List Permutations

This file introduces the List.Perm relation, which is true if two lists are permutations of one another.

Notation

The notation ~ is used for permutation equivalence.

@[simp]

theorem List.Perm.refl {α : Type u_1} (l : List α) : l.Perm l

theorem List.Perm.rfl {α : Type u_1} {l : List α} : l.Perm l

theorem List.Perm.of_eq : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ = l₂ → l₁.Perm l₂

theorem List.Perm.symm {α : Type u_1} {l₁ : List α} {l₂ : List α} (h : l₁.Perm l₂) : l₂.Perm l₁

theorem List.perm_comm {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁.Perm l₂ ↔ l₂.Perm l₁

theorem List.Perm.swap' {α : Type u_1} (x : α) (y : α) {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : (y :: x :: l₁).Perm (x :: y :: l₂)

theorem List.Perm.recOnSwap' {α : Type u_1} {motive : (l₁ l₂ : List α) → l₁.Perm l₂ → Prop} {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) (nil : motive [] [] ⋯) (cons : ∀ (x : α) {l₁ l₂ : List α} (h : l₁.Perm l₂), motive l₁ l₂ h → motive (x :: l₁) (x :: l₂) ⋯) (swap' : ∀ (x y : α) {l₁ l₂ : List α} (h : l₁.Perm l₂), motive l₁ l₂ h → motive (y :: x :: l₁) (x :: y :: l₂) ⋯) (trans : ∀ {l₁ l₂ l₃ : List α} (h₁ : l₁.Perm l₂) (h₂ : l₂.Perm l₃), motive l₁ l₂ h₁ → motive l₂ l₃ h₂ → motive l₁ l₃ ⋯) : motive l₁ l₂ p

Similar to Perm.recOn, but the swap case is generalized to Perm.swap', where the tail of the lists are not necessarily the same.

theorem List.Perm.eqv (α : Type u_1) : Equivalence List.Perm

instance List.isSetoid (α : Type u_1) : Setoid (List α)

Equations

• List.isSetoid α = { r := List.Perm, iseqv := ⋯ }

theorem List.Perm.mem_iff {α : Type u_1} {a : α} {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : a ∈ l₁ ↔ a ∈ l₂

theorem List.Perm.subset {α : Type u_1} {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : l₁ ⊆ l₂

theorem List.Perm.append_right {α : Type u_1} {l₁ : List α} {l₂ : List α} (t₁ : List α) (p : l₁.Perm l₂) : (l₁ ++ t₁).Perm (l₂ ++ t₁)

theorem List.Perm.append_left {α : Type u_1} {t₁ : List α} {t₂ : List α} (l : List α) : t₁.Perm t₂ → (l ++ t₁).Perm (l ++ t₂)

theorem List.Perm.append {α : Type u_1} {l₁ : List α} {l₂ : List α} {t₁ : List α} {t₂ : List α} (p₁ : l₁.Perm l₂) (p₂ : t₁.Perm t₂) : (l₁ ++ t₁).Perm (l₂ ++ t₂)

theorem List.Perm.append_cons {α : Type u_1} (a : α) {h₁ : List α} {h₂ : List α} {t₁ : List α} {t₂ : List α} (p₁ : h₁.Perm h₂) (p₂ : t₁.Perm t₂) : (h₁ ++ a :: t₁).Perm (h₂ ++ a :: t₂)

@[simp]

theorem List.perm_middle {α : Type u_1} {a : α} {l₁ : List α} {l₂ : List α} : (l₁ ++ a :: l₂).Perm (a :: (l₁ ++ l₂))

@[simp]

theorem List.perm_append_singleton {α : Type u_1} (a : α) (l : List α) : (l ++ [a]).Perm (a :: l)

theorem List.perm_append_comm {α : Type u_1} {l₁ : List α} {l₂ : List α} : (l₁ ++ l₂).Perm (l₂ ++ l₁)

theorem List.concat_perm {α : Type u_1} (l : List α) (a : α) : (l.concat a).Perm (a :: l)

theorem List.Perm.length_eq {α : Type u_1} {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : l₁.length = l₂.length

theorem List.Perm.eq_nil {α : Type u_1} {l : List α} (p : l.Perm []) : l = []

theorem List.Perm.nil_eq {α : Type u_1} {l : List α} (p : [].Perm l) : [] = l

@[simp]

theorem List.perm_nil {α : Type u_1} {l₁ : List α} : l₁.Perm [] ↔ l₁ = []

@[simp]

theorem List.nil_perm {α : Type u_1} {l₁ : List α} : [].Perm l₁ ↔ l₁ = []

theorem List.not_perm_nil_cons {α : Type u_1} (x : α) (l : List α) : ¬[].Perm (x :: l)

@[simp]

theorem List.reverse_perm {α : Type u_1} (l : List α) : l.reverse.Perm l

theorem List.perm_cons_append_cons {α : Type u_1} {l : List α} {l₁ : List α} {l₂ : List α} (a : α) (p : l.Perm (l₁ ++ l₂)) : (a :: l).Perm (l₁ ++ a :: l₂)

@[simp]

theorem List.perm_replicate {α : Type u_1} {n : Nat} {a : α} {l : List α} : l.Perm (List.replicate n a) ↔ l = List.replicate n a

@[simp]

theorem List.replicate_perm {α : Type u_1} {n : Nat} {a : α} {l : List α} : (List.replicate n a).Perm l ↔ List.replicate n a = l

@[simp]

theorem List.perm_singleton {α : Type u_1} {a : α} {l : List α} : l.Perm [a] ↔ l = [a]

@[simp]

theorem List.singleton_perm {α : Type u_1} {a : α} {l : List α} : [a].Perm l ↔ [a] = l

theorem List.Perm.eq_singleton {α : Type u_1} {a : α} {l : List α} : l.Perm [a] → l = [a]

Alias of the forward direction of List.perm_singleton.

theorem List.Perm.singleton_eq {α : Type u_1} {a : α} {l : List α} : [a].Perm l → [a] = l

Alias of the forward direction of List.singleton_perm.

theorem List.singleton_perm_singleton {α : Type u_1} {a : α} {b : α} : [a].Perm [b] ↔ a = b

theorem List.perm_cons_erase {α : Type u_1} [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) : l.Perm (a :: l.erase a)

theorem List.Perm.filterMap {α : Type u_1} {β : Type u_2} (f : α → Option β) {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : (List.filterMap f l₁).Perm (List.filterMap f l₂)

theorem List.Perm.map {α : Type u_1} {β : Type u_2} (f : α → β) {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : (List.map f l₁).Perm (List.map f l₂)

theorem List.Perm.pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p✝ a → β) {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) {H₁ : ∀ (a : α), a ∈ l₁ → p✝ a} {H₂ : ∀ (a : α), a ∈ l₂ → p✝ a} : (List.pmap f l₁ H₁).Perm (List.pmap f l₂ H₂)

theorem List.Perm.filter {α : Type u_1} (p : α → Bool) {l₁ : List α} {l₂ : List α} (s : l₁.Perm l₂) : (List.filter p l₁).Perm (List.filter p l₂)

theorem List.filter_append_perm {α : Type u_1} (p : α → Bool) (l : List α) : (List.filter p l ++ List.filter (fun (x : α) => !p x) l).Perm l

theorem List.exists_perm_sublist {α : Type u_1} {l₁ : List α} {l₂ : List α} {l₂' : List α} (s : l₁.Sublist l₂) (p : l₂.Perm l₂') : ∃ (l₁' : List α), l₁'.Perm l₁ ∧ l₁'.Sublist l₂'

theorem List.Perm.sizeOf_eq_sizeOf {α : Type u_1} [SizeOf α] {l₁ : List α} {l₂ : List α} (h : l₁.Perm l₂) : sizeOf l₁ = sizeOf l₂

theorem List.nil_subperm {α : Type u_1} {l : List α} : [].Subperm l

theorem List.Perm.subperm_left {α : Type u_1} {l : List α} {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : l.Subperm l₁ ↔ l.Subperm l₂

theorem List.Perm.subperm_right {α : Type u_1} {l₁ : List α} {l₂ : List α} {l : List α} (p : l₁.Perm l₂) : l₁.Subperm l ↔ l₂.Subperm l

theorem List.Sublist.subperm {α : Type u_1} {l₁ : List α} {l₂ : List α} (s : l₁.Sublist l₂) : l₁.Subperm l₂

theorem List.Perm.subperm {α : Type u_1} {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : l₁.Subperm l₂

theorem List.Subperm.refl {α : Type u_1} (l : List α) : l.Subperm l

theorem List.Subperm.trans {α : Type u_1} {l₁ : List α} {l₂ : List α} {l₃ : List α} (s₁₂ : l₁.Subperm l₂) (s₂₃ : l₂.Subperm l₃) : l₁.Subperm l₃

theorem List.Subperm.cons_right {α : Type u_1} {l : List α} {l' : List α} (x : α) (h : l.Subperm l') : l.Subperm (x :: l')

theorem List.Subperm.length_le {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁.Subperm l₂ → l₁.length ≤ l₂.length

theorem List.Subperm.perm_of_length_le {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁.Subperm l₂ → l₂.length ≤ l₁.length → l₁.Perm l₂

theorem List.Subperm.antisymm {α : Type u_1} {l₁ : List α} {l₂ : List α} (h₁ : l₁.Subperm l₂) (h₂ : l₂.Subperm l₁) : l₁.Perm l₂

theorem List.Subperm.subset {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁.Subperm l₂ → l₁ ⊆ l₂

theorem List.Subperm.filter {α : Type u_1} (p : α → Bool) ⦃l : List α⦄ ⦃l' : List α⦄ (h : l.Subperm l') : (List.filter p l).Subperm (List.filter p l')

@[simp]

theorem List.singleton_subperm_iff {α : Type u_1} {l : List α} {a : α} : [a].Subperm l ↔ a ∈ l

theorem List.Sublist.exists_perm_append {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁.Sublist l₂ → ∃ (l : List α), l₂.Perm (l₁ ++ l)

theorem List.Perm.countP_eq {α : Type u_1} (p : α → Bool) {l₁ : List α} {l₂ : List α} (s : l₁.Perm l₂) : List.countP p l₁ = List.countP p l₂

theorem List.Subperm.countP_le {α : Type u_1} (p : α → Bool) {l₁ : List α} {l₂ : List α} : l₁.Subperm l₂ → List.countP p l₁ ≤ List.countP p l₂

theorem List.Perm.countP_congr {α : Type u_1} {l₁ : List α} {l₂ : List α} (s : l₁.Perm l₂) {p : α → Bool} {p' : α → Bool} (hp : ∀ (x : α), x ∈ l₁ → p x = p' x) : List.countP p l₁ = List.countP p' l₂

theorem List.countP_eq_countP_filter_add {α : Type u_1} (l : List α) (p : α → Bool) (q : α → Bool) : List.countP p l = List.countP p (List.filter q l) + List.countP p (List.filter (fun (a : α) => !q a) l)

theorem List.Perm.count_eq {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) (a : α) : List.count a l₁ = List.count a l₂

theorem List.Subperm.count_le {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} (s : l₁.Subperm l₂) (a : α) : List.count a l₁ ≤ List.count a l₂

theorem List.Perm.foldl_eq' {β : Type u_1} {α : Type u_2} {f : β → α → β} {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) (comm : ∀ (x : α), x ∈ l₁ → ∀ (y : α), y ∈ l₁ → ∀ (z : β), f (f z x) y = f (f z y) x) (init : β) : List.foldl f init l₁ = List.foldl f init l₂

theorem List.Perm.rec_heq {α : Type u_1} {β : List α → Sort u_2} {f : (a : α) → (l : List α) → β l → β (a :: l)} {b : β []} {l : List α} {l' : List α} (hl : l.Perm l') (f_congr : ∀ {a : α} {l l' : List α} {b : β l} {b' : β l'}, l.Perm l' → HEq b b' → HEq (f a l b) (f a l' b')) (f_swap : ∀ {a a' : α} {l : List α} {b : β l}, HEq (f a (a' :: l) (f a' l b)) (f a' (a :: l) (f a l b))) : HEq (List.rec b f l) (List.rec b f l')

theorem List.perm_inv_core {α : Type u_1} {a : α} {l₁ : List α} {l₂ : List α} {r₁ : List α} {r₂ : List α} : (l₁ ++ a :: r₁).Perm (l₂ ++ a :: r₂) → (l₁ ++ r₁).Perm (l₂ ++ r₂)

Lemma used to destruct perms element by element.

theorem List.Perm.cons_inv {α : Type u_1} {a : α} {l₁ : List α} {l₂ : List α} : (a :: l₁).Perm (a :: l₂) → l₁.Perm l₂

@[simp]

theorem List.perm_cons {α : Type u_1} (a : α) {l₁ : List α} {l₂ : List α} : (a :: l₁).Perm (a :: l₂) ↔ l₁.Perm l₂

theorem List.perm_append_left_iff {α : Type u_1} {l₁ : List α} {l₂ : List α} (l : List α) : (l ++ l₁).Perm (l ++ l₂) ↔ l₁.Perm l₂

theorem List.perm_append_right_iff {α : Type u_1} {l₁ : List α} {l₂ : List α} (l : List α) : (l₁ ++ l).Perm (l₂ ++ l) ↔ l₁.Perm l₂

theorem List.subperm_cons {α : Type u_1} (a : α) {l₁ : List α} {l₂ : List α} : (a :: l₁).Subperm (a :: l₂) ↔ l₁.Subperm l₂

theorem List.cons_subperm_of_not_mem_of_mem {α : Type u_1} {a : α} {l₁ : List α} {l₂ : List α} (h₁ : ¬a ∈ l₁) (h₂ : a ∈ l₂) (s : l₁.Subperm l₂) : (a :: l₁).Subperm l₂

Weaker version of Subperm.cons_left

theorem List.subperm_append_left {α : Type u_1} {l₁ : List α} {l₂ : List α} (l : List α) : (l ++ l₁).Subperm (l ++ l₂) ↔ l₁.Subperm l₂

theorem List.subperm_append_right {α : Type u_1} {l₁ : List α} {l₂ : List α} (l : List α) : (l₁ ++ l).Subperm (l₂ ++ l) ↔ l₁.Subperm l₂

theorem List.Subperm.exists_of_length_lt {α : Type u_1} {l₁ : List α} {l₂ : List α} (s : l₁.Subperm l₂) (h : l₁.length < l₂.length) : ∃ (a : α), (a :: l₁).Subperm l₂

theorem List.subperm_of_subset : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁.Nodup → l₁ ⊆ l₂ → l₁.Subperm l₂

theorem List.perm_ext_iff_of_nodup {α : Type u_1} {l₁ : List α} {l₂ : List α} (d₁ : l₁.Nodup) (d₂ : l₂.Nodup) : l₁.Perm l₂ ↔ ∀ (a : α), a ∈ l₁ ↔ a ∈ l₂

theorem List.Nodup.perm_iff_eq_of_sublist {α : Type u_1} {l₁ : List α} {l₂ : List α} {l : List α} (d : l.Nodup) (s₁ : l₁.Sublist l) (s₂ : l₂.Sublist l) : l₁.Perm l₂ ↔ l₁ = l₂

theorem List.Perm.erase {α : Type u_1} [DecidableEq α] (a : α) {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : (l₁.erase a).Perm (l₂.erase a)

theorem List.subperm_cons_erase {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : l.Subperm (a :: l.erase a)

theorem List.erase_subperm {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : (l.erase a).Subperm l

theorem List.Subperm.erase {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} (a : α) (h : l₁.Subperm l₂) : (l₁.erase a).Subperm (l₂.erase a)

theorem List.Perm.diff_right {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} (t : List α) (h : l₁.Perm l₂) : (l₁.diff t).Perm (l₂.diff t)

theorem List.Perm.diff_left {α : Type u_1} [DecidableEq α] (l : List α) {t₁ : List α} {t₂ : List α} (h : t₁.Perm t₂) : l.diff t₁ = l.diff t₂

theorem List.Perm.diff {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} {t₁ : List α} {t₂ : List α} (hl : l₁.Perm l₂) (ht : t₁.Perm t₂) : (l₁.diff t₁).Perm (l₂.diff t₂)

theorem List.Subperm.diff_right {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} (h : l₁.Subperm l₂) (t : List α) : (l₁.diff t).Subperm (l₂.diff t)

theorem List.erase_cons_subperm_cons_erase {α : Type u_1} [DecidableEq α] (a : α) (b : α) (l : List α) : ((a :: l).erase b).Subperm (a :: l.erase b)

theorem List.subperm_cons_diff {α : Type u_1} [DecidableEq α] {a : α} {l₁ : List α} {l₂ : List α} : ((a :: l₁).diff l₂).Subperm (a :: l₁.diff l₂)

theorem List.subset_cons_diff {α : Type u_1} [DecidableEq α] {a : α} {l₁ : List α} {l₂ : List α} : (a :: l₁).diff l₂ ⊆ a :: l₁.diff l₂

theorem List.cons_perm_iff_perm_erase {α : Type u_1} [DecidableEq α] {a : α} {l₁ : List α} {l₂ : List α} : (a :: l₁).Perm l₂ ↔ a ∈ l₂ ∧ l₁.Perm (l₂.erase a)

theorem List.perm_iff_count {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} : l₁.Perm l₂ ↔ ∀ (a : α), List.count a l₁ = List.count a l₂

theorem List.subperm_append_diff_self_of_count_le {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} (h : ∀ (x : α), x ∈ l₁ → List.count x l₁ ≤ List.count x l₂) : (l₁ ++ l₂.diff l₁).Perm l₂

The list version of add_tsub_cancel_of_le for multisets.

theorem List.subperm_ext_iff {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} : l₁.Subperm l₂ ↔ ∀ (x : α), x ∈ l₁ → List.count x l₁ ≤ List.count x l₂

The list version of Multiset.le_iff_count.

theorem List.isSubperm_iff {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} : l₁.isSubperm l₂ = true ↔ l₁.Subperm l₂

instance List.decidableSubperm {α : Type u_1} [DecidableEq α] : DecidableRel fun (x x_1 : List α) => x.Subperm x_1

Equations

• x✝.decidableSubperm x = decidable_of_iff (x✝.isSubperm x = true) ⋯

theorem List.Subperm.cons_left {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} (h : l₁.Subperm l₂) (x : α) (hx : List.count x l₁ < List.count x l₂) : (x :: l₁).Subperm l₂

theorem List.isPerm_iff {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} : l₁.isPerm l₂ = true ↔ l₁.Perm l₂

instance List.decidablePerm {α : Type u_1} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁.Perm l₂)

Equations

• l₁.decidablePerm l₂ = decidable_of_iff (l₁.isPerm l₂ = true) ⋯

theorem List.Perm.insert {α : Type u_1} [DecidableEq α] (a : α) {l₁ : List α} {l₂ : List α} (p : l₁.Perm l₂) : (List.insert a l₁).Perm (List.insert a l₂)

theorem List.perm_insert_swap {α : Type u_1} [DecidableEq α] (x : α) (y : α) (l : List α) : (List.insert x (List.insert y l)).Perm (List.insert y (List.insert x l))

theorem List.perm_insertNth {α : Type u_1} (x : α) (l : List α) {n : Nat} (h : n ≤ l.length) : (List.insertNth n x l).Perm (x :: l)

theorem List.Perm.union_right {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} (t₁ : List α) (h : l₁.Perm l₂) : (l₁ ∪ t₁).Perm (l₂ ∪ t₁)

theorem List.Perm.union_left {α : Type u_1} [DecidableEq α] (l : List α) {t₁ : List α} {t₂ : List α} (h : t₁.Perm t₂) : (l ∪ t₁).Perm (l ∪ t₂)

theorem List.Perm.union {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} {t₁ : List α} {t₂ : List α} (p₁ : l₁.Perm l₂) (p₂ : t₁.Perm t₂) : (l₁ ∪ t₁).Perm (l₂ ∪ t₂)

theorem List.Perm.inter_right {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} (t₁ : List α) : l₁.Perm l₂ → (l₁ ∩ t₁).Perm (l₂ ∩ t₁)

theorem List.Perm.inter_left {α : Type u_1} [DecidableEq α] (l : List α) {t₁ : List α} {t₂ : List α} (p : t₁.Perm t₂) : l ∩ t₁ = l ∩ t₂

theorem List.Perm.inter {α : Type u_1} [DecidableEq α] {l₁ : List α} {l₂ : List α} {t₁ : List α} {t₂ : List α} (p₁ : l₁.Perm l₂) (p₂ : t₁.Perm t₂) : (l₁ ∩ t₁).Perm (l₂ ∩ t₂)

theorem List.Perm.pairwise_iff {α : Type u_1} {R : α → α → Prop} (S : ∀ {x y : α}, R x y → R y x) {l₁ : List α} {l₂ : List α} (_p : l₁.Perm l₂) : List.Pairwise R l₁ ↔ List.Pairwise R l₂

theorem List.Pairwise.perm {α : Type u_1} {R : α → α → Prop} {l : List α} {l' : List α} (hR : List.Pairwise R l) (hl : l.Perm l') (hsymm : ∀ {x y : α}, R x y → R y x) : List.Pairwise R l'

theorem List.Perm.pairwise {α : Type u_1} {R : α → α → Prop} {l : List α} {l' : List α} (hl : l.Perm l') (hR : List.Pairwise R l) (hsymm : ∀ {x y : α}, R x y → R y x) : List.Pairwise R l'

theorem List.Perm.nodup_iff {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁.Perm l₂ → (l₁.Nodup ↔ l₂.Nodup)

theorem List.Perm.join {α : Type u_1} {l₁ : List (List α)} {l₂ : List (List α)} (h : l₁.Perm l₂) : l₁.join.Perm l₂.join

theorem List.Perm.bind_right {α : Type u_1} {β : Type u_2} {l₁ : List α} {l₂ : List α} (f : α → List β) (p : l₁.Perm l₂) : (l₁.bind f).Perm (l₂.bind f)

theorem List.Perm.join_congr {α : Type u_1} {l₁ : List (List α)} {l₂ : List (List α)} : List.Forall₂ (fun (x x_1 : List α) => x.Perm x_1) l₁ l₂ → l₁.join.Perm l₂.join

theorem List.Perm.eraseP {α : Type u_1} (f : α → Bool) {l₁ : List α} {l₂ : List α} (H : List.Pairwise (fun (a b : α) => f a = true → f b = true → False) l₁) (p : l₁.Perm l₂) : (List.eraseP f l₁).Perm (List.eraseP f l₂)

theorem List.perm_merge {α : Type u_1} (s : α → α → Bool) (l : List α) (r : List α) : (List.merge s l r).Perm (l ++ r)

theorem List.Perm.merge {α : Type u_1} {l₁ : List α} {l₂ : List α} {r₁ : List α} {r₂ : List α} (s₁ : α → α → Bool) (s₂ : α → α → Bool) (hl : l₁.Perm l₂) (hr : r₁.Perm r₂) : (List.merge s₁ l₁ r₁).Perm (List.merge s₂ l₂ r₂)