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

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

instance Sum.instReflLiftRel_mathlib {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Refl r] [Std.Refl s] : Std.Refl (LiftRel r s)

instance Sum.instIrreflLiftRel_mathlib {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Irrefl r] [Std.Irrefl s] : Std.Irrefl (LiftRel r s)

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 b c : α ⊕ β} : LiftRel r s a b → LiftRel r s b c → LiftRel r s a c

instance Sum.instIsTransLiftRel {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) (LiftRel r s)

instance Sum.instAntisymmLiftRel_mathlib {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Antisymm r] [Std.Antisymm s] : Std.Antisymm (LiftRel r s)

instance Sum.instReflLex_mathlib {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Refl r] [Std.Refl s] : Std.Refl (Lex r s)

instance Sum.instIrreflLex_mathlib {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Irrefl r] [Std.Irrefl s] : Std.Irrefl (Lex r s)

instance Sum.instIsTransLex {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) (Lex r s)

instance Sum.instAntisymmLex_mathlib {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Antisymm r] [Std.Antisymm s] : Std.Antisymm (Lex r s)

instance Sum.instTotalLex_mathlib {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Total r] [Std.Total s] : Std.Total (Lex r s)

instance Sum.instTrichotomousLex_mathlib {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) [Std.Trichotomous r] [Std.Trichotomous s] : Std.Trichotomous (Lex r s)

instance Sum.instIsWellOrderLex {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) (Lex r s)

Disjoint sum of two orders

@[implicit_reducible]

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

@[implicit_reducible]

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

theorem Sum.le_def {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a b : α ⊕ β} : a ≤ b ↔ LiftRel (fun (x1 x2 : α) => x1 ≤ x2) (fun (x1 x2 : β) => x1 ≤ x2) a b

theorem Sum.lt_def {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a b : α ⊕ β} : a < b ↔ LiftRel (fun (x1 x2 : α) => x1 < x2) (fun (x1 x2 : β) => x1 < x2) a b

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[implicit_reducible]

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

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

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

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

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

@[implicit_reducible]

instance Sum.instPartialOrder {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] : PartialOrder (α ⊕ β)

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

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

@[simp]

theorem Sum.noMinOrder_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] : NoMinOrder (α ⊕ β) ↔ NoMinOrder α ∧ NoMinOrder β

@[simp]

theorem Sum.noMaxOrder_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] : NoMaxOrder (α ⊕ β) ↔ NoMaxOrder α ∧ NoMaxOrder β

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

@[simp]

theorem Sum.denselyOrdered_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] : DenselyOrdered (α ⊕ β) ↔ DenselyOrdered α ∧ DenselyOrdered β

@[simp]

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

@[simp]

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

Linear sum of two orders

def Sum.Lex.«term_⊕ₗ_» : Lean.TrailingParserDescr

The linear sum of two orders

@[reducible, match_pattern, inline]

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

Lexicographical Sum.inl. Only used for pattern matching.

@[reducible, match_pattern, inline]

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

Lexicographical Sum.inr. Only used for pattern matching.

@[implicit_reducible]

instance Sum.Lex.LE {α : Type u_1} {β : Type u_2} [LE α] [LE β] : LE (_root_.Lex (α ⊕ β))

The linear/lexicographical ≤ on a sum.

@[implicit_reducible]

instance Sum.Lex.LT {α : Type u_1} {β : Type u_2} [LT α] [LT β] : LT (_root_.Lex (α ⊕ β))

The linear/lexicographical < on a sum.

@[simp]

theorem Sum.Lex.toLex_le_toLex {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a b : α ⊕ β} : toLex a ≤ toLex b ↔ Lex (fun (x1 x2 : α) => x1 ≤ x2) (fun (x1 x2 : β) => x1 ≤ x2) a b

@[simp]

theorem Sum.Lex.toLex_lt_toLex {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a b : α ⊕ β} : toLex a < toLex b ↔ Lex (fun (x1 x2 : α) => x1 < x2) (fun (x1 x2 : β) => x1 < x2) a b

theorem Sum.Lex.le_def {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a b : _root_.Lex (α ⊕ β)} : a ≤ b ↔ Lex (fun (x1 x2 : α) => x1 ≤ x2) (fun (x1 x2 : β) => x1 ≤ x2) (ofLex a) (ofLex b)

theorem Sum.Lex.lt_def {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a b : _root_.Lex (α ⊕ β)} : a < b ↔ Lex (fun (x1 x2 : α) => x1 < x2) (fun (x1 x2 : β) => x1 < x2) (ofLex a) (ofLex b)

theorem Sum.Lex.inl_le_inl_iff {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a b : α} : toLex (inl a) ≤ toLex (inl b) ↔ a ≤ b

theorem Sum.Lex.inr_le_inr_iff {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a b : β} : toLex (inr a) ≤ toLex (inr b) ↔ a ≤ b

theorem Sum.Lex.inl_lt_inl_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a b : α} : toLex (inl a) < toLex (inl b) ↔ a < b

theorem Sum.Lex.inr_lt_inr_iff {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a b : β} : toLex (inr a) < toLex (inr b) ↔ a < b

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

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

theorem Sum.Lex.not_inr_le_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] {a : α} {b : β} : ¬toLex (inr b) ≤ toLex (inl a)

theorem Sum.Lex.not_inr_lt_inl {α : Type u_1} {β : Type u_2} [LT α] [LT β] {a : α} {b : β} : ¬toLex (inr b) < toLex (inl a)

def Sum.Lex.toLexRelIsoLT {α : Type u_1} {β : Type u_2} [LT α] [LT β] : (Lex (fun (x1 x2 : α) => x1 < x2) fun (x1 x2 : β) => x1 < x2) ≃r fun (x1 x2 : _root_.Lex (α ⊕ β)) => x1 < x2

toLex promoted to a RelIso between < relations.

@[simp]

theorem Sum.Lex.toLexRelIsoLT_coe {α : Type u_1} {β : Type u_2} [LT α] [LT β] : ⇑toLexRelIsoLT = ⇑toLex

@[simp]

theorem Sum.Lex.toLexRelIsoLT_symm_coe {α : Type u_1} {β : Type u_2} [LT α] [LT β] : ⇑toLexRelIsoLT.symm = ⇑ofLex

def Sum.Lex.toLexRelIsoLE {α : Type u_1} {β : Type u_2} [LE α] [LE β] : (Lex (fun (x1 x2 : α) => x1 ≤ x2) fun (x1 x2 : β) => x1 ≤ x2) ≃r fun (x1 x2 : _root_.Lex (α ⊕ β)) => x1 ≤ x2

toLex promoted to a RelIso between ≤ relations.

@[simp]

theorem Sum.Lex.toLexRelIsoLE_coe {α : Type u_1} {β : Type u_2} [LE α] [LE β] : ⇑toLexRelIsoLE = ⇑toLex

@[simp]

theorem Sum.Lex.toLexRelIsoLE_symm_coe {α : Type u_1} {β : Type u_2} [LE α] [LE β] : ⇑toLexRelIsoLE.symm = ⇑ofLex

@[implicit_reducible]

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

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

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

theorem Sum.Lex.inl_mono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] : Monotone (⇑toLex ∘ inl)

theorem Sum.Lex.inr_mono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] : Monotone (⇑toLex ∘ inr)

theorem Sum.Lex.inl_strictMono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] : StrictMono (⇑toLex ∘ inl)

theorem Sum.Lex.inr_strictMono {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] : StrictMono (⇑toLex ∘ inr)

@[implicit_reducible]

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

@[implicit_reducible]

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

@[implicit_reducible]

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

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

@[simp]

theorem Sum.Lex.inl_bot {α : Type u_1} {β : Type u_2} [LE α] [OrderBot α] [LE β] : toLex (inl ⊥) = ⊥

@[implicit_reducible]

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

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

@[simp]

theorem Sum.Lex.inr_top {α : Type u_1} {β : Type u_2} [LE α] [LE β] [OrderTop β] : toLex (inr ⊤) = ⊤

@[implicit_reducible]

instance Sum.Lex.boundedOrder {α : Type u_1} {β : Type u_2} [LE α] [LE β] [OrderBot α] [OrderTop β] : BoundedOrder (_root_.Lex (α ⊕ β))

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

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

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

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

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 (_root_.Lex (α ⊕ β))

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 (_root_.Lex (α ⊕ β))

Order isomorphisms

def OrderIso.sumCongr {α₁ : Type u_4} {α₂ : Type u_5} {β₁ : Type u_6} {β₂ : Type u_7} [LE α₁] [LE α₂] [LE β₁] [LE β₂] (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : α₁ ⊕ β₁ ≃o α₂ ⊕ β₂

Equiv.sumCongr promoted to an order isomorphism.

@[simp]

theorem OrderIso.sumCongr_apply {α₁ : Type u_4} {α₂ : Type u_5} {β₁ : Type u_6} {β₂ : Type u_7} [LE α₁] [LE α₂] [LE β₁] [LE β₂] (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) (a✝ : α₁ ⊕ β₁) : (ea.sumCongr eb) a✝ = Sum.map (⇑ea) (⇑eb) a✝

@[simp]

theorem OrderIso.sumCongr_trans {α₁ : Type u_4} {α₂ : Type u_5} {β₁ : Type u_6} {β₂ : Type u_7} {γ₁ : Type u_8} {γ₂ : Type u_9} [LE α₁] [LE α₂] [LE β₁] [LE β₂] [LE γ₁] [LE γ₂] (e₁ : α₁ ≃o β₁) (e₂ : α₂ ≃o β₂) (f₁ : β₁ ≃o γ₁) (f₂ : β₂ ≃o γ₂) : (e₁.sumCongr e₂).trans (f₁.sumCongr f₂) = (e₁.trans f₁).sumCongr (e₂.trans f₂)

@[simp]

theorem OrderIso.sumCongr_symm {α₁ : Type u_4} {α₂ : Type u_5} {β₁ : Type u_6} {β₂ : Type u_7} [LE α₁] [LE α₂] [LE β₁] [LE β₂] (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : (ea.sumCongr eb).symm = ea.symm.sumCongr eb.symm

@[simp]

theorem OrderIso.sumCongr_refl {α : Type u_1} {β : Type u_2} [LE α] [LE β] : (refl α).sumCongr (refl β) = refl (α ⊕ β)

def OrderIso.sumComm (α : Type u_10) (β : Type u_11) [LE α] [LE β] : α ⊕ β ≃o β ⊕ α

Equiv.sumComm promoted to an order isomorphism.

@[simp]

theorem OrderIso.sumComm_apply (α : Type u_10) (β : Type u_11) [LE α] [LE β] (a✝ : α ⊕ β) : (sumComm α β) a✝ = a✝.swap

@[simp]

theorem OrderIso.sumComm_symm (α : Type u_10) (β : Type u_11) [LE α] [LE β] : (sumComm α β).symm = sumComm β α

def OrderIso.sumAssoc (α : Type u_10) (β : Type u_11) (γ : Type u_12) [LE α] [LE β] [LE γ] : (α ⊕ β) ⊕ γ ≃o α ⊕ β ⊕ γ

Equiv.sumAssoc promoted to an order isomorphism.

@[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) : (sumAssoc α β γ) (Sum.inl (Sum.inl a)) = Sum.inl a

@[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) : (sumAssoc α β γ) (Sum.inl (Sum.inr b)) = Sum.inr (Sum.inl b)

@[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) : (sumAssoc α β γ) (Sum.inr c) = Sum.inr (Sum.inr c)

@[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) : (sumAssoc α β γ).symm (Sum.inl a) = Sum.inl (Sum.inl a)

@[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) : (sumAssoc α β γ).symm (Sum.inr (Sum.inl b)) = Sum.inl (Sum.inr b)

@[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) : (sumAssoc α β γ).symm (Sum.inr (Sum.inr c)) = Sum.inr c

def OrderIso.sumDualDistrib (α : Type u_10) (β : Type u_11) [LE α] [LE β] : (α ⊕ β)ᵒᵈ ≃o αᵒᵈ ⊕ βᵒᵈ

orderDual is distributive over ⊕ up to an order isomorphism.

@[simp]

theorem OrderIso.sumDualDistrib_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] (a : α) : (sumDualDistrib α β) (OrderDual.toDual (Sum.inl a)) = Sum.inl (OrderDual.toDual a)

@[simp]

theorem OrderIso.sumDualDistrib_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] (b : β) : (sumDualDistrib α β) (OrderDual.toDual (Sum.inr b)) = Sum.inr (OrderDual.toDual b)

@[simp]

theorem OrderIso.sumDualDistrib_symm_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] (a : α) : (sumDualDistrib α β).symm (Sum.inl (OrderDual.toDual a)) = OrderDual.toDual (Sum.inl a)

@[simp]

theorem OrderIso.sumDualDistrib_symm_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] (b : β) : (sumDualDistrib α β).symm (Sum.inr (OrderDual.toDual b)) = OrderDual.toDual (Sum.inr b)

def OrderIso.sumLexCongr {α₁ : Type u_4} {α₂ : Type u_5} {β₁ : Type u_6} {β₂ : Type u_7} [LE α₁] [LE α₂] [LE β₁] [LE β₂] (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : Lex (α₁ ⊕ β₁) ≃o Lex (α₂ ⊕ β₂)

Equiv.sumCongr promoted to an order isomorphism between lexicographic sums.

@[simp]

theorem OrderIso.sumLexCongr_apply {α₁ : Type u_4} {α₂ : Type u_5} {β₁ : Type u_6} {β₂ : Type u_7} [LE α₁] [LE α₂] [LE β₁] [LE β₂] (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) (a✝ : Lex (α₁ ⊕ β₁)) : (ea.sumLexCongr eb) a✝ = toLex (Sum.map (⇑ea) (⇑eb) (ofLex a✝))

@[simp]

theorem OrderIso.sumLexCongr_trans {α₁ : Type u_4} {α₂ : Type u_5} {β₁ : Type u_6} {β₂ : Type u_7} {γ₁ : Type u_8} {γ₂ : Type u_9} [LE α₁] [LE α₂] [LE β₁] [LE β₂] [LE γ₁] [LE γ₂] (e₁ : α₁ ≃o β₁) (e₂ : α₂ ≃o β₂) (f₁ : β₁ ≃o γ₁) (f₂ : β₂ ≃o γ₂) : (e₁.sumLexCongr e₂).trans (f₁.sumLexCongr f₂) = (e₁.trans f₁).sumLexCongr (e₂.trans f₂)

@[simp]

theorem OrderIso.sumLexCongr_symm {α₁ : Type u_4} {α₂ : Type u_5} {β₁ : Type u_6} {β₂ : Type u_7} [LE α₁] [LE α₂] [LE β₁] [LE β₂] (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : (ea.sumLexCongr eb).symm = ea.symm.sumLexCongr eb.symm

@[simp]

theorem OrderIso.sumLexCongr_refl {α : Type u_1} {β : Type u_2} [LE α] [LE β] : (refl α).sumLexCongr (refl β) = refl (Lex (α ⊕ β))

def OrderIso.sumLexAssoc (α : Type u_10) (β : Type u_11) (γ : Type u_12) [LE α] [LE β] [LE γ] : Lex (Lex (α ⊕ β) ⊕ γ) ≃o Lex (α ⊕ Lex (β ⊕ γ))

Equiv.sumAssoc promoted to an order isomorphism.

@[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) : (sumLexAssoc α β γ) (toLex (Sum.inl (toLex (Sum.inl a)))) = toLex (Sum.inl a)

@[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) : (sumLexAssoc α β γ) (toLex (Sum.inl (toLex (Sum.inr b)))) = toLex (Sum.inr (toLex (Sum.inl b)))

@[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) : (sumLexAssoc α β γ) (toLex (Sum.inr c)) = toLex (Sum.inr (toLex (Sum.inr c)))

@[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) : (sumLexAssoc α β γ).symm (Sum.inl a) = Sum.inl (Sum.inl a)

@[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) : (sumLexAssoc α β γ).symm (Sum.inr (Sum.inl b)) = Sum.inl (Sum.inr b)

@[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) : (sumLexAssoc α β γ).symm (Sum.inr (Sum.inr c)) = Sum.inr c

def OrderIso.sumLexDualAntidistrib (α : Type u_10) (β : Type u_11) [LE α] [LE β] : (Lex (α ⊕ β))ᵒᵈ ≃o Lex (βᵒᵈ ⊕ αᵒᵈ)

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

@[simp]

theorem OrderIso.sumLexDualAntidistrib_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] (a : α) : (sumLexDualAntidistrib α β) (OrderDual.toDual (Sum.inl a)) = Sum.inr (OrderDual.toDual a)

@[simp]

theorem OrderIso.sumLexDualAntidistrib_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] (b : β) : (sumLexDualAntidistrib α β) (OrderDual.toDual (Sum.inr b)) = Sum.inl (OrderDual.toDual b)

@[simp]

theorem OrderIso.sumLexDualAntidistrib_symm_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] (b : β) : (sumLexDualAntidistrib α β).symm (Sum.inl (OrderDual.toDual b)) = OrderDual.toDual (Sum.inr b)

@[simp]

theorem OrderIso.sumLexDualAntidistrib_symm_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] (a : α) : (sumLexDualAntidistrib α β).symm (Sum.inr (OrderDual.toDual a)) = OrderDual.toDual (Sum.inl a)

def OrderIso.sumLexEmpty {α : Type u_1} {β : Type u_2} [LE α] [LE β] [IsEmpty β] : Lex (α ⊕ β) ≃o α

Equiv.sumEmpty as an OrderIso with the lexicographic sum.

def OrderIso.emptySumLex {α : Type u_1} {β : Type u_2} [LE α] [LE β] [IsEmpty β] : Lex (β ⊕ α) ≃o α

Equiv.emptySum as an OrderIso with the lexicographic sum.

@[simp]

theorem OrderIso.sumLexEmpty_apply_inl {α : Type u_1} {β : Type u_2} [LE α] [LE β] [IsEmpty β] (x : α) : sumLexEmpty (toLex (Sum.inl x)) = x

@[simp]

theorem OrderIso.emptySumLex_apply_inr {α : Type u_1} {β : Type u_2} [LE α] [LE β] [IsEmpty β] (x : α) : emptySumLex (toLex (Sum.inr x)) = x

def WithBot.orderIsoPUnitSumLex {α : Type u_1} [LE α] : WithBot α ≃o Lex (PUnit ⊕ α)

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

@[simp]

theorem WithBot.orderIsoPUnitSumLex_bot {α : Type u_1} [LE α] : orderIsoPUnitSumLex ⊥ = toLex (Sum.inl PUnit.unit)

@[simp]

theorem WithBot.orderIsoPUnitSumLex_toLex {α : Type u_1} [LE α] (a : α) : orderIsoPUnitSumLex ↑a = toLex (Sum.inr a)

@[simp]

theorem WithBot.orderIsoPUnitSumLex_symm_inl {α : Type u_1} [LE α] (x : PUnit) : orderIsoPUnitSumLex.symm (toLex (Sum.inl x)) = ⊥

@[simp]

theorem WithBot.orderIsoPUnitSumLex_symm_inr {α : Type u_1} [LE α] (a : α) : orderIsoPUnitSumLex.symm (toLex (Sum.inr a)) = ↑a

def WithTop.orderIsoSumLexPUnit {α : Type u_1} [LE α] : WithTop α ≃o Lex (α ⊕ PUnit)

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

@[simp]

theorem WithTop.orderIsoSumLexPUnit_top {α : Type u_1} [LE α] : orderIsoSumLexPUnit ⊤ = toLex (Sum.inr PUnit.unit)

@[simp]

theorem WithTop.orderIsoSumLexPUnit_toLex {α : Type u_1} [LE α] (a : α) : orderIsoSumLexPUnit ↑a = toLex (Sum.inl a)

@[simp]

theorem WithTop.orderIsoSumLexPUnit_symm_inr {α : Type u_1} [LE α] (x : PUnit) : orderIsoSumLexPUnit.symm (toLex (Sum.inr x)) = ⊤

@[simp]

theorem WithTop.orderIsoSumLexPUnit_symm_inl {α : Type u_1} [LE α] (a : α) : orderIsoSumLexPUnit.symm (toLex (Sum.inl a)) = ↑a