feat(data/nat/pairing): add some nat.pair lemmas (#8740)

src/data/nat/pairing.lean

@@ -125,10 +125,30 @@ end
 end nat
 open nat
 
+section complete_lattice
+
+lemma supr_unpair {α} [complete_lattice α] (f : ℕ → ℕ → α) :
+  (⨆ n : ℕ, f n.unpair.1 n.unpair.2) = ⨆ i j : ℕ, f i j :=
+by rw [← (supr_prod : (⨆ i : ℕ × ℕ, f i.1 i.2) = _), ← nat.surjective_unpair.supr_comp]
+
+lemma infi_unpair {α} [complete_lattice α] (f : ℕ → ℕ → α) :
+  (⨅ n : ℕ, f n.unpair.1 n.unpair.2) = ⨅ i j : ℕ, f i j :=
+supr_unpair (show ℕ → ℕ → order_dual α, from f)
+
+end complete_lattice
+
 namespace set
 
 lemma Union_unpair_prod {α β} {s : ℕ → set α} {t : ℕ → set β} :
   (⋃ n : ℕ, (s n.unpair.fst).prod (t n.unpair.snd)) = (⋃ n, s n).prod (⋃ n, t n) :=
 by { rw [← Union_prod], convert surjective_unpair.Union_comp _, refl }
 
+lemma Union_unpair {α} (f : ℕ → ℕ → set α) :
+  (⋃ n : ℕ, f n.unpair.1 n.unpair.2) = ⋃ i j : ℕ, f i j :=
+supr_unpair f
+
+lemma Inter_unpair {α} (f : ℕ → ℕ → set α) :
+  (⋂ n : ℕ, f n.unpair.1 n.unpair.2) = ⋂ i j : ℕ, f i j :=
+infi_unpair f
+
 end set