chore(order/bounded_lattice): trivial generalizations (#9246)

src/order/bounded_lattice.lean

@@ -556,12 +556,12 @@ instance order_bot [partial_order α] : order_bot (with_bot α) :=
 @[simp] theorem some_le_some [preorder α] {a b : α} :
   @has_le.le (with_bot α) _ (some a) (some b) ↔ a ≤ b := coe_le_coe
 
-theorem coe_le [partial_order α] {a b : α} :
+theorem coe_le [preorder α] {a b : α} :
   ∀ {o : option α}, b ∈ o → ((a : with_bot α) ≤ o ↔ a ≤ b)
 | _ rfl := coe_le_coe
 
 @[norm_cast]
-lemma coe_lt_coe [partial_order α] {a b : α} : (a : with_bot α) < b ↔ a < b := some_lt_some
+lemma coe_lt_coe [preorder α] {a b : α} : (a : with_bot α) < b ↔ a < b := some_lt_some
 
 lemma le_coe_get_or_else [preorder α] : ∀ (a : with_bot α) (b : α), a ≤ a.get_or_else b
 | (some a) b := le_refl a
@@ -782,12 +782,12 @@ instance order_top [partial_order α] : order_top (with_top α) :=
 { le_top := λ a a' h, option.no_confusion h,
   ..with_top.partial_order, .. with_top.has_top }
 
-@[simp, norm_cast] theorem coe_le_coe [partial_order α] {a b : α} :
+@[simp, norm_cast] theorem coe_le_coe [preorder α] {a b : α} :
   (a : with_top α) ≤ b ↔ a ≤ b :=
 ⟨λ h, by rcases h b rfl with ⟨_, ⟨⟩, h⟩; exact h,
  λ h a' e, option.some_inj.1 e ▸ ⟨a, rfl, h⟩⟩
 
-theorem le_coe [partial_order α] {a b : α} :
+theorem le_coe [preorder α] {a b : α} :
   ∀ {o : option α}, a ∈ o →
   (@has_le.le (with_top α) _ o b ↔ a ≤ b)
 | _ rfl := coe_le_coe
@@ -805,15 +805,15 @@ theorem lt_iff_exists_coe [partial_order α] : ∀{a b : with_top α}, a < b ↔
 | none     b := by simp [none_eq_top]
 
 @[norm_cast]
-lemma coe_lt_coe [partial_order α] {a b : α} : (a : with_top α) < b ↔ a < b := some_lt_some
+lemma coe_lt_coe [preorder α] {a b : α} : (a : with_top α) < b ↔ a < b := some_lt_some
 
-lemma coe_lt_top [partial_order α] (a : α) : (a : with_top α) < ⊤ := some_lt_none a
+lemma coe_lt_top [preorder α] (a : α) : (a : with_top α) < ⊤ := some_lt_none a
 
-theorem coe_lt_iff [partial_order α] {a : α} : ∀{x : with_top α}, ↑a < x ↔ (∀b:α, x = ↑b → a < b)
+theorem coe_lt_iff [preorder α] {a : α} : ∀{x : with_top α}, ↑a < x ↔ (∀b:α, x = ↑b → a < b)
 | (some b) := by simp [some_eq_coe, coe_eq_coe, coe_lt_coe]
 | none     := by simp [none_eq_top, coe_lt_top]
 
-lemma not_top_le_coe [partial_order α] (a : α) : ¬ (⊤:with_top α) ≤ ↑a :=
+lemma not_top_le_coe [preorder α] (a : α) : ¬ (⊤:with_top α) ≤ ↑a :=
 λ h, (lt_irrefl ⊤ (lt_of_le_of_lt h (coe_lt_top a))).elim
 
 instance decidable_le [preorder α] [@decidable_rel α (≤)] : @decidable_rel (with_top α) (≤) :=