Counting in lists

This file proves basic properties of List.countP and List.count, which count the number of elements of a list satisfying a predicate and equal to a given element respectively. Their definitions can be found in Std.Data.List.Basic.

@[simp]

theorem List.countP_nil {α : Type u_1} (p : α → Bool) : List.countP p [] = 0

theorem List.countP_go_eq_add {α : Type u_1} (p : α → Bool) {n : Nat} (l : List α) : List.countP.go p l n = n + List.countP.go p l 0

@[simp]

theorem List.countP_cons_of_pos {α : Type u_1} (p : α → Bool) {a : α} (l : List α) (pa : p a = true) : List.countP p (a :: l) = List.countP p l + 1

@[simp]

theorem List.countP_cons_of_neg {α : Type u_1} (p : α → Bool) {a : α} (l : List α) (pa : ¬p a = true) : List.countP p (a :: l) = List.countP p l

theorem List.countP_cons {α : Type u_1} (p : α → Bool) (a : α) (l : List α) : List.countP p (a :: l) = List.countP p l + if p a = true then 1 else 0

theorem List.length_eq_countP_add_countP {α : Type u_1} (p : α → Bool) (l : List α) : List.length l = List.countP p l + List.countP (fun a => decide ¬p a = true) l

theorem List.countP_eq_length_filter {α : Type u_1} (p : α → Bool) (l : List α) : List.countP p l = List.length (List.filter p l)

theorem List.countP_le_length {α : Type u_1} (p : α → Bool) {l : List α} : List.countP p l ≤ List.length l

@[simp]

theorem List.countP_append {α : Type u_1} (p : α → Bool) (l₁ : List α) (l₂ : List α) : List.countP p (l₁ ++ l₂) = List.countP p l₁ + List.countP p l₂

theorem List.countP_pos {α : Type u_1} (p : α → Bool) {l : List α} : 0 < List.countP p l ↔ ∃ a, a ∈ l ∧ p a = true

theorem List.countP_eq_zero {α : Type u_1} (p : α → Bool) {l : List α} : List.countP p l = 0 ↔ ∀ (a : α), a ∈ l → ¬p a = true

theorem List.countP_eq_length {α : Type u_1} (p : α → Bool) {l : List α} : List.countP p l = List.length l ↔ ∀ (a : α), a ∈ l → p a = true

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

theorem List.countP_filter {α : Type u_1} (p : α → Bool) (q : α → Bool) (l : List α) : List.countP p (List.filter q l) = List.countP (fun a => decide (p a = true ∧ q a = true)) l

@[simp]

theorem List.countP_true {α : Type u_1} {l : List α} : List.countP (fun x => true) l = List.length l

@[simp]

theorem List.countP_false {α : Type u_1} {l : List α} : List.countP (fun x => false) l = 0

@[simp]

theorem List.countP_map {α : Type u_2} {β : Type u_1} (p : β → Bool) (f : α → β) (l : List α) : List.countP p (List.map f l) = List.countP (p ∘ f) l

theorem List.countP_mono_left {α : Type u_1} {p : α → Bool} {q : α → Bool} {l : List α} (h : ∀ (x : α), x ∈ l → p x = true → q x = true) : List.countP p l ≤ List.countP q l

theorem List.countP_congr {α : Type u_1} {p : α → Bool} {q : α → Bool} {l : List α} (h : ∀ (x : α), x ∈ l → (p x = true ↔ q x = true)) : List.countP p l = List.countP q l

count

@[simp]

theorem List.count_nil {α : Type u_1} [DecidableEq α] (a : α) : List.count a [] = 0

theorem List.count_cons {α : Type u_1} [DecidableEq α] (a : α) (b : α) (l : List α) : List.count a (b :: l) = List.count a l + if a = b then 1 else 0

@[simp]

theorem List.count_cons_self {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : List.count a (a :: l) = List.count a l + 1

@[simp]

theorem List.count_cons_of_ne {α : Type u_1} [DecidableEq α] {a : α} {b : α} (h : a ≠ b) (l : List α) : List.count a (b :: l) = List.count a l

theorem List.count_tail {α : Type u_1} [DecidableEq α] (l : List α) (a : α) (h : 0 < List.length l) : List.count a (List.tail l) = List.count a l - if a = List.get l { val := 0, isLt := h } then 1 else 0

theorem List.count_le_length {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : List.count a l ≤ List.length l

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

theorem List.count_le_count_cons {α : Type u_1} [DecidableEq α] (a : α) (b : α) (l : List α) : List.count a l ≤ List.count a (b :: l)

theorem List.count_singleton {α : Type u_1} [DecidableEq α] (a : α) : List.count a [a] = 1

theorem List.count_singleton' {α : Type u_1} [DecidableEq α] (a : α) (b : α) : List.count a [b] = if a = b then 1 else 0

@[simp]

theorem List.count_append {α : Type u_1} [DecidableEq α] (a : α) (l₁ : List α) (l₂ : List α) : List.count a (l₁ ++ l₂) = List.count a l₁ + List.count a l₂

theorem List.count_concat {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : List.count a (List.concat l a) = Nat.succ (List.count a l)

theorem List.count_pos_iff_mem {α : Type u_1} [DecidableEq α] {a : α} {l : List α} : 0 < List.count a l ↔ a ∈ l

@[simp]

theorem List.count_eq_zero_of_not_mem {α : Type u_1} [DecidableEq α] {a : α} {l : List α} (h : ¬a ∈ l) : List.count a l = 0

theorem List.not_mem_of_count_eq_zero {α : Type u_1} [DecidableEq α] {a : α} {l : List α} (h : List.count a l = 0) : ¬a ∈ l

theorem List.count_eq_zero {α : Type u_1} [DecidableEq α] {a : α} {l : List α} : List.count a l = 0 ↔ ¬a ∈ l

theorem List.count_eq_length {α : Type u_1} [DecidableEq α] {a : α} {l : List α} : List.count a l = List.length l ↔ ∀ (b : α), b ∈ l → a = b

@[simp]

theorem List.count_replicate_self {α : Type u_1} [DecidableEq α] (a : α) (n : Nat) : List.count a (List.replicate n a) = n

theorem List.count_replicate {α : Type u_1} [DecidableEq α] (a : α) (b : α) (n : Nat) : List.count a (List.replicate n b) = if a = b then n else 0

theorem List.filter_beq' {α : Type u_1} [DecidableEq α] (l : List α) (a : α) : List.filter (fun x => x == a) l = List.replicate (List.count a l) a

theorem List.filter_eq' {α : Type u_1} [DecidableEq α] (l : List α) (a : α) : List.filter (fun x => decide (x = a)) l = List.replicate (List.count a l) a

theorem List.filter_eq {α : Type u_1} [DecidableEq α] (l : List α) (a : α) : List.filter (fun x => decide (a = x)) l = List.replicate (List.count a l) a

theorem List.filter_beq {α : Type u_1} [DecidableEq α] (l : List α) (a : α) : List.filter (fun x => a == x) l = List.replicate (List.count a l) a

theorem List.le_count_iff_replicate_sublist {α : Type u_1} [DecidableEq α] {n : Nat} {a : α} {l : List α} : n ≤ List.count a l ↔ List.Sublist (List.replicate n a) l

theorem List.replicate_count_eq_of_count_eq_length {α : Type u_1} [DecidableEq α] {a : α} {l : List α} (h : List.count a l = List.length l) : List.replicate (List.count a l) a = l

@[simp]

theorem List.count_filter {α : Type u_1} [DecidableEq α] {p : α → Bool} {a : α} {l : List α} (h : p a = true) : List.count a (List.filter p l) = List.count a l

theorem List.count_le_count_map {α : Type u_2} [DecidableEq α] {β : Type u_1} [DecidableEq β] (l : List α) (f : α → β) (x : α) : List.count x l ≤ List.count (f x) (List.map f l)

theorem List.count_erase {α : Type u_1} [DecidableEq α] (a : α) (b : α) (l : List α) : List.count a (List.erase l b) = List.count a l - if a = b then 1 else 0

@[simp]

theorem List.count_erase_self {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : List.count a (List.erase l a) = List.count a l - 1

@[simp]

theorem List.count_erase_of_ne {α : Type u_1} [DecidableEq α] {a : α} {b : α} (ab : a ≠ b) (l : List α) : List.count a (List.erase l b) = List.count a l