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

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theorem Array.lt_toList {α : Type u_1} [LT α] (l₁ l₂ : Array α) : l₁.toList < l₂.toList ↔ l₁ < l₂

@[simp]

theorem Array.le_toList {α : Type u_1} [LT α] (l₁ l₂ : Array α) : l₁.toList ≤ l₂.toList ↔ l₁ ≤ l₂

theorem Array.not_lt_iff_ge {α : Type u_1} [LT α] (l₁ l₂ : List α) : ¬l₁ < l₂ ↔ l₂ ≤ l₁

theorem Array.not_le_iff_gt {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] (l₁ l₂ : List α) : ¬l₁ ≤ l₂ ↔ l₂ < l₁

@[simp]

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

@[simp]

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 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

@[simp]

theorem Array.lex_toList {α : Type u_1} [BEq α] (lt : α → α → Bool) (l₁ l₂ : Array α) : l₁.toList.lex l₂.toList lt = l₁.lex l₂ lt

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

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] (l : Array α) : ¬l < #[]

@[simp]

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

@[simp]

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

@[simp]

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

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

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] {l₁ l₂ l₃ : Array α} (h₁ : l₁ < l₂) (h₂ : l₂ < l₃) : l₁ < l₃

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} [DecidableEq α] [LT α] [DecidableLT α] [i₀ : Std.Irrefl fun (x1 x2 : α) => x1 < x2] [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] {l₁ l₂ l₃ : Array α} (h₁ : l₁ ≤ l₂) (h₂ : l₂ < l₃) : l₁ < l₃

theorem Array.le_trans {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] [Std.Asymm fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] [Trans (fun (x1 x2 : α) => ¬x1 < x2) (fun (x1 x2 : α) => ¬x1 < x2) fun (x1 x2 : α) => ¬x1 < x2] {l₁ l₂ l₃ : Array α} (h₁ : l₁ ≤ l₂) (h₂ : l₂ ≤ l₃) : l₁ ≤ l₃

instance Array.instTransLeOfDecidableEqOfDecidableLTOfIrreflOfAsymmOfAntisymmOfNotLt {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] [Std.Asymm fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] [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.instTransLeOfDecidableEqOfDecidableLTOfIrreflOfAsymmOfAntisymmOfNotLt = { trans := ⋯ }

theorem Array.lt_asymm {α : Type u_1} [LT α] [i : Std.Asymm fun (x1 x2 : α) => x1 < x2] {l₁ l₂ : Array α} (h : l₁ < l₂) : ¬l₂ < l₁

instance Array.instAsymmLtOfDecidableEqOfDecidableLT {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT 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} [DecidableEq α] [LT α] [DecidableLT α] [i : Std.Total fun (x1 x2 : α) => ¬x1 < x2] (l₁ l₂ : Array α) : l₁ ≤ l₂ ∨ l₂ ≤ l₁

@[simp]

theorem Array.not_lt {α : Type u_1} [LT α] {l₁ l₂ : Array α} : ¬l₁ < l₂ ↔ l₂ ≤ l₁

@[simp]

theorem Array.not_le {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Array α} : ¬l₂ ≤ l₁ ↔ l₁ < l₂

theorem Array.le_of_lt {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] [i : Std.Total fun (x1 x2 : α) => ¬x1 < x2] {l₁ l₂ : Array α} (h : l₁ < l₂) : l₁ ≤ l₂

theorem Array.le_iff_lt_or_eq {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] [Std.Total fun (x1 x2 : α) => ¬x1 < x2] {l₁ l₂ : Array α} : l₁ ≤ l₂ ↔ l₁ < l₂ ∨ l₁ = l₂

instance Array.instTotalLeOfDecidableEqOfDecidableLTOfNotLt {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] [Std.Total 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} [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Array α} : (l₁.lex l₂ fun (x1 x2 : α) => decide (x1 < x2)) = true ↔ l₁ < l₂

@[simp]

theorem Array.lex_eq_false_iff_ge {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Array α} : (l₁.lex l₂ fun (x1 x2 : α) => decide (x1 < x2)) = false ↔ l₂ ≤ l₁

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

Equations

• l₁.instDecidableLTOfDecidableEq l₂ = decidable_of_iff ((l₁.lex l₂ 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

• l₁.instDecidableLEOfDecidableEqOfDecidableLT l₂ = decidable_of_iff ((l₂.lex l₁ 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 asymmmetric (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} [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : Array α} : l₁ < l₂ ↔ l₁ = l₂.take l₁.size ∧ l₁.size < l₂.size ∨ ∃ (i : Nat), ∃ (h₁ : i < l₁.size), ∃ (h₂ : i < l₂.size), (∀ (j : Nat) (hj : j < i), l₁[j] = l₂[j]) ∧ l₁[i] < l₂[i]

theorem Array.le_iff_exists {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] [Std.Asymm fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] {l₁ l₂ : Array α} : l₁ ≤ l₂ ↔ l₁ = l₂.take l₁.size ∨ ∃ (i : Nat), ∃ (h₁ : i < l₁.size), ∃ (h₂ : i < l₂.size), (∀ (j : Nat) (hj : j < i), l₁[j] = l₂[j]) ∧ l₁[i] < l₂[i]

theorem Array.append_left_lt {α : Type u_1} [LT α] {l₁ l₂ l₃ : Array α} (h : l₂ < l₃) : l₁ ++ l₂ < l₁ ++ l₃

theorem Array.append_left_le {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] [Std.Asymm fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] {l₁ l₂ l₃ : Array α} (h : l₂ ≤ l₃) : l₁ ++ l₂ ≤ l₁ ++ l₃

theorem Array.le_append_left {α : Type u_1} [LT α] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] {l₁ l₂ : Array α} : l₁ ≤ l₁ ++ l₂

theorem Array.map_lt {α : Type u_1} {β : Type u_2} [LT α] [LT β] {l₁ l₂ : Array α} {f : α → β} (w : ∀ (x y : α), x < y → f x < f y) (h : l₁ < l₂) : Array.map f l₁ < Array.map f l₂

theorem Array.map_le {α : Type u_1} {β : Type u_2} [DecidableEq α] [LT α] [DecidableLT α] [DecidableEq β] [LT β] [DecidableLT β] [Std.Irrefl fun (x1 x2 : α) => x1 < x2] [Std.Asymm fun (x1 x2 : α) => x1 < x2] [Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] [Std.Irrefl fun (x1 x2 : β) => x1 < x2] [Std.Asymm fun (x1 x2 : β) => x1 < x2] [Std.Antisymm fun (x1 x2 : β) => ¬x1 < x2] {l₁ l₂ : Array α} {f : α → β} (w : ∀ (x y : α), x < y → f x < f y) (h : l₁ ≤ l₂) : Array.map f l₁ ≤ Array.map f l₂