Prefixes, suffixes, infixes

This file proves properties about

• List.isPrefix: l₁ is a prefix of l₂ if l₂ starts with l₁.
• List.isSuffix: l₁ is a suffix of l₂ if l₂ ends with l₁.
• List.isInfix: l₁ is an infix of l₂ if l₁ is a prefix of some suffix of l₂.
• List.inits: The list of prefixes of a list.
• List.tails: The list of prefixes of a list.
• insert on lists

All those (except insert) are defined in Mathlib.Data.List.Defs.

Notation

• l₁ <+: l₂: l₁ is a prefix of l₂.
• l₁ <:+ l₂: l₁ is a suffix of l₂.
• l₁ <:+: l₂: l₁ is an infix of l₂.

prefix, suffix, infix

theorem List.prefix_rfl {α : Type u_1} {l : List α} : l <+: l

theorem List.suffix_rfl {α : Type u_1} {l : List α} : l <:+ l

theorem List.infix_rfl {α : Type u_1} {l : List α} : l <:+: l

theorem List.prefix_concat {α : Type u_1} (a : α) (l : List α) : l <+: l.concat a

theorem List.prefix_concat_iff {α : Type u_1} {l₁ : List α} {l₂ : List α} {a : α} : l₁ <+: l₂ ++ [a] ↔ l₁ = l₂ ++ [a] ∨ l₁ <+: l₂

theorem List.IsSuffix.reverse : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+ l₂ → l₁.reverse <+: l₂.reverse

Alias of the reverse direction of List.reverse_prefix.

theorem List.IsPrefix.reverse : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <+: l₂ → l₁.reverse <:+ l₂.reverse

Alias of the reverse direction of List.reverse_suffix.

theorem List.IsInfix.reverse : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+: l₂ → l₁.reverse <:+: l₂.reverse

Alias of the reverse direction of List.reverse_infix.

@[deprecated List.IsSuffix.reverse]

theorem List.isSuffix.reverse : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+ l₂ → l₁.reverse <+: l₂.reverse

Alias of the reverse direction of List.reverse_prefix.

---

Alias of the reverse direction of List.reverse_prefix.

@[deprecated List.IsPrefix.reverse]

theorem List.isPrefix.reverse : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <+: l₂ → l₁.reverse <:+ l₂.reverse

Alias of the reverse direction of List.reverse_suffix.

---

Alias of the reverse direction of List.reverse_suffix.

@[deprecated List.IsInfix.reverse]

theorem List.isInfix.reverse : ∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+: l₂ → l₁.reverse <:+: l₂.reverse

Alias of the reverse direction of List.reverse_infix.

---

Alias of the reverse direction of List.reverse_infix.

theorem List.eq_nil_of_infix_nil : ∀ {α : Type u_1} {l : List α}, l <:+: [] → l = []

Alias of the forward direction of List.infix_nil.

theorem List.eq_nil_of_prefix_nil : ∀ {α : Type u_1} {l : List α}, l <+: [] → l = []

Alias of the forward direction of List.prefix_nil.

theorem List.eq_nil_of_suffix_nil : ∀ {α : Type u_1} {l : List α}, l <:+ [] → l = []

Alias of the forward direction of List.suffix_nil.

@[deprecated List.IsInfix.eq_of_length]

theorem List.eq_of_infix_of_length_eq {α : Type u_1} {l₁ : List α} {l₂ : List α} (h : l₁ <:+: l₂) : l₁.length = l₂.length → l₁ = l₂

@[deprecated List.IsPrefix.eq_of_length]

theorem List.eq_of_prefix_of_length_eq {α : Type u_1} {l₁ : List α} {l₂ : List α} (h : l₁ <+: l₂) : l₁.length = l₂.length → l₁ = l₂

@[deprecated List.IsSuffix.eq_of_length]

theorem List.eq_of_suffix_of_length_eq {α : Type u_1} {l₁ : List α} {l₂ : List α} (h : l₁ <:+ l₂) : l₁.length = l₂.length → l₁ = l₂

theorem List.dropSlice_sublist {α : Type u_1} (n : ℕ) (m : ℕ) (l : List α) : (List.dropSlice n m l).Sublist l

theorem List.dropSlice_subset {α : Type u_1} (n : ℕ) (m : ℕ) (l : List α) : List.dropSlice n m l ⊆ l

theorem List.mem_of_mem_dropSlice {α : Type u_1} {n : ℕ} {m : ℕ} {l : List α} {a : α} (h : a ∈ List.dropSlice n m l) : a ∈ l

theorem List.takeWhile_prefix {α : Type u_1} {l : List α} (p : α → Bool) : List.takeWhile p l <+: l

theorem List.dropWhile_suffix {α : Type u_1} {l : List α} (p : α → Bool) : List.dropWhile p l <:+ l

theorem List.dropLast_prefix {α : Type u_1} (l : List α) : l.dropLast <+: l

theorem List.tail_suffix {α : Type u_1} (l : List α) : l.tail <:+ l

theorem List.dropLast_sublist {α : Type u_1} (l : List α) : l.dropLast.Sublist l

theorem List.drop_sublist_drop_left {α : Type u_1} (l : List α) {m : ℕ} {n : ℕ} (h : m ≤ n) : (List.drop n l).Sublist (List.drop m l)

theorem List.dropLast_subset {α : Type u_1} (l : List α) : l.dropLast ⊆ l

theorem List.tail_subset {α : Type u_1} (l : List α) : l.tail ⊆ l

theorem List.mem_of_mem_dropLast {α : Type u_1} {l : List α} {a : α} (h : a ∈ l.dropLast) : a ∈ l

theorem List.mem_of_mem_tail {α : Type u_1} {l : List α} {a : α} (h : a ∈ l.tail) : a ∈ l

theorem List.Sublist.drop {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁.Sublist l₂ → ∀ (n : ℕ), (List.drop n l₁).Sublist (List.drop n l₂)

theorem List.prefix_iff_eq_append {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁ <+: l₂ ↔ l₁ ++ List.drop l₁.length l₂ = l₂

theorem List.suffix_iff_eq_append {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁ <:+ l₂ ↔ List.take (l₂.length - l₁.length) l₂ ++ l₁ = l₂

theorem List.prefix_iff_eq_take {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁ <+: l₂ ↔ l₁ = List.take l₁.length l₂

theorem List.prefix_take_iff {α : Type u_1} {x : List α} {y : List α} {n : ℕ} : x <+: List.take n y ↔ x <+: y ∧ x.length ≤ n

theorem List.concat_get_prefix {α : Type u_1} {x : List α} {y : List α} (h : x <+: y) (hl : x.length < y.length) : x ++ [y.get ⟨x.length, hl⟩] <+: y

theorem List.suffix_iff_eq_drop {α : Type u_1} {l₁ : List α} {l₂ : List α} : l₁ <:+ l₂ ↔ l₁ = List.drop (l₂.length - l₁.length) l₂

instance List.decidablePrefix {α : Type u_1} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁ <+: l₂)

Equations

• [].decidablePrefix x = isTrue ⋯
• (a :: l₁).decidablePrefix [] = isFalse ⋯
• (a :: l₁).decidablePrefix (b :: l₂) = if h : a = b then decidable_of_decidable_of_iff ⋯ else isFalse ⋯

instance List.decidableSuffix {α : Type u_1} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁ <:+ l₂)

Equations

• [].decidableSuffix x = isTrue ⋯
• (a :: l₁).decidableSuffix [] = isFalse ⋯
• x.decidableSuffix (b :: l₂) = decidable_of_decidable_of_iff ⋯

instance List.decidableInfix {α : Type u_1} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁ <:+: l₂)

Equations

• [].decidableInfix x = isTrue ⋯
• (a :: l₁).decidableInfix [] = isFalse ⋯
• x.decidableInfix (b :: l₂) = decidable_of_decidable_of_iff ⋯

theorem List.prefix_take_le_iff {α : Type u_1} {m : ℕ} {n : ℕ} {L : List α} (hm : m < L.length) : List.take m L <+: List.take n L ↔ m ≤ n

@[deprecated List.cons_prefix_cons]

theorem List.cons_prefix_iff {α : Type u_1} {l₁ : List α} {l₂ : List α} {a : α} {b : α} : a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂

theorem List.IsPrefix.map {α : Type u_1} {β : Type u_2} {l₁ : List α} {l₂ : List α} (h : l₁ <+: l₂) (f : α → β) : List.map f l₁ <+: List.map f l₂

theorem List.IsPrefix.filterMap {α : Type u_1} {β : Type u_2} {l₁ : List α} {l₂ : List α} (h : l₁ <+: l₂) (f : α → Option β) : List.filterMap f l₁ <+: List.filterMap f l₂

@[deprecated List.IsPrefix.filterMap]

theorem List.IsPrefix.filter_map {α : Type u_1} {β : Type u_2} {l₁ : List α} {l₂ : List α} (h : l₁ <+: l₂) (f : α → Option β) : List.filterMap f l₁ <+: List.filterMap f l₂

Alias of List.IsPrefix.filterMap.

theorem List.IsPrefix.reduceOption {α : Type u_1} {l₁ : List (Option α)} {l₂ : List (Option α)} (h : l₁ <+: l₂) : l₁.reduceOption <+: l₂.reduceOption

instance List.instIsPartialOrderIsPrefix {α : Type u_1} : IsPartialOrder (List α) fun (x x_1 : List α) => x <+: x_1

Equations

• ⋯ = ⋯

instance List.instIsPartialOrderIsSuffix {α : Type u_1} : IsPartialOrder (List α) fun (x x_1 : List α) => x <:+ x_1

Equations

• ⋯ = ⋯

instance List.instIsPartialOrderIsInfix {α : Type u_1} : IsPartialOrder (List α) fun (x x_1 : List α) => x <:+: x_1

Equations

• ⋯ = ⋯

@[simp]

theorem List.mem_inits {α : Type u_1} (s : List α) (t : List α) : s ∈ t.inits ↔ s <+: t

@[simp]

theorem List.mem_tails {α : Type u_1} (s : List α) (t : List α) : s ∈ t.tails ↔ s <:+ t

theorem List.inits_cons {α : Type u_1} (a : α) (l : List α) : (a :: l).inits = [] :: List.map (fun (t : List α) => a :: t) l.inits

theorem List.tails_cons {α : Type u_1} (a : α) (l : List α) : (a :: l).tails = (a :: l) :: l.tails

@[simp]

theorem List.inits_append {α : Type u_1} (s : List α) (t : List α) : (s ++ t).inits = s.inits ++ List.map (fun (l : List α) => s ++ l) t.inits.tail

@[simp]

theorem List.tails_append {α : Type u_1} (s : List α) (t : List α) : (s ++ t).tails = List.map (fun (l : List α) => l ++ t) s.tails ++ t.tails.tail

theorem List.inits_eq_tails {α : Type u_1} (l : List α) : l.inits = (List.map List.reverse l.reverse.tails).reverse

theorem List.tails_eq_inits {α : Type u_1} (l : List α) : l.tails = (List.map List.reverse l.reverse.inits).reverse

theorem List.inits_reverse {α : Type u_1} (l : List α) : l.reverse.inits = (List.map List.reverse l.tails).reverse

theorem List.tails_reverse {α : Type u_1} (l : List α) : l.reverse.tails = (List.map List.reverse l.inits).reverse

theorem List.map_reverse_inits {α : Type u_1} (l : List α) : List.map List.reverse l.inits = l.reverse.tails.reverse

theorem List.map_reverse_tails {α : Type u_1} (l : List α) : List.map List.reverse l.tails = l.reverse.inits.reverse

@[simp]

theorem List.length_tails {α : Type u_1} (l : List α) : l.tails.length = l.length + 1

@[simp]

theorem List.length_inits {α : Type u_1} (l : List α) : l.inits.length = l.length + 1

@[simp]

theorem List.getElem_tails {α : Type u_1} (l : List α) (n : ℕ) (h : n < l.tails.length) : l.tails[n] = List.drop n l

theorem List.get_tails {α : Type u_1} (l : List α) (n : Fin l.tails.length) : l.tails.get n = List.drop (↑n) l

@[simp]

theorem List.getElem_inits {α : Type u_1} (l : List α) (n : ℕ) (h : n < l.inits.length) : l.inits[n] = List.take n l

theorem List.get_inits {α : Type u_1} (l : List α) (n : Fin l.inits.length) : l.inits.get n = List.take (↑n) l

insert

theorem List.insert_eq_ite {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : insert a l = if a ∈ l then l else a :: l

@[simp]

theorem List.suffix_insert {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : l <:+ List.insert a l

theorem List.infix_insert {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : l <:+: List.insert a l

theorem List.sublist_insert {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : l.Sublist (List.insert a l)

theorem List.subset_insert {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : l ⊆ List.insert a l

theorem List.mem_of_mem_suffix {α : Type u_1} {l₁ : List α} {l₂ : List α} {a : α} (hx : a ∈ l₁) (hl : l₁ <:+ l₂) : a ∈ l₂

theorem List.IsPrefix.ne_nil {α : Type u_1} {x : List α} {y : List α} (h : x <+: y) (hx : x ≠ []) : y ≠ []

theorem List.IsPrefix.getElem {α : Type u_1} {x : List α} {y : List α} (h : x <+: y) {n : ℕ} (hn : n < x.length) : x[n] = y[n]

theorem List.IsPrefix.get_eq {α : Type u_1} {x : List α} {y : List α} (h : x <+: y) {n : ℕ} (hn : n < x.length) : x.get ⟨n, hn⟩ = y.get ⟨n, ⋯⟩

theorem List.IsPrefix.head_eq {α : Type u_1} {x : List α} {y : List α} (h : x <+: y) (hx : x ≠ []) : x.head hx = y.head ⋯