Lexicographic order on Pi types

This file defines the lexicographic order for Pi types. a is less than b if a i = b i for all i up to some point k, and a k < b k.

Notation

• Πₗ i, α i: Pi type equipped with the lexicographic order. Type synonym of Π i, α i.

See also

Related files are:

• Data.Finset.Colex: Colexicographic order on finite sets.
• Data.List.Lex: Lexicographic order on lists.
• Data.Sigma.Order: Lexicographic order on Σₗ i, α i.
• Data.PSigma.Order: Lexicographic order on Σₗ' i, α i.
• Data.Prod.Lex: Lexicographic order on α × β.

def Pi.Lex {ι : Type u_1} {β : ι → Type u_2} (r : ι → ι → Prop) (s : {i : ι} → β i → β i → Prop) (x : (i : ι) → β i) (y : (i : ι) → β i) : Prop

The lexicographic relation on Π i : ι, β i, where ι is ordered by r, and each β i is ordered by s.

Equations

• Pi.Lex r s x y = ∃ (i : ι), (∀ (j : ι), r j i → x j = y j) ∧ s (x i) (y i)

@[simp]

theorem Pi.toLex_apply {ι : Type u_1} {β : ι → Type u_2} (x : (i : ι) → β i) (i : ι) : toLex x i = x i

@[simp]

theorem Pi.ofLex_apply {ι : Type u_1} {β : ι → Type u_2} (x : Lex ((i : ι) → β i)) (i : ι) : ofLex x i = x i

theorem Pi.lex_lt_of_lt_of_preorder {ι : Type u_1} {β : ι → Type u_2} [(i : ι) → Preorder (β i)] {r : ι → ι → Prop} (hwf : WellFounded r) {x : (i : ι) → β i} {y : (i : ι) → β i} (hlt : x < y) : ∃ (i : ι), (∀ (j : ι), r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i

theorem Pi.lex_lt_of_lt {ι : Type u_1} {β : ι → Type u_2} [(i : ι) → PartialOrder (β i)] {r : ι → ι → Prop} (hwf : WellFounded r) {x : (i : ι) → β i} {y : (i : ι) → β i} (hlt : x < y) : Pi.Lex r (fun (x : ι) (x1 x2 : β x) => x1 < x2) x y

theorem Pi.isTrichotomous_lex {ι : Type u_1} {β : ι → Type u_2} (r : ι → ι → Prop) (s : {i : ι} → β i → β i → Prop) [∀ (i : ι), IsTrichotomous (β i) s] (wf : WellFounded r) : IsTrichotomous ((i : ι) → β i) (Pi.Lex r s)

instance Pi.instLTLexForall {ι : Type u_1} {β : ι → Type u_2} [LT ι] [(a : ι) → LT (β a)] : LT (Lex ((i : ι) → β i))

Equations

• Pi.instLTLexForall = { lt := Pi.Lex (fun (x1 x2 : ι) => x1 < x2) fun (x : ι) (x1 x2 : β x) => x1 < x2 }

instance Pi.Lex.isStrictOrder {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [(a : ι) → PartialOrder (β a)] : IsStrictOrder (Lex ((i : ι) → β i)) fun (x1 x2 : Lex ((i : ι) → β i)) => x1 < x2

Equations

• ⋯ = ⋯

instance Pi.instPartialOrderLexForallOfLinearOrder {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [(a : ι) → PartialOrder (β a)] : PartialOrder (Lex ((i : ι) → β i))

Equations

• Pi.instPartialOrderLexForallOfLinearOrder = partialOrderOfSO fun (x1 x2 : Lex ((i : ι) → β i)) => x1 < x2

noncomputable instance Pi.instLinearOrderLexForallOfIsWellOrderLt {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(a : ι) → LinearOrder (β a)] : LinearOrder (Lex ((i : ι) → β i))

Πₗ i, α i is a linear order if the original order is well-founded.

Equations

• Pi.instLinearOrderLexForallOfIsWellOrderLt = linearOrderOfSTO fun (x1 x2 : Lex ((i : ι) → (fun (i : ι) => β i) i)) => x1 < x2

theorem Pi.toLex_monotone {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → PartialOrder (β i)] : Monotone ⇑toLex

theorem Pi.toLex_strictMono {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → PartialOrder (β i)] : StrictMono ⇑toLex

@[simp]

theorem Pi.lt_toLex_update_self_iff {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i} : toLex x < toLex (Function.update x i a) ↔ x i < a

@[simp]

theorem Pi.toLex_update_lt_self_iff {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i} : toLex (Function.update x i a) < toLex x ↔ a < x i

@[simp]

theorem Pi.le_toLex_update_self_iff {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i} : toLex x ≤ toLex (Function.update x i a) ↔ x i ≤ a

@[simp]

theorem Pi.toLex_update_le_self_iff {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i} : toLex (Function.update x i a) ≤ toLex x ↔ a ≤ x i

instance Pi.instOrderBotLexForallOfIsWellOrderLt {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(a : ι) → PartialOrder (β a)] [(a : ι) → OrderBot (β a)] : OrderBot (Lex ((a : ι) → β a))

Equations

• Pi.instOrderBotLexForallOfIsWellOrderLt = OrderBot.mk ⋯

instance Pi.instOrderTopLexForallOfIsWellOrderLt {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(a : ι) → PartialOrder (β a)] [(a : ι) → OrderTop (β a)] : OrderTop (Lex ((a : ι) → β a))

Equations

• Pi.instOrderTopLexForallOfIsWellOrderLt = OrderTop.mk ⋯

instance Pi.instBoundedOrderLexForallOfIsWellOrderLt {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [(a : ι) → PartialOrder (β a)] [(a : ι) → BoundedOrder (β a)] : BoundedOrder (Lex ((a : ι) → β a))

Equations

• Pi.instBoundedOrderLexForallOfIsWellOrderLt = BoundedOrder.mk

instance Pi.instDenselyOrderedLexForall {ι : Type u_1} {β : ι → Type u_2} [Preorder ι] [(i : ι) → LT (β i)] [∀ (i : ι), DenselyOrdered (β i)] : DenselyOrdered (Lex ((i : ι) → β i))

Equations

• ⋯ = ⋯

theorem Pi.Lex.noMaxOrder' {ι : Type u_1} {β : ι → Type u_2} [Preorder ι] [(i : ι) → LT (β i)] (i : ι) [NoMaxOrder (β i)] : NoMaxOrder (Lex ((i : ι) → β i))

instance Pi.instNoMaxOrderLexForallOfIsWellOrderLtOfNonempty {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [Nonempty ι] [(i : ι) → PartialOrder (β i)] [∀ (i : ι), NoMaxOrder (β i)] : NoMaxOrder (Lex ((i : ι) → β i))

Equations

• ⋯ = ⋯

instance Pi.instNoMinOrderLexForallOfIsWellOrderLtOfNonempty {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun (x1 x2 : ι) => x1 < x2] [Nonempty ι] [(i : ι) → PartialOrder (β i)] [∀ (i : ι), NoMinOrder (β i)] : NoMinOrder (Lex ((i : ι) → β i))

Equations

• ⋯ = ⋯

theorem Pi.lex_desc {ι : Type u_1} {α : Type u_3} [Preorder ι] [DecidableEq ι] [Preorder α] {f : ι → α} {i : ι} {j : ι} (h₁ : i ≤ j) (h₂ : f j < f i) : toLex (f ∘ ⇑(Equiv.swap i j)) < toLex f

If we swap two strictly decreasing values in a function, then the result is lexicographically smaller than the original function.