Lexicographic ordering

@[simp]

theorem List.lex_lt {α : Type u_1} [LT α] (l₁ l₂ : List α) : List.Lex (fun (x1 x2 : α) => x1 < x2) l₁ l₂ ↔ l₁ < l₂

@[simp]

theorem List.not_lex_lt {α : Type u_1} [LT α] (l₁ l₂ : List α) : ¬List.Lex (fun (x1 x2 : α) => x1 < x2) l₁ l₂ ↔ l₂ ≤ l₁

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

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

theorem List.lex_irrefl {α : Type u_1} {r : α → α → Prop} (irrefl : ∀ (x : α), ¬r x x) (l : List α) : ¬List.Lex r l l

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

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

@[simp]

theorem List.not_lex_nil {α✝ : Type u_1} {r : α✝ → α✝ → Prop} {l : List α✝} : ¬List.Lex r l []

@[simp]

theorem List.not_lt_nil {α : Type u_1} [LT α] (l : List α) : ¬l < []

@[simp]

theorem List.nil_le {α : Type u_1} [LT α] (l : List α) : [] ≤ l

@[simp]

theorem List.not_nil_lex_iff {α✝ : Type u_1} {r : α✝ → α✝ → Prop} {l : List α✝} : ¬List.Lex r [] l ↔ l = []

@[simp]

theorem List.le_nil {α : Type u_1} [LT α] (l : List α) : l ≤ [] ↔ l = []

@[simp]

theorem List.nil_lex_cons {α✝ : Type u_1} {r : α✝ → α✝ → Prop} {a : α✝} {l : List α✝} : List.Lex r [] (a :: l)

@[simp]

theorem List.nil_lt_cons {α : Type u_1} [LT α] (a : α) (l : List α) : [] < a :: l

theorem List.cons_lex_cons_iff {α✝ : Type u_1} {r : α✝ → α✝ → Prop} {a : α✝} {l₁ : List α✝} {b : α✝} {l₂ : List α✝} : List.Lex r (a :: l₁) (b :: l₂) ↔ r a b ∨ a = b ∧ List.Lex r l₁ l₂

theorem List.cons_lt_cons_iff {α : Type u_1} [LT α] {a b : α} {l₁ l₂ : List α} : a :: l₁ < b :: l₂ ↔ a < b ∨ a = b ∧ l₁ < l₂

@[simp]

theorem List.cons_lt_cons_self {α : Type u_1} {a : α} [LT α] [i₀ : Std.Irrefl fun (x1 x2 : α) => x1 < x2] {l₁ l₂ : List α} : a :: l₁ < a :: l₂ ↔ l₁ < l₂

theorem List.not_cons_lex_cons_iff {α : Type u_1} {r : α → α → Prop} [DecidableEq α] [DecidableRel r] {a b : α} {l₁ l₂ : List α} : ¬List.Lex r (a :: l₁) (b :: l₂) ↔ ¬r a b ∧ a ≠ b ∨ ¬r a b ∧ ¬List.Lex r l₁ l₂

theorem List.cons_le_cons_iff {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] {a b : α} {l₁ l₂ : List α} : a :: l₁ ≤ b :: l₂ ↔ a < b ∨ a = b ∧ l₁ ≤ l₂

theorem List.not_lt_of_cons_le_cons {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] {a b : α} {l₁ l₂ : List α} (h : a :: l₁ ≤ b :: l₂) : ¬b < a

theorem List.le_of_cons_le_cons {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] {a : α} {l₁ l₂ : List α} (h : a :: l₁ ≤ a :: l₂) : l₁ ≤ l₂

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

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

theorem List.lex_trans {α : Type u_1} {l₁ l₂ l₃ : List α} {r : α → α → Prop} (lt_trans : ∀ {x y z : α}, r x y → r y z → r x z) (h₁ : List.Lex r l₁ l₂) (h₂ : List.Lex r l₂ l₃) : List.Lex r l₁ l₃

theorem List.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 : List α} (h₁ : l₁ < l₂) (h₂ : l₂ < l₃) : l₁ < l₃

instance List.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 : List α) => x1 < x2) (fun (x1 x2 : List α) => x1 < x2) fun (x1 x2 : List α) => x1 < x2

Equations

• List.instTransLt = { trans := ⋯ }

@[reducible, inline, deprecated List.le_antisymm (since := "2024-12-13")]

abbrev List.lt_antisymm {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] [i : Std.Antisymm fun (x1 x2 : α) => ¬x1 < x2] {as bs : List α} (h₁ : as ≤ bs) (h₂ : bs ≤ as) : as = bs

Equations

• @List.lt_antisymm = @List.le_antisymm

theorem List.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 : List α} (h₁ : l₁ ≤ l₂) (h₂ : l₂ < l₃) : l₁ < l₃

theorem List.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 : List α} (h₁ : l₁ ≤ l₂) (h₂ : l₂ ≤ l₃) : l₁ ≤ l₃

instance List.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 : List α) => x1 ≤ x2) (fun (x1 x2 : List α) => x1 ≤ x2) fun (x1 x2 : List α) => x1 ≤ x2

Equations

• List.instTransLeOfDecidableEqOfDecidableLTOfIrreflOfAsymmOfAntisymmOfNotLt = { trans := ⋯ }

theorem List.lex_asymm {α : Type u_1} {r : α → α → Prop} (h : ∀ {x y : α}, r x y → ¬r y x) {l₁ l₂ : List α} : List.Lex r l₁ l₂ → ¬List.Lex r l₂ l₁

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

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

theorem List.not_lex_total {α : Type u_1} [DecidableEq α] {r : α → α → Prop} [DecidableRel r] (h : ∀ (x y : α), ¬r x y ∨ ¬r y x) (l₁ l₂ : List α) : ¬List.Lex r l₁ l₂ ∨ ¬List.Lex r l₂ l₁

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

instance List.instTotalLeOfDecidableEqOfDecidableLTOfNotLt {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] [Std.Total fun (x1 x2 : α) => ¬x1 < x2] : Std.Total fun (x1 x2 : List α) => x1 ≤ x2

@[simp]

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

@[simp]

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

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

theorem List.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 : List α} : l₁ ≤ l₂ ↔ l₁ < l₂ ∨ l₁ = l₂

theorem List.lex_eq_decide_lex {α : Type u_1} {l₁ l₂ : List α} [DecidableEq α] (lt : α → α → Bool) : l₁.lex l₂ lt = decide (List.Lex (fun (x y : α) => lt x y = true) l₁ l₂)

@[simp]

theorem List.lex_eq_true_iff_lex {α : Type u_1} {l₁ l₂ : List α} [DecidableEq α] (lt : α → α → Bool) : l₁.lex l₂ lt = true ↔ List.Lex (fun (x y : α) => lt x y = true) l₁ l₂

Variant of lex_eq_true_iff using an arbitrary comparator.

@[simp]

theorem List.lex_eq_false_iff_not_lex {α : Type u_1} {l₁ l₂ : List α} [DecidableEq α] (lt : α → α → Bool) : l₁.lex l₂ lt = false ↔ ¬List.Lex (fun (x y : α) => lt x y = true) l₁ l₂

Variant of lex_eq_false_iff using an arbitrary comparator.

@[simp]

theorem List.lex_eq_true_iff_lt {α : Type u_1} [DecidableEq α] [LT α] [DecidableLT α] {l₁ l₂ : List α} : (l₁.lex l₂ fun (x1 x2 : α) => decide (x1 < x2)) = true ↔ l₁ < l₂

@[simp]

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

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

theorem List.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 : List α} : l₁ ≤ l₂ ↔ l₁ = List.take l₁.length l₂ ∨ ∃ (i : Nat), ∃ (h₁ : i < l₁.length), ∃ (h₂ : i < l₂.length), (∀ (j : Nat) (hj : j < i), l₁[j] = l₂[j]) ∧ l₁[i] < l₂[i]

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

theorem List.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 : List α} (h : l₂ ≤ l₃) : l₁ ++ l₂ ≤ l₁ ++ l₃

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

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

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

theorem List.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 : List α} {f : α → β} (w : ∀ (x y : α), x < y → f x < f y) (h : l₁ ≤ l₂) : List.map f l₁ ≤ List.map f l₂