[Merged by Bors] - feat(group_theory/perm/basic): mul_left is a monoid hom
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YaelDillies
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YaelDillies
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Dec 11, 2022
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| import logic.function.iterate | ||
| import group_theory.perm.basic | ||
| import algebra.group_power.lemmas |
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algebra.hom.iterate and group_theory.perm.basic are neighbor files. It makes more sense to have equiv.perm.coe_pow in group_theory.perm.basic and I have an upcoming PR that will make it refl, so algebra.hom.iterate won't even be needed anymore!
src/group_theory/perm/basic.lean
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| section add_group | ||
| variables [add_group α] (a b : α) | ||
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| @[simp] lemma add_left_zero : equiv.add_left (0 : α) = 1 := ext zero_add | ||
| @[simp] lemma add_right_zero : equiv.add_right (0 : α) = 1 := ext add_zero | ||
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| @[simp] lemma add_left_add : equiv.add_left (a + b) = equiv.add_left a * equiv.add_left b := | ||
| ext $ add_assoc _ _ | ||
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| @[simp] lemma add_right_add : equiv.add_right (a + b) = equiv.add_right b * equiv.add_right a := | ||
| ext $ λ _, (add_assoc _ _ _).symm | ||
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| @[simp] lemma add_left_neg : equiv.add_left (-a) = (equiv.add_left a)⁻¹ := equiv.coe_inj.1 rfl | ||
| @[simp] lemma add_right_neg : equiv.add_right (-a) = (equiv.add_right a)⁻¹ := equiv.coe_inj.1 rfl | ||
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| @[simp] lemma add_left_nsmul (n : ℕ) : equiv.add_left (n • a) = equiv.add_left a ^ n := | ||
| map_nsmul (⟨equiv.add_left, add_left_zero, add_left_add⟩ : α →+ additive (perm α)) _ _ | ||
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| @[simp] lemma add_right_nsmul (n : ℕ) : equiv.add_right (n • a) = equiv.add_right a ^ n := | ||
| @add_left_nsmul αᵃᵒᵖ _ _ _ | ||
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| @[simp] lemma add_left_zsmul (n : ℤ) : equiv.add_left (n • a) = equiv.add_left a ^ n := | ||
| map_zsmul (⟨equiv.add_left, add_left_zero, add_left_add⟩ : α →+ additive (perm α)) _ _ | ||
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| @[simp] lemma add_right_zsmul (n : ℤ) : equiv.add_right (n • a) = equiv.add_right a ^ n := | ||
| @add_left_zsmul αᵃᵒᵖ _ _ _ | ||
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| end add_group |
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to_additive gave up on me throughout, because perm α is a multiplicative group and to_additive isn't smart enough to insert additive in the right places.
src/group_theory/perm/basic.lean
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| @[simp, to_additive] lemma mul_right_zpow : ∀ n : ℤ, equiv.mul_right (a ^ n) = equiv.mul_right a ^ n | ||
| | (int.of_nat n) := by simp | ||
| | (int.neg_succ_of_nat n) := by simp |
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Unexplicably, not a single proof works for the four of mul_left_pow/mul_right_pow/mul_left_zpow/mul_right_zpow.
eric-wieser
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Dec 11, 2022
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Vierkantor
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Jan 5, 2023
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✌️ YaelDillies can now approve this pull request. To approve and merge a pull request, simply reply with |
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mathlib4 approved, so bors merge |
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`equiv.mul_left` is a monoid homomorphism.
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Pull request successfully merged into master. Build succeeded: |
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mul_left is a monoid hommul_left is a monoid hom
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equiv.mul_leftis a monoid homomorphism.Match leanprover-community/mathlib4#1348