85 lines
3.2 KiB
Text
85 lines
3.2 KiB
Text
/-
|
||
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
|
||
Module: logic.quantifiers
|
||
Authors: Leonardo de Moura, Jeremy Avigad
|
||
|
||
Universal and existential quantifiers. See also init.logic.
|
||
-/
|
||
|
||
open inhabited nonempty
|
||
|
||
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} (H : ∀x, φ x ↔ ψ x) : (∀x, φ x) ↔ (∀x, ψ x) :=
|
||
iff.intro
|
||
(assume Hl, take x, iff.elim_left (H x) (Hl x))
|
||
(assume Hr, take x, iff.elim_right (H x) (Hr x))
|
||
|
||
theorem exists_congr {A : Type} {φ ψ : A → Prop} (H : ∀x, φ x ↔ ψ x) : (∃x, φ x) ↔ (∃x, ψ x) :=
|
||
iff.intro
|
||
(assume Hex, obtain w Pw, from Hex,
|
||
exists.intro w (iff.elim_left (H w) Pw))
|
||
(assume Hex, obtain w Qw, from Hex,
|
||
exists.intro w (iff.elim_right (H w) Qw))
|
||
|
||
theorem forall_true_iff_true (A : Type) : (∀x : A, true) ↔ true :=
|
||
iff.intro (assume H, trivial) (assume H, take x, trivial)
|
||
|
||
theorem forall_p_iff_p (A : Type) [H : inhabited A] (p : Prop) : (∀x : A, p) ↔ p :=
|
||
iff.intro (assume Hl, inhabited.destruct H (take x, Hl x)) (assume Hr, take x, Hr)
|
||
|
||
theorem exists_p_iff_p (A : Type) [H : inhabited A] (p : Prop) : (∃x : A, p) ↔ p :=
|
||
iff.intro
|
||
(assume Hl, obtain a Hp, from Hl, Hp)
|
||
(assume Hr, inhabited.destruct H (take a, exists.intro a Hr))
|
||
|
||
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.elim_left (H x)) (take x, and.elim_right (H x)))
|
||
(assume H, take x, and.intro (and.elim_left H x) (and.elim_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
|
||
(assume H, obtain (w : A) (Hw : φ w ∨ ψ w), from H,
|
||
or.elim Hw
|
||
(assume Hw1 : φ w, or.inl (exists.intro w Hw1))
|
||
(assume Hw2 : ψ w, or.inr (exists.intro w Hw2)))
|
||
(assume H, or.elim H
|
||
(assume H1, obtain (w : A) (Hw : φ w), from H1,
|
||
exists.intro w (or.inl Hw))
|
||
(assume H2, obtain (w : A) (Hw : ψ w), from H2,
|
||
exists.intro w (or.inr Hw)))
|
||
|
||
theorem nonempty_of_exists {A : Type} {P : A → Prop} (H : ∃x, P x) : nonempty A :=
|
||
obtain w Hw, from H, nonempty.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) :=
|
||
decidable.rec_on H
|
||
(λ pa, inl (λ x heq, eq.rec_on (eq.rec_on heq rfl) pa))
|
||
(λ npa, inr (λ h, absurd (h a rfl) npa))
|
||
|
||
definition decidable_exists_eq [instance] : decidable (∃ x, x = a ∧ P x) :=
|
||
decidable.rec_on H
|
||
(λ pa, inl (exists.intro a (and.intro rfl pa)))
|
||
(λ npa, inr (λ h,
|
||
obtain (w : A) (Hw : w = a ∧ P w), from h,
|
||
absurd (and.rec_on Hw (λ h₁ h₂, h₁ ▸ h₂)) npa))
|
||
end
|