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

library/logic/quantifiers.lean

@@ -83,6 +83,41 @@ 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