diff --git a/library/data/sum.lean b/library/data/sum.lean
index f5479b8abd..3296b87ac9 100644
--- a/library/data/sum.lean
+++ b/library/data/sum.lean
@@ -8,7 +8,7 @@ Authors: Leonardo de Moura, Jeremy Avigad
 The sum type, aka disjoint union.
 -/
 import logic.connectives
-open inhabited decidable eq.ops
+open inhabited eq.ops
 
 namespace sum
   notation A ⊎ B := sum A B
@@ -19,15 +19,17 @@ namespace sum
   end low_precedence_plus
 
   variables {A B : Type}
-  variables {a a₁ a₂ : A} {b b₁ b₂ : B}
 
-  theorem inl_neq_inr : inl a ≠ inr b :=
+  definition inl_ne_inr (a : A) (b : B) : inl a ≠ inr b :=
   assume H, no_confusion H
 
-  theorem inl_inj : intro_left B a₁ = intro_left B a₂ → a₁ = a₂ :=
+  definition inr_ne_inl (b : B) (a : A) : inr b ≠ inl a :=
+  assume H, no_confusion H
+
+  definition inl_inj {a₁ a₂ : A} : intro_left B a₁ = intro_left B a₂ → a₁ = a₂ :=
   assume H, no_confusion H (λe, e)
 
-  theorem inr_inj : intro_right A b₁ = intro_right A b₂ → b₁ = b₂ :=
+  definition inr_inj {b₁ b₂ : B} : intro_right A b₁ = intro_right A b₂ → b₁ = b₂ :=
   assume H, no_confusion H (λe, e)
 
   protected definition is_inhabited_left [instance] : inhabited A → inhabited (A + B) :=
@@ -36,29 +38,17 @@ namespace sum
   protected definition is_inhabited_right [instance] : inhabited B → inhabited (A + B) :=
   assume H : inhabited B, inhabited.mk (inr (default B))
 
-  protected definition has_eq_decidable [instance] :
-    decidable_eq A → decidable_eq B → decidable_eq (A + B) :=
-  assume (H₁ : decidable_eq A) (H₂ : decidable_eq B),
-  take s₁ s₂ : A + B,
-    rec_on s₁
-      (take a₁, show decidable (inl a₁ = s₂), from
-        rec_on s₂
-          (take a₂, show decidable (inl a₁ = inl a₂), from
-            decidable.rec_on (H₁ a₁ a₂)
-              (assume Heq : a₁ = a₂, decidable.inl (Heq ▸ rfl))
-              (assume Hne : a₁ ≠ a₂, decidable.inr (mt inl_inj Hne)))
-          (take b₂,
-            have H₃ : (inl a₁ = inr b₂) ↔ false,
-              from iff.intro inl_neq_inr (assume H₄, !false.rec H₄),
-            show decidable (inl a₁ = inr b₂), from decidable_of_decidable_of_iff _ (iff.symm H₃)))
-      (take b₁, show decidable (inr b₁ = s₂), from
-        rec_on s₂
-          (take a₂,
-            have H₃ : (inr b₁ = inl a₂) ↔ false,
-              from iff.intro (assume H₄, inl_neq_inr (H₄⁻¹)) (assume H₄, !false.rec H₄),
-            show decidable (inr b₁ = inl a₂), from decidable_of_decidable_of_iff _ (iff.symm H₃))
-          (take b₂, show decidable (inr b₁ = inr 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))))
+  protected definition has_eq_decidable [instance] (h₁ : decidable_eq A) (h₂ : decidable_eq B) : ∀ s₁ s₂ : A + B, decidable (s₁ = s₂),
+  has_eq_decidable (inl a₁) (inl a₂) :=
+    match h₁ a₁ a₂ with
+      decidable.inl hp := decidable.inl (hp ▸ rfl),
+      decidable.inr hn := decidable.inr (λ he, absurd (inl_inj he) hn)
+    end,
+  has_eq_decidable (inl a₁) (inr b₂) := decidable.inr (λ e, sum.no_confusion e),
+  has_eq_decidable (inr b₁) (inl a₂) := decidable.inr (λ e, sum.no_confusion e),
+  has_eq_decidable (inr b₁) (inr b₂) :=
+    match h₂ b₁ b₂ with
+      decidable.inl hp := decidable.inl (hp ▸ rfl),
+      decidable.inr hn := decidable.inr (λ he, absurd (inr_inj he) hn)
+    end
 end sum