Documentation

Init.Data.List.Nat.Modify

modifyHead #

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@[simp]
theorem List.length_modifyHead {α : Type u_1} {f : α → α} {l : List α} :
(modifyHead f l).length = l.length
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theorem List.modifyHead_eq_set {α : Type u_1} [Inhabited α] (f : α → α) (l : List α) :
modifyHead f l = l.set 0 (f (l[0]?.getD default))
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@[simp]
theorem List.modifyHead_eq_nil_iff {α : Type u_1} {f : α → α} {l : List α} :
modifyHead f l = [] ↔ l = []
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@[simp]
theorem List.modifyHead_modifyHead {α : Type u_1} {l : List α} {f g : α → α} :
modifyHead g (modifyHead f l) = modifyHead (g ∘ f) l
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theorem List.getElem_modifyHead {α : Type u_1} {l : List α} {f : α → α} {n : Nat} (h : n < (modifyHead f l).length) :
(modifyHead f l)[n] = if h' : n = 0 then f l[0] else l[n]
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@[simp]
theorem List.getElem_modifyHead_zero {α : Type u_1} {l : List α} {f : α → α} {h : 0 < (modifyHead f l).length} :
(modifyHead f l)[0] = f l[0]
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@[simp]
theorem List.getElem_modifyHead_succ {α : Type u_1} {l : List α} {f : α → α} {n : Nat} (h : n + 1 < (modifyHead f l).length) :
(modifyHead f l)[n + 1] = l[n + 1]
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theorem List.getElem?_modifyHead {α : Type u_1} {l : List α} {f : α → α} {n : Nat} :
(modifyHead f l)[n]? = if n = 0 then Option.map f l[n]? else l[n]?
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@[simp]
theorem List.getElem?_modifyHead_zero {α : Type u_1} {l : List α} {f : α → α} :
(modifyHead f l)[0]? = Option.map f l[0]?
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@[simp]
theorem List.getElem?_modifyHead_succ {α : Type u_1} {l : List α} {f : α → α} {n : Nat} :
(modifyHead f l)[n + 1]? = l[n + 1]?
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@[simp]
theorem List.head_modifyHead {α : Type u_1} (f : α → α) (l : List α) (h : modifyHead f l ≠ []) :
(modifyHead f l).head h = f (l.head ⋯)
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@[simp]
theorem List.head?_modifyHead {α : Type u_1} {l : List α} {f : α → α} :
(modifyHead f l).head? = Option.map f l.head?
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@[simp]
theorem List.tail_modifyHead {α : Type u_1} {f : α → α} {l : List α} :
(modifyHead f l).tail = l.tail
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@[simp]
theorem List.take_modifyHead {α : Type u_1} {f : α → α} {l : List α} {n : Nat} :
take n (modifyHead f l) = modifyHead f (take n l)
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@[simp]
theorem List.drop_modifyHead_of_pos {α : Type u_1} {f : α → α} {l : List α} {n : Nat} (h : 0 < n) :
drop n (modifyHead f l) = drop n l
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theorem List.eraseIdx_modifyHead_zero {α : Type u_1} {f : α → α} {l : List α} :
(modifyHead f l).eraseIdx 0 = l.eraseIdx 0
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@[simp]
theorem List.eraseIdx_modifyHead_of_pos {α : Type u_1} {f : α → α} {l : List α} {n : Nat} (h : 0 < n) :
(modifyHead f l).eraseIdx n = modifyHead f (l.eraseIdx n)
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@[simp]
theorem List.modifyHead_id {α : Type u_1} :
modifyHead id = id
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@[simp]
theorem List.modifyHead_dropLast {α : Type u_1} {l : List α} {f : α → α} :
modifyHead f l.dropLast = (modifyHead f l).dropLast

modifyTailIdx #

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@[simp]
theorem List.modifyTailIdx_id {α : Type u_1} (n : Nat) (l : List α) :
modifyTailIdx id n l = l
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theorem List.eraseIdx_eq_modifyTailIdx {α : Type u_1} (n : Nat) (l : List α) :
l.eraseIdx n = modifyTailIdx tail n l
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@[simp]
theorem List.length_modifyTailIdx {α : Type u_1} (f : List α → List α) (H : ∀ (l : List α), (f l).length = l.length) (n : Nat) (l : List α) :
(modifyTailIdx f n l).length = l.length
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theorem List.modifyTailIdx_add {α : Type u_1} (f : List α → List α) (n : Nat) (l₁ l₂ : List α) :
modifyTailIdx f (l₁.length + n) (l₁ ++ l₂) = l₁ ++ modifyTailIdx f n l₂
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theorem List.modifyTailIdx_eq_take_drop {α : Type u_1} (f : List α → List α) (H : f [] = []) (n : Nat) (l : List α) :
modifyTailIdx f n l = take n l ++ f (drop n l)
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theorem List.exists_of_modifyTailIdx {α : Type u_1} (f : List α → List α) {n : Nat} {l : List α} (h : n ≤ l.length) :
∃ (l₁ : List α), ∃ (l₂ : List α), l = l₁ ++ l₂ ∧ l₁.length = n ∧ modifyTailIdx f n l = l₁ ++ f l₂
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theorem List.modifyTailIdx_modifyTailIdx {α : Type u_1} {f g : List α → List α} (m n : Nat) (l : List α) :
modifyTailIdx g (m + n) (modifyTailIdx f n l) = modifyTailIdx (fun (l : List α) => modifyTailIdx g m (f l)) n l
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theorem List.modifyTailIdx_modifyTailIdx_le {α : Type u_1} {f g : List α → List α} (m n : Nat) (l : List α) (h : n ≤ m) :
modifyTailIdx g m (modifyTailIdx f n l) = modifyTailIdx (fun (l : List α) => modifyTailIdx g (m - n) (f l)) n l
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theorem List.modifyTailIdx_modifyTailIdx_self {α : Type u_1} {f g : List α → List α} (n : Nat) (l : List α) :
modifyTailIdx g n (modifyTailIdx f n l) = modifyTailIdx (g ∘ f) n l

modify #

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@[simp]
theorem List.modify_nil {α : Type u_1} (f : α → α) (n : Nat) :
modify f n [] = []
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@[simp]
theorem List.modify_zero_cons {α : Type u_1} (f : α → α) (a : α) (l : List α) :
modify f 0 (a :: l) = f a :: l
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@[simp]
theorem List.modify_succ_cons {α : Type u_1} (f : α → α) (a : α) (l : List α) (n : Nat) :
modify f (n + 1) (a :: l) = a :: modify f n l
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theorem List.modifyHead_eq_modify_zero {α : Type u_1} (f : α → α) (l : List α) :
modifyHead f l = modify f 0 l
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@[simp]
theorem List.modify_eq_nil_iff {α : Type u_1} {f : α → α} {n : Nat} {l : List α} :
modify f n l = [] ↔ l = []
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theorem List.getElem?_modify {α : Type u_1} (f : α → α) (n : Nat) (l : List α) (m : Nat) :
(modify f n l)[m]? = (fun (a : α) => if n = m then f a else a) <$> l[m]?
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@[simp]
theorem List.length_modify {α : Type u_1} (f : α → α) (n : Nat) (l : List α) :
(modify f n l).length = l.length
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@[simp]
theorem List.getElem?_modify_eq {α : Type u_1} (f : α → α) (n : Nat) (l : List α) :
(modify f n l)[n]? = f <$> l[n]?
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@[simp]
theorem List.getElem?_modify_ne {α : Type u_1} (f : α → α) {m n : Nat} (l : List α) (h : m ≠ n) :
(modify f m l)[n]? = l[n]?
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theorem List.getElem_modify {α : Type u_1} (f : α → α) (n : Nat) (l : List α) (m : Nat) (h : m < (modify f n l).length) :
(modify f n l)[m] = if n = m then f l[m] else l[m]
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@[simp]
theorem List.getElem_modify_eq {α : Type u_1} (f : α → α) (n : Nat) (l : List α) (h : n < (modify f n l).length) :
(modify f n l)[n] = f l[n]
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@[simp]
theorem List.getElem_modify_ne {α : Type u_1} (f : α → α) {m n : Nat} (l : List α) (h : m ≠ n) (h' : n < (modify f m l).length) :
(modify f m l)[n] = l[n]
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theorem List.modify_eq_self {α : Type u_1} {f : α → α} {n : Nat} {l : List α} (h : l.length ≤ n) :
modify f n l = l
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theorem List.modify_modify_eq {α : Type u_1} (f g : α → α) (n : Nat) (l : List α) :
modify g n (modify f n l) = modify (g ∘ f) n l
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theorem List.modify_modify_ne {α : Type u_1} (f g : α → α) {m n : Nat} (l : List α) (h : m ≠ n) :
modify g n (modify f m l) = modify f m (modify g n l)
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theorem List.modify_eq_set {α : Type u_1} [Inhabited α] (f : α → α) (n : Nat) (l : List α) :
modify f n l = l.set n (f (l[n]?.getD default))
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theorem List.modify_eq_take_drop {α : Type u_1} (f : α → α) (n : Nat) (l : List α) :
modify f n l = take n l ++ modifyHead f (drop n l)
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theorem List.modify_eq_take_cons_drop {α : Type u_1} {f : α → α} {n : Nat} {l : List α} (h : n < l.length) :
modify f n l = take n l ++ f l[n] :: drop (n + 1) l
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theorem List.exists_of_modify {α : Type u_1} (f : α → α) {n : Nat} {l : List α} (h : n < l.length) :
∃ (l₁ : List α), ∃ (a : α), ∃ (l₂ : List α), l = l₁ ++ a :: l₂ ∧ l₁.length = n ∧ modify f n l = l₁ ++ f a :: l₂
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@[simp]
theorem List.modify_id {α : Type u_1} (n : Nat) (l : List α) :
modify id n l = l
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theorem List.take_modify {α : Type u_1} (f : α → α) (n m : Nat) (l : List α) :
take n (modify f m l) = modify f m (take n l)
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theorem List.drop_modify_of_lt {α : Type u_1} (f : α → α) (n m : Nat) (l : List α) (h : n < m) :
drop m (modify f n l) = drop m l
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theorem List.drop_modify_of_ge {α : Type u_1} (f : α → α) (n m : Nat) (l : List α) (h : n ≥ m) :
drop m (modify f n l) = modify f (n - m) (drop m l)
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theorem List.eraseIdx_modify_of_eq {α : Type u_1} (f : α → α) (n : Nat) (l : List α) :
(modify f n l).eraseIdx n = l.eraseIdx n
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theorem List.eraseIdx_modify_of_lt {α : Type u_1} (f : α → α) (i j : Nat) (l : List α) (h : j < i) :
(modify f i l).eraseIdx j = modify f (i - 1) (l.eraseIdx j)
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theorem List.eraseIdx_modify_of_gt {α : Type u_1} (f : α → α) (i j : Nat) (l : List α) (h : j > i) :
(modify f i l).eraseIdx j = modify f i (l.eraseIdx j)
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theorem List.modify_eraseIdx_of_lt {α : Type u_1} (f : α → α) (i j : Nat) (l : List α) (h : j < i) :
modify f j (l.eraseIdx i) = (modify f j l).eraseIdx i
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theorem List.modify_eraseIdx_of_ge {α : Type u_1} (f : α → α) (i j : Nat) (l : List α) (h : j ≥ i) :
modify f j (l.eraseIdx i) = (modify f (j + 1) l).eraseIdx i