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feat(library/logic/quantifiers): add 'the'

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Jeremy Avigad 2015-09-12 08:16:13 -04:00
parent 3035dd7e66
commit 20f6b4c6bd
1 changed files with 56 additions and 21 deletions
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77
library/logic/quantifiers.lean
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@ -14,43 +14,43 @@ 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,
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,
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)) :=
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)) :=
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 :=
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 forall_p_iff_p (A : Type) [H : inhabited A] (p : Prop) : (∀ x : A, p) ↔ p :=
iff.intro (inhabited.destruct H) (λ Hr 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 exists_p_iff_p (A : Type) [H : inhabited A] (p : Prop) : (∃ x : A, p) ↔ p :=
iff.intro (Exists.rec (λ x 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) :=
(∀ 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) :=
(∃ 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)))
(Exists.rec (λ x, or.imp !exists.intro !exists.intro))
(or.rec (exists_imp_exists (λ x, or.inl))
(exists_imp_exists (λ x, or.inr)))
section
open decidable eq.ops
@ -83,19 +83,54 @@ theorem exists_unique.elim {A : Type} {p : A → Prop} {b : Prop}
obtain w Hw, from H2,
H1 w (and.left Hw) (and.right Hw)
theorem exists_unique_of_exists_of_unique {A : Type} {p : A → Prop}
(Hex : ∃ x, p x) (Hunique : ∀ y₁ y₂, p y₁ → p y₂ → y₁ = y₂) :
∃! x, p x :=
obtain x px, from Hex,
exists_unique.intro x px (take y, suppose p y, show y = x, from !Hunique this px)
theorem exists_of_exists_unique {A : Type} {p : A → Prop} (H : ∃! x, p x) :
∃ x, p x :=
obtain x Hx, from H,
exists.intro x (and.left Hx)
theorem unique_of_exists_unique {A : Type} {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)))
/- definite description -/
section
open classical
noncomputable definition the {A : Type} {p : A → Prop} (H : ∃! x, p x) : A :=
some (exists_of_exists_unique H)
theorem the_spec {A : Type} {p : A → Prop} (H : ∃! x, p x) : p (the H) :=
some_spec (exists_of_exists_unique H)
theorem eq_the {A : Type} {p : A → Prop} (H : ∃! x, p x) {y : A} (Hy : p y) :
y = the H :=
unique_of_exists_unique H Hy (the_spec H)
end
/- congruences -/
section
variables {A : Type} {p₁ p₂ : A → Prop} (H : ∀x, p₁ x ↔ p₂ x)
variables {A : Type} {p₁ p₂ : A → Prop} (H : ∀ x, p₁ x ↔ p₂ x)
theorem congr_forall : (∀x, p₁ x) ↔ (∀x, p₂ x) :=
theorem congr_forall : (∀ x, p₁ x) ↔ (∀ x, p₂ x) :=
forall_congr H
theorem congr_exists : (∃x, p₁ x) ↔ (∃x, p₂ x) :=
theorem congr_exists : (∃ x, p₁ x) ↔ (∃ x, p₂ x) :=
exists_congr H
include H
theorem congr_exists_unique : (∃!x, p₁ x) ↔ (∃!x, p₂ x) :=
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