Lemmas about Vector.countP and Vector.count.

countP

@[simp]

theorem Vector.countP_empty {α : Type u_1} (p : α → Bool) : countP p { toArray := #[], size_toArray := ⋯ } = 0

@[simp]

theorem Vector.countP_push_of_pos {α : Type u_1} (p : α → Bool) {n : Nat} {a : α} (l : Vector α n) (pa : p a = true) : countP p (l.push a) = countP p l + 1

@[simp]

theorem Vector.countP_push_of_neg {α : Type u_1} (p : α → Bool) {n : Nat} {a : α} (l : Vector α n) (pa : ¬p a = true) : countP p (l.push a) = countP p l

theorem Vector.countP_push {α : Type u_1} (p : α → Bool) {n : Nat} (a : α) (l : Vector α n) : countP p (l.push a) = countP p l + if p a = true then 1 else 0

@[simp]

theorem Vector.countP_singleton {α : Type u_1} (p : α → Bool) (a : α) : countP p { toArray := #[a], size_toArray := ⋯ } = if p a = true then 1 else 0

theorem Vector.size_eq_countP_add_countP {α : Type u_1} (p : α → Bool) {n : Nat} (l : Vector α n) : n = countP p l + countP (fun (a : α) => decide ¬p a = true) l

theorem Vector.countP_le_size {α : Type u_1} (p : α → Bool) {n : Nat} {l : Vector α n} : countP p l ≤ n

@[simp]

theorem Vector.countP_append {α : Type u_1} (p : α → Bool) {n m : Nat} (l₁ : Vector α n) (l₂ : Vector α m) : countP p (l₁ ++ l₂) = countP p l₁ + countP p l₂

@[simp]

theorem Vector.countP_pos_iff {α✝ : Type u_1} {n✝ : Nat} {l : Vector α✝ n✝} {p : α✝ → Bool} : 0 < countP p l ↔ ∃ (a : α✝), a ∈ l ∧ p a = true

@[simp]

theorem Vector.one_le_countP_iff {α✝ : Type u_1} {n✝ : Nat} {l : Vector α✝ n✝} {p : α✝ → Bool} : 1 ≤ countP p l ↔ ∃ (a : α✝), a ∈ l ∧ p a = true

@[simp]

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

@[simp]

theorem Vector.countP_eq_size {α✝ : Type u_1} {n✝ : Nat} {l : Vector α✝ n✝} {p : α✝ → Bool} : countP p l = l.size ↔ ∀ (a : α✝), a ∈ l → p a = true

@[simp]

theorem Vector.countP_cast {α : Type u_1} {n a✝ : Nat} {h : n = a✝} (p : α → Bool) (l : Vector α n) : countP p (Vector.cast h l) = countP p l

theorem Vector.countP_mkVector {α : Type u_1} (p : α → Bool) (a : α) (n : Nat) : countP p (mkVector n a) = if p a = true then n else 0

theorem Vector.boole_getElem_le_countP {α : Type u_1} {n : Nat} (p : α → Bool) (l : Vector α n) (i : Nat) (h : i < n) : (if p l[i] = true then 1 else 0) ≤ countP p l

theorem Vector.countP_set {α : Type u_1} {n : Nat} (p : α → Bool) (l : Vector α n) (i : Nat) (a : α) (h : i < n) : countP p (l.set i a h) = (countP p l - if p l[i] = true then 1 else 0) + if p a = true then 1 else 0

@[simp]

theorem Vector.countP_true {α : Type u_1} {n : Nat} : (countP fun (x : α) => true) = fun (x : Vector α n) => n

@[simp]

theorem Vector.countP_false {α : Type u_1} {n : Nat} : (countP fun (x : α) => false) = fun (x : Vector α n) => 0

@[simp]

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

@[simp]

theorem Vector.countP_flatten {α : Type u_1} (p : α → Bool) {m n : Nat} (l : Vector (Vector α m) n) : countP p l.flatten = (map (countP p) l).sum

theorem Vector.countP_flatMap {α : Type u_2} {β : Type u_1} {n m : Nat} (p : β → Bool) (l : Vector α n) (f : α → Vector β m) : countP p (l.flatMap f) = (map (countP p ∘ f) l).sum

@[simp]

theorem Vector.countP_reverse {α : Type u_1} (p : α → Bool) {n : Nat} (l : Vector α n) : countP p l.reverse = countP p l

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

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

count

@[simp]

theorem Vector.count_empty {α : Type u_1} [BEq α] (a : α) : count a { toArray := #[], size_toArray := ⋯ } = 0

theorem Vector.count_push {α : Type u_1} [BEq α] {n : Nat} (a b : α) (l : Vector α n) : count a (l.push b) = count a l + if (b == a) = true then 1 else 0

theorem Vector.count_eq_countP {α : Type u_1} [BEq α] {n : Nat} (a : α) (l : Vector α n) : count a l = countP (fun (x : α) => x == a) l

theorem Vector.count_eq_countP' {α : Type u_1} [BEq α] {n : Nat} {a : α} : count a = countP fun (x : α) => x == a

theorem Vector.count_le_size {α : Type u_1} [BEq α] {n : Nat} (a : α) (l : Vector α n) : count a l ≤ n

theorem Vector.count_le_count_push {α : Type u_1} [BEq α] {n : Nat} (a b : α) (l : Vector α n) : count a l ≤ count a (l.push b)

@[simp]

theorem Vector.count_singleton {α : Type u_1} [BEq α] (a b : α) : count a { toArray := #[b], size_toArray := ⋯ } = if (b == a) = true then 1 else 0

@[simp]

theorem Vector.count_append {α : Type u_1} [BEq α] {n m : Nat} (a : α) (l₁ : Vector α n) (l₂ : Vector α m) : count a (l₁ ++ l₂) = count a l₁ + count a l₂

@[simp]

theorem Vector.count_flatten {α : Type u_1} [BEq α] {m n : Nat} (a : α) (l : Vector (Vector α m) n) : count a l.flatten = (map (count a) l).sum

@[simp]

theorem Vector.count_reverse {α : Type u_1} [BEq α] {n : Nat} (a : α) (l : Vector α n) : count a l.reverse = count a l

theorem Vector.boole_getElem_le_count {α : Type u_1} [BEq α] {n : Nat} (a : α) (l : Vector α n) (i : Nat) (h : i < n) : (if (l[i] == a) = true then 1 else 0) ≤ count a l

theorem Vector.count_set {α : Type u_1} [BEq α] {n : Nat} (a b : α) (l : Vector α n) (i : Nat) (h : i < n) : count b (l.set i a h) = (count b l - if (l[i] == b) = true then 1 else 0) + if (a == b) = true then 1 else 0

@[simp]

theorem Vector.count_cast {α : Type u_1} [BEq α] {n a✝ : Nat} {h : n = a✝} {a : α} (l : Vector α n) : count a (Vector.cast h l) = count a l

@[simp]

theorem Vector.count_push_self {α : Type u_1} [BEq α] [LawfulBEq α] {n : Nat} (a : α) (l : Vector α n) : count a (l.push a) = count a l + 1

@[simp]

theorem Vector.count_push_of_ne {α : Type u_1} [BEq α] [LawfulBEq α] {b a : α} {n : Nat} (h : b ≠ a) (l : Vector α n) : count a (l.push b) = count a l

theorem Vector.count_singleton_self {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) : count a { toArray := #[a], size_toArray := ⋯ } = 1

@[simp]

theorem Vector.count_pos_iff {α : Type u_1} [BEq α] [LawfulBEq α] {n : Nat} {a : α} {l : Vector α n} : 0 < count a l ↔ a ∈ l

@[simp]

theorem Vector.one_le_count_iff {α : Type u_1} [BEq α] [LawfulBEq α] {n : Nat} {a : α} {l : Vector α n} : 1 ≤ count a l ↔ a ∈ l

theorem Vector.count_eq_zero_of_not_mem {α : Type u_1} [BEq α] [LawfulBEq α] {n : Nat} {a : α} {l : Vector α n} (h : ¬a ∈ l) : count a l = 0

theorem Vector.not_mem_of_count_eq_zero {α : Type u_1} [BEq α] [LawfulBEq α] {n : Nat} {a : α} {l : Vector α n} (h : count a l = 0) : ¬a ∈ l

theorem Vector.count_eq_zero {α : Type u_1} [BEq α] [LawfulBEq α] {n : Nat} {a : α} {l : Vector α n} : count a l = 0 ↔ ¬a ∈ l

theorem Vector.count_eq_size {α : Type u_1} [BEq α] [LawfulBEq α] {n : Nat} {a : α} {l : Vector α n} : count a l = l.size ↔ ∀ (b : α), b ∈ l → a = b

@[simp]

theorem Vector.count_mkVector_self {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (n : Nat) : count a (mkVector n a) = n

theorem Vector.count_mkVector {α : Type u_1} [BEq α] [LawfulBEq α] (a b : α) (n : Nat) : count a (mkVector n b) = if (b == a) = true then n else 0

theorem Vector.count_le_count_map {α : Type u_2} [BEq α] [LawfulBEq α] {β : Type u_1} {n : Nat} [DecidableEq β] (l : Vector α n) (f : α → β) (x : α) : count x l ≤ count (f x) (map f l)

theorem Vector.count_flatMap {β : Type u_1} {n m : Nat} {α : Type u_2} [BEq β] (l : Vector α n) (f : α → Vector β m) (x : β) : count x (l.flatMap f) = (map (count x ∘ f) l).sum