# Documentation

Mathlib.Data.PSigma.Order

# Lexicographic order on a sigma type #

This file defines the lexicographic order on Σₗ' i, α i. a is less than b if its summand is strictly less than the summand of b or they are in the same summand and a is less than b there.

## Notation #

• Σₗ' i, α i: Sigma type equipped with the lexicographic order. A type synonym of Σ' i, α i.

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.
• Data.Sigma.Order: Lexicographic order on Σₗ i, α i. Basically a twin of this file.
• Data.Prod.Lex: Lexicographic order on α × β.

## TODO #

Define the disjoint order on Σ' i, α i, where x ≤ y only if x.fst = y.fst. Prove that a sigma type is a NoMaxOrder, NoMinOrder, DenselyOrdered when its summands are.

Instances For

The notation Σₗ' i, α i refers to a sigma type which is locally equipped with the lexicographic order.

Instances For
instance PSigma.Lex.le {ι : Type u_1} {α : ιType u_2} [LT ι] [(i : ι) → LE (α i)] :
LE (Σₗ' (i : ι), α i)

The lexicographical ≤ on a sigma type.

instance PSigma.Lex.lt {ι : Type u_1} {α : ιType u_2} [LT ι] [(i : ι) → LT (α i)] :
LT (Σₗ' (i : ι), α i)

The lexicographical < on a sigma type.

instance PSigma.Lex.preorder {ι : Type u_1} {α : ιType u_2} [] [(i : ι) → Preorder (α i)] :
Preorder (Σₗ' (i : ι), α i)
instance PSigma.Lex.partialOrder {ι : Type u_1} {α : ιType u_2} [] [(i : ι) → PartialOrder (α i)] :
PartialOrder (Σₗ' (i : ι), α i)

Dictionary / lexicographic partial_order for dependent pairs.

instance PSigma.Lex.linearOrder {ι : Type u_1} {α : ιType u_2} [] [(i : ι) → LinearOrder (α i)] :
LinearOrder (Σₗ' (i : ι), α i)

Dictionary / lexicographic linear_order for pairs.

instance PSigma.Lex.orderBot {ι : Type u_1} {α : ιType u_2} [] [] [(i : ι) → Preorder (α i)] [OrderBot (α )] :
OrderBot (Σₗ' (i : ι), α i)

The lexicographical linear order on a sigma type.

instance PSigma.Lex.orderTop {ι : Type u_1} {α : ιType u_2} [] [] [(i : ι) → Preorder (α i)] [OrderTop (α )] :
OrderTop (Σₗ' (i : ι), α i)

The lexicographical linear order on a sigma type.

instance PSigma.Lex.boundedOrder {ι : Type u_1} {α : ιType u_2} [] [] [(i : ι) → Preorder (α i)] [OrderBot (α )] [OrderTop (α )] :
BoundedOrder (Σₗ' (i : ι), α i)

The lexicographical linear order on a sigma type.

instance PSigma.Lex.denselyOrdered {ι : Type u_1} {α : ιType u_2} [] [] [∀ (i : ι), Nonempty (α i)] [(i : ι) → Preorder (α i)] [∀ (i : ι), DenselyOrdered (α i)] :
DenselyOrdered (Σₗ' (i : ι), α i)
instance PSigma.Lex.denselyOrdered_of_noMaxOrder {ι : Type u_1} {α : ιType u_2} [] [(i : ι) → Preorder (α i)] [∀ (i : ι), DenselyOrdered (α i)] [∀ (i : ι), NoMaxOrder (α i)] :
DenselyOrdered (Σₗ' (i : ι), α i)
instance PSigma.Lex.densely_ordered_of_noMinOrder {ι : Type u_1} {α : ιType u_2} [] [(i : ι) → Preorder (α i)] [∀ (i : ι), DenselyOrdered (α i)] [∀ (i : ι), NoMinOrder (α i)] :
DenselyOrdered (Σₗ' (i : ι), α i)
instance PSigma.Lex.noMaxOrder_of_nonempty {ι : Type u_1} {α : ιType u_2} [] [(i : ι) → Preorder (α i)] [] [∀ (i : ι), Nonempty (α i)] :
NoMaxOrder (Σₗ' (i : ι), α i)
instance PSigma.Lex.noMinOrder_of_nonempty {ι : Type u_1} {α : ιType u_2} [] [(i : ι) → Preorder (α i)] [] [∀ (i : ι), Nonempty (α i)] :
NoMinOrder (Σₗ' (i : ι), α i)
instance PSigma.Lex.noMaxOrder {ι : Type u_1} {α : ιType u_2} [] [(i : ι) → Preorder (α i)] [∀ (i : ι), NoMaxOrder (α i)] :
NoMaxOrder (Σₗ' (i : ι), α i)
instance PSigma.Lex.noMinOrder {ι : Type u_1} {α : ιType u_2} [] [(i : ι) → Preorder (α i)] [∀ (i : ι), NoMinOrder (α i)] :
NoMinOrder (Σₗ' (i : ι), α i)