feat(library/algebra/order.lean, data/int/{basic,order}.lean): add theorem, correct gt_trans

library/algebra/order.lean

@@ -87,10 +87,13 @@ section
   theorem gt.trans {a b c : A} (H1 : a > b) (H2: b > c) : a > c := lt.trans H2 H1
   calc_trans gt.trans
 
-  theorem ne_of_lt {a b : A} : a < b → a ≠ b :=
-  assume lt_ab : a < b, assume eq_ab : a = b,
+  theorem ne_of_lt {a b : A} (lt_ab : a < b) : a ≠ b :=
+  assume eq_ab : a = b,
   show false, from lt.irrefl b (eq_ab ▸ lt_ab)
 
+  theorem ne_of_gt {a b : A} (gt_ab : a > b) : a ≠ b :=
+  ne.symm (ne_of_lt gt_ab)
+
   theorem lt.asymm {a b : A} (H : a < b) : ¬ b < a :=
   assume H1 : b < a, lt.irrefl _ (lt.trans H H1)
 end

library/data/int/basic.lean

@@ -150,19 +150,19 @@ int.cases_on a
 
 /- int is a quotient of ordered pairs of natural numbers -/
 
-definition equiv (p q : ℕ × ℕ) : Prop :=  pr1 p + pr2 q = pr2 p + pr1 q
+protected definition equiv (p q : ℕ × ℕ) : Prop :=  pr1 p + pr2 q = pr2 p + pr1 q
 
 local notation p `≡` q := equiv p q
 
-theorem equiv.refl {p : ℕ × ℕ} : p ≡ p := !add.comm
+protected theorem equiv.refl {p : ℕ × ℕ} : p ≡ p := !add.comm
 
-theorem equiv.symm {p q : ℕ × ℕ} (H : p ≡ q) : q ≡ p :=
+protected theorem equiv.symm {p q : ℕ × ℕ} (H : p ≡ q) : q ≡ p :=
 calc
   pr1 q + pr2 p = pr2 p + pr1 q : !add.comm
     ... = pr1 p + pr2 q         : H⁻¹
     ... = pr2 q + pr1 p         : !add.comm
 
-theorem equiv.trans {p q r : ℕ × ℕ} (H1 : p ≡ q) (H2 : q ≡ r) : p ≡ r :=
+protected theorem equiv.trans {p q r : ℕ × ℕ} (H1 : p ≡ q) (H2 : q ≡ r) : p ≡ r :=
 have H3 : pr1 p + pr2 r + pr2 q = pr2 p + pr1 r + pr2 q, from
   calc
    pr1 p + pr2 r + pr2 q = pr1 p + pr2 q + pr2 r : by simp
@@ -172,10 +172,10 @@ have H3 : pr1 p + pr2 r + pr2 q = pr2 p + pr1 r + pr2 q, from
     ... = pr2 p + pr1 r + pr2 q                  : by simp,
 show pr1 p + pr2 r = pr2 p + pr1 r, from add.cancel_right H3
 
-theorem equiv_equiv : is_equivalence equiv :=
+protected theorem equiv_equiv : is_equivalence equiv :=
 is_equivalence.mk @equiv.refl @equiv.symm @equiv.trans
 
-theorem equiv_cases {p q : ℕ × ℕ} (H : equiv p q) :
+protected theorem equiv_cases {p q : ℕ × ℕ} (H : equiv p q) :
     (pr1 p ≥ pr2 p ∧ pr1 q ≥ pr2 q) ∨ (pr1 p < pr2 p ∧ pr1 q < pr2 q) :=
 or.elim (@le_or_gt (pr2 p) (pr1 p))
   (assume H1: pr1 p ≥ pr2 p,
@@ -185,7 +185,7 @@ or.elim (@le_or_gt (pr2 p) (pr1 p))
     have H2 : pr2 p + pr1 q < pr2 p + pr2 q, from H ▸ add_lt_add_right H1 (pr2 q),
     or.inr (and.intro H1 (lt_of_add_lt_add_left H2)))
 
-theorem equiv_of_eq {p q : ℕ × ℕ} (H : p = q) : p ≡ q := H ▸ equiv.refl
+protected theorem equiv_of_eq {p q : ℕ × ℕ} (H : p = q) : p ≡ q := H ▸ equiv.refl
 
 calc_trans equiv.trans
 calc_refl equiv.refl

library/data/int/order.lean

@@ -259,7 +259,8 @@ section port_algebra
   show decidable (b < a), from _
 
   theorem ge.trans: ∀{a b c : ℤ}, a ≥ b → b ≥ c → a ≥ c := @algebra.ge.trans _ _
-  theorem gt.trans: ∀{a b c : ℤ}, a ≥ b → b ≥ c → a ≥ c := @algebra.ge.trans _ _
+  theorem gt.trans: ∀{a b c : ℤ}, a > b → b > c → a > c := @algebra.gt.trans _ _
+  theorem ne_of_gt: ∀{a b : ℤ}, a > b → a ≠ b := @algebra.ne_of_gt _ _
   theorem gt_of_gt_of_ge : ∀{a b c : ℤ}, a > b → b ≥ c → a > c := @algebra.gt_of_gt_of_ge _ _
   theorem gt_of_ge_of_gt : ∀{a b c : ℤ}, a ≥ b → b > c → a > c := @algebra.gt_of_ge_of_gt _ _
   theorem lt.asymm : ∀{a b : ℤ}, a < b → ¬ b < a := @algebra.lt.asymm _ _