Documentation

Mathlib.Data.List.Nodup

Lists with no duplicates #

List.Nodup is defined in Data/List/Basic. In this file we prove various properties of this predicate.

@[simp]
theorem List.forall_mem_ne {α : Type u} {a : α} {l : List α} :
(a'l, ¬a = a') al
@[simp]
theorem List.nodup_nil {α : Type u} :
@[simp]
theorem List.nodup_cons {α : Type u} {a : α} {l : List α} :
List.Nodup (a :: l) al List.Nodup l
theorem List.Pairwise.nodup {α : Type u} {l : List α} {r : ααProp} [IsIrrefl α r] (h : List.Pairwise r l) :
theorem List.rel_nodup {α : Type u} {β : Type v} {r : αβProp} (hr : Relator.BiUnique r) :
(List.Forall₂ r fun (x x_1 : Prop) => x x_1) List.Nodup List.Nodup
theorem List.Nodup.cons {α : Type u} {l : List α} {a : α} (ha : al) (hl : List.Nodup l) :
theorem List.nodup_singleton {α : Type u} (a : α) :
theorem List.Nodup.of_cons {α : Type u} {l : List α} {a : α} (h : List.Nodup (a :: l)) :
theorem List.Nodup.not_mem {α : Type u} {l : List α} {a : α} (h : List.Nodup (a :: l)) :
al
theorem List.not_nodup_cons_of_mem {α : Type u} {l : List α} {a : α} :
a l¬List.Nodup (a :: l)
theorem List.Nodup.sublist {α : Type u} {l₁ : List α} {l₂ : List α} :
List.Sublist l₁ l₂List.Nodup l₂List.Nodup l₁
theorem List.not_nodup_pair {α : Type u} (a : α) :
theorem List.nodup_iff_sublist {α : Type u} {l : List α} :
List.Nodup l ∀ (a : α), ¬List.Sublist [a, a] l
@[deprecated List.nodup_iff_injective_get]
theorem List.nodup_iff_nthLe_inj {α : Type u} {l : List α} :
List.Nodup l ∀ (i j : ) (h₁ : i < List.length l) (h₂ : j < List.length l), List.nthLe l i h₁ = List.nthLe l j h₂i = j
theorem List.Nodup.get_inj_iff {α : Type u} {l : List α} (h : List.Nodup l) {i : Fin (List.length l)} {j : Fin (List.length l)} :
List.get l i = List.get l j i = j
@[deprecated List.Nodup.get_inj_iff]
theorem List.Nodup.nthLe_inj_iff {α : Type u} {l : List α} (h : List.Nodup l) {i : } {j : } (hi : i < List.length l) (hj : j < List.length l) :
List.nthLe l i hi = List.nthLe l j hj i = j
theorem List.nodup_iff_get?_ne_get? {α : Type u} {l : List α} :
List.Nodup l ∀ (i j : ), i < jj < List.length lList.get? l i List.get? l j
theorem List.Nodup.ne_singleton_iff {α : Type u} {l : List α} (h : List.Nodup l) (x : α) :
l [x] l = [] ∃ y ∈ l, y x
theorem List.not_nodup_of_get_eq_of_ne {α : Type u} (xs : List α) (n : Fin (List.length xs)) (m : Fin (List.length xs)) (h : List.get xs n = List.get xs m) (hne : n m) :
@[deprecated List.not_nodup_of_get_eq_of_ne]
theorem List.nthLe_eq_of_ne_imp_not_nodup {α : Type u} (xs : List α) (n : ) (m : ) (hn : n < List.length xs) (hm : m < List.length xs) (h : List.nthLe xs n hn = List.nthLe xs m hm) (hne : n m) :
theorem List.get_indexOf {α : Type u} [DecidableEq α] {l : List α} (H : List.Nodup l) (i : Fin (List.length l)) :
List.indexOf (List.get l i) l = i
@[simp, deprecated List.get_indexOf]
theorem List.nthLe_index_of {α : Type u} [DecidableEq α] {l : List α} (H : List.Nodup l) (n : ) (h : n < List.length l) :
theorem List.nodup_iff_count_le_one {α : Type u} [DecidableEq α] {l : List α} :
List.Nodup l ∀ (a : α), List.count a l 1
theorem List.nodup_iff_count_eq_one {α : Type u} {l : List α} [DecidableEq α] :
List.Nodup l al, List.count a l = 1
theorem List.nodup_replicate {α : Type u} (a : α) {n : } :
@[simp]
theorem List.count_eq_one_of_mem {α : Type u} [DecidableEq α] {a : α} {l : List α} (d : List.Nodup l) (h : a l) :
theorem List.count_eq_of_nodup {α : Type u} [DecidableEq α] {a : α} {l : List α} (d : List.Nodup l) :
List.count a l = if a l then 1 else 0
theorem List.Nodup.of_append_left {α : Type u} {l₁ : List α} {l₂ : List α} :
List.Nodup (l₁ ++ l₂)List.Nodup l₁
theorem List.Nodup.of_append_right {α : Type u} {l₁ : List α} {l₂ : List α} :
List.Nodup (l₁ ++ l₂)List.Nodup l₂
theorem List.nodup_append {α : Type u} {l₁ : List α} {l₂ : List α} :
List.Nodup (l₁ ++ l₂) List.Nodup l₁ List.Nodup l₂ List.Disjoint l₁ l₂
theorem List.disjoint_of_nodup_append {α : Type u} {l₁ : List α} {l₂ : List α} (d : List.Nodup (l₁ ++ l₂)) :
List.Disjoint l₁ l₂
theorem List.Nodup.append {α : Type u} {l₁ : List α} {l₂ : List α} (d₁ : List.Nodup l₁) (d₂ : List.Nodup l₂) (dj : List.Disjoint l₁ l₂) :
List.Nodup (l₁ ++ l₂)
theorem List.nodup_append_comm {α : Type u} {l₁ : List α} {l₂ : List α} :
List.Nodup (l₁ ++ l₂) List.Nodup (l₂ ++ l₁)
theorem List.nodup_middle {α : Type u} {a : α} {l₁ : List α} {l₂ : List α} :
List.Nodup (l₁ ++ a :: l₂) List.Nodup (a :: (l₁ ++ l₂))
theorem List.Nodup.of_map {α : Type u} {β : Type v} (f : αβ) {l : List α} :
theorem List.Nodup.map_on {α : Type u} {β : Type v} {l : List α} {f : αβ} (H : xl, yl, f x = f yx = y) (d : List.Nodup l) :
theorem List.inj_on_of_nodup_map {α : Type u} {β : Type v} {f : αβ} {l : List α} (d : List.Nodup (List.map f l)) ⦃x : α :
x l∀ ⦃y : α⦄, y lf x = f yx = y
theorem List.nodup_map_iff_inj_on {α : Type u} {β : Type v} {f : αβ} {l : List α} (d : List.Nodup l) :
List.Nodup (List.map f l) xl, yl, f x = f yx = y
theorem List.Nodup.map {α : Type u} {β : Type v} {l : List α} {f : αβ} (hf : Function.Injective f) :
theorem List.nodup_map_iff {α : Type u} {β : Type v} {f : αβ} {l : List α} (hf : Function.Injective f) :
@[simp]
theorem List.Nodup.of_attach {α : Type u} {l : List α} :

Alias of the forward direction of List.nodup_attach.

theorem List.Nodup.attach {α : Type u} {l : List α} :

Alias of the reverse direction of List.nodup_attach.

theorem List.Nodup.pmap {α : Type u} {β : Type v} {p : αProp} {f : (a : α) → p aβ} {l : List α} {H : al, p a} (hf : ∀ (a : α) (ha : p a) (b : α) (hb : p b), f a ha = f b hba = b) (h : List.Nodup l) :
theorem List.Nodup.filter {α : Type u} (p : αBool) {l : List α} :
@[simp]
theorem List.Nodup.erase_eq_filter {α : Type u} [DecidableEq α] {l : List α} (d : List.Nodup l) (a : α) :
List.erase l a = List.filter (fun (x : α) => decide (x a)) l
theorem List.Nodup.erase {α : Type u} {l : List α} [DecidableEq α] (a : α) :
theorem List.Nodup.diff {α : Type u} {l₁ : List α} {l₂ : List α} [DecidableEq α] :
List.Nodup l₁List.Nodup (List.diff l₁ l₂)
theorem List.Nodup.mem_erase_iff {α : Type u} {l : List α} {a : α} {b : α} [DecidableEq α] (d : List.Nodup l) :
a List.erase l b a b a l
theorem List.Nodup.not_mem_erase {α : Type u} {l : List α} {a : α} [DecidableEq α] (h : List.Nodup l) :
aList.erase l a
theorem List.nodup_join {α : Type u} {L : List (List α)} :
List.Nodup (List.join L) (lL, List.Nodup l) List.Pairwise List.Disjoint L
theorem List.nodup_bind {α : Type u} {β : Type v} {l₁ : List α} {f : αList β} :
List.Nodup (List.bind l₁ f) (xl₁, List.Nodup (f x)) List.Pairwise (fun (a b : α) => List.Disjoint (f a) (f b)) l₁
theorem List.Nodup.product {α : Type u} {β : Type v} {l₁ : List α} {l₂ : List β} (d₁ : List.Nodup l₁) (d₂ : List.Nodup l₂) :
List.Nodup (l₁ ×ˢ l₂)
theorem List.Nodup.sigma {α : Type u} {l₁ : List α} {σ : αType u_1} {l₂ : (a : α) → List (σ a)} (d₁ : List.Nodup l₁) (d₂ : ∀ (a : α), List.Nodup (l₂ a)) :
theorem List.Nodup.filterMap {α : Type u} {β : Type v} {l : List α} {f : αOption β} (h : ∀ (a a' : α), bf a, b f a'a = a') :
theorem List.Nodup.concat {α : Type u} {l : List α} {a : α} (h : al) (h' : List.Nodup l) :
theorem List.Nodup.insert {α : Type u} {l : List α} {a : α} [DecidableEq α] (h : List.Nodup l) :
theorem List.Nodup.union {α : Type u} {l₂ : List α} [DecidableEq α] (l₁ : List α) (h : List.Nodup l₂) :
List.Nodup (l₁ l₂)
theorem List.Nodup.inter {α : Type u} {l₁ : List α} [DecidableEq α] (l₂ : List α) :
List.Nodup l₁List.Nodup (l₁ l₂)
theorem List.Nodup.diff_eq_filter {α : Type u} [DecidableEq α] {l₁ : List α} {l₂ : List α} :
List.Nodup l₁List.diff l₁ l₂ = List.filter (fun (x : α) => decide (xl₂)) l₁
theorem List.Nodup.mem_diff_iff {α : Type u} {l₁ : List α} {l₂ : List α} {a : α} [DecidableEq α] (hl₁ : List.Nodup l₁) :
a List.diff l₁ l₂ a l₁ al₂
theorem List.Nodup.set {α : Type u} {l : List α} {n : } {a : α} :
List.Nodup lalList.Nodup (List.set l n a)
theorem List.Nodup.map_update {α : Type u} {β : Type v} [DecidableEq α] {l : List α} (hl : List.Nodup l) (f : αβ) (x : α) (y : β) :
List.map (Function.update f x y) l = if x l then List.set (List.map f l) (List.indexOf x l) y else List.map f l
theorem List.Nodup.pairwise_of_forall_ne {α : Type u} {l : List α} {r : ααProp} (hl : List.Nodup l) (h : al, bl, a br a b) :
theorem List.Nodup.pairwise_of_set_pairwise {α : Type u} {l : List α} {r : ααProp} (hl : List.Nodup l) (h : Set.Pairwise {x : α | x l} r) :
@[simp]
theorem List.Nodup.pairwise_coe {α : Type u} {l : List α} {r : ααProp} [IsSymm α r] (hl : List.Nodup l) :
Set.Pairwise {a : α | a l} r List.Pairwise r l
theorem List.Nodup.take_eq_filter_mem {α : Type u} [DecidableEq α] {l : List α} {n : } :
List.Nodup lList.take n l = List.filter (fun (x : α) => decide (x List.take n l)) l