feat(data/finset/basic): Add coe_pair lemmas (#16235)

Adds `coe_pair` lemmas to go with `coe_singleton` and `coe_eq_singleton` (and fixes a couple of style issues).

Author: linesthatinterlace

src/data/finset/basic.lean

@@ -446,9 +446,8 @@ singleton_injective.eq_iff
 @[simp, norm_cast] lemma coe_singleton (a : α) : (({a} : finset α) : set α) = {a} :=
 by { ext, simp }
 
-@[simp, norm_cast] lemma coe_eq_singleton {α : Type*} {s : finset α} {a : α} :
-  (s : set α) = {a} ↔ s = {a} :=
-by rw [←finset.coe_singleton, finset.coe_inj]
+@[simp, norm_cast] lemma coe_eq_singleton {s : finset α} {a : α} : (s : set α) = {a} ↔ s = {a} :=
+by rw [←coe_singleton, coe_inj]
 
 lemma eq_singleton_iff_unique_mem {s : finset α} {a : α} :
   s = {a} ↔ a ∈ s ∧ ∀ x ∈ s, x = a :=
@@ -611,8 +610,7 @@ lemma mem_of_mem_insert_of_ne (h : b ∈ insert a s) : b ≠ a → b ∈ s := (m
 lemma eq_of_not_mem_of_mem_insert (ha : b ∈ insert a s) (hb : b ∉ s) : b = a :=
 (mem_insert.1 ha).resolve_right hb
 
-@[simp] theorem cons_eq_insert {α} [decidable_eq α] (a s h) : @cons α a s h = insert a s :=
-ext $ λ a, by simp
+@[simp] theorem cons_eq_insert (a s h) : @cons α a s h = insert a s := ext $ λ a, by simp
 
 @[simp, norm_cast] lemma coe_insert (a : α) (s : finset α) :
   ↑(insert a s) = (insert a s : set α) :=
@@ -636,8 +634,13 @@ insert_eq_of_mem $ mem_singleton_self _
 theorem insert.comm (a b : α) (s : finset α) : insert a (insert b s) = insert b (insert a s) :=
 ext $ λ x, by simp only [mem_insert, or.left_comm]
 
-theorem pair_comm (a b : α) : ({a, b} : finset α) = {b, a} :=
-insert.comm a b ∅
+@[simp, norm_cast] lemma coe_pair {a b : α} :
+  (({a, b} : finset α) : set α) = {a, b} := by { ext, simp }
+
+@[simp, norm_cast] lemma coe_eq_pair {s : finset α} {a b : α} :
+  (s : set α) = {a, b} ↔ s = {a, b} := by rw [←coe_pair, coe_inj]
+
+theorem pair_comm (a b : α) : ({a, b} : finset α) = {b, a} := insert.comm a b ∅
 
 @[simp] theorem insert_idem (a : α) (s : finset α) : insert a (insert a s) = insert a s :=
 ext $ λ x, by simp only [mem_insert, or.assoc.symm, or_self]