chore(builtin/kernel): remove \bowtie as notation for transitivity

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

examples/lean/ex1.lean

@@ -19,8 +19,8 @@ axiom H2 : b = e
 -- Proof that (h a b) = (h c e)
 theorem T1 : (h a b) = (h c e) :=
     or_elim H1
-        (λ C1, congrh ((and_eliml C1) ⋈ (and_elimr C1)) H2)
+        (λ C1, congrh (trans (and_eliml C1) (and_elimr C1)) H2)
-        (λ C2, congrh ((and_elimr C2) ⋈ (and_eliml C2)) H2)
+        (λ C2, congrh (trans (and_elimr C2) (and_eliml C2)) H2)
 
 -- We can use theorem T1 to prove other theorems
 theorem T2 : (h a (h a b)) = (h a (h c e)) :=

src/builtin/kernel.lean

@@ -156,9 +156,6 @@ theorem symm {A : TypeU} {a b : A} (H : a = b) : b = a
 theorem trans {A : TypeU} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c
 := subst H1 H2
 
-infixl 100 >< : trans
-infixl 100 ⋈  : trans
-
 theorem ne_symm {A : TypeU} {a b : A} (H : a ≠ b) : b ≠ a
 := assume H1 : b = a, H (symm H1)
 
@@ -239,13 +236,13 @@ theorem or_falsel (a : Bool) : a ∨ false ↔ a
            (assume H, or_introl H false)
 
 theorem or_falser (a : Bool) : false ∨ a ↔ a
-:= (or_comm false a) ⋈ (or_falsel a)
+:= trans (or_comm false a) (or_falsel a)
 
 theorem or_truel (a : Bool) : true ∨ a ↔ true
 := eqt_intro (case (λ x : Bool, true ∨ x) trivial trivial a)
 
 theorem or_truer (a : Bool) : a ∨ true ↔ true
-:= (or_comm a true) ⋈ (or_truel a)
+:= trans (or_comm a true) (or_truel a)
 
 theorem or_tauto (a : Bool) : a ∨ ¬ a ↔ true
 := eqt_intro (em a)
@@ -274,7 +271,7 @@ theorem and_falsel (a : Bool) : a ∧ false ↔ false
            (assume H, false_elim (a ∧ false) H)
 
 theorem and_falser (a : Bool) : false ∧ a ↔ false
-:= (and_comm false a) ⋈ (and_falsel a)
+:= trans (and_comm false a) (and_falsel a)
 
 theorem and_absurd (a : Bool) : a ∧ ¬ a ↔ false
 := boolext (assume H, absurd (and_eliml H) (and_elimr H))

tests/lean/elab7.lean.expected.out

@@ -9,7 +9,8 @@
   Assumed: H
   Assumed: a
 eta (F2 a) : (λ x : B, F2 a x) = F2 a
-funext (λ a : A, symm (eta (F1 a)) ⋈ (funext (λ b : B, H a b) ⋈ eta (F2 a))) : (λ x : A, F1 x) = (λ x : A, F2 x)
+funext (λ a : A, trans (symm (eta (F1 a))) (trans (funext (λ b : B, H a b)) (eta (F2 a)))) :
+    (λ x : A, F1 x) = (λ x : A, F2 x)
 funext (λ a : A, funext (λ b : B, H a b)) : (λ (x : A) (x::1 : B), F1 x x::1) = (λ (x : A) (x::1 : B), F2 x x::1)
   Proved: T1
   Proved: T2

tests/lean/induction1.lean.expected.out

@@ -9,6 +9,6 @@ theorem Comm2 : ∀ n m : ℕ, n + m = m + n :=
     λ n : ℕ,
       Induction
           (λ x : ℕ, n + x = x + n)
-          (Nat::add_zeror n ⋈ symm (Nat::add_zerol n))
+          (trans (Nat::add_zeror n) (symm (Nat::add_zerol n)))
           (λ (m : ℕ) (iH : n + m = m + n),
-             Nat::add_succr n m ⋈ subst (refl (n + m + 1)) iH ⋈ symm (Nat::add_succl m n))
+             trans (trans (Nat::add_succr n m) (subst (refl (n + m + 1)) iH)) (symm (Nat::add_succl m n)))

tests/lean/induction2.lean.expected.out

@@ -9,7 +9,7 @@ Failed to solve
         Induction
     with arguments:
         ?M::3
-        λ m : ℕ, Nat::add_zerol m ⋈ symm (Nat::add_zeror m)
+        λ m : ℕ, trans (Nat::add_zerol m) (symm (Nat::add_zeror m))
         λ (n : ℕ) (iH : (?M::3[lift:0:1]) n) (m : ℕ),
           @trans ℕ
                  (n + 1 + m)

tests/lean/scope.lean.expected.out

@@ -15,7 +15,7 @@ definition g (A : Type) (f : A → A → A) (x y : A) : A := f y x
 theorem f_eq_g (A : Type) (f : A → A → A) (H1 : ∀ x y : A, f x y = f y x) : f = g A f :=
     funext (λ x : A,
               funext (λ y : A,
-                        let L1 : f x y = f y x := H1 x y, L2 : f y x = g A f x y := refl (g A f x y) in L1 ⋈ L2))
+                        let L1 : f x y = f y x := H1 x y, L2 : f y x = g A f x y := refl (g A f x y) in trans L1 L2))
 theorem Inj (A B : Type)
     (h : A → B)
     (hinv : B → A)
@@ -26,6 +26,6 @@ theorem Inj (A B : Type)
         L2 : hinv (h x) = x := Inv x,
         L3 : hinv (h y) = y := Inv y,
         L4 : x = hinv (h x) := symm L2,
-        L5 : x = hinv (h y) := L4 ⋈ L1
+        L5 : x = hinv (h y) := trans L4 L1
-    in L5 ⋈ L3
+    in trans L5 L3
 10

tests/lean/tst6.lean.expected.out

@@ -56,12 +56,12 @@ theorem Example2 (a b c d : N) (H : @eq N a b ∧ @eq N b c ∨ @eq N a d ∧ @e
   Set: lean::pp::implicit
 theorem Example3 (a b c d e : N) (H : a = b ∧ b = e ∧ b = c ∨ a = d ∧ d = c) : h a b = h c b :=
     or_elim H
-            (λ H1 : a = b ∧ b = e ∧ b = c, congrH (and_eliml H1 ⋈ and_elimr (and_elimr H1)) (refl b))
+            (λ H1 : a = b ∧ b = e ∧ b = c, congrH (trans (and_eliml H1) (and_elimr (and_elimr H1))) (refl b))
-            (λ H1 : a = d ∧ d = c, congrH (and_eliml H1 ⋈ and_elimr H1) (refl b))
+            (λ H1 : a = d ∧ d = c, congrH (trans (and_eliml H1) (and_elimr H1)) (refl b))
   Proved: Example4
   Set: lean::pp::implicit
 theorem Example4 (a b c d e : N) (H : a = b ∧ b = e ∧ b = c ∨ a = d ∧ d = c) : h a c = h c a :=
     or_elim H
             (λ H1 : a = b ∧ b = e ∧ b = c,
-               let AeqC := and_eliml H1 ⋈ and_elimr (and_elimr H1) in congrH AeqC (symm AeqC))
+               let AeqC := trans (and_eliml H1) (and_elimr (and_elimr H1)) in congrH AeqC (symm AeqC))
-            (λ H1 : a = d ∧ d = c, let AeqC := and_eliml H1 ⋈ and_elimr H1 in congrH AeqC (symm AeqC))
+            (λ H1 : a = d ∧ d = c, let AeqC := trans (and_eliml H1) (and_elimr H1) in congrH AeqC (symm AeqC))