Lexicographic order

This file defines the lexicographic relation for pairs of orders, partial orders and linear orders.

Main declarations

• Prod.Lex.<pre/partial/linear>Order: Instances lifting the orders on α and β to α ×ₗ β.

Notation

• α ×ₗ β: α × β equipped with the lexicographic order

See also

Related files are:

• Data.Finset.CoLex: Colexicographic order on finite sets.
• Data.List.Lex: Lexicographic order on lists.
• Data.Pi.Lex: Lexicographic order on Πₗ i, α i.
• Order.PSigma: Lexicographic order on Σ' i, α i.
• Order.Sigma: Lexicographic order on Σ i, α i.

TODO

Some lemmas could be automatically generated with to_dual. See [https://github.com/leanprover-community/mathlib4/pull/37939#discussion_r3367855484]

def Prod.Lex.«term_×ₗ_» : Lean.TrailingParserDescr

A type synonym to equip a type with its lexicographic order.

Equations

• Prod.Lex.«term_×ₗ_» = Lean.ParserDescr.trailingNode `Prod.Lex.«term_×ₗ_» 35 0 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ×ₗ ") (Lean.ParserDescr.cat `term 34))

@[instance_reducible]

instance Prod.Lex.instLE (α : Type u_3) (β : Type u_4) [LT α] [LE β] : LE (Lex (α × β))

Dictionary / lexicographic ordering on pairs.

Equations

• Prod.Lex.instLE α β = { le := Prod.Lex (fun (x1 x2 : α) => x1 < x2) fun (x1 x2 : β) => x1 ≤ x2 }

@[instance_reducible]

instance Prod.Lex.instLT (α : Type u_3) (β : Type u_4) [LT α] [LT β] : LT (Lex (α × β))

Equations

• Prod.Lex.instLT α β = { lt := Prod.Lex (fun (x1 x2 : α) => x1 < x2) fun (x1 x2 : β) => x1 < x2 }

theorem Prod.Lex.toLex_le_toLex {α : Type u_1} {β : Type u_2} [LT α] [LE β] {x y : α × β} : toLex x ≤ toLex y ↔ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 ≤ y.2

theorem Prod.Lex.toLex_lt_toLex {α : Type u_1} {β : Type u_2} [LT α] [LT β] {x y : α × β} : toLex x < toLex y ↔ x.1 < y.1 ∨ x.1 = y.1 ∧ x.2 < y.2

theorem Prod.Lex.le_iff {α : Type u_1} {β : Type u_2} [LT α] [LE β] {x y : Lex (α × β)} : x ≤ y ↔ (ofLex x).1 < (ofLex y).1 ∨ (ofLex x).1 = (ofLex y).1 ∧ (ofLex x).2 ≤ (ofLex y).2

theorem Prod.Lex.lt_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] {x y : Lex (α × β)} : x < y ↔ (ofLex x).1 < (ofLex y).1 ∨ (ofLex x).1 = (ofLex y).1 ∧ (ofLex x).2 < (ofLex y).2

instance Prod.Lex.instWellFoundedLTLex {α : Type u_1} {β : Type u_2} [LT α] [LT β] [WellFoundedLT α] [WellFoundedLT β] : WellFoundedLT (Lex (α × β))

@[instance_reducible]

instance Prod.Lex.instWellFoundedRelationLexOfWellFoundedLT {α : Type u_1} {β : Type u_2} [LT α] [LT β] [WellFoundedLT α] [WellFoundedLT β] : WellFoundedRelation (Lex (α × β))

Equations

• Prod.Lex.instWellFoundedRelationLexOfWellFoundedLT = WellFoundedLT.toWellFoundedRelation

@[instance_reducible]

instance Prod.Lex.instPreorder (α : Type u_3) (β : Type u_4) [Preorder α] [Preorder β] : Preorder (Lex (α × β))

Dictionary / lexicographic preorder for pairs.

Equations

• Prod.Lex.instPreorder α β = { toLE := Prod.Lex.instLE α β, toLT := Prod.Lex.instLT α β, le_refl := ⋯, le_trans := ⋯, lt_iff_le_not_ge := ⋯ }

theorem Prod.Lex.monotone_fst {α : Type u_1} {β : Type u_2} [Preorder α] [LE β] (t c : Lex (α × β)) (h : t ≤ c) : (ofLex t).1 ≤ (ofLex c).1

See also monotone_fst_ofLex for a version stated in terms of Monotone.

theorem Prod.Lex.monotone_fst_ofLex {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] : Monotone fun (x : Lex (α × β)) => (ofLex x).1

theorem WCovBy.fst_ofLex {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a b : Lex (α × β)} (h : a ⩿ b) : (ofLex a).1 ⩿ (ofLex b).1

theorem Prod.Lex.toLex_covBy_toLex_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a₁ a₂ : α} {b₁ b₂ : β} : toLex (a₁, b₁) ⋖ toLex (a₂, b₂) ↔ a₁ = a₂ ∧ b₁ ⋖ b₂ ∨ a₁ ⋖ a₂ ∧ IsMax b₁ ∧ IsMin b₂

theorem Prod.Lex.covBy_iff {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a b : Lex (α × β)} : a ⋖ b ↔ (ofLex a).1 = (ofLex b).1 ∧ (ofLex a).2 ⋖ (ofLex b).2 ∨ (ofLex a).1 ⋖ (ofLex b).1 ∧ IsMax (ofLex a).2 ∧ IsMin (ofLex b).2

theorem Prod.Lex.toLex_le_toLex' {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] {x y : α × β} : toLex x ≤ toLex y ↔ x.1 ≤ y.1 ∧ (x.1 = y.1 → x.2 ≤ y.2)

Variant of Prod.Lex.toLex_le_toLex for partial orders.

theorem Prod.Lex.toLex_lt_toLex' {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] {x y : α × β} : toLex x < toLex y ↔ x.1 ≤ y.1 ∧ (x.1 = y.1 → x.2 < y.2)

Variant of Prod.Lex.toLex_lt_toLex for partial orders.

theorem Prod.Lex.le_iff' {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] {x y : Lex (α × β)} : x ≤ y ↔ (ofLex x).1 ≤ (ofLex y).1 ∧ ((ofLex x).1 = (ofLex y).1 → (ofLex x).2 ≤ (ofLex y).2)

Variant of Prod.Lex.le_iff for partial orders.

theorem Prod.Lex.lt_iff' {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] {x y : Lex (α × β)} : x < y ↔ (ofLex x).1 ≤ (ofLex y).1 ∧ ((ofLex x).1 = (ofLex y).1 → (ofLex x).2 < (ofLex y).2)

Variant of Prod.Lex.lt_iff for partial orders.

theorem Prod.Lex.toLex_mono {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] : Monotone ⇑toLex

theorem Prod.Lex.toLex_strictMono {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] : StrictMono ⇑toLex

@[instance_reducible]

instance Prod.Lex.instPartialOrder (α : Type u_3) (β : Type u_4) [PartialOrder α] [PartialOrder β] : PartialOrder (Lex (α × β))

Dictionary / lexicographic partial order for pairs.

Equations

• Prod.Lex.instPartialOrder α β = { toPreorder := Prod.Lex.instPreorder α β, le_antisymm := ⋯ }

@[instance_reducible]

instance Prod.Lex.instOrdLexProd {α : Type u_1} {β : Type u_2} [Ord α] [Ord β] : Ord (Lex (α × β))

Equations

• Prod.Lex.instOrdLexProd = { compare := fun (a a_1 : Lex (α × β)) => compareLex (compareOn fun (x : α × β) => x.1) (compareOn fun (x : α × β) => x.2) a a_1 }

theorem Prod.Lex.compare_def {α : Type u_1} {β : Type u_2} [Ord α] [Ord β] : compare = compareLex (compareOn fun (x : Lex (α × β)) => (ofLex x).1) (compareOn fun (x : Lex (α × β)) => (ofLex x).2)

theorem lexOrd_eq {α : Type u_1} {β : Type u_2} [Ord α] [Ord β] : lexOrd = Prod.Lex.instOrdLexProd

theorem Ord.lex_eq {α : Type u_1} {β : Type u_2} [oα : Ord α] [oβ : Ord β] : oα.lex oβ = Prod.Lex.instOrdLexProd

instance Prod.Lex.instOrientedOrdLex {α : Type u_1} {β : Type u_2} [Ord α] [Ord β] [Std.OrientedOrd α] [Std.OrientedOrd β] : Std.OrientedOrd (Lex (α × β))

instance Prod.Lex.instTransOrdLex {α : Type u_1} {β : Type u_2} [Ord α] [Ord β] [Std.TransOrd α] [Std.TransOrd β] : Std.TransOrd (Lex (α × β))

@[instance_reducible]

instance Prod.Lex.instLinearOrder (α : Type u_3) (β : Type u_4) [LinearOrder α] [LinearOrder β] : LinearOrder (Lex (α × β))

Dictionary / lexicographic linear order for pairs.

Equations

• One or more equations did not get rendered due to their size.

@[instance_reducible]

instance Prod.Lex.orderBot {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] [OrderBot α] [OrderBot β] : OrderBot (Lex (α × β))

Equations

• Prod.Lex.orderBot = { bot := toLex ⊥, bot_le := ⋯ }

@[instance_reducible]

instance Prod.Lex.orderTop {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] [OrderTop α] [OrderTop β] : OrderTop (Lex (α × β))

Equations

• Prod.Lex.orderTop = { top := toLex ⊤, le_top := ⋯ }

@[instance_reducible]

instance Prod.Lex.boundedOrder {α : Type u_1} {β : Type u_2} [PartialOrder α] [Preorder β] [BoundedOrder α] [BoundedOrder β] : BoundedOrder (Lex (α × β))

Equations

• Prod.Lex.boundedOrder = { toOrderTop := Prod.Lex.orderTop, toOrderBot := Prod.Lex.orderBot }

instance Prod.Lex.instDenselyOrderedLex {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] : DenselyOrdered (Lex (α × β))

instance Prod.Lex.instDenselyOrderedLexOfNoMinOrder {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [NoMinOrder β] [DenselyOrdered β] : DenselyOrdered (Lex (α × β))

instance Prod.Lex.instDenselyOrderedLexOfNoMaxOrder {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [NoMaxOrder β] [DenselyOrdered β] : DenselyOrdered (Lex (α × β))

instance Prod.Lex.noMaxOrder_of_left {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [NoMaxOrder α] : NoMaxOrder (Lex (α × β))

instance Prod.Lex.noMinOrder_of_left {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [NoMinOrder α] : NoMinOrder (Lex (α × β))

instance Prod.Lex.noMaxOrder_of_right {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [NoMaxOrder β] : NoMaxOrder (Lex (α × β))

instance Prod.Lex.noMinOrder_of_right {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [NoMinOrder β] : NoMinOrder (Lex (α × β))