lean2/library/logic/quantifiers.lean

/-
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

Universal and existential quantifiers. See also init.logic.
-/
import .connectives
open inhabited nonempty

theorem exists_imp_distrib {A : Type} {B : Prop} {P : A → Prop} : ((∃ a : A, P a) → B) ↔ (∀ a : A, P a → B) :=
iff.intro (λ e x H, e (exists.intro x H)) Exists.rec

theorem forall_iff_not_exists {A : Type} {P : A → Prop} : (¬ ∃ a : A, P a) ↔ ∀ a : A, ¬ P a :=
exists_imp_distrib

theorem not_forall_not_of_exists {A : Type} {p : A → Prop} (H : ∃x, p x) : ¬∀x, ¬p x :=
assume H1 : ∀x, ¬p x,
  obtain (w : A) (Hw : p w), from H,
  absurd Hw (H1 w)

theorem not_exists_not_of_forall {A : Type} {p : A → Prop} (H2 : ∀x, p x) : ¬∃x, ¬p x :=
assume H1 : ∃x, ¬p x,
  obtain (w : A) (Hw : ¬p w), from H1,
  absurd (H2 w) Hw

theorem forall_congr {A : Type} {φ ψ : A → Prop} : (∀x, φ x ↔ ψ x) → ((∀x, φ x) ↔ (∀x, ψ x)) :=
forall_iff_forall

theorem exists_congr {A : Type} {φ ψ : A → Prop} : (∀x, φ x ↔ ψ x) → ((∃x, φ x) ↔ (∃x, ψ x)) :=
exists_iff_exists

theorem forall_true_iff_true (A : Type) : (∀x : A, true) ↔ true :=
iff_true_intro (λH, trivial)

theorem forall_p_iff_p (A : Type) [H : inhabited A] (p : Prop) : (∀x : A, p) ↔ p :=
iff.intro (inhabited.destruct H) (λHr x, Hr)

theorem exists_p_iff_p (A : Type) [H : inhabited A] (p : Prop) : (∃x : A, p) ↔ p :=
iff.intro (Exists.rec (λx Hp, Hp)) (inhabited.destruct H exists.intro)

theorem forall_and_distribute {A : Type} (φ ψ : A → Prop) :
  (∀x, φ x ∧ ψ x) ↔ (∀x, φ x) ∧ (∀x, ψ x) :=
iff.intro
  (assume H, and.intro (take x, and.left (H x)) (take x, and.right (H x)))
  (assume H x, and.intro (and.left H x) (and.right H x))

theorem exists_or_distribute {A : Type} (φ ψ : A → Prop) :
  (∃x, φ x ∨ ψ x) ↔ (∃x, φ x) ∨ (∃x, ψ x) :=
iff.intro
  (Exists.rec (λx, or.imp !exists.intro !exists.intro))
  (or.rec (exists_imp_exists (λx, or.inl))
          (exists_imp_exists (λx, or.inr)))

theorem nonempty_of_exists {A : Type} {P : A → Prop} : (∃x, P x) → nonempty A :=
Exists.rec (λw H, intro w)

section
  open decidable eq.ops

  variables {A : Type} (P : A → Prop) (a : A) [H : decidable (P a)]
  include H

  definition decidable_forall_eq [instance] : decidable (∀ x, x = a → P x) :=
  if pa : P a then inl (λ x heq, eq.substr heq pa)
  else inr (not.mto (λH, H a rfl) pa)

  definition decidable_exists_eq [instance] : decidable (∃ x, x = a ∧ P x) :=
  if pa : P a then inl (exists.intro a (and.intro rfl pa))
  else inr (Exists.rec (λh, and.rec (λheq, eq.substr heq pa)))
end

/- exists_unique -/

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

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

theorem exists_unique.intro {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, p y → y = w) :
  ∃!x, p x :=
exists.intro w (and.intro H1 H2)

theorem exists_unique.elim {A : Type} {p : A → Prop} {b : Prop}
    (H2 : ∃!x, p x) (H1 : ∀x, p x → (∀y, p y → y = x) → b) : b :=
obtain w Hw, from H2,
H1 w (and.left Hw) (and.right Hw)

/- congruences -/

section
  variables {A : Type} {p₁ p₂ : A → Prop} (H : ∀x, p₁ x ↔ p₂ x)

  theorem congr_forall : (∀x, p₁ x) ↔ (∀x, p₂ x) :=
  forall_congr H

  theorem congr_exists : (∃x, p₁ x) ↔ (∃x, p₂ x) :=
  exists_congr H

  include H
  theorem congr_exists_unique : (∃!x, p₁ x) ↔ (∃!x, p₂ x) :=
   congr_exists (λx, congr_and (H x) (congr_forall
     (λy, congr_imp (H y) iff.rfl)))
end