chore(CountableDenseLinearOrder): Encodable -> Countable (#10308)

Mathlib/Order/CountableDenseLinearOrder.lean

@@ -199,8 +199,9 @@ open PartialIso
 -- variable (α β)
 
 /-- Any countable linear order embeds in any nontrivial dense linear order. -/
-theorem embedding_from_countable_to_dense [Encodable α] [DenselyOrdered β] [Nontrivial β] :
+theorem embedding_from_countable_to_dense [Countable α] [DenselyOrdered β] [Nontrivial β] :
     Nonempty (α ↪o β) := by
+  cases nonempty_encodable α
   rcases exists_pair_lt β with ⟨x, y, hxy⟩
   cases' exists_between hxy with a ha
   haveI : Nonempty (Set.Ioo x y) := ⟨⟨a, ha⟩⟩
@@ -217,15 +218,17 @@ theorem embedding_from_countable_to_dense [Encodable α] [DenselyOrdered β] [No
 #align order.embedding_from_countable_to_dense Order.embedding_from_countable_to_dense
 
 /-- Any two countable dense, nonempty linear orders without endpoints are order isomorphic. -/
-theorem iso_of_countable_dense [Encodable α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α]
-    [Nonempty α] [Encodable β] [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β] :
-    Nonempty (α ≃o β) :=
+theorem iso_of_countable_dense [Countable α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α]
+    [Nonempty α] [Countable β] [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β] :
+    Nonempty (α ≃o β) := by
+  cases nonempty_encodable α
+  cases nonempty_encodable β
   let to_cofinal : Sum α β → Cofinal (PartialIso α β) := fun p ↦
     Sum.recOn p (definedAtLeft β) (definedAtRight α)
   let our_ideal : Ideal (PartialIso α β) := idealOfCofinals default to_cofinal
   let F a := funOfIdeal a our_ideal (cofinal_meets_idealOfCofinals _ to_cofinal (Sum.inl a))
   let G b := invOfIdeal b our_ideal (cofinal_meets_idealOfCofinals _ to_cofinal (Sum.inr b))
-  ⟨OrderIso.ofCmpEqCmp (fun a ↦ (F a).val) (fun b ↦ (G b).val) fun a b ↦ by
+  exact ⟨OrderIso.ofCmpEqCmp (fun a ↦ (F a).val) (fun b ↦ (G b).val) fun a b ↦ by
       rcases (F a).prop with ⟨f, hf, ha⟩
       rcases (G b).prop with ⟨g, hg, hb⟩
       rcases our_ideal.directed _ hf _ hg with ⟨m, _, fm, gm⟩