refactor(library): replace decidable_eq with abbreviation

library/data/bool.lean

@@ -139,8 +139,8 @@ namespace bool
   inhabited.mk ff
 
   theorem has_decidable_eq [protected] [instance] : decidable_eq bool :=
-  decidable_eq.intro (λ (a b : bool),
+  take a b : bool,
     rec_on a
       (rec_on b (inl rfl) (inr ff_ne_tt))
-      (rec_on b (inr (ne.symm ff_ne_tt)) (inl rfl)))
+      (rec_on b (inr (ne.symm ff_ne_tt)) (inl rfl))
 end bool

library/data/int/basic.lean

@@ -206,7 +206,7 @@ exists_intro (pr1 (rep a))
 definition of_nat (n : ℕ) : ℤ := psub (pair n 0)
 
 theorem has_decidable_eq [instance] [protected] : decidable_eq ℤ :=
-decidable_eq.intro (λ (a b : ℤ), _)
+take a b : ℤ, _
 
 opaque_hint (hiding int)
 coercion of_nat

library/data/nat/basic.lean

@@ -106,7 +106,7 @@ induction_on n
   (take k IH H, IH (succ_inj H))
 
 theorem has_decidable_eq [instance] [protected] : decidable_eq ℕ :=
-decidable_eq.intro (λ (n m : ℕ),
+take n m : ℕ,
 have general : ∀n, decidable (n = m), from
   rec_on m
     (take n,
@@ -123,7 +123,7 @@ have general : ∀n, decidable (n = m), from
               have H1 : succ n' ≠ succ m', from
                 assume Heq, absurd (succ_inj Heq) Hne,
               inr H1))),
-general n)
+general n
 
 theorem two_step_induction_on {P : ℕ → Prop} (a : ℕ) (H1 : P 0) (H2 : P 1)
     (H3 : ∀ (n : ℕ) (IH1 : P n) (IH2 : P (succ n)), P (succ (succ n))) : P a :=

library/data/option.lean

@@ -39,12 +39,12 @@ namespace option
   inhabited.mk none
 
   theorem has_decidable_eq [protected] [instance] {A : Type} (H : decidable_eq A) : decidable_eq (option A) :=
-  decidable_eq.intro (λ (o₁ o₂ : option A),
+  take o₁ o₂ : option A,
     rec_on o₁
       (rec_on o₂ (inl rfl) (take a₂, (inr (none_ne_some a₂))))
       (take a₁ : A, rec_on o₂
         (inr (ne.symm (none_ne_some a₁)))
         (take a₂ : A, decidable.rec_on (H a₁ a₂)
           (assume Heq : a₁ = a₂, inl (Heq ▸ rfl))
-          (assume Hne : a₁ ≠ a₂, inr (assume Hn : some a₁ = some a₂, absurd (equal Hn) Hne)))))
+          (assume Hne : a₁ ≠ a₂, inr (assume Hn : some a₁ = some a₂, absurd (equal Hn) Hne))))
 end option

library/data/prod.lean

@@ -49,11 +49,11 @@ section
   inhabited.destruct H1 (λa, inhabited.destruct H2 (λb, inhabited.mk (pair a b)))
 
   theorem has_decidable_eq [protected] [instance] (H1 : decidable_eq A) (H2 : decidable_eq B) : decidable_eq (A × B) :=
-  decidable_eq.intro (λ (u v : A × B),
+  take u v : A × B,
     have H3 : u = v ↔ (pr1 u = pr1 v) ∧ (pr2 u = pr2 v), from
       iff.intro
         (assume H, H ▸ and.intro rfl rfl)
         (assume H, and.elim H (assume H4 H5, equal H4 H5)),
-    decidable_iff_equiv _ (iff.symm H3))
+    decidable_iff_equiv _ (iff.symm H3)
 end
 end prod

library/data/subtype.lean

@@ -42,9 +42,9 @@ section
   inhabited.mk (tag a H)
 
   theorem has_decidable_eq [protected] [instance] (H : decidable_eq A) : decidable_eq {x, P x} :=
-  decidable_eq.intro (λ (a1 a2 : {x, P x}),
+  take a1 a2 : {x, P x},
     have H1 : (a1 = a2) ↔ (elt_of a1 = elt_of a2), from
       iff.intro (assume H, eq.subst H rfl) (assume H, equal H),
-    decidable_iff_equiv _ (iff.symm H1))
+    decidable_iff_equiv _ (iff.symm H1)
 end
 end subtype

library/data/sum.lean

@@ -57,7 +57,7 @@ namespace sum
 
   theorem has_eq_decidable [protected] [instance] {A B : Type} (H1 : decidable_eq A) (H2 : decidable_eq B) :
        decidable_eq (A ⊎ B) :=
-  decidable_eq.intro (λ (s1 s2 : A ⊎ B),
+  take s1 s2 : A ⊎ B,
     rec_on s1
       (take a1, show decidable (inl B a1 = s2), from
         rec_on s2
@@ -78,5 +78,5 @@ namespace sum
           (take b2, show decidable (inr A b1 = inr A b2), from
             decidable.rec_on (H2 b1 b2)
               (assume Heq : b1 = b2, decidable.inl (Heq ▸ rfl))
-              (assume Hne : b1 ≠ b2, decidable.inr (mt inr_inj Hne)))))
+              (assume Hne : b1 ≠ b2, decidable.inr (mt inr_inj Hne))))
 end sum

library/data/unit.lean

@@ -19,5 +19,5 @@ namespace unit
   inhabited.mk ⋆
 
   theorem has_decidable_eq [protected] [instance] : decidable_eq unit :=
-  decidable_eq.intro (λ (a b : unit), inl (equal a b))
+  take (a b : unit), inl (equal a b)
 end unit

library/logic/classes/decidable.lean

@@ -96,8 +96,4 @@ decidable_iff_equiv Ha (eq_to_iff H)
 
 end decidable
 
-inductive decidable_eq [class] (A : Type) : Type :=
-intro : (Π x y : A, decidable (x = y)) → decidable_eq A
-
-theorem of_decidable_eq [instance] [coercion] {A : Type} (H : decidable_eq A) (x y : A) : decidable (x = y) :=
-decidable_eq.rec (λ H, H) H x y
+abbreviation decidable_eq (A : Type) := Π (a b : A), decidable (a = b)

tests/lean/interactive/coe.input

@@ -1,3 +1,3 @@
 VISIT coe.lean
 WAIT
-INFO 5 33
+INFO 5 16

tests/lean/interactive/coe.input.expected.out

@@ -1,13 +1,13 @@
 -- BEGINWAIT
 -- ENDWAIT
 -- BEGININFO
--- TYPE|5|33
+-- TYPE|5|16
 decidable (eq a b)
 -- ACK
--- SYNTH|5|33
+-- SYNTH|5|16
 int.has_decidable_eq a b
 -- ACK
--- SYMBOL|5|33
+-- SYMBOL|5|16
 _
 -- ACK
 -- ENDINFO

tests/lean/interactive/coe.lean

@@ -2,7 +2,7 @@ import data.int
 open int
 
 theorem has_decidable_eq [instance] [protected] : decidable_eq ℤ :=
-decidable_eq.intro (λ (a b : ℤ), _)
+take (a b : ℤ), _
 
 variable n : nat
 variable i : int

tests/lean/run/whnfinst.lean

@@ -9,5 +9,6 @@ section
 
   theorem tst1 (H : Πx y, decidable (R x y)) (a b c : A) : decidable (R a b ∧ R b a)
 
-  theorem tst2 (H : decidable_bin_rel R) (a b c : A) : decidable (R a b ∧ R b a)
+  theorem tst2 (H : decidable_bin_rel R) (a b c : A) : decidable (R a b ∧ R b a ∨ R b b) :=
+  _
 end