# 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.

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 : ιType u_2} (r : ιιProp) (s : {i : ι} → β iβ iProp) (x : (i : ι) → β i) (y : (i : ι) → β i) :

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 ix j = y j) s (x i) (y i)
Instances For

The notation Πₗ i, α i refers to a pi type equipped with the lexicographic order.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem Pi.toLex_apply {ι : Type u_1} {β : ιType u_2} (x : (i : ι) → β i) (i : ι) :
toLex x i = x i
@[simp]
theorem Pi.ofLex_apply {ι : Type u_1} {β : ιType 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 : ιType u_2} [(i : ι) → Preorder (β i)] {r : ιιProp} (hwf : ) {x : (i : ι) → β i} {y : (i : ι) → β i} (hlt : x < y) :
∃ (i : ι), (∀ (j : ι), r j ix j y j y j x j) x i < y i
theorem Pi.lex_lt_of_lt {ι : Type u_1} {β : ιType u_2} [(i : ι) → PartialOrder (β i)] {r : ιιProp} (hwf : ) {x : (i : ι) → β i} {y : (i : ι) → β i} (hlt : x < y) :
Pi.Lex r (fun (i : ι) (x x_1 : β i) => x < x_1) x y
theorem Pi.isTrichotomous_lex {ι : Type u_1} {β : ιType u_2} (r : ιιProp) (s : {i : ι} → β iβ iProp) [∀ (i : ι), IsTrichotomous (β i) s] (wf : ) :
IsTrichotomous ((i : ι) → β i) (Pi.Lex r s)
instance Pi.instLTLexForall {ι : Type u_1} {β : ιType u_2} [LT ι] [(a : ι) → LT (β a)] :
LT (Lex ((i : ι) → β i))
Equations
• Pi.instLTLexForall = { lt := Pi.Lex (fun (x x_1 : ι) => x < x_1) fun (x : ι) (x_1 x_2 : β x) => x_1 < x_2 }
instance Pi.Lex.isStrictOrder {ι : Type u_1} {β : ιType u_2} [] [(a : ι) → PartialOrder (β a)] :
IsStrictOrder (Lex ((i : ι) → β i)) fun (x x_1 : Lex ((i : ι) → β i)) => x < x_1
Equations
• =
instance Pi.instPartialOrderLexForallOfLinearOrder {ι : Type u_1} {β : ιType u_2} [] [(a : ι) → PartialOrder (β a)] :
PartialOrder (Lex ((i : ι) → β i))
Equations
• Pi.instPartialOrderLexForallOfLinearOrder = partialOrderOfSO fun (x x_1 : Lex ((i : ι) → β i)) => x < x_1
noncomputable instance Pi.instLinearOrderLexForallOfIsWellOrderLt {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(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 (x x_1 : Lex ((i : ι) → (fun (i : ι) => β i) i)) => x < x_1
theorem Pi.toLex_monotone {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(i : ι) → PartialOrder (β i)] :
Monotone toLex
theorem Pi.toLex_strictMono {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(i : ι) → PartialOrder (β i)] :
StrictMono toLex
@[simp]
theorem Pi.lt_toLex_update_self_iff {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(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 : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(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 : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(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 : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(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 : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(a : ι) → PartialOrder (β a)] [(a : ι) → OrderBot (β a)] :
OrderBot (Lex ((a : ι) → β a))
Equations
• Pi.instOrderBotLexForallOfIsWellOrderLt =
instance Pi.instOrderTopLexForallOfIsWellOrderLt {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(a : ι) → PartialOrder (β a)] [(a : ι) → OrderTop (β a)] :
OrderTop (Lex ((a : ι) → β a))
Equations
• Pi.instOrderTopLexForallOfIsWellOrderLt =
instance Pi.instBoundedOrderLexForallOfIsWellOrderLt {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [(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 : ιType u_2} [] [(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 : ιType u_2} [] [(i : ι) → LT (β i)] (i : ι) [NoMaxOrder (β i)] :
NoMaxOrder (Lex ((i : ι) → β i))
instance Pi.instNoMaxOrderLexForallOfIsWellOrderLtOfNonempty {ι : Type u_1} {β : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [] [(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 : ιType u_2} [] [IsWellOrder ι fun (x x_1 : ι) => x < x_1] [] [(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} [] [] [] {f : ια} {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.