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feat: Commute and IsCoprime are Std.Symm #41612

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feat: Commute and IsCoprime are Std.Symm #41612

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4 changes: 4 additions & 0 deletions Mathlib/Algebra/Group/Commute/Defs.lean
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Expand Up @@ -76,6 +76,10 @@ protected theorem symm_iff {a b : S} : Commute a b ↔ Commute b a :=
instance : @Std.Refl S Commute :=
⟨Commute.refl⟩

@[to_additive]
instance : @Std.Symm S Commute where
symm _ _ := .symm

-- This instance is useful for `Finset.noncommProd`
@[to_additive]
instance on_refl {f : G → S} : Std.Refl fun a b => Commute (f a) (f b) :=
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3 changes: 3 additions & 0 deletions Mathlib/RingTheory/Coprime/Basic.lean
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Expand Up @@ -55,6 +55,9 @@ theorem IsCoprime.symm (H : IsCoprime x y) : IsCoprime y x :=
theorem isCoprime_comm : IsCoprime x y ↔ IsCoprime y x :=
⟨IsCoprime.symm, IsCoprime.symm⟩

instance : @Std.Symm R IsCoprime where
symm _ _ := .symm

theorem isCoprime_self : IsCoprime x x ↔ IsUnit x :=
⟨fun ⟨a, b, h⟩ => .of_mul_eq_one (a + b) <| by rwa [mul_comm, add_mul], fun h =>
let ⟨b, hb⟩ := isUnit_iff_exists_inv'.1 h
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