Lexicographic ordering

@[simp]

theorem List.lt_toArray {α : Type u_1} [LT α] {l₁ l₂ : List α} : l₁.toArray < l₂.toArray ↔ l₁ < l₂

@[simp]

theorem List.le_toArray {α : Type u_1} [LT α] {l₁ l₂ : List α} : l₁.toArray ≤ l₂.toArray ↔ l₁ ≤ l₂

@[simp]

theorem Array.lt_toList {α : Type u_1} [LT α] {xs ys : Array α} : xs.toList < ys.toList ↔ xs < ys

@[simp]

theorem Array.le_toList {α : Type u_1} [LT α] {xs ys : Array α} : xs.toList ≤ ys.toList ↔ xs ≤ ys

theorem Array.not_lt_iff_ge {α : Type u_1} [LT α] {xs ys : Array α} : ¬xs < ys ↔ ys ≤ xs

theorem Array.not_le_iff_gt {α : Type u_1} [LT α] {xs ys : Array α} : ¬xs ≤ ys ↔ ys < xs

@[simp]

theorem Array.lex_empty {α : Type u_1} [BEq α] {lt : α → α → Bool} {xs : Array α} : xs.lex #[] lt = false

@[simp]

theorem List.lex_toArray {α : Type u_1} [BEq α] {lt : α → α → Bool} {l₁ l₂ : List α} : l₁.toArray.lex l₂.toArray lt = l₁.lex l₂ lt

theorem Array.singleton_lex_singleton {α : Type u_1} {a b : α} [BEq α] {lt : α → α → Bool} : #[a].lex #[b] lt = lt a b

@[simp]

theorem Array.lex_toList {α : Type u_1} [BEq α] {lt : α → α → Bool} {xs ys : Array α} : xs.toList.lex ys.toList lt = xs.lex ys lt

instance Array.instIsLinearOrderOfLawfulOrderLT {α : Type u_1} [LT α] [LE α] [Std.LawfulOrderLT α] [Std.IsLinearOrder α] : Std.IsLinearOrder (Array α)

theorem Array.lt_irrefl {α : Type u_1} [LT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] (xs : Array α) : ¬xs < xs

instance Array.ltIrrefl {α : Type u_1} [LT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] : Std.Irrefl fun (x1 x2 : Array α) => x1 < x2

@[simp]

theorem Array.not_lt_empty {α : Type u_1} [LT α] (xs : Array α) : ¬xs < #[]

@[simp]

theorem Array.empty_le {α : Type u_1} [LT α] (xs : Array α) : #[] ≤ xs

@[simp]

theorem Array.le_empty {α : Type u_1} [LT α] {xs : Array α} : xs ≤ #[] ↔ xs = #[]

@[simp]

theorem Array.empty_lt_push {α : Type u_1} [LT α] (xs : Array α) (a : α) : #[] < xs.push a

theorem Array.le_refl {α : Type u_1} [LT α] [i₀ : Std.Irrefl fun (x1 x2 : α) => x1 < x2] (xs : Array α) : xs ≤ xs

instance Array.instReflLeOfIrreflLt {α : Type u_1} [LT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] : Std.Refl fun (x1 x2 : Array α) => x1 ≤ x2

theorem Array.lt_trans {α : Type u_1} [LT α] [i₁ : Trans (fun (x1 x2 : α) => x1 < x2) (fun (x1 x2 : α) => x1 < x2) fun (x1 x2 : α) => x1 < x2] {xs ys zs : Array α} (h₁ : xs < ys) (h₂ : ys < zs) : xs < zs

instance Array.instTransLt {α : Type u_1} [LT α] [Trans (fun (x1 x2 : α) => x1 < x2) (fun (x1 x2 : α) => x1 < x2) fun (x1 x2 : α) => x1 < x2] : Trans (fun (x1 x2 : Array α) => x1 < x2) (fun (x1 x2 : Array α) => x1 < x2) fun (x1 x2 : Array α) => x1 < x2

Equations

• Array.instTransLt = { trans := ⋯ }

theorem Array.lt_of_le_of_lt {α : Type u_1} [LE α] [LT α] [Std.LawfulOrderLT α] [Std.IsLinearOrder α] {xs ys zs : Array α} (h₁ : xs ≤ ys) (h₂ : ys < zs) : xs < zs

@[deprecated Array.lt_of_le_of_lt (since := "2025-08-01")]

theorem Array.lt_of_le_of_lt' {α : Type u_1} [LT α] [i₁ : Std.Asymm fun (x1 x2 : α) => x1 < x2] [i₂ : Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] [i₃ : Trans (fun (x1 x2 : α) => ¬x1 < x2) (fun (x1 x2 : α) => ¬x1 < x2) fun (x1 x2 : α) => ¬x1 < x2] {xs ys zs : Array α} (h₁ : xs ≤ ys) (h₂ : ys < zs) : xs < zs

theorem Array.le_trans {α : Type u_1} [LE α] [LT α] [Std.LawfulOrderLT α] [Std.IsLinearOrder α] {xs ys zs : Array α} (h₁ : xs ≤ ys) (h₂ : ys ≤ zs) : xs ≤ zs

@[deprecated Array.le_trans (since := "2025-08-01")]

theorem Array.le_trans' {α : Type u_1} [LT α] [i₁ : Std.Asymm fun (x1 x2 : α) => x1 < x2] [i₂ : Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] [i₃ : Trans (fun (x1 x2 : α) => ¬x1 < x2) (fun (x1 x2 : α) => ¬x1 < x2) fun (x1 x2 : α) => ¬x1 < x2] {xs ys zs : Array α} (h₁ : xs ≤ ys) (h₂ : ys ≤ zs) : xs ≤ zs

instance Array.instTransLeOfLawfulOrderLTOfIsLinearOrder {α : Type u_1} [LE α] [LT α] [Std.LawfulOrderLT α] [Std.IsLinearOrder α] : Trans (fun (x1 x2 : Array α) => x1 ≤ x2) (fun (x1 x2 : Array α) => x1 ≤ x2) fun (x1 x2 : Array α) => x1 ≤ x2

Equations

• Array.instTransLeOfLawfulOrderLTOfIsLinearOrder = { trans := ⋯ }

theorem Array.lt_asymm {α : Type u_1} [LT α] [i : Std.Asymm fun (x1 x2 : α) => x1 < x2] {xs ys : Array α} (h : xs < ys) : ¬ys < xs

instance Array.instAsymmLt {α : Type u_1} [LT α] [Std.Asymm fun (x1 x2 : α) => x1 < x2] : Std.Asymm fun (x1 x2 : Array α) => x1 < x2

theorem Array.le_total {α : Type u_1} [LT α] [i : Std.Asymm fun (x1 x2 : α) => x1 < x2] (xs ys : Array α) : xs ≤ ys ∨ ys ≤ xs

@[simp]

theorem Array.not_lt {α : Type u_1} [LT α] {xs ys : Array α} : ¬xs < ys ↔ ys ≤ xs

@[simp]

theorem Array.not_le {α : Type u_1} [LT α] {xs ys : Array α} : ¬ys ≤ xs ↔ xs < ys

theorem Array.le_of_lt {α : Type u_1} [LT α] [i : Std.Asymm fun (x1 x2 : α) => x1 < x2] {xs ys : Array α} (h : xs < ys) : xs ≤ ys

theorem Array.le_iff_lt_or_eq {α : Type u_1} [LT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] [Std.Asymm fun (x1 x2 : α) => x1 < x2] {xs ys : Array α} : xs ≤ ys ↔ xs < ys ∨ xs = ys

theorem Array.le_antisymm {α : Type u_1} [LT α] [LE α] [Std.IsLinearOrder α] [Std.LawfulOrderLT α] {xs ys : Array α} : xs ≤ ys → ys ≤ xs → xs = ys

instance Array.instTotalLeOfAsymmLt {α : Type u_1} [LT α] [Std.Asymm fun (x1 x2 : α) => x1 < x2] : Std.Total fun (x1 x2 : Array α) => x1 ≤ x2

@[simp]

theorem Array.lex_eq_true_iff_lt {α : Type u_1} [BEq α] [LawfulBEq α] [LT α] [DecidableLT α] {xs ys : Array α} : (xs.lex ys fun (x1 x2 : α) => decide (x1 < x2)) = true ↔ xs < ys

@[simp]

theorem Array.lex_eq_false_iff_ge {α : Type u_1} [BEq α] [LawfulBEq α] [LT α] [DecidableLT α] {xs ys : Array α} : (xs.lex ys fun (x1 x2 : α) => decide (x1 < x2)) = false ↔ ys ≤ xs

instance Array.instDecidableLTOfDecidableEq {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] : DecidableLT (Array α)

Equations

• xs.instDecidableLTOfDecidableEq ys = decidable_of_iff ((xs.lex ys fun (x1 x2 : α) => decide (x1 < x2)) = true) ⋯

instance Array.instDecidableLEOfDecidableEqOfDecidableLT {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] : DecidableLE (Array α)

Equations

• xs.instDecidableLEOfDecidableEqOfDecidableLT ys = decidable_of_iff ((ys.lex xs fun (x1 x2 : α) => decide (x1 < x2)) = false) ⋯

theorem Array.lex_eq_true_iff_exists {α : Type u_1} {l₁ l₂ : Array α} [BEq α] (lt : α → α → Bool) : l₁.lex l₂ lt = true ↔ (l₁.isEqv (l₂.take l₁.size) fun (x1 x2 : α) => x1 == x2) = true ∧ l₁.size < l₂.size ∨ ∃ (i : Nat), ∃ (h₁ : i < l₁.size), ∃ (h₂ : i < l₂.size), (∀ (j : Nat) (hj : j < i), (l₁[j] == l₂[j]) = true) ∧ lt l₁[i] l₂[i] = true

l₁ is lexicographically less than l₂ if either

• l₁ is pairwise equivalent under · == · to l₂.take l₁.size, and l₁ is shorter than l₂ or
• there exists an index i such that
  • for all j < i, l₁[j] == l₂[j] and
  • l₁[i] < l₂[i]

theorem Array.lex_eq_false_iff_exists {α : Type u_1} {l₁ l₂ : Array α} [BEq α] [PartialEquivBEq α] (lt : α → α → Bool) (lt_irrefl : ∀ (x y : α), (x == y) = true → lt x y = false) (lt_asymm : ∀ (x y : α), lt x y = true → lt y x = false) (lt_antisymm : ∀ (x y : α), lt x y = false → lt y x = false → (x == y) = true) : l₁.lex l₂ lt = false ↔ (l₂.isEqv (l₁.take l₂.size) fun (x1 x2 : α) => x1 == x2) = true ∨ ∃ (i : Nat), ∃ (h₁ : i < l₁.size), ∃ (h₂ : i < l₂.size), (∀ (j : Nat) (hj : j < i), (l₁[j] == l₂[j]) = true) ∧ lt l₂[i] l₁[i] = true

l₁ is not lexicographically less than l₂ (which you might think of as "l₂ is lexicographically greater than or equal to l₁"") if either

• l₁ is pairwise equivalent under · == · to l₂.take l₁.length or
• there exists an index i such that
  • for all j < i, l₁[j] == l₂[j] and
  • l₂[i] < l₁[i]

This formulation requires that == and lt are compatible in the following senses:

• == is symmetric (we unnecessarily further assume it is transitive, to make use of the existing typeclasses)
• lt is irreflexive with respect to == (i.e. if x == y then lt x y = false
• lt is asymmetric (i.e. lt x y = true → lt y x = false)
• lt is antisymmetric with respect to == (i.e. lt x y = false → lt y x = false → x == y)

theorem Array.lt_iff_exists {α : Type u_1} [LT α] {xs ys : Array α} : xs < ys ↔ xs = ys.take xs.size ∧ xs.size < ys.size ∨ ∃ (i : Nat), ∃ (h₁ : i < xs.size), ∃ (h₂ : i < ys.size), (∀ (j : Nat) (hj : j < i), xs[j] = ys[j]) ∧ xs[i] < ys[i]

theorem Array.le_iff_exists {α : Type u_1} [LT α] [Std.Asymm fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] {xs ys : Array α} : xs ≤ ys ↔ xs = ys.take xs.size ∨ ∃ (i : Nat), ∃ (h₁ : i < xs.size), ∃ (h₂ : i < ys.size), (∀ (j : Nat) (hj : j < i), xs[j] = ys[j]) ∧ xs[i] < ys[i]

theorem Array.append_left_lt {α : Type u_1} [LT α] {xs ys zs : Array α} (h : ys < zs) : xs ++ ys < xs ++ zs

theorem Array.append_left_le {α : Type u_1} [LT α] [Std.Asymm fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] {xs ys zs : Array α} (h : ys ≤ zs) : xs ++ ys ≤ xs ++ zs

theorem Array.le_append_left {α : Type u_1} [LT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] {xs ys : Array α} : xs ≤ xs ++ ys

theorem Array.map_lt {α : Type u_1} {β : Type u_2} [LT α] [LT β] {xs ys : Array α} {f : α → β} (w : ∀ (x y : α), x < y → f x < f y) (h : xs < ys) : map f xs < map f ys

theorem Array.map_le {α : Type u_1} {β : Type u_2} [LT α] [LT β] [Std.Asymm fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] [Std.Asymm fun (x1 x2 : β) => x1 < x2] [Std.Antisymm fun (x1 x2 : β) => ¬x1 < x2] {xs ys : Array α} {f : α → β} (w : ∀ (x y : α), x < y → f x < f y) (h : xs ≤ ys) : map f xs ≤ map f ys