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[Merged by Bors] - feat(algebra/divisibility): is_refl and is_trans instances for divisibility #14240

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[Merged by Bors] - feat(algebra/divisibility): is_refl and is_trans instances for divisibility #14240

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add instances
vihdzp May 19, 2022
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fix
vihdzp May 19, 2022
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vihdzp May 19, 2022
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12 changes: 6 additions & 6 deletions src/algebra/divisibility.lean
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Expand Up @@ -60,6 +60,8 @@ local attribute [simp] mul_assoc mul_comm mul_left_comm

alias dvd_trans ← has_dvd.dvd.trans

instance : is_trans α (∣) := ⟨λ a b c, dvd_trans⟩

@[simp] theorem dvd_mul_right (a b : α) : a ∣ a * b := dvd.intro b rfl

theorem dvd_mul_of_dvd_left (h : a ∣ b) (c : α) : a ∣ b * c :=
Expand Down Expand Up @@ -89,13 +91,11 @@ section monoid

variables [monoid α]

@[refl, simp] theorem dvd_refl (a : α) : a ∣ a :=
dvd.intro 1 (mul_one _)

lemma dvd_rfl {a : α} : a ∣ a :=
dvd_refl a
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@[refl, simp] theorem dvd_refl (a : α) : a ∣ a := dvd.intro 1 (mul_one a)
theorem dvd_rfl : ∀ {a : α}, a ∣ a := dvd_refl
instance : is_refl α (∣) := ⟨dvd_refl⟩

theorem one_dvd (a : α) : 1 ∣ a := dvd.intro a (one_mul _)
theorem one_dvd (a : α) : 1 ∣ a := dvd.intro a (one_mul a)

end monoid

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