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[Merged by Bors] - chore(data/equiv/basic): arrow_congr preserves properties of binary operators #4759

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[Merged by Bors] - chore(data/equiv/basic): arrow_congr preserves properties of binary operators #4759

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chore(data/equiv/basic): arrow_congr preserves properties of binary o…
eric-wieser Oct 23, 2020
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Update src/data/equiv/basic.lean
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Update basic.lean
eric-wieser Oct 24, 2020
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Update basic.lean
eric-wieser Oct 24, 2020
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Prove `semiconj₂` versions, add `equiv.semiconj_conj`
urkud Oct 25, 2020
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21 changes: 21 additions & 0 deletions src/data/equiv/basic.lean
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Expand Up @@ -372,6 +372,27 @@ rfl
(arrow_congr e₁ e₂).symm = arrow_congr e₁.symm e₂.symm :=
rfl

section binary_op

variables (f : α → α → α) (e : α ≃ β)
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instance [is_associative α f] : is_associative β (equiv.arrow_congr e (equiv.arrow_congr e e) f) :=
⟨λ x y z, by simp [@is_associative.assoc _ f]⟩

instance [is_idempotent α f] : is_idempotent β (equiv.arrow_congr e (equiv.arrow_congr e e) f) :=
⟨λ x, by simp [@is_idempotent.idempotent _ f]⟩

instance [is_left_cancel α f] : is_left_cancel β (equiv.arrow_congr e (equiv.arrow_congr e e) f) :=
⟨λ x y z hx, equiv.injective e.symm $
by { simp at hx, rw @is_left_cancel.left_cancel _ f _ _ _ _ hx }⟩

instance [is_right_cancel α f] :
is_right_cancel β (equiv.arrow_congr e (equiv.arrow_congr e e) f) :=
⟨λ x y z hx, equiv.injective e.symm $
by { simp at hx, rw @is_right_cancel.right_cancel _ f _ _ _ _ hx }⟩

end binary_op

/--
A version of `equiv.arrow_congr` in `Type`, rather than `Sort`.

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