add Countable instance for decidable Sigma types

theories/countable.v

@@ -340,3 +340,21 @@ Next Obligation.
   intros T ?? t.
   by rewrite <-(right_id_L [] _ (gen_tree_to_list _)), gen_tree_of_to_list.
 Qed.
+
+(** ** Decidable Sigma *)
+Section sigma.
+  Context `{Countable A} (P : A → Prop).
+  Context `{∀ x, Decision (P x)} `{∀ x, ProofIrrel (P x)}.
+
+  Program Instance countable_sig : Countable { x : A | P x } :=
+    {| encode := λ x, encode (`x);
+       decode := λ p, x ← decode p;
+                      match decide (P x) return option { x | P x } with
+                      | left H => Some (x ↾ H)
+                      | right _ => None
+                      end |}.
+  Next Obligation.
+    intros []. rewrite decode_encode; simpl. by erewrite decide_left.
+  Qed.
+
+End sigma.