refactor(library/standard): cleanup notation

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

library/standard/bool.lean

@@ -116,6 +116,6 @@ using decidable
 theorem decidable_eq [instance] (a b : bool) : decidable (a = b)
 := bool_rec
     (bool_rec (inl (refl '0)) (inr b0_ne_b1) b)
-    (bool_rec (inr (not_eq_symm b0_ne_b1)) (inl (refl '1)) b)
+    (bool_rec (inr (ne_symm b0_ne_b1)) (inl (refl '1)) b)
     a
 end

library/standard/classical.lean

@@ -23,13 +23,13 @@ theorem prop_complete_swapped (a : Prop) : a = false ∨ a = true
         a
 
 theorem not_true : (¬true) = false
-:= have aux : ¬ (¬true) = true, from
+:= have aux : (¬true) ≠ true, from
      assume H : (¬true) = true,
        absurd_not_true (H⁻¹ ▸ trivial),
    resolve_right (prop_complete (¬true)) aux
 
 theorem not_false : (¬false) = true
-:= have aux : ¬ (¬false) = false, from
+:= have aux : (¬false) ≠ false, from
      assume H : (¬false) = false,
         H ▸ not_false_trivial,
    resolve_right (prop_complete_swapped (¬false)) aux

library/standard/logic.lean

@@ -196,11 +196,8 @@ theorem a_neq_a_elim {A : Type} {a : A} (H : a ≠ a) : false
 theorem ne_irrefl {A : Type} {a : A} (H : a ≠ a) : false
 := H (refl a)
 
-theorem not_eq_symm {A : Type} {a b : A} (H : ¬ a = b) : ¬ b = a
-:= assume H1 : b = a, H (H1⁻¹)
-
 theorem ne_symm {A : Type} {a b : A} (H : a ≠ b) : b ≠ a
-:= not_eq_symm H
+:= assume H1 : b = a, H (H1⁻¹)
 
 theorem eq_ne_trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b ≠ c) : a ≠ c
 := H1⁻¹ ▸ H2