Orders on a sum type #

This file defines the disjoint sum and the linear (aka lexicographic) sum of two orders and provides relation instances for Sum.LiftRel and Sum.Lex.

We declare the disjoint sum of orders as the default set of instances. The linear order goes on a type synonym.

Main declarations #

Sum.LE, Sum.LT: Disjoint sum of orders.
Sum.Lex.LE, Sum.Lex.LT: Lexicographic/linear sum of orders.

Notation #

α ⊕ₗ β: The linear sum of α and β.

Unbundled relation classes #

source

theorem Sum.LiftRel.refl {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsRefl α r] [IsRefl β s] (x : α ⊕ β) :

Sum.LiftRel r s x x

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instance Sum.instIsReflSumLiftRel {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsRefl α r] [IsRefl β s] :

IsRefl (α ⊕ β) (Sum.LiftRel r s)

Equations

@Sum.instIsReflSumLiftRel = @Sum.instIsReflSumLiftRel.proof_1

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instance Sum.instIsIrreflSumLiftRel {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsIrrefl α r] [IsIrrefl β s] :

IsIrrefl (α ⊕ β) (Sum.LiftRel r s)

Equations

@Sum.instIsIrreflSumLiftRel = @Sum.instIsIrreflSumLiftRel.proof_1

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theorem Sum.LiftRel.trans {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsTrans α r] [IsTrans β s] {a : α ⊕ β} {b : α ⊕ β} {c : α ⊕ β} :

Sum.LiftRel r s a b → Sum.LiftRel r s b c → Sum.LiftRel r s a c

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instance Sum.instIsTransSumLiftRel {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsTrans α r] [IsTrans β s] :

IsTrans (α ⊕ β) (Sum.LiftRel r s)

Equations

@Sum.instIsTransSumLiftRel = @Sum.instIsTransSumLiftRel.proof_1

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instance Sum.instIsAntisymmSumLiftRel {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsAntisymm α r] [IsAntisymm β s] :

IsAntisymm (α ⊕ β) (Sum.LiftRel r s)

Equations

@Sum.instIsAntisymmSumLiftRel = @Sum.instIsAntisymmSumLiftRel.proof_1

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instance Sum.instIsReflSumLex {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsRefl α r] [IsRefl β s] :

IsRefl (α ⊕ β) (Sum.Lex r s)

Equations

@Sum.instIsReflSumLex = @Sum.instIsReflSumLex.proof_1

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instance Sum.instIsIrreflSumLex {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsIrrefl α r] [IsIrrefl β s] :

IsIrrefl (α ⊕ β) (Sum.Lex r s)

Equations

@Sum.instIsIrreflSumLex = @Sum.instIsIrreflSumLex.proof_1

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instance Sum.instIsTransSumLex {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsTrans α r] [IsTrans β s] :

IsTrans (α ⊕ β) (Sum.Lex r s)

Equations

@Sum.instIsTransSumLex = @Sum.instIsTransSumLex.proof_1

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instance Sum.instIsAntisymmSumLex {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsAntisymm α r] [IsAntisymm β s] :

IsAntisymm (α ⊕ β) (Sum.Lex r s)

Equations

@Sum.instIsAntisymmSumLex = @Sum.instIsAntisymmSumLex.proof_1

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instance Sum.instIsTotalSumLex {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsTotal α r] [IsTotal β s] :

IsTotal (α ⊕ β) (Sum.Lex r s)

Equations

@Sum.instIsTotalSumLex = @Sum.instIsTotalSumLex.proof_1

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instance Sum.instIsTrichotomousSumLex {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsTrichotomous α r] [IsTrichotomous β s] :

IsTrichotomous (α ⊕ β) (Sum.Lex r s)

Equations

@Sum.instIsTrichotomousSumLex = @Sum.instIsTrichotomousSumLex.proof_1

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instance Sum.instIsWellOrderSumLex {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [IsWellOrder α r] [IsWellOrder β s] :

IsWellOrder (α ⊕ β) (Sum.Lex r s)

Equations

@Sum.instIsWellOrderSumLex = @Sum.instIsWellOrderSumLex.proof_1

Disjoint sum of two orders #

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instance Sum.instLESum {α : Type u_1} {β : Type u_2} [LE α] [LE β] :

LE (α ⊕ β)

Equations

Sum.instLESum = { le := Sum.LiftRel (fun x x_1 => x ≤ x_1) fun x x_1 => x ≤ x_1 }

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instance Sum.instLTSum {α : Type u_1} {β : Type u_2} [LT α] [LT β] :

LT (α ⊕ β)

Equations

Sum.instLTSum = { lt := Sum.LiftRel (fun x x_1 => x < x_1) fun x x_1 => x < x_1 }

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theorem Sum.le_def {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : α ⊕ β} {b : α ⊕ β} :

a ≤ b ↔ Sum.LiftRel (fun x x_1 => x ≤ x_1) (fun x x_1 => x ≤ x_1) a b

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theorem Sum.lt_def {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : α ⊕ β} {b : α ⊕ β} :

a < b ↔ Sum.LiftRel (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) a b

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@[simp]

theorem Sum.inl_le_inl_iff {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : α} {b : α} :

Sum.inl a ≤ Sum.inl b ↔ a ≤ b

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@[simp]

theorem Sum.inr_le_inr_iff {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : β} {b : β} :

Sum.inr a ≤ Sum.inr b ↔ a ≤ b

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@[simp]

theorem Sum.inl_lt_inl_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : α} {b : α} :

Sum.inl a < Sum.inl b ↔ a < b

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@[simp]

theorem Sum.inr_lt_inr_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : β} {b : β} :

Sum.inr a < Sum.inr b ↔ a < b

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@[simp]

theorem Sum.not_inl_le_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : α} {b : β} :

¬Sum.inl b ≤ Sum.inr a

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@[simp]

theorem Sum.not_inl_lt_inr {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : α} {b : β} :

¬Sum.inl b < Sum.inr a

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@[simp]

theorem Sum.not_inr_le_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : α} {b : β} :

¬Sum.inr b ≤ Sum.inl a

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@[simp]

theorem Sum.not_inr_lt_inl {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : α} {b : β} :

¬Sum.inr b < Sum.inl a

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instance Sum.instPreorderSum {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

Preorder (α ⊕ β)

Equations

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

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theorem Sum.inl_mono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

Monotone Sum.inl

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theorem Sum.inr_mono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

Monotone Sum.inr

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theorem Sum.inl_strictMono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

StrictMono Sum.inl

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theorem Sum.inr_strictMono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

StrictMono Sum.inr

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instance Sum.instPartialOrderSum {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] :

PartialOrder (α ⊕ β)

Equations

Sum.instPartialOrderSum = let src := Sum.instPreorderSum; PartialOrder.mk (_ : ∀ (x x_1 : α ⊕ β), Sum.LiftRel (fun x x_2 => x ≤ x_2) (fun x x_2 => x ≤ x_2) x x_1 → x_1 ≤ x → x = x_1)

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instance Sum.noMinOrder {α : Type u_1} {β : Type u_2} [LT α] [LT β] [NoMinOrder α] [NoMinOrder β] :

NoMinOrder (α ⊕ β)

Equations

@Sum.noMinOrder = @Sum.noMinOrder.proof_1

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instance Sum.noMaxOrder {α : Type u_1} {β : Type u_2} [LT α] [LT β] [NoMaxOrder α] [NoMaxOrder β] :

NoMaxOrder (α ⊕ β)

Equations

@Sum.noMaxOrder = @Sum.noMaxOrder.proof_1

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@[simp]

theorem Sum.noMinOrder_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] :

NoMinOrder (α ⊕ β) ↔ NoMinOrder α ∧ NoMinOrder β

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@[simp]

theorem Sum.noMaxOrder_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] :

NoMaxOrder (α ⊕ β) ↔ NoMaxOrder α ∧ NoMaxOrder β

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instance Sum.denselyOrdered {α : Type u_1} {β : Type u_2} [LT α] [LT β] [DenselyOrdered α] [DenselyOrdered β] :

DenselyOrdered (α ⊕ β)

Equations

@Sum.denselyOrdered = @Sum.denselyOrdered.proof_1

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@[simp]

theorem Sum.denselyOrdered_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] :

DenselyOrdered (α ⊕ β) ↔ DenselyOrdered α ∧ DenselyOrdered β

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@[simp]

theorem Sum.swap_le_swap_iff {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : α ⊕ β} {b : α ⊕ β} :

Sum.swap a ≤ Sum.swap b ↔ a ≤ b

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@[simp]

theorem Sum.swap_lt_swap_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : α ⊕ β} {b : α ⊕ β} :

Sum.swap a < Sum.swap b ↔ a < b

Linear sum of two orders #

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def Sum.Lex.«term_⊕ₗ_» :

Lean.TrailingParserDescr

The linear sum of two orders

Equations

Sum.Lex.«term_⊕ₗ_» = Lean.ParserDescr.trailingNode `Sum.Lex.term_⊕ₗ_ 30 0 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⊕ₗ ") (Lean.ParserDescr.cat `term 29))

Instances For

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@[match_pattern, inline, reducible]

abbrev Sum.inlₗ {α : Type u_1} {β : Type u_2} (x : α) :

Lex (α ⊕ β)

Lexicographical Sum.inl. Only used for pattern matching.

Equations

Sum.inlₗ x = ↑toLex (Sum.inl x)

Instances For

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@[match_pattern, inline, reducible]

abbrev Sum.inrₗ {α : Type u_1} {β : Type u_2} (x : β) :

Lex (α ⊕ β)

Lexicographical Sum.inr. Only used for pattern matching.

Equations

Sum.inrₗ x = ↑toLex (Sum.inr x)

Instances For

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instance Sum.Lex.LE {α : Type u_1} {β : Type u_2} [LE α] [LE β] :

LE (Lex (α ⊕ β))

The linear/lexicographical ≤ on a sum.

Equations

Sum.Lex.LE = { le := Sum.Lex (fun x x_1 => x ≤ x_1) fun x x_1 => x ≤ x_1 }

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instance Sum.Lex.LT {α : Type u_1} {β : Type u_2} [LT α] [LT β] :

LT (Lex (α ⊕ β))

The linear/lexicographical < on a sum.

Equations

Sum.Lex.LT = { lt := Sum.Lex (fun x x_1 => x < x_1) fun x x_1 => x < x_1 }

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@[simp]

theorem Sum.Lex.toLex_le_toLex {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : α ⊕ β} {b : α ⊕ β} :

↑toLex a ≤ ↑toLex b ↔ Sum.Lex (fun x x_1 => x ≤ x_1) (fun x x_1 => x ≤ x_1) a b

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@[simp]

theorem Sum.Lex.toLex_lt_toLex {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : α ⊕ β} {b : α ⊕ β} :

↑toLex a < ↑toLex b ↔ Sum.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) a b

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theorem Sum.Lex.le_def {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : Lex (α ⊕ β)} {b : Lex (α ⊕ β)} :

a ≤ b ↔ Sum.Lex (fun x x_1 => x ≤ x_1) (fun x x_1 => x ≤ x_1) (↑ofLex a) (↑ofLex b)

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theorem Sum.Lex.lt_def {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : Lex (α ⊕ β)} {b : Lex (α ⊕ β)} :

a < b ↔ Sum.Lex (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (↑ofLex a) (↑ofLex b)

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theorem Sum.Lex.inl_le_inl_iff {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : α} {b : α} :

↑toLex (Sum.inl a) ≤ ↑toLex (Sum.inl b) ↔ a ≤ b

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theorem Sum.Lex.inr_le_inr_iff {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : β} {b : β} :

↑toLex (Sum.inr a) ≤ ↑toLex (Sum.inr b) ↔ a ≤ b

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theorem Sum.Lex.inl_lt_inl_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : α} {b : α} :

↑toLex (Sum.inl a) < ↑toLex (Sum.inl b) ↔ a < b

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theorem Sum.Lex.inr_lt_inr_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : β} {b : β} :

↑toLex (Sum.inr a) < ↑toLex (Sum.inr b) ↔ a < b

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theorem Sum.Lex.inl_le_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] (a : α) (b : β) :

↑toLex (Sum.inl a) ≤ ↑toLex (Sum.inr b)

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theorem Sum.Lex.inl_lt_inr {α : Type u_1} {β : Type u_2} [LT α] [LT β] (a : α) (b : β) :

↑toLex (Sum.inl a) < ↑toLex (Sum.inr b)

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theorem Sum.Lex.not_inr_le_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : α} {b : β} :

¬↑toLex (Sum.inr b) ≤ ↑toLex (Sum.inl a)

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theorem Sum.Lex.not_inr_lt_inl {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : α} {b : β} :

¬↑toLex (Sum.inr b) < ↑toLex (Sum.inl a)

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instance Sum.Lex.preorder {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

Preorder (Lex (α ⊕ β))

Equations

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

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theorem Sum.Lex.toLex_mono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

Monotone ↑toLex

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theorem Sum.Lex.toLex_strictMono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

StrictMono ↑toLex

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theorem Sum.Lex.inl_mono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

Monotone (↑toLex ∘ Sum.inl)

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theorem Sum.Lex.inr_mono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

Monotone (↑toLex ∘ Sum.inr)

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theorem Sum.Lex.inl_strictMono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

StrictMono (↑toLex ∘ Sum.inl)

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theorem Sum.Lex.inr_strictMono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] :

StrictMono (↑toLex ∘ Sum.inr)

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instance Sum.Lex.partialOrder {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] :

PartialOrder (Lex (α ⊕ β))

Equations

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

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instance Sum.Lex.linearOrder {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] :

LinearOrder (Lex (α ⊕ β))

Equations

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

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instance Sum.Lex.orderBot {α : Type u_1} {β : Type u_2} [LE α] [OrderBot α] [LE β] :

OrderBot (Lex (α ⊕ β))

The lexicographical bottom of a sum is the bottom of the left component.

Equations

Sum.Lex.orderBot = OrderBot.mk (_ : ∀ (a : Lex (α ⊕ β)), ⊥ ≤ a)

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@[simp]

theorem Sum.Lex.inl_bot {α : Type u_1} {β : Type u_2} [LE α] [OrderBot α] [LE β] :

↑toLex (Sum.inl ⊥) = ⊥

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instance Sum.Lex.orderTop {α : Type u_1} {β : Type u_2} [LE α] [LE β] [OrderTop β] :

OrderTop (Lex (α ⊕ β))

The lexicographical top of a sum is the top of the right component.

Equations

Sum.Lex.orderTop = OrderTop.mk (_ : ∀ (a : Lex (α ⊕ β)), a ≤ ⊤)

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@[simp]

theorem Sum.Lex.inr_top {α : Type u_1} {β : Type u_2} [LE α] [LE β] [OrderTop β] :

↑toLex (Sum.inr ⊤) = ⊤

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instance Sum.Lex.boundedOrder {α : Type u_1} {β : Type u_2} [LE α] [LE β] [OrderBot α] [OrderTop β] :

BoundedOrder (Lex (α ⊕ β))

Equations

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

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instance Sum.Lex.noMinOrder {α : Type u_1} {β : Type u_2} [LT α] [LT β] [NoMinOrder α] [NoMinOrder β] :

NoMinOrder (Lex (α ⊕ β))

Equations

@Sum.Lex.noMinOrder = @Sum.Lex.noMinOrder.proof_1

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instance Sum.Lex.noMaxOrder {α : Type u_1} {β : Type u_2} [LT α] [LT β] [NoMaxOrder α] [NoMaxOrder β] :

NoMaxOrder (Lex (α ⊕ β))

Equations

@Sum.Lex.noMaxOrder = @Sum.Lex.noMaxOrder.proof_1

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instance Sum.Lex.noMinOrder_of_nonempty {α : Type u_1} {β : Type u_2} [LT α] [LT β] [NoMinOrder α] [Nonempty α] :

NoMinOrder (Lex (α ⊕ β))

Equations

@Sum.Lex.noMinOrder_of_nonempty = @Sum.Lex.noMinOrder_of_nonempty.proof_1

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instance Sum.Lex.noMaxOrder_of_nonempty {α : Type u_1} {β : Type u_2} [LT α] [LT β] [NoMaxOrder β] [Nonempty β] :

NoMaxOrder (Lex (α ⊕ β))

Equations

@Sum.Lex.noMaxOrder_of_nonempty = @Sum.Lex.noMaxOrder_of_nonempty.proof_1

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instance Sum.Lex.denselyOrdered_of_noMaxOrder {α : Type u_1} {β : Type u_2} [LT α] [LT β] [DenselyOrdered α] [DenselyOrdered β] [NoMaxOrder α] :

DenselyOrdered (Lex (α ⊕ β))

Equations

@Sum.Lex.denselyOrdered_of_noMaxOrder = @Sum.Lex.denselyOrdered_of_noMaxOrder.proof_1

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instance Sum.Lex.denselyOrdered_of_noMinOrder {α : Type u_1} {β : Type u_2} [LT α] [LT β] [DenselyOrdered α] [DenselyOrdered β] [NoMinOrder β] :

DenselyOrdered (Lex (α ⊕ β))

Equations

@Sum.Lex.denselyOrdered_of_noMinOrder = @Sum.Lex.denselyOrdered_of_noMinOrder.proof_1

Order isomorphisms #

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@[simp]

theorem OrderIso.sumComm_apply (α : Type u_5) (β : Type u_6) [LE α] [LE β] :

∀ (a : α ⊕ β), ↑(OrderIso.sumComm α β) a = Sum.swap a

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def OrderIso.sumComm (α : Type u_5) (β : Type u_6) [LE α] [LE β] :

α ⊕ β ≃o β ⊕ α

Equiv.sumComm promoted to an order isomorphism.

Equations

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

Instances For

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@[simp]

theorem OrderIso.sumComm_symm (α : Type u_5) (β : Type u_6) [LE α] [LE β] :

OrderIso.symm (OrderIso.sumComm α β) = OrderIso.sumComm β α

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def OrderIso.sumAssoc (α : Type u_5) (β : Type u_6) (γ : Type u_7) [LE α] [LE β] [LE γ] :

(α ⊕ β) ⊕ γ ≃o α ⊕ β ⊕ γ

Equiv.sumAssoc promoted to an order isomorphism.

Equations

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

Instances For

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@[simp]

theorem OrderIso.sumAssoc_apply_inl_inl {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (a : α) :

↑(OrderIso.sumAssoc α β γ) (Sum.inl (Sum.inl a)) = Sum.inl a

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@[simp]

theorem OrderIso.sumAssoc_apply_inl_inr {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (b : β) :

↑(OrderIso.sumAssoc α β γ) (Sum.inl (Sum.inr b)) = Sum.inr (Sum.inl b)

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@[simp]

theorem OrderIso.sumAssoc_apply_inr {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (c : γ) :

↑(OrderIso.sumAssoc α β γ) (Sum.inr c) = Sum.inr (Sum.inr c)

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@[simp]

theorem OrderIso.sumAssoc_symm_apply_inl {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (a : α) :

↑(OrderIso.symm (OrderIso.sumAssoc α β γ)) (Sum.inl a) = Sum.inl (Sum.inl a)

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@[simp]

theorem OrderIso.sumAssoc_symm_apply_inr_inl {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (b : β) :

↑(OrderIso.symm (OrderIso.sumAssoc α β γ)) (Sum.inr (Sum.inl b)) = Sum.inl (Sum.inr b)

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@[simp]

theorem OrderIso.sumAssoc_symm_apply_inr_inr {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (c : γ) :

↑(OrderIso.symm (OrderIso.sumAssoc α β γ)) (Sum.inr (Sum.inr c)) = Sum.inr c

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def OrderIso.sumDualDistrib (α : Type u_5) (β : Type u_6) [LE α] [LE β] :

(α ⊕ β)ᵒᵈ ≃o αᵒᵈ ⊕ βᵒᵈ

orderDual is distributive over ⊕ up to an order isomorphism.

Equations

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

Instances For

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@[simp]

theorem OrderIso.sumDualDistrib_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] (a : α) :

↑(OrderIso.sumDualDistrib α β) (↑OrderDual.toDual (Sum.inl a)) = Sum.inl (↑OrderDual.toDual a)

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@[simp]

theorem OrderIso.sumDualDistrib_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] (b : β) :

↑(OrderIso.sumDualDistrib α β) (↑OrderDual.toDual (Sum.inr b)) = Sum.inr (↑OrderDual.toDual b)

source

@[simp]

theorem OrderIso.sumDualDistrib_symm_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] (a : α) :

↑(OrderIso.symm (OrderIso.sumDualDistrib α β)) (Sum.inl (↑OrderDual.toDual a)) = ↑OrderDual.toDual (Sum.inl a)

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@[simp]

theorem OrderIso.sumDualDistrib_symm_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] (b : β) :

↑(OrderIso.symm (OrderIso.sumDualDistrib α β)) (Sum.inr (↑OrderDual.toDual b)) = ↑OrderDual.toDual (Sum.inr b)

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def OrderIso.sumLexAssoc (α : Type u_5) (β : Type u_6) (γ : Type u_7) [LE α] [LE β] [LE γ] :

Lex (Lex (α ⊕ β) ⊕ γ) ≃o Lex (α ⊕ Lex (β ⊕ γ))

Equiv.SumAssoc promoted to an order isomorphism.

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@[simp]

theorem OrderIso.sumLexAssoc_apply_inl_inl {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (a : α) :

↑(OrderIso.sumLexAssoc α β γ) (↑toLex (Sum.inl (↑toLex (Sum.inl a)))) = ↑toLex (Sum.inl a)

source

@[simp]

theorem OrderIso.sumLexAssoc_apply_inl_inr {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (b : β) :

↑(OrderIso.sumLexAssoc α β γ) (↑toLex (Sum.inl (↑toLex (Sum.inr b)))) = ↑toLex (Sum.inr (↑toLex (Sum.inl b)))

source

@[simp]

theorem OrderIso.sumLexAssoc_apply_inr {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (c : γ) :

↑(OrderIso.sumLexAssoc α β γ) (↑toLex (Sum.inr c)) = ↑toLex (Sum.inr (↑toLex (Sum.inr c)))

source

@[simp]

theorem OrderIso.sumLexAssoc_symm_apply_inl {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (a : α) :

↑(OrderIso.symm (OrderIso.sumLexAssoc α β γ)) (Sum.inl a) = Sum.inl (Sum.inl a)

source

@[simp]

theorem OrderIso.sumLexAssoc_symm_apply_inr_inl {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (b : β) :

↑(OrderIso.symm (OrderIso.sumLexAssoc α β γ)) (Sum.inr (Sum.inl b)) = Sum.inl (Sum.inr b)

source

@[simp]

theorem OrderIso.sumLexAssoc_symm_apply_inr_inr {α : Type u_1} {β : Type u_2} {γ : Type u_3} [LE α] [LE β] [LE γ] (c : γ) :

↑(OrderIso.symm (OrderIso.sumLexAssoc α β γ)) (Sum.inr (Sum.inr c)) = Sum.inr c

source

def OrderIso.sumLexDualAntidistrib (α : Type u_5) (β : Type u_6) [LE α] [LE β] :

(Lex (α ⊕ β))ᵒᵈ ≃o Lex (βᵒᵈ ⊕ αᵒᵈ)

OrderDual is antidistributive over ⊕ₗ up to an order isomorphism.

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@[simp]

theorem OrderIso.sumLexDualAntidistrib_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] (a : α) :

↑(OrderIso.sumLexDualAntidistrib α β) (↑OrderDual.toDual (Sum.inl a)) = Sum.inr (↑OrderDual.toDual a)

source

@[simp]

theorem OrderIso.sumLexDualAntidistrib_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] (b : β) :

↑(OrderIso.sumLexDualAntidistrib α β) (↑OrderDual.toDual (Sum.inr b)) = Sum.inl (↑OrderDual.toDual b)

source

@[simp]

theorem OrderIso.sumLexDualAntidistrib_symm_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] (b : β) :

↑(OrderIso.symm (OrderIso.sumLexDualAntidistrib α β)) (Sum.inl (↑OrderDual.toDual b)) = ↑OrderDual.toDual (Sum.inr b)

source

@[simp]

theorem OrderIso.sumLexDualAntidistrib_symm_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] (a : α) :

↑(OrderIso.symm (OrderIso.sumLexDualAntidistrib α β)) (Sum.inr (↑OrderDual.toDual a)) = ↑OrderDual.toDual (Sum.inl a)

source

def WithBot.orderIsoPUnitSumLex {α : Type u_1} [LE α] :

WithBot α ≃o Lex (PUnit.{u_5 + 1} ⊕ α)

WithBot α is order-isomorphic to PUnit ⊕ₗ α, by sending ⊥ to Unit and ↑a to a.

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@[simp]

theorem WithBot.orderIsoPUnitSumLex_bot {α : Type u_1} [LE α] :

↑WithBot.orderIsoPUnitSumLex ⊥ = ↑toLex (Sum.inl PUnit.unit)

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@[simp]

theorem WithBot.orderIsoPUnitSumLex_toLex {α : Type u_1} [LE α] (a : α) :

↑WithBot.orderIsoPUnitSumLex ↑a = ↑toLex (Sum.inr a)

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@[simp]

theorem WithBot.orderIsoPUnitSumLex_symm_inl {α : Type u_1} [LE α] (x : PUnit.{u_5 + 1} ) :

↑(OrderIso.symm WithBot.orderIsoPUnitSumLex) (↑toLex (Sum.inl x)) = ⊥

source

@[simp]

theorem WithBot.orderIsoPUnitSumLex_symm_inr {α : Type u_1} [LE α] (a : α) :

↑(OrderIso.symm WithBot.orderIsoPUnitSumLex) (↑toLex (Sum.inr a)) = ↑a

source

def WithTop.orderIsoSumLexPUnit {α : Type u_1} [LE α] :

WithTop α ≃o Lex (α ⊕ PUnit.{u_5 + 1} )

WithTop α is order-isomorphic to α ⊕ₗ PUnit, by sending ⊤ to Unit and ↑a to a.

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@[simp]

theorem WithTop.orderIsoSumLexPUnit_top {α : Type u_1} [LE α] :

↑WithTop.orderIsoSumLexPUnit ⊤ = ↑toLex (Sum.inr PUnit.unit)

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@[simp]

theorem WithTop.orderIsoSumLexPUnit_toLex {α : Type u_1} [LE α] (a : α) :

↑WithTop.orderIsoSumLexPUnit ↑a = ↑toLex (Sum.inl a)

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@[simp]

theorem WithTop.orderIsoSumLexPUnit_symm_inr {α : Type u_1} [LE α] (x : PUnit.{u_5 + 1} ) :

↑(OrderIso.symm WithTop.orderIsoSumLexPUnit) (↑toLex (Sum.inr x)) = ⊤

source

@[simp]

theorem WithTop.orderIsoSumLexPUnit_symm_inl {α : Type u_1} [LE α] (a : α) :

↑(OrderIso.symm WithTop.orderIsoSumLexPUnit) (↑toLex (Sum.inl a)) = ↑a