⬝`:75 := trans
end
using logic

theorem trans_refl_right {A : Type} {x y : A} (p : x = y) : p = p ⬝ (refl y)
:= refl p

theorem trans_refl_left {A : Type} {x y : A} (p : x = y) : p = (refl x) ⬝ p
:= path_rec (trans_refl_right (refl x)) p

theorem refl_symm {A : Type} (x : A) : (refl x)⁻¹ = (refl x)
:= refl (refl x)

theorem refl_trans {A : Type} (x : A) : (refl x) ⬝ (refl x) = (refl x)
:= refl (refl x)

theorem trans_symm {A : Type} {x y : A} (p : x = y) : p ⬝ p⁻¹ = refl x
:= have q : (refl x) ⬝ (refl x)⁻¹ = refl x, from
     ((refl_symm x)⁻¹)*(refl_trans x),
   path_rec q p

theorem symm_trans {A : Type} {x y : A} (p : x = y) : p⁻¹ ⬝ p = refl y
:= have q : (refl x)⁻¹ ⬝ (refl x) = refl x, from
     ((refl_symm x)⁻¹)*(refl_trans x),
   path_rec q p

theorem symm_symm {A : Type} {x y : A} (p : x = y) : (p⁻¹)⁻¹ = p
:= have q : ((refl x)⁻¹)⁻¹ = refl x, from
     refl (refl x),
   path_rec q p

theorem trans_assoc {A : Type} {x y z w : A} (p : x = y) (q : y = z) (r : z = w) : p ⬝ (q ⬝ r) = (p ⬝ q) ⬝ r
:= have e1 : (p ⬝ q) ⬝ (refl z) = p ⬝ q, from
     (trans_refl_right (p ⬝ q))⁻¹,
   have e2 : q ⬝ (refl z) = q, from
     (trans_refl_right q)⁻¹,
   have e3 : p ⬝ (q ⬝ (refl z)) = p ⬝ q, from
     e2*(refl (p ⬝ (q ⬝ (refl z)))),
   path_rec (e3 ⬝ e1⁻¹) r

definition ap {A : Type} {B : Type} (f : A → B) {a b : A} (p : a = b) : f a = f b
:= p*(refl (f a))

theorem ap_refl {A : Type} {B : Type} (f : A → B) (a : A) : ap f (refl a) = refl (f a)
:= refl (refl (f a))

section
  parameters {A : Type} {B : Type} {C : Type}
  parameters (f : A → B) (g : B → C)
  parameters (x y z : A) (p : x = y) (q : y = z)

  theorem ap_trans_dist : ap f (p ⬝ q) = (ap f p) ⬝ (ap f q)
  := have e1 : ap f (p ⬝ refl y) = (ap f p) ⬝ (ap f (refl y)), from refl _,
     path_rec e1 q

  theorem ap_inv_dist : ap f (p⁻¹) = (ap f p)⁻¹
  := have e1 : ap f ((refl x)⁻¹) = (ap f (refl x))⁻¹, from refl _,
     path_rec e1 p

  theorem ap_compose : ap g (ap f p) = ap (g∘f) p
  := have e1 : ap g (ap f (refl x)) = ap (g∘f) (refl x), from refl _,
     path_rec e1 p

  theorem ap_id : ap id p = p
  := have e1 : ap id (refl x) = (refl x), from refl (refl x),
     path_rec e1 p
end

section
  parameters {A : Type} {B : A → Type} (f : Π x, B x)
  definition D [private] (x y : A) (p : x = y) := p*(f x) = f y
  definition d [private] (x : A) : D x x (refl x)
  := refl (f x)

  theorem apd {a b : A} (p : a = b) : p*(f a) = f b
  := path_rec (d a) p
end


abbreviation homotopy {A : Type} {P : A → Type} (f g : Π x, P x)
:= Π x, f x = g x

namespace logic
  infix `∼`:50 := homotopy
end
using logic

notation `assume` binders `,` r:(scoped f, f) := r
notation `take`   binders `,` r:(scoped f, f) := r

section
  parameters {A : Type} {B : Type}

  theorem hom_refl (f : A → B) : f ∼ f
  := take x, refl (f x)

  theorem hom_symm {f g : A → B} : f ∼ g → g ∼ f
  := assume h, take x, (h x)⁻¹

  theorem hom_trans {f g h : A → B} : f ∼ g → g ∼ h → f ∼ h
  := assume h1 h2, take x, (h1 x) ⬝ (h2 x)

  theorem hom_fun {f g : A → B} {x y : A} (H : f ∼ g) (p : x = y) : (H x) ⬝ (ap g p) = (ap f p) ⬝ (H y)
  := have e1 : (H x) ⬝ (refl (g x)) = (refl (f x)) ⬝ (H x), from
       calc  (H x) ⬝ (refl (g x)) = H x                 : (trans_refl_right (H x))⁻¹
                     ...         = (refl (f x)) ⬝ (H x) : trans_refl_left (H x),
     have e2 : (H x) ⬝ (ap g (refl x)) = (ap f (refl x)) ⬝ (H x), from
       calc (H x) ⬝ (ap g (refl x)) = (H x) ⬝ (refl (g x))   : {ap_refl g x}
                             ...   = (refl (f x)) ⬝ (H x)    : e1
                             ...   = (ap f (refl x)) ⬝ (H x) : {symm (ap_refl f x)},
     path_rec e2 p


end

definition loop_space (A : Type) (a : A) := a = a
notation `Ω` `(` A `,` a `)` := loop_space A a

definition loop2d_space (A : Type) (a : A) := (refl a) = (refl a)
notation `Ω²` `(` A `,` a `)` := loop2d_space A a

inductive empty : Type

theorem empty_elim (c : Type) (H : empty) : c
:= empty_rec (λ e, c) H

definition not (A : Type) := A → empty
prefix `¬`:40 := not

theorem not_intro {a : Type} (H : a → empty) : ¬ a
:= H

theorem not_elim {a : Type} (H1 : ¬ a) (H2 : a) : empty
:= H1 H2

theorem absurd {a : Type} (H1 : a) (H2 : ¬ a) : empty
:= H2 H1

theorem mt {a b : Type} (H1 : a → b) (H2 : ¬ b) : ¬ a
:= assume Ha : a, absurd (H1 Ha) H2

theorem contrapos {a b : Type} (H : a → b) : ¬ b → ¬ a
:= assume Hnb : ¬ b, mt H Hnb

theorem absurd_elim {a : Type} (b : Type) (H1 : a) (H2 : ¬ a) : b
:= empty_elim b (absurd H1 H2)

inductive unit : Type :=
| star : unit

notation `⋆`:max := star

theorem absurd_not_unit (H : ¬ unit) : empty
:= absurd star H

theorem not_empty_trivial : ¬ empty
:= assume H : empty, H

theorem upun (x : unit) : x = ⋆
:= unit_rec (refl ⋆) x

inductive product (A : Type) (B : Type) : Type :=
| pair : A → B → product A B

infixr `×`:30 := product
infixr `∧`:30 := product

notation `(` h `,` t:(foldl `,` (e r, pair r e) h) `)` := t

definition pr1 {A : Type} {B : Type} (p : A × B) : A
:= product_rec (λ a b, a) p

definition pr2 {A : Type} {B : Type} (p : A × B) : B
:= product_rec (λ a b, b) p

theorem uppt {A : Type} {B : Type} (p : A × B) : (pr1 p, pr2 p) = p
:= product_rec (λ x y, refl (x, y)) p

inductive sum (A : Type) (B : Type) : Type :=
| inl : A → sum A B
| inr : B → sum A B

namespace logic
  infixr `+`:25 := sum
end
using logic
infixr `∨`:25 := sum

theorem sum_elim {a : Type} {b : Type} {c : Type} (H1 : a + b) (H2 : a → c) (H3 : b → c) : c
:= sum_rec H2 H3 H1

theorem resolve_right {a : Type} {b : Type} (H1 : a + b) (H2 : ¬ a) : b
:= sum_elim H1 (assume Ha, absurd_elim b Ha H2) (assume Hb, Hb)

theorem resolve_left {a : Type} {b : Type} (H1 : a + b) (H2 : ¬ b) : a
:= sum_elim H1 (assume Ha, Ha) (assume Hb, absurd_elim a Hb H2)

theorem sum_flip {a : Type} {b : Type} (H : a + b) : b + a
:= sum_elim H (assume Ha, inr b Ha) (assume Hb, inl a Hb)

inductive bool : Type :=
| true  : bool
| false : bool

theorem bool_cases (p : bool) : p = true ∨ p = false
:= bool_rec (inl _ (refl true)) (inr _ (refl false)) p

inductive Sigma {A : Type} (B : A → Type) : Type :=
| sigma_intro : Π a, B a → Sigma B

notation `Σ` binders `,` r:(scoped P, Sigma P) := r

definition dpr1 {A : Type} {B : A → Type} (p : Σ x, B x) : A
:= Sigma_rec (λ a b, a) p

definition dpr2 {A : Type} {B : A → Type} (p : Σ x, B x) : B (dpr1 p)
:= Sigma_rec (λ a b, b) p