Lemmas about Array.eraseP, Array.erase, and Array.eraseIdx.

eraseP

@[simp]

theorem Array.eraseP_empty {α✝ : Type u_1} {p : α✝ → Bool} : #[].eraseP p = #[]

theorem Array.eraseP_of_forall_mem_not {α : Type u_1} {p : α → Bool} {l : Array α} (h : ∀ (a : α), a ∈ l → ¬p a = true) : l.eraseP p = l

theorem Array.eraseP_of_forall_getElem_not {α : Type u_1} {p : α → Bool} {l : Array α} (h : ∀ (i : Nat) (h : i < l.size), ¬p l[i] = true) : l.eraseP p = l

@[simp]

theorem Array.eraseP_eq_empty_iff {α : Type u_1} {xs : Array α} {p : α → Bool} : xs.eraseP p = #[] ↔ xs = #[] ∨ ∃ (x : α), p x = true ∧ xs = #[x]

theorem Array.eraseP_ne_empty_iff {α : Type u_1} {xs : Array α} {p : α → Bool} : xs.eraseP p ≠ #[] ↔ xs ≠ #[] ∧ ∀ (x : α), p x = true → xs ≠ #[x]

theorem Array.exists_of_eraseP {α : Type u_1} {p : α → Bool} {l : Array α} {a : α} (hm : a ∈ l) (hp : p a = true) : ∃ (a : α), ∃ (l₁ : Array α), ∃ (l₂ : Array α), (∀ (b : α), b ∈ l₁ → ¬p b = true) ∧ p a = true ∧ l = l₁.push a ++ l₂ ∧ l.eraseP p = l₁ ++ l₂

theorem Array.exists_or_eq_self_of_eraseP {α : Type u_1} (p : α → Bool) (l : Array α) : l.eraseP p = l ∨ ∃ (a : α), ∃ (l₁ : Array α), ∃ (l₂ : Array α), (∀ (b : α), b ∈ l₁ → ¬p b = true) ∧ p a = true ∧ l = l₁.push a ++ l₂ ∧ l.eraseP p = l₁ ++ l₂

@[simp]

theorem Array.size_eraseP_of_mem {α : Type u_1} {a : α} {p : α → Bool} {l : Array α} (al : a ∈ l) (pa : p a = true) : (l.eraseP p).size = l.size - 1

theorem Array.size_eraseP {α : Type u_1} {p : α → Bool} {l : Array α} : (l.eraseP p).size = if l.any p = true then l.size - 1 else l.size

theorem Array.size_eraseP_le {α : Type u_1} {p : α → Bool} (l : Array α) : (l.eraseP p).size ≤ l.size

theorem Array.le_size_eraseP {α : Type u_1} {p : α → Bool} (l : Array α) : l.size - 1 ≤ (l.eraseP p).size

theorem Array.mem_of_mem_eraseP {α : Type u_1} {p : α → Bool} {a : α} {l : Array α} : a ∈ l.eraseP p → a ∈ l

@[simp]

theorem Array.mem_eraseP_of_neg {α : Type u_1} {p : α → Bool} {a : α} {l : Array α} (pa : ¬p a = true) : a ∈ l.eraseP p ↔ a ∈ l

@[simp]

theorem Array.eraseP_eq_self_iff {α : Type u_1} {p : α → Bool} {l : Array α} : l.eraseP p = l ↔ ∀ (a : α), a ∈ l → ¬p a = true

theorem Array.eraseP_map {β : Type u_1} {α : Type u_2} {p : α → Bool} (f : β → α) (l : Array β) : (map f l).eraseP p = map f (l.eraseP (p ∘ f))

theorem Array.eraseP_filterMap {α : Type u_1} {β : Type u_2} {p : β → Bool} (f : α → Option β) (l : Array α) : (filterMap f l).eraseP p = filterMap f (l.eraseP fun (x : α) => match f x with | some y => p y | none => false)

theorem Array.eraseP_filter {α : Type u_1} {p : α → Bool} (f : α → Bool) (l : Array α) : (filter f l).eraseP p = filter f (l.eraseP fun (x : α) => p x && f x)

theorem Array.eraseP_append_left {α : Type u_1} {p : α → Bool} {a : α} (pa : p a = true) {l₁ : Array α} (l₂ : Array α) (h : a ∈ l₁) : (l₁ ++ l₂).eraseP p = l₁.eraseP p ++ l₂

theorem Array.eraseP_append_right {α : Type u_1} {p : α → Bool} {l₁ : Array α} (l₂ : Array α) (h : ∀ (b : α), b ∈ l₁ → ¬p b = true) : (l₁ ++ l₂).eraseP p = l₁ ++ l₂.eraseP p

theorem Array.eraseP_append {α : Type u_1} {p : α → Bool} (l₁ l₂ : Array α) : (l₁ ++ l₂).eraseP p = if l₁.any p = true then l₁.eraseP p ++ l₂ else l₁ ++ l₂.eraseP p

theorem Array.eraseP_mkArray {α : Type u_1} (n : Nat) (a : α) (p : α → Bool) : (mkArray n a).eraseP p = if p a = true then mkArray (n - 1) a else mkArray n a

@[simp]

theorem Array.eraseP_mkArray_of_pos {α : Type u_1} {p : α → Bool} {n : Nat} {a : α} (h : p a = true) : (mkArray n a).eraseP p = mkArray (n - 1) a

@[simp]

theorem Array.eraseP_mkArray_of_neg {α : Type u_1} {p : α → Bool} {n : Nat} {a : α} (h : ¬p a = true) : (mkArray n a).eraseP p = mkArray n a

theorem Array.eraseP_eq_iff {α : Type u_1} {l' : Array α} {p : α → Bool} {l : Array α} : l.eraseP p = l' ↔ (∀ (a : α), a ∈ l → ¬p a = true) ∧ l = l' ∨ ∃ (a : α), ∃ (l₁ : Array α), ∃ (l₂ : Array α), (∀ (b : α), b ∈ l₁ → ¬p b = true) ∧ p a = true ∧ l = l₁.push a ++ l₂ ∧ l' = l₁ ++ l₂

theorem Array.eraseP_comm {α : Type u_1} {p q : α → Bool} {l : Array α} (h : ∀ (a : α), a ∈ l → ¬p a = true ∨ ¬q a = true) : (l.eraseP p).eraseP q = (l.eraseP q).eraseP p

erase

theorem Array.erase_of_not_mem {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l : Array α} (h : ¬a ∈ l) : l.erase a = l

theorem Array.erase_eq_eraseP' {α : Type u_1} [BEq α] (a : α) (l : Array α) : l.erase a = l.eraseP fun (x : α) => x == a

theorem Array.erase_eq_eraseP {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (l : Array α) : l.erase a = l.eraseP fun (x : α) => a == x

@[simp]

theorem Array.erase_eq_empty_iff {α : Type u_1} [BEq α] [LawfulBEq α] {xs : Array α} {a : α} : xs.erase a = #[] ↔ xs = #[] ∨ xs = #[a]

theorem Array.erase_ne_empty_iff {α : Type u_1} [BEq α] [LawfulBEq α] {xs : Array α} {a : α} : xs.erase a ≠ #[] ↔ xs ≠ #[] ∧ xs ≠ #[a]

theorem Array.exists_erase_eq {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l : Array α} (h : a ∈ l) : ∃ (l₁ : Array α), ∃ (l₂ : Array α), ¬a ∈ l₁ ∧ l = l₁.push a ++ l₂ ∧ l.erase a = l₁ ++ l₂

@[simp]

theorem Array.size_erase_of_mem {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l : Array α} (h : a ∈ l) : (l.erase a).size = l.size - 1

theorem Array.size_erase {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (l : Array α) : (l.erase a).size = if a ∈ l then l.size - 1 else l.size

theorem Array.size_erase_le {α : Type u_1} [BEq α] (a : α) (l : Array α) : (l.erase a).size ≤ l.size

theorem Array.le_size_erase {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (l : Array α) : l.size - 1 ≤ (l.erase a).size

theorem Array.mem_of_mem_erase {α : Type u_1} [BEq α] {a b : α} {l : Array α} (h : a ∈ l.erase b) : a ∈ l

@[simp]

theorem Array.mem_erase_of_ne {α : Type u_1} [BEq α] [LawfulBEq α] {a b : α} {l : Array α} (ab : a ≠ b) : a ∈ l.erase b ↔ a ∈ l

@[simp]

theorem Array.erase_eq_self_iff {α : Type u_1} [BEq α] {a : α} [LawfulBEq α] {l : Array α} : l.erase a = l ↔ ¬a ∈ l

theorem Array.erase_filter {α : Type u_1} [BEq α] {a : α} [LawfulBEq α] (f : α → Bool) (l : Array α) : (filter f l).erase a = filter f (l.erase a)

theorem Array.erase_append_left {α : Type u_1} [BEq α] {a : α} [LawfulBEq α] {l₁ : Array α} (l₂ : Array α) (h : a ∈ l₁) : (l₁ ++ l₂).erase a = l₁.erase a ++ l₂

theorem Array.erase_append_right {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l₁ : Array α} (l₂ : Array α) (h : ¬a ∈ l₁) : (l₁ ++ l₂).erase a = l₁ ++ l₂.erase a

theorem Array.erase_append {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {l₁ l₂ : Array α} : (l₁ ++ l₂).erase a = if a ∈ l₁ then l₁.erase a ++ l₂ else l₁ ++ l₂.erase a

theorem Array.erase_mkArray {α : Type u_1} [BEq α] [LawfulBEq α] (n : Nat) (a b : α) : (mkArray n a).erase b = if (b == a) = true then mkArray (n - 1) a else mkArray n a

theorem Array.erase_comm {α : Type u_1} [BEq α] [LawfulBEq α] (a b : α) (l : Array α) : (l.erase a).erase b = (l.erase b).erase a

theorem Array.erase_eq_iff {α : Type u_1} [BEq α] {l' : Array α} [LawfulBEq α] {a : α} {l : Array α} : l.erase a = l' ↔ ¬a ∈ l ∧ l = l' ∨ ∃ (l₁ : Array α), ∃ (l₂ : Array α), ¬a ∈ l₁ ∧ l = l₁.push a ++ l₂ ∧ l' = l₁ ++ l₂

@[simp]

theorem Array.erase_mkArray_self {α : Type u_1} [BEq α] {n : Nat} [LawfulBEq α] {a : α} : (mkArray n a).erase a = mkArray (n - 1) a

@[simp]

theorem Array.erase_mkArray_ne {α : Type u_1} [BEq α] {n : Nat} [LawfulBEq α] {a b : α} (h : (!b == a) = true) : (mkArray n a).erase b = mkArray n a

eraseIdx

theorem Array.eraseIdx_eq_take_drop_succ {α : Type u_1} (l : Array α) (i : Nat) (h : i < l.size) : l.eraseIdx i h = l.take i ++ l.drop (i + 1)

theorem Array.getElem?_eraseIdx {α : Type u_1} (l : Array α) (i : Nat) (h : i < l.size) (j : Nat) : (l.eraseIdx i h)[j]? = if j < i then l[j]? else l[j + 1]?

theorem Array.getElem?_eraseIdx_of_lt {α : Type u_1} (l : Array α) (i : Nat) (h : i < l.size) (j : Nat) (h' : j < i) : (l.eraseIdx i h)[j]? = l[j]?

theorem Array.getElem?_eraseIdx_of_ge {α : Type u_1} (l : Array α) (i : Nat) (h : i < l.size) (j : Nat) (h' : i ≤ j) : (l.eraseIdx i h)[j]? = l[j + 1]?

theorem Array.getElem_eraseIdx {α : Type u_1} (l : Array α) (i : Nat) (h : i < l.size) (j : Nat) (h' : j < (l.eraseIdx i h).size) : (l.eraseIdx i h)[j] = if h'' : j < i then l[j] else l[j + 1]

@[simp]

theorem Array.eraseIdx_eq_empty_iff {α : Type u_1} {l : Array α} {i : Nat} {h : i < l.size} : l.eraseIdx i h = #[] ↔ l.size = 1 ∧ i = 0

theorem Array.eraseIdx_ne_empty_iff {α : Type u_1} {l : Array α} {i : Nat} {h : i < l.size} : l.eraseIdx i h ≠ #[] ↔ 2 ≤ l.size

theorem Array.mem_of_mem_eraseIdx {α : Type u_1} {l : Array α} {i : Nat} {h : i < l.size} {a : α} : a ∈ l.eraseIdx i h → a ∈ l

theorem Array.eraseIdx_append_of_lt_size {α : Type u_1} {l : Array α} {k : Nat} (hk : k < l.size) (l' : Array α) (h : k < (l ++ l').size) : (l ++ l').eraseIdx k h = l.eraseIdx k hk ++ l'

theorem Array.eraseIdx_append_of_length_le {α : Type u_1} {l : Array α} {k : Nat} (hk : l.size ≤ k) (l' : Array α) (h : k < (l ++ l').size) : (l ++ l').eraseIdx k h = l ++ l'.eraseIdx (k - l.size) ⋯

theorem Array.eraseIdx_mkArray {α : Type u_1} {n : Nat} {a : α} {k : Nat} {h : k < (mkArray n a).size} : (mkArray n a).eraseIdx k h = mkArray (n - 1) a

theorem Array.mem_eraseIdx_iff_getElem {α : Type u_1} {x : α} {l : Array α} {k : Nat} {h : k < l.size} : x ∈ l.eraseIdx k h ↔ ∃ (i : Nat), ∃ (w : i < l.size), i ≠ k ∧ l[i] = x

theorem Array.mem_eraseIdx_iff_getElem? {α : Type u_1} {x : α} {l : Array α} {k : Nat} {h : k < l.size} : x ∈ l.eraseIdx k h ↔ ∃ (i : Nat), i ≠ k ∧ l[i]? = some x

theorem Array.erase_eq_eraseIdx_of_idxOf {α : Type u_1} [BEq α] [LawfulBEq α] (l : Array α) (a : α) (i : Nat) (w : idxOf a l = i) (h : i < l.size) : l.erase a = l.eraseIdx i h

theorem Array.getElem_eraseIdx_of_lt {α : Type u_1} (l : Array α) (i : Nat) (w : i < l.size) (j : Nat) (h : j < (l.eraseIdx i w).size) (h' : j < i) : (l.eraseIdx i w)[j] = l[j]

theorem Array.getElem_eraseIdx_of_ge {α : Type u_1} (l : Array α) (i : Nat) (w : i < l.size) (j : Nat) (h : j < (l.eraseIdx i w).size) (h' : i ≤ j) : (l.eraseIdx i w)[j] = l[j + 1]

theorem Array.eraseIdx_set_eq {α : Type u_1} {l : Array α} {i : Nat} {a : α} {h : i < l.size} : (l.set i a h).eraseIdx i ⋯ = l.eraseIdx i h

theorem Array.eraseIdx_set_lt {α : Type u_1} {l : Array α} {i : Nat} {w : i < l.size} {j : Nat} {a : α} (h : j < i) : (l.set i a w).eraseIdx j ⋯ = (l.eraseIdx j ⋯).set (i - 1) a ⋯

theorem Array.eraseIdx_set_gt {α : Type u_1} {l : Array α} {i j : Nat} {a : α} (h : i < j) {w : j < l.size} : (l.set i a ⋯).eraseIdx j ⋯ = (l.eraseIdx j w).set i a ⋯

@[simp]

theorem Array.set_getElem_succ_eraseIdx_succ {α : Type u_1} {l : Array α} {i : Nat} (h : i + 1 < l.size) : (l.eraseIdx (i + 1) h).set i l[i + 1] ⋯ = l.eraseIdx i ⋯