mathlib documentation

group_theory.perm.basic

The group of permutations (self-equivalences) of a type α #

This file defines the group structure on equiv.perm α.

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@[protected, instance]
def equiv.perm.perm_group {α : Type u} :
group (equiv.perm α)
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theorem equiv.perm.mul_apply {α : Type u} (f g : equiv.perm α) (x : α) :
(⇑f * g) x = ⇑f (⇑g x)
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theorem equiv.perm.one_apply {α : Type u} (x : α) :
⇑1 x = x
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@[simp]
theorem equiv.perm.inv_apply_self {α : Type u} (f : equiv.perm α) (x : α) :
⇑f⁻¹ (⇑f x) = x
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@[simp]
theorem equiv.perm.apply_inv_self {α : Type u} (f : equiv.perm α) (x : α) :
⇑f (⇑f⁻¹ x) = x
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theorem equiv.perm.one_def {α : Type u} :
1 = equiv.refl α
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theorem equiv.perm.mul_def {α : Type u} (f g : equiv.perm α) :
f * g = equiv.trans g f
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theorem equiv.perm.inv_def {α : Type u} (f : equiv.perm α) :
f⁻¹ = equiv.symm f
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@[simp]
theorem equiv.perm.coe_mul {α : Type u} (f g : equiv.perm α) :
⇑f * g = ⇑f ∘ ⇑g
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@[simp]
theorem equiv.perm.coe_one {α : Type u} :
⇑1 = id
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theorem equiv.perm.eq_inv_iff_eq {α : Type u} {f : equiv.perm α} {x y : α} :
x = ⇑f⁻¹ y ↔ ⇑f x = y
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theorem equiv.perm.inv_eq_iff_eq {α : Type u} {f : equiv.perm α} {x y : α} :
⇑f⁻¹ x = y ↔ x = ⇑f y
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theorem equiv.perm.zpow_apply_comm {α : Type u_1} (σ : equiv.perm α) (m n : ℤ) {x : α} :
⇑(σ ^ m) (⇑(σ ^ n) x) = ⇑(σ ^ n) (⇑(σ ^ m) x)

Lemmas about mixing perm with equiv. Because we have multiple ways to express equiv.refl, equiv.symm, and equiv.trans, we want simp lemmas for every combination. The assumption made here is that if you're using the group structure, you want to preserve it after simp.

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@[simp]
theorem equiv.perm.trans_one {α : Sort u_1} {β : Type u_2} (e : α ≃ β) :
e.trans 1 = e
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@[simp]
theorem equiv.perm.mul_refl {α : Type u} (e : equiv.perm α) :
e * equiv.refl α = e
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@[simp]
theorem equiv.perm.one_symm {α : Type u} :
equiv.symm 1 = 1
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@[simp]
theorem equiv.perm.refl_inv {α : Type u} :
(equiv.refl α)⁻¹ = 1
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@[simp]
theorem equiv.perm.one_trans {α : Type u_1} {β : Sort u_2} (e : α ≃ β) :
equiv.trans 1 e = e
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@[simp]
theorem equiv.perm.refl_mul {α : Type u} (e : equiv.perm α) :
(equiv.refl α) * e = e
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@[simp]
theorem equiv.perm.inv_trans_self {α : Type u} (e : equiv.perm α) :
equiv.trans e⁻¹ e = 1
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@[simp]
theorem equiv.perm.mul_symm {α : Type u} (e : equiv.perm α) :
e * equiv.symm e = 1
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@[simp]
theorem equiv.perm.self_trans_inv {α : Type u} (e : equiv.perm α) :
equiv.trans e e⁻¹ = 1
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@[simp]
theorem equiv.perm.symm_mul {α : Type u} (e : equiv.perm α) :
(equiv.symm e) * e = 1

Lemmas about equiv.perm.sum_congr re-expressed via the group structure.

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@[simp]
theorem equiv.perm.sum_congr_mul {α : Type u_1} {β : Type u_2} (e : equiv.perm α) (f : equiv.perm β) (g : equiv.perm α) (h : equiv.perm β) :
(e.sum_congr f) * g.sum_congr h = (e * g).sum_congr (f * h)
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@[simp]
theorem equiv.perm.sum_congr_inv {α : Type u_1} {β : Type u_2} (e : equiv.perm α) (f : equiv.perm β) :
(e.sum_congr f)⁻¹ = e⁻¹.sum_congr f⁻¹
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@[simp]
theorem equiv.perm.sum_congr_one {α : Type u_1} {β : Type u_2} :
1.sum_congr 1 = 1
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@[simp]
theorem equiv.perm.sum_congr_hom_apply (α : Type u_1) (β : Type u_2) (a : equiv.perm α × equiv.perm β) :
⇑(equiv.perm.sum_congr_hom α β) a = a.fst.sum_congr a.snd
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def equiv.perm.sum_congr_hom (α : Type u_1) (β : Type u_2) :
equiv.perm α × equiv.perm β →* equiv.perm (α ⊕ β)

equiv.perm.sum_congr as a monoid_hom, with its two arguments bundled into a single prod.

This is particularly useful for its monoid_hom.range projection, which is the subgroup of permutations which do not exchange elements between α and β.

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theorem equiv.perm.sum_congr_hom_injective {α : Type u_1} {β : Type u_2} :
function.injective ⇑(equiv.perm.sum_congr_hom α β)
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@[simp]
theorem equiv.perm.sum_congr_swap_one {α : Type u_1} {β : Type u_2} [decidable_eq α] [decidable_eq β] (i j : α) :
(equiv.swap i j).sum_congr 1 = equiv.swap (sum.inl i) (sum.inl j)
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@[simp]
theorem equiv.perm.sum_congr_one_swap {α : Type u_1} {β : Type u_2} [decidable_eq α] [decidable_eq β] (i j : β) :
1.sum_congr (equiv.swap i j) = equiv.swap (sum.inr i) (sum.inr j)

Lemmas about equiv.perm.sigma_congr_right re-expressed via the group structure.

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@[simp]
theorem equiv.perm.sigma_congr_right_mul {α : Type u_1} {β : α → Type u_2} (F G : Π (a : α), equiv.perm (β a)) :
(equiv.perm.sigma_congr_right F) * equiv.perm.sigma_congr_right G = equiv.perm.sigma_congr_right (F * G)
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@[simp]
theorem equiv.perm.sigma_congr_right_inv {α : Type u_1} {β : α → Type u_2} (F : Π (a : α), equiv.perm (β a)) :
(equiv.perm.sigma_congr_right F)⁻¹ = equiv.perm.sigma_congr_right (λ (a : α), (F a)⁻¹)
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@[simp]
theorem equiv.perm.sigma_congr_right_one {α : Type u_1} {β : α → Type u_2} :
equiv.perm.sigma_congr_right 1 = 1
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@[simp]
theorem equiv.perm.sigma_congr_right_hom_apply {α : Type u_1} (β : α → Type u_2) (F : Π (a : α), equiv.perm ((λ (a : α), β a) a)) :
⇑(equiv.perm.sigma_congr_right_hom β) F = equiv.perm.sigma_congr_right F
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def equiv.perm.sigma_congr_right_hom {α : Type u_1} (β : α → Type u_2) :
(Π (a : α), equiv.perm (β a)) →* equiv.perm (Σ (a : α), β a)

equiv.perm.sigma_congr_right as a monoid_hom.

This is particularly useful for its monoid_hom.range projection, which is the subgroup of permutations which do not exchange elements between fibers.

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theorem equiv.perm.sigma_congr_right_hom_injective {α : Type u_1} {β : α → Type u_2} :
function.injective ⇑(equiv.perm.sigma_congr_right_hom β)
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@[simp]
theorem equiv.perm.subtype_congr_hom_apply {α : Type u} (p : α → Prop) [decidable_pred p] (pair : equiv.perm {a // p a} × equiv.perm {a // ¬p a}) :
⇑(equiv.perm.subtype_congr_hom p) pair = pair.fst.subtype_congr pair.snd
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def equiv.perm.subtype_congr_hom {α : Type u} (p : α → Prop) [decidable_pred p] :
equiv.perm {a // p a} × equiv.perm {a // ¬p a} →* equiv.perm α

equiv.perm.subtype_congr as a monoid_hom.

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theorem equiv.perm.subtype_congr_hom_injective {α : Type u} (p : α → Prop) [decidable_pred p] :
function.injective ⇑(equiv.perm.subtype_congr_hom p)
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@[simp]
theorem equiv.perm.perm_congr_eq_mul {α : Type u} (e p : equiv.perm α) :
⇑(equiv.perm_congr e) p = (e * p) * e⁻¹

If e is also a permutation, we can write perm_congr completely in terms of the group structure.

Lemmas about equiv.perm.extend_domain re-expressed via the group structure.

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@[simp]
theorem equiv.perm.extend_domain_one {α : Type u} {β : Type v} {p : β → Prop} [decidable_pred p] (f : α ≃ subtype p) :
1.extend_domain f = 1
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@[simp]
theorem equiv.perm.extend_domain_inv {α : Type u} {β : Type v} (e : equiv.perm α) {p : β → Prop} [decidable_pred p] (f : α ≃ subtype p) :
(e.extend_domain f)⁻¹ = e⁻¹.extend_domain f
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@[simp]
theorem equiv.perm.extend_domain_mul {α : Type u} {β : Type v} {p : β → Prop} [decidable_pred p] (f : α ≃ subtype p) (e e' : equiv.perm α) :
(e.extend_domain f) * e'.extend_domain f = (e * e').extend_domain f
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@[simp]
theorem equiv.perm.extend_domain_hom_apply {α : Type u} {β : Type v} {p : β → Prop} [decidable_pred p] (f : α ≃ subtype p) (e : equiv.perm α) :
⇑(equiv.perm.extend_domain_hom f) e = e.extend_domain f
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def equiv.perm.extend_domain_hom {α : Type u} {β : Type v} {p : β → Prop} [decidable_pred p] (f : α ≃ subtype p) :
equiv.perm α →* equiv.perm β

extend_domain as a group homomorphism

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theorem equiv.perm.extend_domain_hom_injective {α : Type u} {β : Type v} {p : β → Prop} [decidable_pred p] (f : α ≃ subtype p) :
function.injective ⇑(equiv.perm.extend_domain_hom f)
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@[simp]
theorem equiv.perm.extend_domain_eq_one_iff {α : Type u} {β : Type v} {p : β → Prop} [decidable_pred p] {e : equiv.perm α} {f : α ≃ subtype p} :
e.extend_domain f = 1 ↔ e = 1
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def equiv.perm.subtype_perm {α : Type u} (f : equiv.perm α) {p : α → Prop} (h : ∀ (x : α), p x ↔ p (⇑f x)) :
equiv.perm {x // p x}

If the permutation f fixes the subtype {x // p x}, then this returns the permutation on {x // p x} induced by f.

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@[simp]
theorem equiv.perm.subtype_perm_apply {α : Type u} (f : equiv.perm α) {p : α → Prop} (h : ∀ (x : α), p x ↔ p (⇑f x)) (x : {x // p x}) :
⇑(f.subtype_perm h) x = ⟨⇑f ↑x, _⟩
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@[simp]
theorem equiv.perm.subtype_perm_one {α : Type u} (p : α → Prop) (h : ∀ (x : α), p x ↔ p (⇑1 x)) :
1.subtype_perm h = 1
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def equiv.perm.of_subtype {α : Type u} {p : α → Prop} [decidable_pred p] :
equiv.perm (subtype p) →* equiv.perm α

The inclusion map of permutations on a subtype of α into permutations of α, fixing the other points.

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theorem equiv.perm.of_subtype_subtype_perm {α : Type u} {f : equiv.perm α} {p : α → Prop} [decidable_pred p] (h₁ : ∀ (x : α), p x ↔ p (⇑f x)) (h₂ : ∀ (x : α), ⇑f x ≠ x → p x) :
⇑equiv.perm.of_subtype (f.subtype_perm h₁) = f
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theorem equiv.perm.of_subtype_apply_of_mem {α : Type u} {p : α → Prop} [decidable_pred p] (f : equiv.perm (subtype p)) {x : α} (hx : p x) :
⇑(⇑equiv.perm.of_subtype f) x = ↑(⇑f ⟨x, hx⟩)
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@[simp]
theorem equiv.perm.of_subtype_apply_coe {α : Type u} {p : α → Prop} [decidable_pred p] (f : equiv.perm (subtype p)) (x : subtype p) :
⇑(⇑equiv.perm.of_subtype f) ↑x = ↑(⇑f x)
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theorem equiv.perm.of_subtype_apply_of_not_mem {α : Type u} {p : α → Prop} [decidable_pred p] (f : equiv.perm (subtype p)) {x : α} (hx : ¬p x) :
⇑(⇑equiv.perm.of_subtype f) x = x
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theorem equiv.perm.mem_iff_of_subtype_apply_mem {α : Type u} {p : α → Prop} [decidable_pred p] (f : equiv.perm (subtype p)) (x : α) :
p x ↔ p (⇑(⇑equiv.perm.of_subtype f) x)
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@[simp]
theorem equiv.perm.subtype_perm_of_subtype {α : Type u} {p : α → Prop} [decidable_pred p] (f : equiv.perm (subtype p)) :
(⇑equiv.perm.of_subtype f).subtype_perm _ = f
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@[simp]
theorem equiv.perm.default_perm {n : Type u_1} :
default = 1
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@[simp]
theorem equiv.perm.subtype_equiv_subtype_perm_apply_coe {α : Type u} (p : α → Prop) [decidable_pred p] (f : equiv.perm (subtype p)) :
↑(⇑(equiv.perm.subtype_equiv_subtype_perm p) f) = ⇑equiv.perm.of_subtype f
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@[simp]
theorem equiv.perm.subtype_equiv_subtype_perm_symm_apply {α : Type u} (p : α → Prop) [decidable_pred p] (f : {f // ∀ (a : α), ¬p a → ⇑f a = a}) :
⇑((equiv.perm.subtype_equiv_subtype_perm p).symm) f = ↑f.subtype_perm _
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@[protected]
def equiv.perm.subtype_equiv_subtype_perm {α : Type u} (p : α → Prop) [decidable_pred p] :
equiv.perm (subtype p) ≃ {f // ∀ (a : α), ¬p a → ⇑f a = a}

Permutations on a subtype are equivalent to permutations on the original type that fix pointwise the rest.

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theorem equiv.perm.subtype_equiv_subtype_perm_apply_of_mem {α : Type u_1} {p : α → Prop} [decidable_pred p] (f : equiv.perm (subtype p)) {a : α} (h : p a) :
⇑(⇑(equiv.perm.subtype_equiv_subtype_perm p) f) a = ↑(⇑f ⟨a, h⟩)
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theorem equiv.perm.subtype_equiv_subtype_perm_apply_of_not_mem {α : Type u_1} {p : α → Prop} [decidable_pred p] (f : equiv.perm (subtype p)) {a : α} (h : ¬p a) :
⇑(⇑(equiv.perm.subtype_equiv_subtype_perm p) f) a = a
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noncomputable def equiv.perm.via_embedding {α : Type u} {β : Type v} (e : equiv.perm α) (ι : α ↪ β) :
equiv.perm β

Noncomputable version of equiv.perm.via_fintype_embedding that does not assume fintype

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theorem equiv.perm.via_embedding_apply {α : Type u} {β : Type v} (e : equiv.perm α) (ι : α ↪ β) (x : α) :
⇑(e.via_embedding ι) (⇑ι x) = ⇑ι (⇑e x)
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theorem equiv.perm.via_embedding_apply_of_not_mem {α : Type u} {β : Type v} (e : equiv.perm α) (ι : α ↪ β) (x : β) (hx : x ∉ set.range ⇑ι) :
⇑(e.via_embedding ι) x = x
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noncomputable def equiv.perm.via_embedding_hom {α : Type u} {β : Type v} (ι : α ↪ β) :
equiv.perm α →* equiv.perm β

via_embedding as a group homomorphism

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theorem equiv.perm.via_embedding_hom_apply {α : Type u} {β : Type v} (e : equiv.perm α) (ι : α ↪ β) :
⇑(equiv.perm.via_embedding_hom ι) e = e.via_embedding ι
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theorem equiv.perm.via_embedding_hom_injective {α : Type u} {β : Type v} (ι : α ↪ β) :
function.injective ⇑(equiv.perm.via_embedding_hom ι)
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@[simp]
theorem equiv.swap_inv {α : Type u} [decidable_eq α] (x y : α) :
(equiv.swap x y)⁻¹ = equiv.swap x y
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@[simp]
theorem equiv.swap_mul_self {α : Type u} [decidable_eq α] (i j : α) :
(equiv.swap i j) * equiv.swap i j = 1
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theorem equiv.swap_mul_eq_mul_swap {α : Type u} [decidable_eq α] (f : equiv.perm α) (x y : α) :
(equiv.swap x y) * f = f * equiv.swap (⇑f⁻¹ x) (⇑f⁻¹ y)
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theorem equiv.mul_swap_eq_swap_mul {α : Type u} [decidable_eq α] (f : equiv.perm α) (x y : α) :
f * equiv.swap x y = (equiv.swap (⇑f x) (⇑f y)) * f
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theorem equiv.swap_apply_apply {α : Type u} [decidable_eq α] (f : equiv.perm α) (x y : α) :
equiv.swap (⇑f x) (⇑f y) = (f * equiv.swap x y) * f⁻¹
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@[simp]
theorem equiv.swap_mul_self_mul {α : Type u} [decidable_eq α] (i j : α) (σ : equiv.perm α) :
(equiv.swap i j) * (equiv.swap i j) * σ = σ

Left-multiplying a permutation with swap i j twice gives the original permutation.

This specialization of swap_mul_self is useful when using cosets of permutations.

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@[simp]
theorem equiv.mul_swap_mul_self {α : Type u} [decidable_eq α] (i j : α) (σ : equiv.perm α) :
(σ * equiv.swap i j) * equiv.swap i j = σ

Right-multiplying a permutation with swap i j twice gives the original permutation.

This specialization of swap_mul_self is useful when using cosets of permutations.

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@[simp]
theorem equiv.swap_mul_involutive {α : Type u} [decidable_eq α] (i j : α) :
function.involutive (has_mul.mul (equiv.swap i j))

A stronger version of mul_right_injective

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@[simp]
theorem equiv.mul_swap_involutive {α : Type u} [decidable_eq α] (i j : α) :
function.involutive (λ (_x : equiv.perm α), _x * equiv.swap i j)

A stronger version of mul_left_injective

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@[simp]
theorem equiv.swap_eq_one_iff {α : Type u} [decidable_eq α] {i j : α} :
equiv.swap i j = 1 ↔ i = j
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theorem equiv.swap_mul_eq_iff {α : Type u} [decidable_eq α] {i j : α} {σ : equiv.perm α} :
(equiv.swap i j) * σ = σ ↔ i = j
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theorem equiv.mul_swap_eq_iff {α : Type u} [decidable_eq α] {i j : α} {σ : equiv.perm α} :
σ * equiv.swap i j = σ ↔ i = j
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theorem equiv.swap_mul_swap_mul_swap {α : Type u} [decidable_eq α] {x y z : α} (hwz : x ≠ y) (hxz : x ≠ z) :
((equiv.swap y z) * equiv.swap x y) * equiv.swap y z = equiv.swap z x