feat: linear order is isomorphic to lexicographic sum of two intervals

Mathlib.lean

@@ -4134,6 +4134,7 @@ import Mathlib.Order.Hom.Basic
 import Mathlib.Order.Hom.Bounded
 import Mathlib.Order.Hom.CompleteLattice
 import Mathlib.Order.Hom.Lattice
+import Mathlib.Order.Hom.Lex
 import Mathlib.Order.Hom.Order
 import Mathlib.Order.Hom.Set
 import Mathlib.Order.Ideal

Mathlib/Data/Sum/Order.lean

@@ -483,6 +483,18 @@ namespace OrderIso
 
 variable [LE α] [LE β] [LE γ] (a : α) (b : β) (c : γ)
 
+/-- `Equiv.sumCongr` promoted to an order isomorphism. -/
+@[simps! apply]
+def sumCongr {α₁ α₂ β₁ β₂ : Type*} [LE α₁] [LE α₂] [LE β₁] [LE β₂]
+    (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : α₁ ⊕ β₁ ≃o α₂ ⊕ β₂ where
+  toEquiv := .sumCongr ea eb
+  map_rel_iff' := by rintro (a | a) (b | b) <;> simp
+
+@[simp]
+theorem sumCongr_symm {α₁ α₂ β₁ β₂ : Type*} [LE α₁] [LE α₂] [LE β₁] [LE β₂]
+    (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : (sumCongr ea eb).symm = sumCongr ea.symm eb.symm :=
+  rfl
+
 /-- `Equiv.sumComm` promoted to an order isomorphism. -/
 @[simps! apply]
 def sumComm (α β : Type*) [LE α] [LE β] : α ⊕ β ≃o β ⊕ α :=
@@ -552,7 +564,19 @@ theorem sumDualDistrib_symm_inl : (sumDualDistrib α β).symm (inl (toDual a)) =
 theorem sumDualDistrib_symm_inr : (sumDualDistrib α β).symm (inr (toDual b)) = toDual (inr b) :=
   rfl
 
-/-- `Equiv.SumAssoc` promoted to an order isomorphism. -/
+/-- `Equiv.sumCongr` promoted to an order isomorphism. -/
+@[simps! apply]
+def sumLexCongr {α₁ α₂ β₁ β₂ : Type*} [LE α₁] [LE α₂] [LE β₁] [LE β₂]
+    (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : α₁ ⊕ₗ β₁ ≃o α₂ ⊕ₗ β₂ where
+  toEquiv := ofLex.trans ((Equiv.sumCongr ea eb).trans toLex)
+  map_rel_iff' := by simp_rw [Lex.forall]; rintro (a | a) (b | b) <;> simp
+
+@[simp]
+theorem sumLexCongr_symm {α₁ α₂ β₁ β₂ : Type*} [LE α₁] [LE α₂] [LE β₁] [LE β₂]
+    (ea : α₁ ≃o α₂) (eb : β₁ ≃o β₂) : (sumLexCongr ea eb).symm = sumLexCongr ea.symm eb.symm :=
+  rfl
+
+/-- `Equiv.sumAssoc` promoted to an order isomorphism. -/
 def sumLexAssoc (α β γ : Type*) [LE α] [LE β] [LE γ] : (α ⊕ₗ β) ⊕ₗ γ ≃o α ⊕ₗ β ⊕ₗ γ :=
   { Equiv.sumAssoc α β γ with
     map_rel_iff' := fun {a b} =>

Mathlib/Logic/Equiv/Set.lean

@@ -148,7 +148,7 @@ def setProdEquivSigma {α β : Type*} (s : Set (α × β)) :
   right_inv := fun ⟨_, _, _⟩ => rfl
 
 /-- The subtypes corresponding to equal sets are equivalent. -/
-@[simps! apply]
+@[simps! apply symm_apply]
 def setCongr {α : Type*} {s t : Set α} (h : s = t) : s ≃ t :=
   subtypeEquivProp h
 

Mathlib/Order/Hom/Lex.lean

@@ -0,0 +1,177 @@
+/-
+Copyright (c) 2025 Violeta Hernández Palacios. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Violeta Hernández Palacios
+-/
+import Mathlib.Data.Sum.Order
+import Mathlib.Order.Hom.Set
+import Mathlib.Order.RelIso.Set
+
+/-!
+# Lexicographic order and order isomorphisms
+
+The main result in this file is the following: if `α` is a linear order and `x : α`, then `α` is
+order isomorphic to either of `Iio x ⊕ₗ Ici x` or `Iic x ⊕ₗ Ioi x`.
+-/
+
+open Set
+
+variable {α : Type*}
+
+/-! ### Relation isomorphism -/
+
+namespace RelIso
+
+variable {r : α → α → Prop} {x y : α} [IsTrans α r] [IsTrichotomous α r] [DecidableRel r]
+
+variable (r x) in
+/-- A relation is isomorphic to the lexicographic sum of elements less than `x` and elements not
+less than `x`. -/
+-- The explicit type signature stops `simp` from complaining.
+def sumLexComplLeft :
+    @Sum.Lex {y // r y x} {y // ¬ r y x} (Subrel r {y | r y x}) (Subrel r {y | ¬ r y x}) ≃r r where
+  toEquiv := Equiv.sumCompl (r · x)
+  map_rel_iff' := by
+    rintro (⟨a, ha⟩ | ⟨a, ha⟩) (⟨b, hb⟩ | ⟨b, hb⟩)
+    · simp
+    · simpa using trans_trichotomous_right ha hb
+    · simpa using fun h ↦ ha <| trans h hb
+    · simp
+
+@[simp]
+theorem sumLexComplLeft_apply_inl (a : {y // r y x}) : sumLexComplLeft r x (Sum.inl a) = a :=
+  rfl
+
+@[simp]
+theorem sumLexComplLeft_apply_inr (a : {y // ¬ r y x}) : sumLexComplLeft r x (Sum.inr a) = a :=
+  rfl
+
+theorem sumLexComplLeft_symm_apply_of_mem (h : r y x) :
+    (sumLexComplLeft r x).symm y = Sum.inl ⟨y, h⟩ :=
+  Equiv.sumCompl_apply_symm_of_pos (r · x) _ h
+
+theorem sumLexComplLeft_symm_apply_of_not_mem (h : ¬ r y x) :
+    (sumLexComplLeft r x).symm y = Sum.inr ⟨y, h⟩ :=
+  Equiv.sumCompl_apply_symm_of_neg (r · x) _ h
+
+@[simp]
+theorem sumLexComplLeft_symm_apply (a : {y // r y x}) :
+    (sumLexComplLeft r x).symm a = Sum.inl a :=
+  sumLexComplLeft_symm_apply_of_mem a.2
+
+@[simp]
+theorem sumLexComplLeft_symm_apply_neg (a : {y // ¬ r y x}) :
+    (sumLexComplLeft r x).symm a = Sum.inr a :=
+  sumLexComplLeft_symm_apply_of_not_mem a.2
+
+variable (r x) in
+/-- A relation is isomorphic to the lexicographic sum of elements not greater than `x` and elements
+greater than `x`. -/
+-- The explicit type signature stops `simp` from complaining.
+def sumLexComplRight :
+    @Sum.Lex {y // ¬ r x y} {y // r x y} (Subrel r {y | ¬ r x y}) (Subrel r {y | r x y}) ≃r r where
+  toEquiv := (Equiv.sumComm _ _).trans <| Equiv.sumCompl (r x ·)
+  map_rel_iff' := by
+    rintro (⟨a, ha⟩ | ⟨a, ha⟩) (⟨b, hb⟩ | ⟨b, hb⟩)
+    · simp
+    · simpa using trans_trichotomous_left ha hb
+    · simpa using fun h ↦ hb <| trans ha h
+    · simp
+
+@[simp]
+theorem sumLexComplRight_apply_inl (a : {y // ¬ r x y}) : sumLexComplRight r x (Sum.inl a) = a :=
+  rfl
+
+@[simp]
+theorem sumLexComplRight_apply_inr (a : {y // r x y}) : sumLexComplRight r x (Sum.inr a) = a :=
+  rfl
+
+theorem sumLexComplRight_symm_apply_of_mem (h : r x y) :
+    (sumLexComplRight r x).symm y = Sum.inr ⟨y, h⟩ := by
+  simp [sumLexComplRight, h]
+
+theorem sumLexComplRight_symm_apply_of_not_mem (h : ¬ r x y) :
+    (sumLexComplRight r x).symm y = Sum.inl ⟨y, h⟩ := by
+  simp [sumLexComplRight, h]
+
+@[simp]
+theorem sumLexComplRight_symm_apply (a : {y // r x y}) :
+    (sumLexComplRight r x).symm a = Sum.inr a :=
+  sumLexComplRight_symm_apply_of_mem a.2
+
+@[simp]
+theorem sumLexComplRight_symm_apply_neg (a : {y // ¬ r x y}) :
+    (sumLexComplRight r x).symm a = Sum.inl a :=
+  sumLexComplRight_symm_apply_of_not_mem a.2
+
+end RelIso
+
+/-! ### Order isomorphism -/
+
+namespace OrderIso
+
+variable [LinearOrder α] {x y : α}
+
+variable (x) in
+/-- A linear order is isomorphic to the lexicographic sum of elements less than `x` and elements
+greater or equal to `x`. -/
+def sumLexIioIci : Iio x ⊕ₗ Ici x ≃o α :=
+  (sumLexCongr (refl _) (setCongr (Ici x) {y | ¬ y < x} (by ext; simp))).trans <|
+    ofRelIsoLT (RelIso.sumLexComplLeft (· < ·) x)
+
+@[simp]
+theorem sumLexIioIci_apply_inl (a : Iio x) : sumLexIioIci x (Sum.inl a) = a :=
+  rfl
+
+@[simp]
+theorem sumLexIioIci_apply_inr (a : Ici x) : sumLexIioIci x (Sum.inr a) = a :=
+  rfl
+
+theorem sumLexIioIci_symm_apply_of_lt (h : y < x) :
+    (sumLexIioIci x).symm y = Sum.inl ⟨y, h⟩ := by
+  rw [symm_apply_eq, sumLexIioIci_apply_inl]
+
+theorem sumLexIioIci_symm_apply_of_le {y : α} (h : x ≤ y) :
+    (sumLexIioIci x).symm y = Sum.inr ⟨y, h⟩ := by
+  rw [symm_apply_eq, sumLexIioIci_apply_inr]
+
+@[simp]
+theorem sumLexIioIci_symm_apply_Iio (a : Iio x) : (sumLexIioIci x).symm a = Sum.inl a :=
+  sumLexIioIci_symm_apply_of_lt a.2
+
+@[simp]
+theorem sumLexIioIci_symm_apply_Ici (a : Ici x) : (sumLexIioIci x).symm a = Sum.inr a :=
+  sumLexIioIci_symm_apply_of_le a.2
+
+variable (x) in
+/-- A linear order is isomorphic to the lexicographic sum of elements less or equal to `x` and
+elements greater than `x`. -/
+def sumLexIicIoi : Iic x ⊕ₗ Ioi x ≃o α :=
+  (sumLexCongr (setCongr (Iic x) {y | ¬ x < y} (by ext; simp)) (refl _)).trans <|
+    ofRelIsoLT (RelIso.sumLexComplRight (· < ·) x)
+
+@[simp]
+theorem sumLexIicIoi_apply_inl (a : Iic x) : sumLexIicIoi x (Sum.inl a) = a :=
+  rfl
+
+@[simp]
+theorem sumLexIicIoi_apply_inr (a : Ioi x) : sumLexIicIoi x (Sum.inr a) = a :=
+  rfl
+
+theorem sumLexIicIoi_symm_apply_of_le (h : y ≤ x) :
+    (sumLexIicIoi x).symm y = Sum.inl ⟨y, h⟩ := by
+  rw [symm_apply_eq, sumLexIicIoi_apply_inl]
+
+theorem sumLexIicIoi_symm_apply_of_lt {y : α} (h : x < y) :
+    (sumLexIicIoi x).symm y = Sum.inr ⟨y, h⟩ := by
+  rw [symm_apply_eq, sumLexIicIoi_apply_inr]
+
+@[simp]
+theorem sumLexIicIoi_symm_apply_Iic (a : Iic x) : (sumLexIicIoi x).symm a = Sum.inl a :=
+  sumLexIicIoi_symm_apply_of_le a.2
+
+@[simp]
+theorem sumLexIicIoi_symm_apply_Ioi (a : Ioi x) : (sumLexIicIoi x).symm a = Sum.inr a :=
+  sumLexIicIoi_symm_apply_of_lt a.2
+
+end OrderIso

Mathlib/Order/Hom/Set.lean

@@ -62,6 +62,7 @@ open Set
 variable [Preorder α]
 
 /-- Order isomorphism between two equal sets. -/
+@[simps! apply symm_apply]
 def setCongr (s t : Set α) (h : s = t) :
     s ≃o t where
   toEquiv := Equiv.setCongr h