Orders on a sigma type

This file defines two orders on a sigma type:

• The disjoint sum of orders. a is less b iff a and b are in the same summand and a is less than b there.
• The lexicographical order. 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.

We make the disjoint sum of orders the default set of instances. The lexicographic order goes on a type synonym.

Notation

• _root_.Lex (Sigma α): Sigma type equipped with the lexicographic order. Type synonym of Σ i, α i.

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

TODO

Upgrade Equiv.sigma_congr_left, Equiv.sigma_congr, Equiv.sigma_assoc, Equiv.sigma_prod_of_equiv, Equiv.sigma_equiv_prod, ... to order isomorphisms.

Disjoint sum of orders on Sigma

ink

inductive Sigma.le {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → LE (α i)] (_a : (i : ι) × α i) (_b : (i : ι) × α i) : Prop

• fiber: ∀ {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → LE (α i)] (i : ι) (a b : α i), a ≤ b → Sigma.le { fst := i, snd := a } { fst := i, snd := b }

Disjoint sum of orders. ⟨i, a⟩ ≤ ⟨j, b⟩≤ ⟨j, b⟩ iff i = j and a ≤ b≤ b.

ink

inductive Sigma.lt {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → LT (α i)] (_a : (i : ι) × α i) (_b : (i : ι) × α i) : Prop

• fiber: ∀ {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → LT (α i)] (i : ι) (a b : α i), a < b → Sigma.lt { fst := i, snd := a } { fst := i, snd := b }

Disjoint sum of orders. ⟨i, a⟩ < ⟨j, b⟩ iff i = j and a < b.

ink

instance Sigma.LE {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → LE (α i)] : LE ((i : ι) × α i)

Equations

• Sigma.LE = { le := Sigma.le }

ink

instance Sigma.LT {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → LT (α i)] : LT ((i : ι) × α i)

Equations

• Sigma.LT = { lt := Sigma.lt }

ink

@[simp]

theorem Sigma.mk_le_mk_iff {ι : Type u_2} {α : ι → Type u_1} [inst : (i : ι) → LE (α i)] {i : ι} {a : α i} {b : α i} : { fst := i, snd := a } ≤ { fst := i, snd := b } ↔ a ≤ b

ink

@[simp]

theorem Sigma.mk_lt_mk_iff {ι : Type u_2} {α : ι → Type u_1} [inst : (i : ι) → LT (α i)] {i : ι} {a : α i} {b : α i} : { fst := i, snd := a } < { fst := i, snd := b } ↔ a < b

ink

theorem Sigma.le_def {ι : Type u_2} {α : ι → Type u_1} [inst : (i : ι) → LE (α i)] {a : (i : ι) × α i} {b : (i : ι) × α i} : a ≤ b ↔ ∃ h, h ▸ a.snd ≤ b.snd

ink

theorem Sigma.lt_def {ι : Type u_2} {α : ι → Type u_1} [inst : (i : ι) → LT (α i)] {a : (i : ι) × α i} {b : (i : ι) × α i} : a < b ↔ ∃ h, h ▸ a.snd < b.snd

ink

instance Sigma.preorder {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → Preorder (α i)] : Preorder ((i : ι) × α i)

Equations

• Sigma.preorder = let src := Sigma.LE; let src_1 := Sigma.LT; Preorder.mk (_ : ∀ (x : (i : ι) × α i), x ≤ x) (_ : ∀ (a b c : (i : ι) × α i), a ≤ b → b ≤ c → a ≤ c)

ink

instance Sigma.instPartialOrderSigma {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → PartialOrder (α i)] : PartialOrder ((i : ι) × α i)

Equations

• Sigma.instPartialOrderSigma = let src := Sigma.preorder; PartialOrder.mk (_ : ∀ (a b : (i : ι) × α i), a ≤ b → b ≤ a → a = b)

ink

instance Sigma.instDenselyOrderedSigmaLTToLT {ι : Type u_1} {α : ι → Type u_2} [inst : (i : ι) → Preorder (α i)] [inst : ∀ (i : ι), DenselyOrdered (α i)] : DenselyOrdered ((i : ι) × α i)

Equations

• @Sigma.instDenselyOrderedSigmaLTToLT = @Sigma.instDenselyOrderedSigmaLTToLT.proof_1

Lexicographical order on Sigma

ink

def Sigma.Lex.«Σₗ_,_» : Lean.ParserDescr

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

Equations

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

ink

instance Sigma.Lex.LE {ι : Type u_1} {α : ι → Type u_2} [inst : LT ι] [inst : (i : ι) → LE (α i)] : LE (Lex ((i : ι) × α i))

The lexicographical ≤≤ on a sigma type.

Equations

• Sigma.Lex.LE = { le := Sigma.Lex (fun x x_1 => x < x_1) fun x x_1 x_2 => x_1 ≤ x_2 }

ink

instance Sigma.Lex.LT {ι : Type u_1} {α : ι → Type u_2} [inst : LT ι] [inst : (i : ι) → LT (α i)] : LT (Lex ((i : ι) × α i))

The lexicographical < on a sigma type.

Equations

• Sigma.Lex.LT = { lt := Sigma.Lex (fun x x_1 => x < x_1) fun x x_1 x_2 => x_1 < x_2 }

ink

theorem Sigma.Lex.le_def {ι : Type u_1} {α : ι → Type u_2} [inst : LT ι] [inst : (i : ι) → LE (α i)] {a : Lex ((i : ι) × α i)} {b : Lex ((i : ι) × α i)} : a ≤ b ↔ a.fst < b.fst ∨ ∃ h, h ▸ a.snd ≤ b.snd

ink

theorem Sigma.Lex.lt_def {ι : Type u_1} {α : ι → Type u_2} [inst : LT ι] [inst : (i : ι) → LT (α i)] {a : Lex ((i : ι) × α i)} {b : Lex ((i : ι) × α i)} : a < b ↔ a.fst < b.fst ∨ ∃ h, h ▸ a.snd < b.snd

ink

instance Sigma.Lex.preorder {ι : Type u_1} {α : ι → Type u_2} [inst : Preorder ι] [inst : (i : ι) → Preorder (α i)] : Preorder (Lex ((i : ι) × α i))

The lexicographical preorder on a sigma type.

Equations

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

ink

instance Sigma.Lex.partialOrder {ι : Type u_1} {α : ι → Type u_2} [inst : Preorder ι] [inst : (i : ι) → PartialOrder (α i)] : PartialOrder (Lex ((i : ι) × α i))

The lexicographical partial order on a sigma type.

Equations

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

ink

instance Sigma.Lex.linearOrder {ι : Type u_1} {α : ι → Type u_2} [inst : LinearOrder ι] [inst : (i : ι) → LinearOrder (α i)] : LinearOrder (Lex ((i : ι) × α i))

The lexicographical linear order on a sigma type.

Equations

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

ink

instance Sigma.Lex.orderBot {ι : Type u_1} {α : ι → Type u_2} [inst : PartialOrder ι] [inst : OrderBot ι] [inst : (i : ι) → Preorder (α i)] [inst : OrderBot (α ⊥)] : OrderBot (Lex ((i : ι) × α i))

The lexicographical linear order on a sigma type.

Equations

• Sigma.Lex.orderBot = OrderBot.mk (_ : ∀ (x : Lex ((i : ι) × α i)), ⊥ ≤ x)

ink

instance Sigma.Lex.orderTop {ι : Type u_1} {α : ι → Type u_2} [inst : PartialOrder ι] [inst : OrderTop ι] [inst : (i : ι) → Preorder (α i)] [inst : OrderTop (α ⊤)] : OrderTop (Lex ((i : ι) × α i))

The lexicographical linear order on a sigma type.

Equations

• Sigma.Lex.orderTop = OrderTop.mk (_ : ∀ (x : Lex ((i : ι) × α i)), x ≤ ⊤)

ink

instance Sigma.Lex.boundedOrder {ι : Type u_1} {α : ι → Type u_2} [inst : PartialOrder ι] [inst : BoundedOrder ι] [inst : (i : ι) → Preorder (α i)] [inst : OrderBot (α ⊥)] [inst : OrderTop (α ⊤)] : BoundedOrder (Lex ((i : ι) × α i))

The lexicographical linear order on a sigma type.

Equations

• Sigma.Lex.boundedOrder = let src := Sigma.Lex.orderBot; let src_1 := Sigma.Lex.orderTop; BoundedOrder.mk

ink

instance Sigma.Lex.denselyOrdered {ι : Type u_1} {α : ι → Type u_2} [inst : Preorder ι] [inst : DenselyOrdered ι] [inst : ∀ (i : ι), Nonempty (α i)] [inst : (i : ι) → Preorder (α i)] [inst : ∀ (i : ι), DenselyOrdered (α i)] : DenselyOrdered (Lex ((i : ι) × α i))

Equations

• @Sigma.Lex.denselyOrdered = @Sigma.Lex.denselyOrdered.proof_1

ink

instance Sigma.Lex.denselyOrdered_of_noMaxOrder {ι : Type u_1} {α : ι → Type u_2} [inst : Preorder ι] [inst : (i : ι) → Preorder (α i)] [inst : ∀ (i : ι), DenselyOrdered (α i)] [inst : ∀ (i : ι), NoMaxOrder (α i)] : DenselyOrdered (Lex ((i : ι) × α i))

Equations

• @Sigma.Lex.denselyOrdered_of_noMaxOrder = @Sigma.Lex.denselyOrdered_of_noMaxOrder.proof_1

ink

instance Sigma.Lex.denselyOrdered_of_noMinOrder {ι : Type u_1} {α : ι → Type u_2} [inst : Preorder ι] [inst : (i : ι) → Preorder (α i)] [inst : ∀ (i : ι), DenselyOrdered (α i)] [inst : ∀ (i : ι), NoMinOrder (α i)] : DenselyOrdered (Lex ((i : ι) × α i))

Equations

• @Sigma.Lex.denselyOrdered_of_noMinOrder = @Sigma.Lex.denselyOrdered_of_noMinOrder.proof_1

ink

instance Sigma.Lex.noMaxOrder_of_nonempty {ι : Type u_1} {α : ι → Type u_2} [inst : Preorder ι] [inst : (i : ι) → Preorder (α i)] [inst : NoMaxOrder ι] [inst : ∀ (i : ι), Nonempty (α i)] : NoMaxOrder (Lex ((i : ι) × α i))

Equations

• @Sigma.Lex.noMaxOrder_of_nonempty = @Sigma.Lex.noMaxOrder_of_nonempty.proof_1

ink

instance Sigma.Lex.noMinOrder_of_nonempty {ι : Type u_1} {α : ι → Type u_2} [inst : Preorder ι] [inst : (i : ι) → Preorder (α i)] [inst : NoMinOrder ι] [inst : ∀ (i : ι), Nonempty (α i)] : NoMinOrder (Lex ((i : ι) × α i))

Equations

• @Sigma.Lex.noMinOrder_of_nonempty = @Sigma.Lex.noMinOrder_of_nonempty.proof_1

ink

instance Sigma.Lex.noMaxOrder {ι : Type u_1} {α : ι → Type u_2} [inst : Preorder ι] [inst : (i : ι) → Preorder (α i)] [inst : ∀ (i : ι), NoMaxOrder (α i)] : NoMaxOrder (Lex ((i : ι) × α i))

Equations

• @Sigma.Lex.noMaxOrder = @Sigma.Lex.noMaxOrder.proof_1

ink

instance Sigma.Lex.noMinOrder {ι : Type u_1} {α : ι → Type u_2} [inst : Preorder ι] [inst : (i : ι) → Preorder (α i)] [inst : ∀ (i : ι), NoMinOrder (α i)] : NoMinOrder (Lex ((i : ι) × α i))

Equations

• @Sigma.Lex.noMinOrder = @Sigma.Lex.noMinOrder.proof_1