fix(frontends/lean/pp): make sure pp and parser are using the same precedences

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

src/frontends/lean/notation.h

@@ -5,9 +5,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 */
 #pragma once
+#include <limits>
 namespace lean {
 constexpr unsigned g_eq_precedence    = 50;
 constexpr unsigned g_arrow_precedence = 25;
+constexpr unsigned g_app_precedence   = std::numeric_limits<unsigned>::max();
 class environment;
 class io_state;
 void init_builtin_notation(environment const & env, io_state & st);

src/frontends/lean/parser.cpp

@@ -123,9 +123,6 @@ static unsigned g_level_plus_prec = 10;
 static unsigned g_level_cup_prec  = 5;
 // ==========================================
 
-// precedence (aka binding power) for function application
-constexpr unsigned app_lbp = std::numeric_limits<unsigned>::max();
-
 // A name that can't be created by the user.
 // It is used as placeholder for parsing A -> B expressions which
 // are syntax sugar for (Pi (_ : A), B)
@@ -825,7 +822,7 @@ class parser::imp {
                         if (imp_args[i]) {
                             args.push_back(save(mk_placeholder(), pos()));
                         } else {
-                            args.push_back(parse_expr(app_lbp));
+                            args.push_back(parse_expr(g_app_precedence));
                         }
                     }
                     return mk_app(args);
@@ -1312,20 +1309,20 @@ class parser::imp {
             name const & id = curr_name();
             auto it = m_local_decls.find(id);
             if (it != m_local_decls.end()) {
-                return app_lbp;
+                return g_app_precedence;
             } else {
                 optional<unsigned> prec = get_lbp(m_env, id);
                 if (prec)
                     return *prec;
                 else
-                    return app_lbp;
+                    return g_app_precedence;
             }
         }
         case scanner::token::Eq : return g_eq_precedence;
         case scanner::token::Arrow : return g_arrow_precedence;
         case scanner::token::LeftParen: case scanner::token::NatVal: case scanner::token::DecimalVal:
         case scanner::token::StringVal: case scanner::token::Type: case scanner::token::Placeholder:
-            return app_lbp;
+            return g_app_precedence;
         default:
             return 0;
         }

src/frontends/lean/pp.cpp

@@ -431,8 +431,10 @@ class pp_fn {
             return g_eq_precedence;
         } else if (is_arrow(e)) {
             return g_arrow_precedence;
-        } else {
+        } else if (is_lambda(e) || is_pi(e) || is_let(e)) {
             return 0;
+        } else {
+            return g_app_precedence;
         }
     }
 

tests/lean/apply_tac1.lean.expected.out

@@ -7,4 +7,4 @@
   Assumed: b
   Assumed: Ax2
   Proved: T2
-Theorem T2 (a : ℤ) : (P a a) ⇒ (f a a) := Discharge (λ H : P a a, Ax1 a a H)
+Theorem T2 (a : ℤ) : P a a ⇒ f a a := Discharge (λ H : P a a, Ax1 a a H)

tests/lean/arith1.lean.expected.out

@@ -12,8 +12,8 @@
 n + m
 n + x + m
   Set: lean::pp::coercion
-(nat_to_int n) + x + (nat_to_int m) + (nat_to_int 10)
+nat_to_int n + x + nat_to_int m + nat_to_int 10
-x + (nat_to_int n) + (nat_to_int m) + (nat_to_int 10)
+x + nat_to_int n + nat_to_int m + nat_to_int 10
-(nat_to_int (n + m + 10)) + x
+nat_to_int (n + m + 10) + x
   Set: lean::pp::notation
 Int::add (nat_to_int (Nat::add (Nat::add n m) 10)) x

tests/lean/arith2.lean.expected.out

@@ -8,9 +8,9 @@
   Assumed: n
 x + i + 1 + n
   Set: lean::pp::coercion
-x + (int_to_real i) + (nat_to_real 1) + (nat_to_real n)
+x + int_to_real i + nat_to_real 1 + nat_to_real n
-x * (int_to_real i) + x
+x * int_to_real i + x
-x - (int_to_real i) + x - x ≥ (nat_to_real 0)
+x - int_to_real i + x - x ≥ nat_to_real 0
 x < x
 x ≤ x
 x > x

tests/lean/arith4.lean.expected.out

@@ -3,7 +3,7 @@
   Assumed: x
 sin x
 sin (x + -1 * (π / 2))
-(sin x) / (sin (x + -1 * (π / 2)))
+sin x / sin (x + -1 * (π / 2))
-(sin (x + -1 * (π / 2))) / (sin x)
+sin (x + -1 * (π / 2)) / sin x
-1 / (sin (x + -1 * (π / 2)))
+1 / sin (x + -1 * (π / 2))
-1 / (sin x)
+1 / sin x

tests/lean/arith5.lean.expected.out

@@ -1,9 +1,9 @@
   Set: pp::colors
   Set: pp::unicode
   Assumed: x
-(1 + -1 * (exp (-2 * x))) / (2 * (exp (-1 * x)))
+(1 + -1 * exp (-2 * x)) / (2 * exp (-1 * x))
-(1 + (exp (-2 * x))) / (2 * (exp (-1 * x)))
+(1 + exp (-2 * x)) / (2 * exp (-1 * x))
-(1 + -1 * (exp (-2 * x))) / (2 * (exp (-1 * x))) / ((1 + (exp (-2 * x))) / (2 * (exp (-1 * x))))
+(1 + -1 * exp (-2 * x)) / (2 * exp (-1 * x)) / ((1 + exp (-2 * x)) / (2 * exp (-1 * x)))
-(1 + (exp (-2 * x))) / (2 * (exp (-1 * x))) / ((1 + -1 * (exp (-2 * x))) / (2 * (exp (-1 * x))))
+(1 + exp (-2 * x)) / (2 * exp (-1 * x)) / ((1 + -1 * exp (-2 * x)) / (2 * exp (-1 * x)))
-1 / ((1 + (exp (-2 * x))) / (2 * (exp (-1 * x))))
+1 / ((1 + exp (-2 * x)) / (2 * exp (-1 * x)))
-1 / ((1 + -1 * (exp (-2 * x))) / (2 * (exp (-1 * x))))
+1 / ((1 + -1 * exp (-2 * x)) / (2 * exp (-1 * x)))

tests/lean/bad5.lean.expected.out

@@ -6,5 +6,5 @@
   Assumed: H1
   Assumed: H2
   Proved: T1
-Axiom H2 : (g a) > 0
+Axiom H2 : g a > 0
-Theorem T1 : (g b) > 0 := SubstP (λ x : ℤ, (g x) > 0) H2 H1
+Theorem T1 : g b > 0 := SubstP (λ x : ℤ, g x > 0) H2 H1

tests/lean/cast1.lean.expected.out

@@ -9,7 +9,7 @@
 Error (line: 12, pos: 6) type mismatch at application
     f m v1
 Function type:
-    Π (n : ℕ), (vector ℤ n) → ℤ
+    Π (n : ℕ), vector ℤ n → ℤ
 Arguments types:
     m : ℕ
     v1 : vector ℤ (m + 0)

tests/lean/coercion2.lean.expected.out

@@ -28,5 +28,5 @@ h (h r s) r : S
 g (g (t2r a) b) (t2r a)
 h (h r s) r
   Set: lean::pp::notation
-(t2r a) ++ b ++ (t2r a)
+t2r a ++ b ++ t2r a
 r ++ s ++ r

tests/lean/conv.lean.expected.out

@@ -2,7 +2,7 @@
   Set: pp::unicode
   Defined: id
   Assumed: p
-λ x : id ℤ, x : (id ℤ) → (id ℤ)
+λ x : id ℤ, x : id ℤ → id ℤ
   Assumed: f
 p f : Bool
   Defined: c
@@ -11,9 +11,9 @@ p f : Bool
 g a : Bool
   Defined: c2
   Assumed: b
-c2::explicit : Π (T : Type), (Type 3) → T → (Type 3)
+c2::explicit : Π (T : Type), Type 3 → T → Type 3
   Assumed: g2
-g2 : (c2 (Type 2) b) → Bool
+g2 : c2 (Type 2) b → Bool
   Assumed: a2
 g2 a2 : Bool
-λ x : c2 (Type 1) b, g2 x : (c2 (Type 1) b) → Bool
+λ x : c2 (Type 1) b, g2 x : c2 (Type 1) b → Bool

tests/lean/elab4.lean.expected.out

@@ -22,7 +22,7 @@ Definition R::explicit (A A' : Type)
     (B : A → Type)
     (B' : A' → Type)
     (H : (Π x : A, B x) = (Π x : A', B' x))
-    (a : A) : (B a) = (B' (C (D H) a)) :=
+    (a : A) : B a = B' (C (D H) a) :=
     R H a
 Theorem R2 (A1 A2 B1 B2 : Type) (H : eq::explicit Type (A1 → B1) (A2 → B2)) (a : A1) : eq::explicit Type B1 B2 :=
     R::explicit A1 A2 (λ x : A1, B1) (λ x : A2, B2) H a

tests/lean/elab5.lean.expected.out

@@ -22,7 +22,7 @@ Definition R::explicit (A A' : Type)
     (B : A → Type)
     (B' : A' → Type)
     (H : (Π x : A, B x) = (Π x : A', B' x))
-    (a : A) : (B a) = (B' (C (D H) a)) :=
+    (a : A) : B a = B' (C (D H) a) :=
     R H a
 Theorem R2 (A1 A2 B1 B2 : Type) (H : eq::explicit Type (A1 → B1) (A2 → B2)) (a : A1) : eq::explicit Type B1 B2 :=
     R::explicit A1 A2 (λ x : A1, B1) (λ x : A2, B2) H a

tests/lean/elab7.lean.expected.out

@@ -8,7 +8,7 @@
   Assumed: F2
   Assumed: H
   Assumed: a
-Eta (F2 a) : (λ x : B, F2 a x) == (F2 a)
+Eta (F2 a) : (λ x : B, F2 a x) == F2 a
 Abst (λ a : A, Trans (Symm (Eta (F1 a))) (Trans (Abst (λ b : B, H a b)) (Eta (F2 a)))) :
     (λ x : A, F1 x) == (λ x : A, F2 x)
 Abst (λ a : A, Abst (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)

tests/lean/eq1.lean.expected.out

@@ -3,4 +3,4 @@
   Assumed: i
 i = 0 : Bool
   Set: lean::pp::coercion
-i = (nat_to_int 0) : Bool
+i = nat_to_int 0 : Bool

tests/lean/ho.lean.expected.out

@@ -6,5 +6,5 @@
   Assumed: H1
   Assumed: H2
   Proved: T1
-Axiom H2 : (g a) > 0
+Axiom H2 : g a > 0
-Theorem T1 : (g b) > 0 := Subst H2 H1
+Theorem T1 : g b > 0 := Subst H2 H1

tests/lean/implicit1.lean.expected.out

@@ -1,13 +1,13 @@
   Set: pp::colors
   Set: pp::unicode
   Assumed: f
-∀ a : ℤ, (f a a) > 0
+∀ a : ℤ, f a a > 0
-∀ a b : ℤ, (f a b) > 0
+∀ a b : ℤ, f a b > 0
   Assumed: g
-∀ (a : ℤ) (b : ℝ), (g a b) > 0
+∀ (a : ℤ) (b : ℝ), g a b > 0
-∀ a b : ℤ, (g a (f a b)) > 0
+∀ a b : ℤ, g a (f a b) > 0
   Set: lean::pp::coercion
-∀ a b : ℤ, (g a (int_to_real (f a b))) > (nat_to_int 0)
+∀ a b : ℤ, g a (int_to_real (f a b)) > nat_to_int 0
 λ a : ℕ, a + 1
 λ a b : ℕ, a + b
 λ a b c : ℤ, a + c + b

tests/lean/interactive/t7.lean.expected.out

@@ -5,7 +5,7 @@
 #   Assumed: Ax
 #   Assumed: a
 # Proof state:
-x : ℤ ⊢ (q a x) ⇒ (p x)
+x : ℤ ⊢ q a x ⇒ p x
 ## Proof state:
 H : q a x, x : ℤ ⊢ p x
 ## Proof state:

tests/lean/let1.lean.expected.out

@@ -11,4 +11,4 @@ let a : ℕ := 10 in a + 1 : ℕ
 30
 let a : ℤ := 20 in a + 10 : ℤ
   Set: lean::pp::coercion
-let a : ℤ := nat_to_int 20 in a + (nat_to_int 10)
+let a : ℤ := nat_to_int 20 in a + nat_to_int 10

tests/lean/norm1.lean.expected.out

@@ -8,7 +8,7 @@
 λ (f : N → N) (y : N), g (f a)
 λ (a : N) (f : N → N) (g : (N → N) → N → N) (h : N → N → N) (z : N), h (g (λ x : N, f (f x)) (f a)) (f a)
 λ (a b : N) (g : Bool → N) (y : Bool), g (a == b)
-λ (a : Type) (b : a → Type) (g : (Type U) → Bool) (y : Type U), g (Π x : a, b x)
+λ (a : Type) (b : a → Type) (g : Type U → Bool) (y : Type U), g (Π x : a, b x)
 λ (f : N → N) (y z : N), g (f a)
 λ (f g : N → N) (y z : N), g (f a)
 λ (f : N → N) (y z : N), P (f a) y z

tests/lean/scope.lean.expected.out

@@ -11,22 +11,16 @@
   Proved: f_eq_g
   Proved: Inj
 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 : Π x y : A, (f x y) = (f y x)) : f = (g A f) :=
+Theorem f_eq_g (A : Type) (f : A → A → A) (H1 : Π x y : A, f x y = f y x) : f = g A f :=
     Abst (λ x : A,
             Abst (λ y : A,
-                    let L1 : (f x y) = (f y x) := H1 x y,
+                    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))
-                        L2 : (f y x) = (g A f x y) := Refl (g A f x y)
+Theorem Inj (A B : Type) (h : A → B) (hinv : B → A) (Inv : Π x : A, hinv (h x) = x) (x y : A) (H : h x = h y) : x =
-                    in Trans L1 L2))
+    y :=
-Theorem Inj (A B : Type)
+    let L1 : hinv (h x) = hinv (h y) := Congr2 hinv H,
-    (h : A → B)
+        L2 : hinv (h x) = x := Inv x,
-    (hinv : B → A)
+        L3 : hinv (h y) = y := Inv y,
-    (Inv : Π x : A, (hinv (h x)) = x)
+        L4 : x = hinv (h x) := Symm L2,
-    (x y : A)
+        L5 : x = hinv (h y) := Trans L4 L1
-    (H : (h x) = (h y)) : x = y :=
-    let L1 : (hinv (h x)) = (hinv (h y)) := Congr2 hinv H,
-        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)) := Trans L4 L1
     in Trans L5 L3
 10

tests/lean/tactic10.lean.expected.out

@@ -5,6 +5,6 @@
   Proved: T1
   Defined: h
   Proved: T2
-Theorem T2 (a b : Bool) : (h a b) ⇒ (f b) ⇒ a :=
+Theorem T2 (a b : Bool) : h a b ⇒ f b ⇒ a :=
     Discharge
         (λ H : a ∨ b, Discharge (λ H::1 : ¬ b, DisjCases H (λ H : a, H) (λ H : b, AbsurdImpAny b a H H::1)))

tests/lean/tactic13.lean.expected.out

@@ -3,7 +3,7 @@
   Assumed: f
   Proved: T1
   Proved: T2
-Theorem T1 (a b c : ℤ) (H1 : a = b) (H2 : a = c) : (f (f a a) b) = (f (f b c) a) :=
+Theorem T1 (a b c : ℤ) (H1 : a = b) (H2 : a = c) : f (f a a) b = f (f b c) a :=
     Congr (Congr (Refl f) (Congr (Congr (Refl f) H1) H2)) (Symm H1)
-Theorem T2 (a b c : ℤ) (H1 : a = b) (H2 : a = c) : (f (f a c)) = (f (f b a)) :=
+Theorem T2 (a b c : ℤ) (H1 : a = b) (H2 : a = c) : f (f a c) = f (f b a) :=
     Congr (Refl f) (Congr (Congr (Refl f) H1) (Symm H2))

tests/lean/tactic14.lean.expected.out

@@ -1,5 +1,5 @@
   Set: pp::colors
   Set: pp::unicode
   Proved: T1
-Theorem T1 (a b : ℤ) (f : ℤ → ℤ) (assumption : a = b) : (f (f a)) = (f (f b)) :=
+Theorem T1 (a b : ℤ) (f : ℤ → ℤ) (assumption : a = b) : f (f a) = f (f b) :=
     Congr (Refl f) (Congr (Refl f) assumption)

tests/lean/tactic9.lean.expected.out

@@ -2,5 +2,5 @@
   Set: pp::unicode
   Defined: f
   Proved: T
-Theorem T (a b : Bool) : a ∨ b ⇒ (f b) ⇒ a :=
+Theorem T (a b : Bool) : a ∨ b ⇒ f b ⇒ a :=
     Discharge (λ H : a ∨ b, Discharge (λ H::1 : f b, DisjCases H (λ H : a, H) (λ H : b, AbsurdImpAny b a H H::1)))

tests/lean/tst1.lean.expected.out

@@ -33,7 +33,7 @@ update (const three ⊥) two ⊤ : vector Bool three
 select::explicit : Π (A : Type) (n : N) (v : vector A n) (i : N), i < n → A
 
 map type ---> 
-map::explicit : Π (A B C : Type) (n : N), (A → B → C) → (vector A n) → (vector B n) → (vector C n)
+map::explicit : Π (A B C : Type) (n : N), (A → B → C) → vector A n → vector B n → vector C n
 
 map normal form --> 
 λ (A B C : Type)

tests/lean/tst11.lean.expected.out

@@ -6,7 +6,7 @@
 ⊤
   Assumed: a
 a ⊕ a ⊕ a
-Subst::explicit : Π (A : Type U) (a b : A) (P : A → Bool), (P a) → a == b → (P b)
+Subst::explicit : Π (A : Type U) (a b : A) (P : A → Bool), P a → a == b → P b
   Proved: EM2
 EM2 : Π a : Bool, a ∨ ¬ a
 EM2 a : a ∨ ¬ a

tests/lean/tst12.lean.expected.out

@@ -5,5 +5,5 @@ let x := ⊤,
     y := ⊤,
     z := x ∧ y,
     f := λ arg1 arg2 : Bool, arg1 ∧ arg2 ⇔ arg2 ∧ arg1 ⇔ arg1 ∨ arg2 ∨ arg2
-in (f x y) ∨ z
+in f x y ∨ z
 ⊤

tests/lean/tst13.lean.expected.out

@@ -1,4 +1,4 @@
   Set: pp::colors
   Set: pp::unicode
 λ x x : Bool, x
-let x := ⊤, y := ⊤, z := x ∧ y, f := λ x y : Bool, x ∧ y ⇔ y ∧ x ⇔ x ∨ y ∨ y in (f x y) ∨ z
+let x := ⊤, y := ⊤, z := x ∧ y, f := λ x y : Bool, x ∧ y ⇔ y ∧ x ⇔ x ∨ y ∨ y in f x y ∨ z

tests/lean/tst15.lean.expected.out

@@ -3,7 +3,7 @@
   Assumed: x
 x : Type U+3 ⊔ M+2 ⊔ 3
   Assumed: f
-f : (Type U+10) → Type
+f : Type U+10 → Type
 f x : Type
 Type 4 : Type 5
 x : Type U+3 ⊔ M+2 ⊔ 3
@@ -14,9 +14,9 @@ Type U ⊔ M : Type U+1 ⊔ M+1
 Type U ⊔ M ⊔ 3 : Type U+1 ⊔ M+1 ⊔ 4
 Type U+1 ⊔ M ⊔ 3
 Type U+1 ⊔ M ⊔ 3 : Type U+2 ⊔ M+1 ⊔ 4
-(Type U) → (Type 5)
+Type U → Type 5
-(Type U) → (Type 5) : Type U+1 ⊔ 6
+Type U → Type 5 : Type U+1 ⊔ 6
-(Type M ⊔ 3) → (Type U+5) : Type M+1 ⊔ 4 ⊔ U+6
+Type M ⊔ 3 → Type U+5 : Type M+1 ⊔ 4 ⊔ U+6
-(Type M ⊔ 3) → (Type U) → (Type 5)
+Type M ⊔ 3 → Type U → Type 5
-(Type M ⊔ 3) → (Type U) → (Type 5) : Type M+1 ⊔ 6 ⊔ U+1
+Type M ⊔ 3 → Type U → Type 5 : Type M+1 ⊔ 6 ⊔ U+1
 Type U

tests/lean/tst16.lean.expected.out

@@ -1,17 +1,17 @@
   Set: pp::colors
   Set: pp::unicode
   Assumed: f
-∀ a b : Type, (f a) = (f b)
+∀ a b : Type, f a = f b
   Assumed: g
-∀ (a b : Type) (c : Bool), g c ((f a) = (f b))
+∀ (a b : Type) (c : Bool), g c (f a = f b)
-∃ (a b : Type) (c : Bool), g c ((f a) = (f b))
+∃ (a b : Type) (c : Bool), g c (f a = f b)
-∀ (a b : Type) (c : Bool), (g c (f a)) = (f b) ⇒ (f a)
+∀ (a b : Type) (c : Bool), g c (f a) = f b ⇒ f a
-∀ (a b : Type) (c : Bool), g c ((f a) = (f b)) : Bool
+∀ (a b : Type) (c : Bool), g c (f a = f b) : Bool
-∀ (a b : Type) (c : Bool), g c ((f a) = (f b))
+∀ (a b : Type) (c : Bool), g c (f a = f b)
-∀ a b : Type, (f a) = (f b)
+∀ a b : Type, f a = f b
-∃ a b : Type, (f a) = (f b) ∧ (f a)
+∃ a b : Type, f a = f b ∧ f a
-∃ a b : Type, (f a) = (f b) ∨ (f b)
+∃ a b : Type, f a = f b ∨ f b
   Assumed: a
-(f a) ∨ (f a)
+f a ∨ f a
-(f a) = a ∨ (f a)
+f a = a ∨ f a
-(f a) = a ∧ (f a)
+f a = a ∧ f a

tests/lean/tst17.lean.expected.out

@@ -2,9 +2,8 @@
   Set: pp::unicode
   Assumed: f
   Assumed: g
-∀ a b : Type, ∃ c : Type, (g a b) = (f c)
+∀ a b : Type, ∃ c : Type, g a b = f c
-∀ a b : Type, ∃ c : Type, (g a b) = (f c) : Bool
+∀ a b : Type, ∃ c : Type, g a b = f c : Bool
 (λ a : Type,
-   (λ b : Type, if ((λ x : Type, if ((g a b) == (f x)) ⊥ ⊤) == (λ x : Type, ⊤)) ⊥ ⊤) ==
+   (λ b : Type, if ((λ x : Type, if (g a b == f x) ⊥ ⊤) == (λ x : Type, ⊤)) ⊥ ⊤) == (λ x : Type, ⊤)) ==
-   (λ x : Type, ⊤)) ==
 (λ x : Type, ⊤)

tests/lean/tst4.lean.expected.out

@@ -10,8 +10,7 @@ f::explicit ((N → N) → N → N) (λ x : N → N, x) (λ y : N → N, y)
   Assumed: EqNice
 EqNice::explicit N n1 n2
 f::explicit N n1 n2 : N
-Congr::explicit :
+Congr::explicit : Π (A : Type U) (B : A → Type U) (f g : Π x : A, B x) (a b : A), f == g → a == b → f a == g b
-    Π (A : Type U) (B : A → (Type U)) (f g : Π x : A, B x) (a b : A), f == g → a == b → (f a) == (g b)
 f::explicit N n1 n2
   Assumed: a
   Assumed: b

tests/lean/tst6.lean.expected.out

@@ -17,10 +17,10 @@ Theorem CongrH {a1 a2 b1 b2 : N} (H1 : eq::explicit N a1 b1) (H2 : eq::explicit
         b2
         (Congr::explicit N (λ x : N, N → N) h h a1 b1 (Refl::explicit (N → N → N) h) H1)
         H2
-Theorem CongrH::explicit (a1 a2 b1 b2 : N) (H1 : a1 = b1) (H2 : a2 = b2) : (h a1 a2) = (h b1 b2) := CongrH H1 H2
+Theorem CongrH::explicit (a1 a2 b1 b2 : N) (H1 : a1 = b1) (H2 : a2 = b2) : h a1 a2 = h b1 b2 := CongrH H1 H2
   Set: lean::pp::implicit
-Theorem CongrH {a1 a2 b1 b2 : N} (H1 : a1 = b1) (H2 : a2 = b2) : (h a1 a2) = (h b1 b2) := Congr (Congr (Refl h) H1) H2
+Theorem CongrH {a1 a2 b1 b2 : N} (H1 : a1 = b1) (H2 : a2 = b2) : h a1 a2 = h b1 b2 := Congr (Congr (Refl h) H1) H2
-Theorem CongrH::explicit (a1 a2 b1 b2 : N) (H1 : a1 = b1) (H2 : a2 = b2) : (h a1 a2) = (h b1 b2) := CongrH H1 H2
+Theorem CongrH::explicit (a1 a2 b1 b2 : N) (H1 : a1 = b1) (H2 : a2 = b2) : h a1 a2 = h b1 b2 := CongrH H1 H2
   Proved: Example1
   Set: lean::pp::implicit
 Theorem Example1 (a b c d : N)
@@ -31,7 +31,7 @@ Theorem Example1 (a b c d : N)
     DisjCases::explicit
         (eq::explicit N a b ∧ eq::explicit N b c)
         (eq::explicit N a d ∧ eq::explicit N d c)
-        ((h a b) == (h c b))
+        (h a b == h c b)
         H
         (λ H1 : eq::explicit N a b ∧ eq::explicit N b c,
            CongrH::explicit
@@ -103,14 +103,14 @@ Theorem Example2 (a b c d : N)
                (Refl::explicit N b))
   Proved: Example3
   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) :=
+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 :=
     DisjCases
         H
         (λ H1 : a = b ∧ b = e ∧ b = c, CongrH (Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1))) (Refl b))
         (λ H1 : a = d ∧ d = c, CongrH (Trans (Conjunct1 H1) (Conjunct2 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) :=
+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 :=
     DisjCases
         H
         (λ H1 : a = b ∧ b = e ∧ b = c,