/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
-/
prelude
import init.datatypes

universe variables u v w

attribute [reducible]
definition id {A : Type u} (a : A) : A :=
a

definition flip {A : Type u} {B : Type v} {C : Type w} (f : A → B → C) : B → A → C :=
λ b a, f a b

/- TODO(Leo): for new elaborator
- Support for partially applied recursors (use eta-expansion)
- checkpoints when processing calc.
-/

/- implication -/

definition implies (a b : Prop) := a → b

attribute [trans]
lemma implies.trans {p q r : Prop} (h₁ : implies p q) (h₂ : implies q r) : implies p r :=
assume hp, h₂ (h₁ hp)

definition trivial : true := ⟨⟩

definition not (a : Prop) := a → false
prefix `¬` := not

attribute [inline]
definition absurd {a : Prop} {b : Type v} (h₁ : a) (h₂ : ¬a) : b :=
false.rec b (h₂ h₁)

attribute [intro!]
lemma not.intro {a : Prop} (h : a → false) : ¬ a :=
h

theorem mt {a b : Prop} (h₁ : a → b) (h₂ : ¬b) : ¬a :=
assume ha : a, absurd (h₁ ha) h₂

definition implies.resolve {a b : Prop} (h : a → b) (nb : ¬ b) : ¬ a := assume ha, nb (h ha)

/- not -/

theorem not_false : ¬false :=
assume h : false, h

definition non_contradictory (a : Prop) : Prop := ¬¬a

theorem non_contradictory_intro {a : Prop} (ha : a) : ¬¬a :=
assume hna : ¬a, absurd ha hna

/- false -/

theorem false.elim {c : Prop} (h : false) : c :=
false.rec c h

/- eq -/

-- proof irrelevance is built in
theorem proof_irrel {a : Prop} (h₁ h₂ : a) : h₁ = h₂ :=
rfl

attribute [defeq]
definition id.def {A : Type u} (a : A) : id a = a := rfl

-- Remark: we provide the universe levels explicitly to make sure `eq.drec` has the same type of `eq.rec` in the hoTT library
attribute [elab_as_eliminator]
protected theorem {u₁ u₂} eq.drec {A : Type u₂} {a : A} {C : Π (x : A), a = x → Type u₁} (h₁ : C a (eq.refl a)) {b : A} (h₂ : a = b) : C b h₂ :=
eq.rec (λ h₂ : a = a, show C a h₂, from h₁) h₂ h₂

namespace eq
  variables {A : Type u}
  variables {a b c a': A}

  attribute [elab_as_eliminator]
  protected theorem drec_on {a : A} {C : Π (x : A), a = x → Type v} {b : A} (h₂ : a = b) (h₁ : C a (refl a)) : C b h₂ :=
  eq.drec h₁ h₂

  theorem substr {p : A → Prop} (h₁ : b = a) : p a → p b :=
  subst (symm h₁)

  theorem mp {A B : Type u} : (A = B) → A → B :=
  eq.rec_on

  theorem mpr {A B : Type u} : (A = B) → B → A :=
  assume h₁ h₂, eq.rec_on (eq.symm h₁) h₂
end eq

open eq

theorem congr {A : Type u} {B : Type v} {f₁ f₂ : A → B} {a₁ a₂ : A} (h₁ : f₁ = f₂) (h₂ : a₁ = a₂) : f₁ a₁ = f₂ a₂ :=
eq.subst h₁ (eq.subst h₂ rfl)

theorem congr_fun {A : Type u} {B : A → Type v} {f g : Π x, B x} (h : f = g) (a : A) : f a = g a :=
eq.subst h (eq.refl (f a))

theorem congr_arg {A : Type u} {B : Type v} {a₁ a₂ : A} (f : A → B) : a₁ = a₂ → f a₁ = f a₂ :=
congr rfl

section
  variables {A : Type u} {a b c: A}

  theorem trans_rel_left (r : A → A → Prop) (h₁ : r a b) (h₂ : b = c) : r a c :=
  h₂ ▸ h₁

  theorem trans_rel_right (r : A → A → Prop) (h₁ : a = b) (h₂ : r b c) : r a c :=
  symm h₁ ▸ h₂
end

section
  variable {p : Prop}

  theorem of_eq_true (h : p = true) : p :=
  symm h ▸ trivial

  theorem not_of_eq_false (h : p = false) : ¬p :=
  assume hp, h ▸ hp
end

attribute [inline]
definition cast {A B : Type u} (h : A = B) (a : A) : B :=
eq.rec a h

theorem cast_proof_irrel {A B : Type u} (h₁ h₂ : A = B) (a : A) : cast h₁ a = cast h₂ a :=
rfl

theorem cast_eq {A : Type u} (h : A = A) (a : A) : cast h a = a :=
rfl

/- ne -/

attribute [reducible]
definition ne {A : Type u} (a b : A) := ¬(a = b)
attribute [defeq]
definition ne.def {A : Type u} (a b : A) : ne a b = ¬ (a = b) := rfl
notation a ≠ b := ne a b

namespace ne
  variable {A : Type u}
  variables {a b : A}

  theorem intro (h : a = b → false) : a ≠ b := h

  theorem elim (h : a ≠ b) : a = b → false := h

  theorem irrefl (h : a ≠ a) : false := h rfl

  theorem symm (h : a ≠ b) : b ≠ a :=
  assume (h₁ : b = a), h (symm h₁)
end ne

theorem false_of_ne {A : Type u} {a : A} : a ≠ a → false := ne.irrefl

section
  variables {p : Prop}

  theorem ne_false_of_self : p → p ≠ false :=
  assume (hp : p) (heq : p = false), heq ▸ hp

  theorem ne_true_of_not : ¬p → p ≠ true :=
  assume (hnp : ¬p) (heq : p = true), (heq ▸ hnp) trivial

  theorem true_ne_false : ¬true = false :=
  ne_false_of_self trivial
end

infixl ` == `:50 := heq

section
variables {A B C : Type u} {a a' : A} {b b' : B} {c : C}

theorem eq_of_heq (h : a == a') : a = a' :=
have ∀ (A' : Type u) (a' : A') (h₁ : @heq A a A' a') (h₂ : A = A'), (eq.rec_on h₂ a : A') = a', from
  λ (A' : Type u) (a' : A') (h₁ : @heq A a A' a'), heq.rec_on h₁ (λ h₂ : A = A, rfl),
show (eq.rec_on (eq.refl A) a : A) = a', from
  this A a' h (eq.refl A)

theorem heq.elim {A : Type u} {a : A} {p : A → Type v} {b : A} (h₁ : a == b)
: p a → p b := eq.rec_on (eq_of_heq h₁)

theorem heq.subst {p : ∀ T : Type u, T → Prop} : a == b → p A a → p B b :=
heq.rec_on

theorem heq.symm (h : a == b) : b == a :=
heq.rec_on h (heq.refl a)

theorem heq_of_eq (h : a = a') : a == a' :=
eq.subst h (heq.refl a)

theorem heq.trans (h₁ : a == b) (h₂ : b == c) : a == c :=
heq.subst h₂ h₁

theorem heq_of_heq_of_eq (h₁ : a == b) (h₂ : b = b') : a == b' :=
heq.trans h₁ (heq_of_eq h₂)

theorem heq_of_eq_of_heq (h₁ : a = a') (h₂ : a' == b) : a == b :=
heq.trans (heq_of_eq h₁) h₂

definition type_eq_of_heq (h : a == b) : A = B :=
heq.rec_on h (eq.refl A)
end

theorem eq_rec_heq {A : Type u} {C : A → Type v} : ∀ {a a' : A} (h : a = a') (p : C a), (eq.rec_on h p : C a') == p
| a .a rfl p := heq.refl p

theorem heq_of_eq_rec_left {A : Type u} {C : A → Type v} : ∀ {a a' : A} {p₁ : C a} {p₂ : C a'} (e : a = a') (h₂ : (eq.rec_on e p₁ : C a') = p₂), p₁ == p₂
| a .a p₁ p₂ (eq.refl .a) h := eq.rec_on h (heq.refl p₁)

theorem heq_of_eq_rec_right {A : Type u} {C : A → Type v} : ∀ {a a' : A} {p₁ : C a} {p₂ : C a'} (e : a' = a) (h₂ : p₁ = eq.rec_on e p₂), p₁ == p₂
| a .a p₁ p₂ (eq.refl .a) h :=
  have p₁ = p₂, from h,
  this ▸ heq.refl p₁

theorem of_heq_true {a : Prop} (h : a == true) : a :=
of_eq_true (eq_of_heq h)

theorem eq_rec_compose : ∀ {A B C : Type u} (p₁ : B = C) (p₂ : A = B) (a : A), (eq.rec_on p₁ (eq.rec_on p₂ a : B) : C) = eq.rec_on (eq.trans p₂ p₁) a
| A .A .A (eq.refl .A) (eq.refl .A) a := rfl

theorem eq_rec_eq_eq_rec : ∀ {A₁ A₂ : Type u} {p : A₁ = A₂} {a₁ : A₁} {a₂ : A₂}, (eq.rec_on p a₁ : A₂) = a₂ → a₁ = eq.rec_on (eq.symm p) a₂
| A .A rfl a .a rfl := rfl

theorem eq_rec_of_heq_left : ∀ {A₁ A₂ : Type u} {a₁ : A₁} {a₂ : A₂} (h : a₁ == a₂), (eq.rec_on (type_eq_of_heq h) a₁ : A₂) = a₂
| A .A a .a (heq.refl .a) := rfl

theorem eq_rec_of_heq_right {A₁ A₂ : Type u} {a₁ : A₁} {a₂ : A₂} (h : a₁ == a₂) : a₁ = eq.rec_on (eq.symm (type_eq_of_heq h)) a₂ :=
eq_rec_eq_eq_rec (eq_rec_of_heq_left h)

attribute heq.refl [refl]
attribute heq.trans [trans]
attribute heq_of_heq_of_eq [trans]
attribute heq_of_eq_of_heq [trans]
attribute heq.symm [symm]

theorem cast_heq : ∀ {A B : Type u} (h : A = B) (a : A), cast h a == a
| A .A (eq.refl .A) a := heq.refl a

/- and -/

notation a /\ b := and a b
notation a ∧ b  := and a b

variables {a b c d : Prop}

theorem and.elim (h₁ : a ∧ b) (h₂ : a → b → c) : c :=
and.rec h₂ h₁

theorem and.swap : a ∧ b → b ∧ a :=
assume ⟨ha, hb⟩, ⟨hb, ha⟩

/- or -/

notation a \/ b := or a b
notation a ∨ b := or a b

namespace or
  theorem elim (h₁ : a ∨ b) (h₂ : a → c) (h₃ : b → c) : c :=
  or.rec h₂ h₃ h₁
end or

theorem non_contradictory_em (a : Prop) : ¬¬(a ∨ ¬a) :=
assume not_em : ¬(a ∨ ¬a),
  have neg_a : ¬a, from
    assume pos_a : a, absurd (or.inl pos_a) not_em,
  absurd (or.inr neg_a) not_em

theorem or.swap : a ∨ b → b ∨ a := or.rec or.inr or.inl

/- xor -/
definition xor (a b : Prop) := (a ∧ ¬ b) ∨ (b ∧ ¬ a)

/- iff -/

definition iff (a b : Prop) := (a → b) ∧ (b → a)

notation a <-> b := iff a b
notation a ↔ b := iff a b

theorem iff.intro : (a → b) → (b → a) → (a ↔ b) := and.intro

attribute iff.intro [intro!]

theorem iff.elim : ((a → b) → (b → a) → c) → (a ↔ b) → c := and.rec

attribute iff.elim [recursor 5]

theorem iff.elim_left : (a ↔ b) → a → b := and.left

definition iff.mp := @iff.elim_left

theorem iff.elim_right : (a ↔ b) → b → a := and.right

definition iff.mpr := @iff.elim_right

attribute [refl]
theorem iff.refl (a : Prop) : a ↔ a :=
iff.intro (assume h, h) (assume h, h)

theorem iff.rfl {a : Prop} : a ↔ a :=
iff.refl a

attribute [trans]
theorem iff.trans (h₁ : a ↔ b) (h₂ : b ↔ c) : a ↔ c :=
iff.intro
  (assume ha, iff.mp h₂ (iff.mp h₁ ha))
  (assume hc, iff.mpr h₁ (iff.mpr h₂ hc))

attribute [symm]
theorem iff.symm (h : a ↔ b) : b ↔ a :=
iff.intro (iff.elim_right h) (iff.elim_left h)

theorem iff.comm : (a ↔ b) ↔ (b ↔ a) :=
iff.intro iff.symm iff.symm

theorem iff.of_eq {a b : Prop} (h : a = b) : a ↔ b :=
eq.rec_on h iff.rfl

theorem not_iff_not_of_iff (h₁ : a ↔ b) : ¬a ↔ ¬b :=
iff.intro
 (assume (hna : ¬ a) (hb : b), hna (iff.elim_right h₁ hb))
 (assume (hnb : ¬ b) (ha : a), hnb (iff.elim_left h₁ ha))

theorem of_iff_true (h : a ↔ true) : a :=
iff.mp (iff.symm h) trivial

theorem not_of_iff_false : (a ↔ false) → ¬a := iff.mp

theorem iff_true_intro (h : a) : a ↔ true :=
iff.intro
  (λ hl, trivial)
  (λ hr, h)

theorem iff_false_intro (h : ¬a) : a ↔ false :=
iff.intro h (false.rec a)

theorem not_non_contradictory_iff_absurd (a : Prop) : ¬¬¬a ↔ ¬a :=
iff.intro
  (λ (hl : ¬¬¬a) (ha : a), hl (non_contradictory_intro ha))
  absurd

theorem imp_congr (h₁ : a ↔ c) (h₂ : b ↔ d) : (a → b) ↔ (c → d) :=
iff.intro
  (λ hab hc, iff.mp h₂ (hab (iff.mpr h₁ hc)))
  (λ hcd ha, iff.mpr h₂ (hcd (iff.mp h₁ ha)))

theorem imp_congr_ctx (h₁ : a ↔ c) (h₂ : c → (b ↔ d)) : (a → b) ↔ (c → d) :=
iff.intro
  (λ hab hc, have ha : a, from iff.mpr h₁ hc,
             have hb : b, from hab ha,
             iff.mp (h₂ hc) hb)
  (λ hcd ha, have hc : c, from iff.mp h₁ ha,
             have hd : d, from hcd hc,
iff.mpr (h₂ hc) hd)

theorem imp_congr_right (h : a → (b ↔ c)) : (a → b) ↔ (a → c) :=
iff.intro
  (take hab ha, iff.elim_left (h ha) (hab ha))
  (take hab ha, iff.elim_right (h ha) (hab ha))

theorem not_not_intro (ha : a) : ¬¬a :=
assume hna : ¬a, hna ha

theorem not_of_not_not_not (h : ¬¬¬a) : ¬a :=
λ ha, absurd (not_not_intro ha) h

attribute [simp]
theorem not_true : (¬ true) ↔ false :=
iff_false_intro (not_not_intro trivial)

attribute [simp]
theorem not_false_iff : (¬ false) ↔ true :=
iff_true_intro not_false

attribute [congr]
theorem not_congr (h : a ↔ b) : ¬a ↔ ¬b :=
iff.intro (λ h₁ h₂, h₁ (iff.mpr h h₂)) (λ h₁ h₂, h₁ (iff.mp h h₂))

attribute [simp]
theorem ne_self_iff_false {A : Type u} (a : A) : (not (a = a)) ↔ false :=
iff.intro false_of_ne false.elim

attribute [simp]
theorem eq_self_iff_true {A : Type u} (a : A) : (a = a) ↔ true :=
iff_true_intro rfl

attribute [simp]
theorem heq_self_iff_true {A : Type u} (a : A) : (a == a) ↔ true :=
iff_true_intro (heq.refl a)

attribute [simp]
theorem iff_not_self (a : Prop) : (a ↔ ¬a) ↔ false :=
iff_false_intro (λ h,
   have h' : ¬a, from (λ ha, (iff.mp h ha) ha),
   h' (iff.mpr h h'))

attribute [simp]
theorem not_iff_self (a : Prop) : (¬a ↔ a) ↔ false :=
iff_false_intro (λ h,
   have h' : ¬a, from (λ ha, (iff.mpr h ha) ha),
   h' (iff.mp h h'))

attribute [simp]
theorem true_iff_false : (true ↔ false) ↔ false :=
iff_false_intro (λ h, iff.mp h trivial)

attribute [simp]
theorem false_iff_true : (false ↔ true) ↔ false :=
iff_false_intro (λ h, iff.mpr h trivial)

theorem false_of_true_iff_false : (true ↔ false) → false :=
assume h, iff.mp h trivial

/- and simp rules -/
theorem and.imp (hac : a → c) (hbd : b → d) : a ∧ b → c ∧ d :=
assume ⟨ha, hb⟩, ⟨hac ha, hbd hb⟩

attribute [congr]
theorem and_congr (h₁ : a ↔ c) (h₂ : b ↔ d) : (a ∧ b) ↔ (c ∧ d) :=
iff.intro (and.imp (iff.mp h₁) (iff.mp h₂)) (and.imp (iff.mpr h₁) (iff.mpr h₂))

theorem and_congr_right (h : a → (b ↔ c)) : (a ∧ b) ↔ (a ∧ c) :=
iff.intro
  (assume ⟨ha, hb⟩, ⟨ha, iff.elim_left (h ha) hb⟩)
  (assume ⟨ha, hc⟩, ⟨ha, iff.elim_right (h ha) hc⟩)

attribute [simp]
theorem and.comm : a ∧ b ↔ b ∧ a :=
iff.intro and.swap and.swap

attribute [simp]
theorem and.assoc : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
iff.intro
  (assume ⟨⟨ha, hb⟩, hc⟩, ⟨ha, ⟨hb, hc⟩⟩)
  (assume ⟨ha, ⟨hb, hc⟩⟩, ⟨⟨ha, hb⟩, hc⟩)

attribute [simp]
theorem and.left_comm : a ∧ (b ∧ c) ↔ b ∧ (a ∧ c) :=
iff.trans (iff.symm and.assoc) (iff.trans (and_congr and.comm (iff.refl c)) and.assoc)

theorem and_iff_left {a b : Prop} (hb : b) : (a ∧ b) ↔ a :=
iff.intro and.left (λ ha, ⟨ha, hb⟩)

theorem and_iff_right {a b : Prop} (ha : a) : (a ∧ b) ↔ b :=
iff.intro and.right (and.intro ha)

attribute [simp]
theorem and_true (a : Prop) : a ∧ true ↔ a :=
and_iff_left trivial

attribute [simp]
theorem true_and (a : Prop) : true ∧ a ↔ a :=
and_iff_right trivial

attribute [simp]
theorem and_false (a : Prop) : a ∧ false ↔ false :=
iff_false_intro and.right

attribute [simp]
theorem false_and (a : Prop) : false ∧ a ↔ false :=
iff_false_intro and.left

attribute [simp]
theorem not_and_self (a : Prop) : (¬a ∧ a) ↔ false :=
iff_false_intro (λ h, and.elim h (λ h₁ h₂, absurd h₂ h₁))

attribute [simp]
theorem and_not_self (a : Prop) : (a ∧ ¬a) ↔ false :=
iff_false_intro (assume ⟨h₁, h₂⟩, absurd h₁ h₂)

attribute [simp]
theorem and_self (a : Prop) : a ∧ a ↔ a :=
iff.intro and.left (assume h, ⟨h, h⟩)

/- or simp rules -/

theorem or.imp (h₂ : a → c) (h₃ : b → d) : a ∨ b → c ∨ d :=
or.rec (λ h, or.inl (h₂ h)) (λ h, or.inr (h₃ h))

theorem or.imp_left (h : a → b) : a ∨ c → b ∨ c :=
or.imp h id

theorem or.imp_right (h : a → b) : c ∨ a → c ∨ b :=
or.imp id h

attribute [congr]
theorem or_congr (h₁ : a ↔ c) (h₂ : b ↔ d) : (a ∨ b) ↔ (c ∨ d) :=
iff.intro (or.imp (iff.mp h₁) (iff.mp h₂)) (or.imp (iff.mpr h₁) (iff.mpr h₂))

attribute [simp]
theorem or.comm : a ∨ b ↔ b ∨ a := iff.intro or.swap or.swap

attribute [simp]
theorem or.assoc : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
iff.intro
  (or.rec (or.imp_right or.inl) (λ h, or.inr (or.inr h)))
  (or.rec (λ h, or.inl (or.inl h)) (or.imp_left or.inr))

attribute [simp]
theorem or.left_comm : a ∨ (b ∨ c) ↔ b ∨ (a ∨ c) :=
iff.trans (iff.symm or.assoc) (iff.trans (or_congr or.comm (iff.refl c)) or.assoc)

attribute [simp]
theorem or_true (a : Prop) : a ∨ true ↔ true :=
iff_true_intro (or.inr trivial)

attribute [simp]
theorem true_or (a : Prop) : true ∨ a ↔ true :=
iff_true_intro (or.inl trivial)

attribute [simp]
theorem or_false (a : Prop) : a ∨ false ↔ a :=
iff.intro (or.rec id false.elim) or.inl

attribute [simp]
theorem false_or (a : Prop) : false ∨ a ↔ a :=
iff.trans or.comm (or_false a)

attribute [simp]
theorem or_self (a : Prop) : a ∨ a ↔ a :=
iff.intro (or.rec id id) or.inl

/- or resolution rulse -/

definition or.resolve_left {a b : Prop} (h : a ∨ b) (na : ¬ a) : b :=
  or.elim h (λ ha, absurd ha na) id

definition or.neg_resolve_left {a b : Prop} (h : ¬ a ∨ b) (ha : a) : b :=
  or.elim h (λ na, absurd ha na) id

definition or.resolve_right {a b : Prop} (h : a ∨ b) (nb : ¬ b) : a :=
  or.elim h id (λ hb, absurd hb nb)

definition or.neg_resolve_right {a b : Prop} (h : a ∨ ¬ b) (hb : b) : a :=
  or.elim h id (λ nb, absurd hb nb)

/- iff simp rules -/

attribute [simp]
theorem iff_true (a : Prop) : (a ↔ true) ↔ a :=
iff.intro (assume h, iff.mpr h trivial) iff_true_intro

attribute [simp]
theorem true_iff (a : Prop) : (true ↔ a) ↔ a :=
iff.trans iff.comm (iff_true a)

attribute [simp]
theorem iff_false (a : Prop) : (a ↔ false) ↔ ¬ a :=
iff.intro and.left iff_false_intro

attribute [simp]
theorem false_iff (a : Prop) : (false ↔ a) ↔ ¬ a :=
iff.trans iff.comm (iff_false a)

attribute [simp]
theorem iff_self (a : Prop) : (a ↔ a) ↔ true :=
iff_true_intro iff.rfl

attribute [congr]
theorem iff_congr (h₁ : a ↔ c) (h₂ : b ↔ d) : (a ↔ b) ↔ (c ↔ d) :=
and_congr (imp_congr h₁ h₂) (imp_congr h₂ h₁)

/- exists -/

inductive Exists {A : Type u} (p : A → Prop) : Prop
| intro : ∀ (a : A), p a → Exists

attribute Exists.intro [intro]

definition exists.intro := @Exists.intro

notation `exists` binders `, ` r:(scoped P, Exists P) := r
notation `∃` binders `, ` r:(scoped P, Exists P) := r

theorem exists.elim {A : Type u} {p : A → Prop} {b : Prop}
  (h₁ : ∃ x, p x) (h₂ : ∀ (a : A), p a → b) : b :=
Exists.rec h₂ h₁

/- exists unique -/

definition exists_unique {A : Type u} (p : A → Prop) :=
∃ x, p x ∧ ∀ y, p y → y = x

notation `∃!` binders `, ` r:(scoped P, exists_unique P) := r

attribute [intro]
theorem exists_unique.intro {A : Type u} {p : A → Prop} (w : A) (h₁ : p w) (h₂ : ∀ y, p y → y = w) :
  ∃! x, p x :=
exists.intro w ⟨h₁, h₂⟩

attribute [recursor 4]
theorem exists_unique.elim {A : Type u} {p : A → Prop} {b : Prop}
    (h₂ : ∃! x, p x) (h₁ : ∀ x, p x → (∀ y, p y → y = x) → b) : b :=
exists.elim h₂ (λ w hw, h₁ w (and.left hw) (and.right hw))

theorem exists_unique_of_exists_of_unique {A : Type u} {p : A → Prop}
    (hex : ∃ x, p x) (hunique : ∀ y₁ y₂, p y₁ → p y₂ → y₁ = y₂) :  ∃! x, p x :=
exists.elim hex (λ x px, exists_unique.intro x px (take y, suppose p y, hunique y x this px))

theorem exists_of_exists_unique {A : Type u} {p : A → Prop} (h : ∃! x, p x) : ∃ x, p x :=
exists.elim h (λ x hx, ⟨x, and.left hx⟩)

theorem unique_of_exists_unique {A : Type u} {p : A → Prop}
    (h : ∃! x, p x) {y₁ y₂ : A} (py₁ : p y₁) (py₂ : p y₂) : y₁ = y₂ :=
exists_unique.elim h
  (take x, suppose p x,
    assume unique : ∀ y, p y → y = x,
    show y₁ = y₂, from eq.trans (unique _ py₁) (eq.symm (unique _ py₂)))

/- exists, forall, exists unique congruences -/
attribute [congr]
theorem forall_congr {A : Type u} {p q : A → Prop} (h : ∀ a, (p a ↔ q a)) : (∀ a, p a) ↔ ∀ a, q a :=
iff.intro (λ p a, iff.mp (h a) (p a)) (λ q a, iff.mpr (h a) (q a))

theorem exists_imp_exists {A : Type u} {p q : A → Prop} (h : ∀ a, (p a → q a)) (p : ∃ a, p a) : ∃ a, q a :=
exists.elim p (λ a hp, ⟨a, h a hp⟩)

attribute [congr]
theorem exists_congr {A : Type u} {p q : A → Prop} (h : ∀ a, (p a ↔ q a)) : (Exists p) ↔ ∃ a, q a :=
iff.intro
  (exists_imp_exists (λ a, iff.mp (h a)))
  (exists_imp_exists (λ a, iff.mpr (h a)))

attribute [congr]
theorem exists_unique_congr {A : Type u} {p₁ p₂ : A → Prop} (h : ∀ x, p₁ x ↔ p₂ x) : (exists_unique p₁) ↔ (∃! x, p₂ x) := --
exists_congr (λ x, and_congr (h x) (forall_congr (λ y, imp_congr (h y) iff.rfl)))

/- decidable -/

inductive [class] decidable (p : Prop)
| is_false : ¬p → decidable
| is_true :  p → decidable

export decidable (is_true is_false)

attribute [instance]
definition decidable_true : decidable true :=
is_true trivial

attribute [instance]
definition decidable_false : decidable false :=
is_false not_false

-- We use "dependent" if-then-else to be able to communicate the if-then-else condition
-- to the branches
attribute [inline]
definition dite (c : Prop) [h : decidable c] {A : Type u} : (c → A) → (¬ c → A) → A :=
λ t e, decidable.rec_on h e t

/- if-then-else -/

attribute [inline]
definition ite (c : Prop) [h : decidable c] {A : Type u} (t e : A) : A :=
decidable.rec_on h (λ hnc, e) (λ hc, t)

namespace decidable
  variables {p q : Prop}

  definition rec_on_true [h : decidable p] {h₁ : p → Type u} {h₂ : ¬p → Type u} (h₃ : p) (h₄ : h₁ h₃)
      : decidable.rec_on h h₂ h₁ :=
  decidable.rec_on h (λ h, false.rec _ (h h₃)) (λ h, h₄)

  definition rec_on_false [h : decidable p] {h₁ : p → Type u} {h₂ : ¬p → Type u} (h₃ : ¬p) (h₄ : h₂ h₃)
      : decidable.rec_on h h₂ h₁ :=
  decidable.rec_on h (λ h, h₄) (λ h, false.rec _ (h₃ h))

  definition by_cases {q : Type u} [C : decidable p] : (p → q) → (¬p → q) → q := dite _

  theorem em (p : Prop) [decidable p] : p ∨ ¬p := by_cases or.inl or.inr

  theorem by_contradiction [decidable p] (h : ¬p → false) : p :=
  if h₁ : p then h₁ else false.rec _ (h h₁)

  definition to_bool (p : Prop) [h : decidable p] : bool :=
  decidable.cases_on h (λ h₁, bool.ff) (λ h₂, bool.tt)
end decidable

section
  variables {p q : Prop}
  definition  decidable_of_decidable_of_iff (hp : decidable p) (h : p ↔ q) : decidable q :=
  if hp : p then is_true (iff.mp h hp)
  else is_false (iff.mp (not_iff_not_of_iff h) hp)

  definition  decidable_of_decidable_of_eq (hp : decidable p) (h : p = q) : decidable q :=
  decidable_of_decidable_of_iff hp (iff.of_eq h)

  protected definition or.by_cases [decidable p] [decidable q] {A : Type u}
                                   (h : p ∨ q) (h₁ : p → A) (h₂ : q → A) : A :=
  if hp : p then h₁ hp else
    if hq : q then h₂ hq else
      false.rec _ (or.elim h hp hq)
end

section
  variables {p q : Prop}

  attribute [instance]
  definition decidable_and [decidable p] [decidable q] : decidable (p ∧ q) :=
  if hp : p then
    if hq : q then is_true ⟨hp, hq⟩
    else is_false (assume h : p ∧ q, hq (and.right h))
  else is_false (assume h : p ∧ q, hp (and.left h))

  attribute [instance]
  definition decidable_or [decidable p] [decidable q] : decidable (p ∨ q) :=
  if hp : p then is_true (or.inl hp) else
    if hq : q then is_true (or.inr hq) else
      is_false (or.rec hp hq)

  attribute [instance]
  definition decidable_not [decidable p] : decidable (¬p) :=
  if hp : p then is_false (absurd hp) else is_true hp

  attribute [instance]
  definition decidable_implies [decidable p] [decidable q] : decidable (p → q) :=
  if hp : p then
    if hq : q then is_true (assume h, hq)
    else is_false (assume h : p → q, absurd (h hp) hq)
  else is_true (assume h, absurd h hp)

  attribute [instance]
  definition decidable_iff [decidable p] [decidable q] : decidable (p ↔ q) :=
  decidable_and

end

attribute [reducible]
definition decidable_pred {A : Type u} (r : A → Prop) := Π (a   : A), decidable (r a)
attribute [reducible]
definition decidable_rel  {A : Type u} (r : A → A → Prop) := Π (a b : A), decidable (r a b)
attribute [reducible]
definition decidable_eq   (A : Type u) := decidable_rel (@eq A)
attribute [instance]
definition decidable_ne {A : Type u} [decidable_eq A] (a b : A) : decidable (a ≠ b) :=
decidable_implies

theorem bool.ff_ne_tt : ff = tt → false
.

definition is_dec_eq {A : Type u} (p : A → A → bool) : Prop   := ∀ ⦃x y : A⦄, p x y = tt → x = y
definition is_dec_refl {A : Type u} (p : A → A → bool) : Prop := ∀ x, p x x = tt

open decidable
attribute [instance]
protected definition bool.has_decidable_eq : ∀ a b : bool, decidable (a = b)
| ff ff := is_true rfl
| ff tt := is_false bool.ff_ne_tt
| tt ff := is_false (ne.symm bool.ff_ne_tt)
| tt tt := is_true rfl

definition decidable_eq_of_bool_pred {A : Type u} {p : A → A → bool} (h₁ : is_dec_eq p) (h₂ : is_dec_refl p) : decidable_eq A :=
take x y : A,
 if hp : p x y = tt then is_true (h₁ hp)
 else is_false (assume hxy : x = y, absurd (h₂ y) (@eq.rec_on _ _ (λ z, ¬p z y = tt) _ hxy hp))

theorem decidable_eq_inl_refl {A : Type u} [h : decidable_eq A] (a : A) : h a a = is_true (eq.refl a) :=
match (h a a) with
| (is_true e)  := rfl
| (is_false n) := absurd rfl n
end

theorem decidable_eq_inr_neg {A : Type u} [h : decidable_eq A] {a b : A} : Π n : a ≠ b, h a b = is_false n :=
assume n,
match (h a b) with
| (is_true e)   := absurd e n
| (is_false n₁) := proof_irrel n n₁ ▸ eq.refl (is_false n)
end

/- inhabited -/

inductive [class] inhabited (A : Type u)
| mk : A → inhabited

attribute [inline]
protected definition inhabited.value {A : Type u} : inhabited A → A :=
λ h, inhabited.rec (λ a, a) h

attribute [inline]
protected definition inhabited.destruct {A : Type u} {B : Type v} (h₁ : inhabited A) (h₂ : A → B) : B :=
inhabited.rec h₂ h₁

attribute [inline]
definition default (A : Type u) [h : inhabited A] : A :=
inhabited.value h

attribute [inline, irreducible]
definition arbitrary (A : Type u) [h : inhabited A] : A :=
inhabited.value h

attribute [instance]
definition Prop.is_inhabited : inhabited Prop :=
⟨true⟩

attribute [instance]
definition inhabited_fun (A : Type u) {B : Type v} [h : inhabited B] : inhabited (A → B) :=
inhabited.rec_on h (λ b, ⟨λ a, b⟩)

attribute [instance]
definition inhabited_Pi (A : Type u) {B : A → Type v} [Π x, inhabited (B x)] :
  inhabited (Π x, B x) :=
⟨λ a, default (B a)⟩

attribute [inline, instance]
protected definition bool.is_inhabited : inhabited bool :=
⟨ff⟩

attribute [inline, instance]
protected definition pos_num.is_inhabited : inhabited pos_num :=
⟨pos_num.one⟩

attribute [inline, instance]
protected definition num.is_inhabited : inhabited num :=
⟨num.zero⟩

inductive [class] nonempty (A : Type u) : Prop
| intro : A → nonempty

protected definition nonempty.elim {A : Type u} {p : Prop} (h₁ : nonempty A) (h₂ : A → p) : p :=
nonempty.rec h₂ h₁

attribute [instance]
theorem nonempty_of_inhabited {A : Type u} [inhabited A] : nonempty A :=
⟨default A⟩

theorem nonempty_of_exists {A : Type u} {p : A → Prop} : (∃ x, p x) → nonempty A
| ⟨w, h⟩ := ⟨w⟩

/- subsingleton -/

inductive [class] subsingleton (A : Type u) : Prop
| intro : (∀ a b : A, a = b) → subsingleton

protected definition subsingleton.elim {A : Type u} [h : subsingleton A] : ∀ (a b : A), a = b :=
subsingleton.rec (λ p, p) h

protected definition subsingleton.helim {A B : Type u} [h : subsingleton A] (h : A = B) : ∀ (a : A) (b : B), a == b :=
eq.rec_on h (λ a b : A, heq_of_eq (subsingleton.elim a b))

attribute [instance]
definition subsingleton_prop (p : Prop) : subsingleton p :=
⟨λ a b, proof_irrel a b⟩

attribute [instance]
definition subsingleton_decidable (p : Prop) : subsingleton (decidable p) :=
subsingleton.intro (λ d₁,
  match d₁ with
  | (is_true t₁) := (λ d₂,
    match d₂ with
    | (is_true t₂) := eq.rec_on (proof_irrel t₁ t₂) rfl
    | (is_false f₂) := absurd t₁ f₂
    end)
  | (is_false f₁) := (λ d₂,
    match d₂ with
    | (is_true t₂) := absurd t₂ f₁
    | (is_false f₂) := eq.rec_on (proof_irrel f₁ f₂) rfl
    end)
  end)

protected theorem rec_subsingleton {p : Prop} [h : decidable p]
    {h₁ : p → Type u} {h₂ : ¬p → Type u}
    [h₃ : Π (h : p), subsingleton (h₁ h)] [h₄ : Π (h : ¬p), subsingleton (h₂ h)]
  : subsingleton (decidable.rec_on h h₂ h₁) :=
match h with
| (is_true h)  := h₃ h
| (is_false h) := h₄ h
end

theorem if_pos {c : Prop} [h : decidable c] (hc : c) {A : Type u} {t e : A} : (ite c t e) = t :=
match h with
| (is_true  hc)  := rfl
| (is_false hnc) := absurd hc hnc
end

theorem if_neg {c : Prop} [h : decidable c] (hnc : ¬c) {A : Type u} {t e : A} : (ite c t e) = e :=
match h with
| (is_true hc)   := absurd hc hnc
| (is_false hnc) := rfl
end

attribute [simp]
theorem if_t_t (c : Prop) [h : decidable c] {A : Type u} (t : A) : (ite c t t) = t :=
match h with
| (is_true hc)   := rfl
| (is_false hnc) := rfl
end

theorem implies_of_if_pos {c t e : Prop} [decidable c] (h : ite c t e) : c → t :=
assume hc, eq.rec_on (if_pos hc : ite c t e = t) h

theorem implies_of_if_neg {c t e : Prop} [decidable c] (h : ite c t e) : ¬c → e :=
assume hnc, eq.rec_on (if_neg hnc : ite c t e = e) h

theorem if_ctx_congr {A : Type u} {b c : Prop} [dec_b : decidable b] [dec_c : decidable c]
                     {x y u v : A}
                     (h_c : b ↔ c) (h_t : c → x = u) (h_e : ¬c → y = v) :
        ite b x y = ite c u v :=
match dec_b, dec_c with
| (is_false h₁), (is_false h₂) := h_e h₂
| (is_true h₁),  (is_true h₂)  := h_t h₂
| (is_false h₁), (is_true h₂)  := absurd h₂ (iff.mp (not_iff_not_of_iff h_c) h₁)
| (is_true h₁),  (is_false h₂) := absurd h₁ (iff.mpr (not_iff_not_of_iff h_c) h₂)
end

attribute [congr]
theorem if_congr {A : Type u} {b c : Prop} [dec_b : decidable b] [dec_c : decidable c]
                 {x y u v : A}
                 (h_c : b ↔ c) (h_t : x = u) (h_e : y = v) :
        ite b x y = ite c u v :=
@if_ctx_congr A b c dec_b dec_c x y u v h_c (λ h, h_t) (λ h, h_e)

theorem if_ctx_simp_congr {A : Type u} {b c : Prop} [dec_b : decidable b] {x y u v : A}
                        (h_c : b ↔ c) (h_t : c → x = u) (h_e : ¬c → y = v) :
        ite b x y = (@ite c (decidable_of_decidable_of_iff dec_b h_c) A u v) :=
@if_ctx_congr A b c dec_b (decidable_of_decidable_of_iff dec_b h_c) x y u v h_c h_t h_e

attribute [congr]
theorem if_simp_congr {A : Type u} {b c : Prop} [dec_b : decidable b] {x y u v : A}
                 (h_c : b ↔ c) (h_t : x = u) (h_e : y = v) :
        ite b x y = (@ite c (decidable_of_decidable_of_iff dec_b h_c) A u v) :=
@if_ctx_simp_congr A b c dec_b x y u v h_c (λ h, h_t) (λ h, h_e)

attribute [simp]
definition if_true {A : Type u} (t e : A) : (if true then t else e) = t :=
if_pos trivial

attribute [simp]
definition if_false {A : Type u} (t e : A) : (if false then t else e) = e :=
if_neg not_false

theorem if_ctx_congr_prop {b c x y u v : Prop} [dec_b : decidable b] [dec_c : decidable c]
                      (h_c : b ↔ c) (h_t : c → (x ↔ u)) (h_e : ¬c → (y ↔ v)) :
        ite b x y ↔ ite c u v :=
match dec_b, dec_c with
| (is_false h₁), (is_false h₂) := h_e h₂
| (is_true h₁),  (is_true h₂)  := h_t h₂
| (is_false h₁), (is_true h₂)  := absurd h₂ (iff.mp (not_iff_not_of_iff h_c) h₁)
| (is_true h₁),  (is_false h₂) := absurd h₁ (iff.mpr (not_iff_not_of_iff h_c) h₂)
end

attribute [congr]
theorem if_congr_prop {b c x y u v : Prop} [dec_b : decidable b] [dec_c : decidable c]
                      (h_c : b ↔ c) (h_t : x ↔ u) (h_e : y ↔ v) :
        ite b x y ↔ ite c u v :=
if_ctx_congr_prop h_c (λ h, h_t) (λ h, h_e)

theorem if_ctx_simp_congr_prop {b c x y u v : Prop} [dec_b : decidable b]
                               (h_c : b ↔ c) (h_t : c → (x ↔ u)) (h_e : ¬c → (y ↔ v)) :
        ite b x y ↔ (@ite c (decidable_of_decidable_of_iff dec_b h_c) Prop u v) :=
@if_ctx_congr_prop b c x y u v dec_b (decidable_of_decidable_of_iff dec_b h_c) h_c h_t h_e

attribute [congr]
theorem if_simp_congr_prop {b c x y u v : Prop} [dec_b : decidable b]
                           (h_c : b ↔ c) (h_t : x ↔ u) (h_e : y ↔ v) :
        ite b x y ↔ (@ite c (decidable_of_decidable_of_iff dec_b h_c) Prop u v) :=
@if_ctx_simp_congr_prop b c x y u v dec_b h_c (λ h, h_t) (λ h, h_e)

theorem dif_pos {c : Prop} [h : decidable c] (hc : c) {A : Type u} {t : c → A} {e : ¬ c → A} : dite c t e = t hc :=
match h with
| (is_true hc)   := rfl
| (is_false hnc) := absurd hc hnc
end

theorem dif_neg {c : Prop} [h : decidable c] (hnc : ¬c) {A : Type u} {t : c → A} {e : ¬ c → A} : dite c t e = e hnc :=
match h with
| (is_true hc)   := absurd hc hnc
| (is_false hnc) := rfl
end

theorem dif_ctx_congr {A : Type u} {b c : Prop} [dec_b : decidable b] [dec_c : decidable c]
                      {x : b → A} {u : c → A} {y : ¬b → A} {v : ¬c → A}
                      (h_c : b ↔ c)
                      (h_t : ∀ (h : c),    x (iff.mpr h_c h)                      = u h)
                      (h_e : ∀ (h : ¬c),   y (iff.mpr (not_iff_not_of_iff h_c) h) = v h) :
        (@dite b dec_b A x y) = (@dite c dec_c A u v) :=
match dec_b, dec_c with
| (is_false h₁), (is_false h₂) := h_e h₂
| (is_true h₁),  (is_true h₂)  := h_t h₂
| (is_false h₁), (is_true h₂)  := absurd h₂ (iff.mp (not_iff_not_of_iff h_c) h₁)
| (is_true h₁),  (is_false h₂) := absurd h₁ (iff.mpr (not_iff_not_of_iff h_c) h₂)
end

theorem dif_ctx_simp_congr {A : Type u} {b c : Prop} [dec_b : decidable b]
                         {x : b → A} {u : c → A} {y : ¬b → A} {v : ¬c → A}
                         (h_c : b ↔ c)
                         (h_t : ∀ (h : c),    x (iff.mpr h_c h)                      = u h)
                         (h_e : ∀ (h : ¬c),   y (iff.mpr (not_iff_not_of_iff h_c) h) = v h) :
        (@dite b dec_b A x y) = (@dite c (decidable_of_decidable_of_iff dec_b h_c) A u v) :=
@dif_ctx_congr A b c dec_b (decidable_of_decidable_of_iff dec_b h_c) x u y v h_c h_t h_e

-- Remark: dite and ite are "definitionally equal" when we ignore the proofs.
theorem dite_ite_eq (c : Prop) [h : decidable c] {A : Type u} (t : A) (e : A) : dite c (λ h, t) (λ h, e) = ite c t e :=
match h with
| (is_true hc)   := rfl
| (is_false hnc) := rfl
end

definition as_true (c : Prop) [decidable c] : Prop :=
if c then true else false

definition as_false (c : Prop) [decidable c] : Prop :=
if c then false else true

definition of_as_true {c : Prop} [h₁ : decidable c] (h₂ : as_true c) : c :=
match h₁, h₂ with
| (is_true h_c),  h₂ := h_c
| (is_false h_c), h₂ := false.elim h₂
end

-- namespace used to collect congruence rules for "contextual simplification"
namespace contextual
  attribute if_ctx_simp_congr      [congr]
  attribute if_ctx_simp_congr_prop [congr]
  attribute dif_ctx_simp_congr     [congr]
end contextual
