feat(library/standard): add forall and exists theorems

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

library/standard/classical.lean

@@ -143,3 +143,14 @@ theorem a_eq_false (a : Bool) : (a = false) = (¬ a)
 := boolext
      (assume H, eqf_elim H)
      (assume H, eqf_intro H)
+
+theorem not_exists_forall {A : Type} {P : A → Bool} (H : ¬ ∃ x, P x) : ∀ x, ¬ P x
+:= take x, or_elim (em (P x))
+     (assume Hp : P x,   absurd_elim (¬ P x) (exists_intro x Hp) H)
+     (assume Hn : ¬ P x, Hn)
+
+theorem not_forall_exists {A : Type} {P : A → Bool} (H : ¬ ∀ x, P x) : ∃ x, ¬ P x
+:= by_contradiction (assume H1 : ¬ ∃ x, ¬ P x,
+     have H2 : ∀ x, ¬ ¬ P x, from not_exists_forall H1,
+     have H3 : ∀ x, P x, from take x, not_not_elim (H2 x),
+     absurd H3 H)

library/standard/logic.lean

@@ -196,6 +196,16 @@ notation `∃` binders `,` r:(scoped P, Exists P) := r
 theorem exists_elim {A : Type} {P : A → Bool} {B : Bool} (H1 : ∃ x : A, P x) (H2 : ∀ (a : A) (H : P a), B) : B
 := Exists_rec H2 H1
 
+theorem exists_not_forall {A : Type} {P : A → Bool} (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 forall_not_exists {A : Type} {P : A → Bool} (H2 : ∀ x, P x) : ¬ ∃ x, ¬ P x
+:= assume H1 : ∃ x, ¬ P x,
+   obtain (w : A) (Hw : ¬ P w), from H1,
+   absurd (H2 w) Hw
+
 definition exists_unique {A : Type} (p : A → Bool) := ∃ x, p x ∧ ∀ y, y ≠ x → ¬ p y
 
 notation `∃!` binders `,` r:(scoped P, exists_unique P) := r