Lexicographic ordering of lists.

The lexicographic order on List α is defined by L < M iff

• [] < (a :: L) for any a and L,
• (a :: L) < (b :: M) where a < b, or
• (a :: L) < (a :: M) where L < M.

See also

Related files are:

• Mathlib/Data/Finset/Colex.lean: Colexicographic order on finite sets.
• Mathlib/Data/PSigma/Order.lean: Lexicographic order on Σ' i, α i.
• Mathlib/Data/Pi/Lex.lean: Lexicographic order on Πₗ i, α i.
• Mathlib/Data/Sigma/Order.lean: Lexicographic order on Σ i, α i.
• Mathlib/Data/Prod/Lex.lean: Lexicographic order on α × β.

lexicographic ordering

theorem List.lex_cons_iff {α : Type u} {r : α → α → Prop} [IsIrrefl α r] {a : α} {l₁ l₂ : List α} : Lex r (a :: l₁) (a :: l₂) ↔ Lex r l₁ l₂

theorem List.lex_nil_or_eq_nil {α : Type u} {r : α → α → Prop} (l : List α) : Lex r [] l ∨ l = []

@[deprecated List.lex_nil_or_eq_nil (since := "2025-03-14")]

theorem List.Lex.nil_left_or_eq_nil {α : Type u} {r : α → α → Prop} (l : List α) : Lex r [] l ∨ l = []

Alias of List.lex_nil_or_eq_nil.

@[simp]

theorem List.lex_singleton_iff {α : Type u} {r : α → α → Prop} (a b : α) : Lex r [a] [b] ↔ r a b

@[deprecated List.lex_singleton_iff (since := "2025-03-14")]

theorem List.Lex.singleton_iff {α : Type u} {r : α → α → Prop} (a b : α) : Lex r [a] [b] ↔ r a b

Alias of List.lex_singleton_iff.

instance List.Lex.isOrderConnected {α : Type u} (r : α → α → Prop) [IsOrderConnected α r] [IsTrichotomous α r] : IsOrderConnected (List α) (Lex r)

theorem List.Lex.isOrderConnected.aux {α : Type u} (r : α → α → Prop) [IsOrderConnected α r] [IsTrichotomous α r] (x✝ x✝¹ x✝² : List α) : Lex r x✝ x✝² → Lex r x✝ x✝¹ ∨ Lex r x✝¹ x✝²

instance List.Lex.isTrichotomous {α : Type u} (r : α → α → Prop) [IsTrichotomous α r] : IsTrichotomous (List α) (Lex r)

theorem List.Lex.isTrichotomous.aux {α : Type u} (r : α → α → Prop) [IsTrichotomous α r] (x✝ x✝¹ : List α) : Lex r x✝ x✝¹ ∨ x✝ = x✝¹ ∨ Lex r x✝¹ x✝

instance List.Lex.isAsymm {α : Type u} (r : α → α → Prop) [IsAsymm α r] : IsAsymm (List α) (Lex r)

theorem List.Lex.isAsymm.aux {α : Type u} (r : α → α → Prop) [IsAsymm α r] (x✝ x✝¹ : List α) : Lex r x✝ x✝¹ → Lex r x✝¹ x✝ → False

instance List.Lex.decidableRel {α : Type u} [DecidableEq α] (r : α → α → Prop) [DecidableRel r] : DecidableRel (Lex r)

Equations

• List.Lex.decidableRel r x✝ [] = isFalse ⋯
• List.Lex.decidableRel r [] (head :: tail) = isTrue ⋯
• List.Lex.decidableRel r (a :: l₁) (b :: l₂) = decidable_of_iff (r a b ∨ a = b ∧ List.Lex r l₁ l₂) ⋯

theorem List.Lex.append_right {α : Type u} (r : α → α → Prop) {s₁ s₂ : List α} (t : List α) : Lex r s₁ s₂ → Lex r s₁ (s₂ ++ t)

theorem List.Lex.append_left {α : Type u} (R : α → α → Prop) {t₁ t₂ : List α} (h : Lex R t₁ t₂) (s : List α) : Lex R (s ++ t₁) (s ++ t₂)

theorem List.Lex.imp {α : Type u} {r s : α → α → Prop} (H : ∀ (a b : α), r a b → s a b) (l₁ l₂ : List α) : Lex r l₁ l₂ → Lex s l₁ l₂

theorem List.Lex.to_ne {α : Type u} {l₁ l₂ : List α} : Lex (fun (x1 x2 : α) => x1 ≠ x2) l₁ l₂ → l₁ ≠ l₂

theorem Decidable.List.Lex.ne_iff {α : Type u} [DecidableEq α] {l₁ l₂ : List α} (H : l₁.length ≤ l₂.length) : List.Lex (fun (x1 x2 : α) => x1 ≠ x2) l₁ l₂ ↔ l₁ ≠ l₂

theorem List.Lex.ne_iff {α : Type u} {l₁ l₂ : List α} (H : l₁.length ≤ l₂.length) : Lex (fun (x1 x2 : α) => x1 ≠ x2) l₁ l₂ ↔ l₁ ≠ l₂

instance List.instLinearOrder {α : Type u} [LinearOrder α] : LinearOrder (List α)

Equations

• List.instLinearOrder = linearOrderOfSTO (List.Lex fun (x1 x2 : α) => x1 < x2)

instance List.LE' {α : Type u} [LinearOrder α] : LE (List α)

Equations

• List.LE' = List.instLinearOrder.toLE

theorem List.lt_iff_lex_lt {α : Type u} [LT α] (l l' : List α) : l.lt l' ↔ Lex (fun (x1 x2 : α) => x1 < x2) l l'

theorem List.head_le_of_lt {α : Type u} [Preorder α] {a a' : α} {l l' : List α} (h : a' :: l' < a :: l) : a' ≤ a

theorem List.head!_le_of_lt {α : Type u} [Preorder α] [Inhabited α] (l l' : List α) (h : l' < l) (hl' : l' ≠ []) : l'.head! ≤ l.head!

theorem List.cons_le_cons {α : Type u} [LinearOrder α] (a : α) {l l' : List α} (h : l' ≤ l) : a :: l' ≤ a :: l