136 lines
4.8 KiB
Text
136 lines
4.8 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura, Jeremy Avigad
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Universal and existential quantifiers. See also init.logic.
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-/
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import .connectives
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open inhabited nonempty
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theorem exists_imp_distrib {A : Type} {B : Prop} {P : A → Prop} : ((∃ a : A, P a) → B) ↔ (∀ a : A, P a → B) :=
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iff.intro (λ e x H, e (exists.intro x H)) Exists.rec
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theorem forall_iff_not_exists {A : Type} {P : A → Prop} : (¬ ∃ a : A, P a) ↔ ∀ a : A, ¬ P a :=
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exists_imp_distrib
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theorem not_forall_not_of_exists {A : Type} {p : A → Prop} (H : ∃x, p x) : ¬ ∀ x, ¬ p x :=
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assume H1 : ∀ x, ¬ p x,
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obtain (w : A) (Hw : p w), from H,
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absurd Hw (H1 w)
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theorem not_exists_not_of_forall {A : Type} {p : A → Prop} (H2 : ∀x, p x) : ¬ ∃ x, ¬p x :=
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assume H1 : ∃ x, ¬ p x,
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obtain (w : A) (Hw : ¬ p w), from H1,
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absurd (H2 w) Hw
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theorem forall_congr {A : Type} {φ ψ : A → Prop} : (∀ x, φ x ↔ ψ x) → ((∀ x, φ x) ↔ (∀ x, ψ x)) :=
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forall_iff_forall
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theorem exists_congr {A : Type} {φ ψ : A → Prop} : (∀ x, φ x ↔ ψ x) → ((∃ x, φ x) ↔ (∃ x, ψ x)) :=
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exists_iff_exists
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theorem forall_true_iff_true (A : Type) : (∀ x : A, true) ↔ true :=
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iff_true_intro (λH, trivial)
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theorem forall_p_iff_p (A : Type) [H : inhabited A] (p : Prop) : (∀ x : A, p) ↔ p :=
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iff.intro (inhabited.destruct H) (λ Hr x, Hr)
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theorem exists_p_iff_p (A : Type) [H : inhabited A] (p : Prop) : (∃ x : A, p) ↔ p :=
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iff.intro (Exists.rec (λ x Hp, Hp)) (inhabited.destruct H exists.intro)
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theorem forall_and_distribute {A : Type} (φ ψ : A → Prop) :
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(∀ x, φ x ∧ ψ x) ↔ (∀ x, φ x) ∧ (∀ x, ψ x) :=
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iff.intro
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(assume H, and.intro (take x, and.left (H x)) (take x, and.right (H x)))
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(assume H x, and.intro (and.left H x) (and.right H x))
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theorem exists_or_distribute {A : Type} (φ ψ : A → Prop) :
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(∃ x, φ x ∨ ψ x) ↔ (∃ x, φ x) ∨ (∃ x, ψ x) :=
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iff.intro
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(Exists.rec (λ x, or.imp !exists.intro !exists.intro))
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(or.rec (exists_imp_exists (λ x, or.inl))
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(exists_imp_exists (λ x, or.inr)))
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section
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open decidable eq.ops
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variables {A : Type} (P : A → Prop) (a : A) [H : decidable (P a)]
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include H
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definition decidable_forall_eq [instance] : decidable (∀ x, x = a → P x) :=
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if pa : P a then inl (λ x heq, eq.substr heq pa)
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else inr (not.mto (λH, H a rfl) pa)
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definition decidable_exists_eq [instance] : decidable (∃ x, x = a ∧ P x) :=
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if pa : P a then inl (exists.intro a (and.intro rfl pa))
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else inr (Exists.rec (λh, and.rec (λheq, eq.substr heq pa)))
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end
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/- exists_unique -/
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definition exists_unique {A : Type} (p : A → Prop) :=
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∃x, p x ∧ ∀y, p y → y = x
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notation `∃!` binders `,` r:(scoped P, exists_unique P) := r
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theorem exists_unique.intro {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, p y → y = w) :
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∃!x, p x :=
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exists.intro w (and.intro H1 H2)
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theorem exists_unique.elim {A : Type} {p : A → Prop} {b : Prop}
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(H2 : ∃!x, p x) (H1 : ∀x, p x → (∀y, p y → y = x) → b) : b :=
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obtain w Hw, from H2,
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H1 w (and.left Hw) (and.right Hw)
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theorem exists_unique_of_exists_of_unique {A : Type} {p : A → Prop}
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(Hex : ∃ x, p x) (Hunique : ∀ y₁ y₂, p y₁ → p y₂ → y₁ = y₂) :
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∃! x, p x :=
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obtain x px, from Hex,
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exists_unique.intro x px (take y, suppose p y, show y = x, from !Hunique this px)
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theorem exists_of_exists_unique {A : Type} {p : A → Prop} (H : ∃! x, p x) :
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∃ x, p x :=
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obtain x Hx, from H,
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exists.intro x (and.left Hx)
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theorem unique_of_exists_unique {A : Type} {p : A → Prop}
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(H : ∃! x, p x) {y₁ y₂ : A} (py₁ : p y₁) (py₂ : p y₂) :
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y₁ = y₂ :=
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exists_unique.elim H
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(take x, suppose p x,
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assume unique : ∀ y, p y → y = x,
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show y₁ = y₂, from eq.trans (unique _ py₁) (eq.symm (unique _ py₂)))
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/- definite description -/
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section
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open classical
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noncomputable definition the {A : Type} {p : A → Prop} (H : ∃! x, p x) : A :=
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some (exists_of_exists_unique H)
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theorem the_spec {A : Type} {p : A → Prop} (H : ∃! x, p x) : p (the H) :=
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some_spec (exists_of_exists_unique H)
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theorem eq_the {A : Type} {p : A → Prop} (H : ∃! x, p x) {y : A} (Hy : p y) :
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y = the H :=
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unique_of_exists_unique H Hy (the_spec H)
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end
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/- congruences -/
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section
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variables {A : Type} {p₁ p₂ : A → Prop} (H : ∀ x, p₁ x ↔ p₂ x)
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theorem congr_forall : (∀ x, p₁ x) ↔ (∀ x, p₂ x) :=
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forall_congr H
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theorem congr_exists : (∃ x, p₁ x) ↔ (∃ x, p₂ x) :=
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exists_congr H
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include H
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theorem congr_exists_unique : (∃! x, p₁ x) ↔ (∃! x, p₂ x) :=
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congr_exists (λx, congr_and (H x) (congr_forall
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(λy, congr_imp (H y) iff.rfl)))
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end
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