/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies

! This file was ported from Lean 3 source module order.antisymmetrization
! leanprover-community/mathlib commit 3353f661228bd27f632c600cd1a58b874d847c90
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Order.Hom.Basic
import Mathbin.Logic.Relation

/-!
# Turning a preorder into a partial order

> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
> Any changes to this file require a corresponding PR to mathlib4.

This file allows to make a preorder into a partial order by quotienting out the elements `a`, `b`
such that `a ≤ b` and `b ≤ a`.

`antisymmetrization` is a functor from `Preorder` to `PartialOrder`. See `Preorder_to_PartialOrder`.

## Main declarations

* `antisymm_rel`: The antisymmetrization relation. `antisymm_rel r a b` means that `a` and `b` are
  related both ways by `r`.
* `antisymmetrization α r`: The quotient of `α` by `antisymm_rel r`. Even when `r` is just a
  preorder, `antisymmetrization α` is a partial order.
-/

open Function OrderDual

variable {α β : Type _}

section Relation

variable (r : α → α → Prop)

#print AntisymmRel /-
/-- The antisymmetrization relation. -/
def AntisymmRel (a b : α) : Prop :=
  r a b ∧ r b a
#align antisymm_rel AntisymmRel
-/

#print antisymmRel_swap /-
theorem antisymmRel_swap : AntisymmRel (swap r) = AntisymmRel r :=
  funext fun _ => funext fun _ => propext and_comm
#align antisymm_rel_swap antisymmRel_swap
-/

#print antisymmRel_refl /-
@[refl]
theorem antisymmRel_refl [IsRefl α r] (a : α) : AntisymmRel r a a :=
  ⟨refl _, refl _⟩
#align antisymm_rel_refl antisymmRel_refl
-/

variable {r}

#print AntisymmRel.symm /-
@[symm]
theorem AntisymmRel.symm {a b : α} : AntisymmRel r a b → AntisymmRel r b a :=
  And.symm
#align antisymm_rel.symm AntisymmRel.symm
-/

#print AntisymmRel.trans /-
@[trans]
theorem AntisymmRel.trans [IsTrans α r] {a b c : α} (hab : AntisymmRel r a b)
    (hbc : AntisymmRel r b c) : AntisymmRel r a c :=
  ⟨trans hab.1 hbc.1, trans hbc.2 hab.2⟩
#align antisymm_rel.trans AntisymmRel.trans
-/

#print AntisymmRel.decidableRel /-
instance AntisymmRel.decidableRel [DecidableRel r] : DecidableRel (AntisymmRel r) := fun _ _ =>
  And.decidable
#align antisymm_rel.decidable_rel AntisymmRel.decidableRel
-/

#print antisymmRel_iff_eq /-
@[simp]
theorem antisymmRel_iff_eq [IsRefl α r] [IsAntisymm α r] {a b : α} : AntisymmRel r a b ↔ a = b :=
  antisymm_iff
#align antisymm_rel_iff_eq antisymmRel_iff_eq
-/

alias antisymmRel_iff_eq ↔ AntisymmRel.eq _
#align antisymm_rel.eq AntisymmRel.eq

end Relation

section IsPreorder

variable (α) (r : α → α → Prop) [IsPreorder α r]

#print AntisymmRel.setoid /-
/-- The antisymmetrization relation as an equivalence relation. -/
@[simps]
def AntisymmRel.setoid : Setoid α :=
  ⟨AntisymmRel r, antisymmRel_refl _, fun _ _ => AntisymmRel.symm, fun _ _ _ => AntisymmRel.trans⟩
#align antisymm_rel.setoid AntisymmRel.setoid
-/

#print Antisymmetrization /-
/-- The partial order derived from a preorder by making pairwise comparable elements equal. This is
the quotient by `λ a b, a ≤ b ∧ b ≤ a`. -/
def Antisymmetrization : Type _ :=
  Quotient <| AntisymmRel.setoid α r
#align antisymmetrization Antisymmetrization
-/

variable {α}

#print toAntisymmetrization /-
/-- Turn an element into its antisymmetrization. -/
def toAntisymmetrization : α → Antisymmetrization α r :=
  Quotient.mk''
#align to_antisymmetrization toAntisymmetrization
-/

#print ofAntisymmetrization /-
/-- Get a representative from the antisymmetrization. -/
noncomputable def ofAntisymmetrization : Antisymmetrization α r → α :=
  Quotient.out'
#align of_antisymmetrization ofAntisymmetrization
-/

instance [Inhabited α] : Inhabited (Antisymmetrization α r) :=
  Quotient.inhabited _

#print Antisymmetrization.ind /-
@[elab_as_elim]
protected theorem Antisymmetrization.ind {p : Antisymmetrization α r → Prop} :
    (∀ a, p <| toAntisymmetrization r a) → ∀ q, p q :=
  Quot.ind
#align antisymmetrization.ind Antisymmetrization.ind
-/

#print Antisymmetrization.induction_on /-
@[elab_as_elim]
protected theorem Antisymmetrization.induction_on {p : Antisymmetrization α r → Prop}
    (a : Antisymmetrization α r) (h : ∀ a, p <| toAntisymmetrization r a) : p a :=
  Quotient.inductionOn' a h
#align antisymmetrization.induction_on Antisymmetrization.induction_on
-/

#print toAntisymmetrization_ofAntisymmetrization /-
@[simp]
theorem toAntisymmetrization_ofAntisymmetrization (a : Antisymmetrization α r) :
    toAntisymmetrization r (ofAntisymmetrization r a) = a :=
  Quotient.out_eq' _
#align to_antisymmetrization_of_antisymmetrization toAntisymmetrization_ofAntisymmetrization
-/

end IsPreorder

section Preorder

variable {α} [Preorder α] [Preorder β] {a b : α}

/- warning: antisymm_rel.image -> AntisymmRel.image is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Preorder.{u1} α] [_inst_2 : Preorder.{u2} β] {a : α} {b : α}, (AntisymmRel.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) a b) -> (forall {f : α -> β}, (Monotone.{u1, u2} α β _inst_1 _inst_2 f) -> (AntisymmRel.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (f a) (f b)))
but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Preorder.{u2} α] [_inst_2 : Preorder.{u1} β] {a : α} {b : α}, (AntisymmRel.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.533 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.535 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.533 x._@.Mathlib.Order.Antisymmetrization._hyg.535) a b) -> (forall {f : α -> β}, (Monotone.{u2, u1} α β _inst_1 _inst_2 f) -> (AntisymmRel.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.556 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.558 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.556 x._@.Mathlib.Order.Antisymmetrization._hyg.558) (f a) (f b)))
Case conversion may be inaccurate. Consider using '#align antisymm_rel.image AntisymmRel.imageₓ'. -/
theorem AntisymmRel.image {a b : α} (h : AntisymmRel (· ≤ ·) a b) {f : α → β} (hf : Monotone f) :
    AntisymmRel (· ≤ ·) (f a) (f b) :=
  ⟨hf h.1, hf h.2⟩
#align antisymm_rel.image AntisymmRel.image

instance : PartialOrder (Antisymmetrization α (· ≤ ·))
    where
  le a b :=
    Quotient.liftOn₂' a b (· ≤ ·) fun (a₁ a₂ b₁ b₂ : α) h₁ h₂ =>
      propext ⟨fun h => h₁.2.trans <| h.trans h₂.1, fun h => h₁.1.trans <| h.trans h₂.2⟩
  lt a b :=
    Quotient.liftOn₂' a b (· < ·) fun (a₁ a₂ b₁ b₂ : α) h₁ h₂ =>
      propext ⟨fun h => h₁.2.trans_lt <| h.trans_le h₂.1, fun h => h₁.1.trans_lt <| h.trans_le h₂.2⟩
  le_refl a := Quotient.inductionOn' a <| le_refl
  le_trans a b c := Quotient.inductionOn₃' a b c fun a b c => le_trans
  lt_iff_le_not_le a b := Quotient.inductionOn₂' a b fun a b => lt_iff_le_not_le
  le_antisymm a b := Quotient.inductionOn₂' a b fun a b hab hba => Quotient.sound' ⟨hab, hba⟩

/- warning: antisymmetrization_fibration -> antisymmetrization_fibration is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α], Relation.Fibration.{u1, u1} α (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1)) (LT.lt.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)))) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α], Relation.Fibration.{u1, u1} α (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.848 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.850 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.848 x._@.Mathlib.Order.Antisymmetrization._hyg.850) (instIsPreorderLeToLE.{u1} α _inst_1)) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.814 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.816 : α) => LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.814 x._@.Mathlib.Order.Antisymmetrization._hyg.816) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.829 : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.848 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.850 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.848 x._@.Mathlib.Order.Antisymmetrization._hyg.850) (instIsPreorderLeToLE.{u1} α _inst_1)) (x._@.Mathlib.Order.Antisymmetrization._hyg.831 : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.848 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.850 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.848 x._@.Mathlib.Order.Antisymmetrization._hyg.850) (instIsPreorderLeToLE.{u1} α _inst_1)) => LT.lt.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.848 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.850 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.848 x._@.Mathlib.Order.Antisymmetrization._hyg.850) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.848 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.850 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.848 x._@.Mathlib.Order.Antisymmetrization._hyg.850) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.848 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.850 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.848 x._@.Mathlib.Order.Antisymmetrization._hyg.850) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1))) x._@.Mathlib.Order.Antisymmetrization._hyg.829 x._@.Mathlib.Order.Antisymmetrization._hyg.831) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.848 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.850 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.848 x._@.Mathlib.Order.Antisymmetrization._hyg.850) (instIsPreorderLeToLE.{u1} α _inst_1))
Case conversion may be inaccurate. Consider using '#align antisymmetrization_fibration antisymmetrization_fibrationₓ'. -/
theorem antisymmetrization_fibration :
    Relation.Fibration (· < ·) (· < ·) (@toAntisymmetrization α (· ≤ ·) _) :=
  by
  rintro a ⟨b⟩ h
  exact ⟨b, h, rfl⟩
#align antisymmetrization_fibration antisymmetrization_fibration

/- warning: acc_antisymmetrization_iff -> acc_antisymmetrization_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : α}, Iff (Acc.{succ u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (LT.lt.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)))) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) a)) (Acc.{succ u1} α (LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1)) a)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : α}, Iff (Acc.{succ u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.911 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.913 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.911 x._@.Mathlib.Order.Antisymmetrization._hyg.913) (instIsPreorderLeToLE.{u1} α _inst_1)) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.892 : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.911 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.913 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.911 x._@.Mathlib.Order.Antisymmetrization._hyg.913) (instIsPreorderLeToLE.{u1} α _inst_1)) (x._@.Mathlib.Order.Antisymmetrization._hyg.894 : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.911 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.913 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.911 x._@.Mathlib.Order.Antisymmetrization._hyg.913) (instIsPreorderLeToLE.{u1} α _inst_1)) => LT.lt.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.911 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.913 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.911 x._@.Mathlib.Order.Antisymmetrization._hyg.913) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.911 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.913 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.911 x._@.Mathlib.Order.Antisymmetrization._hyg.913) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.911 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.913 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.911 x._@.Mathlib.Order.Antisymmetrization._hyg.913) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1))) x._@.Mathlib.Order.Antisymmetrization._hyg.892 x._@.Mathlib.Order.Antisymmetrization._hyg.894) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.911 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.913 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.911 x._@.Mathlib.Order.Antisymmetrization._hyg.913) (instIsPreorderLeToLE.{u1} α _inst_1) a)) (Acc.{succ u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.928 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.930 : α) => LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.928 x._@.Mathlib.Order.Antisymmetrization._hyg.930) a)
Case conversion may be inaccurate. Consider using '#align acc_antisymmetrization_iff acc_antisymmetrization_iffₓ'. -/
theorem acc_antisymmetrization_iff : Acc (· < ·) (toAntisymmetrization (· ≤ ·) a) ↔ Acc (· < ·) a :=
  acc_liftOn₂'_iff
#align acc_antisymmetrization_iff acc_antisymmetrization_iff

/- warning: well_founded_antisymmetrization_iff -> wellFounded_antisymmetrization_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α], Iff (WellFounded.{succ u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (LT.lt.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1))))) (WellFounded.{succ u1} α (LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1)))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α], Iff (WellFounded.{succ u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.988 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.990 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.988 x._@.Mathlib.Order.Antisymmetrization._hyg.990) (instIsPreorderLeToLE.{u1} α _inst_1)) (LT.lt.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.988 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.990 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.988 x._@.Mathlib.Order.Antisymmetrization._hyg.990) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.988 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.990 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.988 x._@.Mathlib.Order.Antisymmetrization._hyg.990) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.988 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.990 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.988 x._@.Mathlib.Order.Antisymmetrization._hyg.990) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1))))) (WellFounded.{succ u1} α (LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1)))
Case conversion may be inaccurate. Consider using '#align well_founded_antisymmetrization_iff wellFounded_antisymmetrization_iffₓ'. -/
theorem wellFounded_antisymmetrization_iff :
    WellFounded (@LT.lt (Antisymmetrization α (· ≤ ·)) _) ↔ WellFounded (@LT.lt α _) :=
  wellFounded_liftOn₂'_iff
#align well_founded_antisymmetrization_iff wellFounded_antisymmetrization_iff

instance [WellFoundedLT α] : WellFoundedLT (Antisymmetrization α (· ≤ ·)) :=
  ⟨wellFounded_antisymmetrization_iff.2 IsWellFounded.wf⟩

instance [@DecidableRel α (· ≤ ·)] [@DecidableRel α (· < ·)] [IsTotal α (· ≤ ·)] :
    LinearOrder (Antisymmetrization α (· ≤ ·)) :=
  {
    Antisymmetrization.partialOrder with
    le_total := fun a b => Quotient.inductionOn₂' a b <| total_of (· ≤ ·)
    DecidableEq := @Quotient.decidableEq _ (AntisymmRel.setoid _ (· ≤ ·)) AntisymmRel.decidableRel
    decidableLe := fun _ _ => Quotient.liftOn₂'.decidable _ _ _ _
    decidableLt := fun _ _ => Quotient.liftOn₂'.decidable _ _ _ _ }

/- warning: to_antisymmetrization_le_to_antisymmetrization_iff -> toAntisymmetrization_le_toAntisymmetrization_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : α} {b : α}, Iff (LE.le.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1))) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) a) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) b)) (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : α} {b : α}, Iff (LE.le.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1352 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1354 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1352 x._@.Mathlib.Order.Antisymmetrization._hyg.1354) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1352 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1354 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1352 x._@.Mathlib.Order.Antisymmetrization._hyg.1354) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1352 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1354 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1352 x._@.Mathlib.Order.Antisymmetrization._hyg.1354) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1))) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1352 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1354 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1352 x._@.Mathlib.Order.Antisymmetrization._hyg.1354) (instIsPreorderLeToLE.{u1} α _inst_1) a) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1370 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1372 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1370 x._@.Mathlib.Order.Antisymmetrization._hyg.1372) (instIsPreorderLeToLE.{u1} α _inst_1) b)) (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) a b)
Case conversion may be inaccurate. Consider using '#align to_antisymmetrization_le_to_antisymmetrization_iff toAntisymmetrization_le_toAntisymmetrization_iffₓ'. -/
@[simp]
theorem toAntisymmetrization_le_toAntisymmetrization_iff :
    toAntisymmetrization (· ≤ ·) a ≤ toAntisymmetrization (· ≤ ·) b ↔ a ≤ b :=
  Iff.rfl
#align to_antisymmetrization_le_to_antisymmetrization_iff toAntisymmetrization_le_toAntisymmetrization_iff

/- warning: to_antisymmetrization_lt_to_antisymmetrization_iff -> toAntisymmetrization_lt_toAntisymmetrization_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : α} {b : α}, Iff (LT.lt.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1))) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) a) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) b)) (LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1) a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : α} {b : α}, Iff (LT.lt.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1412 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1414 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1412 x._@.Mathlib.Order.Antisymmetrization._hyg.1414) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1412 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1414 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1412 x._@.Mathlib.Order.Antisymmetrization._hyg.1414) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1412 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1414 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1412 x._@.Mathlib.Order.Antisymmetrization._hyg.1414) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1))) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1412 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1414 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1412 x._@.Mathlib.Order.Antisymmetrization._hyg.1414) (instIsPreorderLeToLE.{u1} α _inst_1) a) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1430 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1432 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1430 x._@.Mathlib.Order.Antisymmetrization._hyg.1432) (instIsPreorderLeToLE.{u1} α _inst_1) b)) (LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1) a b)
Case conversion may be inaccurate. Consider using '#align to_antisymmetrization_lt_to_antisymmetrization_iff toAntisymmetrization_lt_toAntisymmetrization_iffₓ'. -/
@[simp]
theorem toAntisymmetrization_lt_toAntisymmetrization_iff :
    toAntisymmetrization (· ≤ ·) a < toAntisymmetrization (· ≤ ·) b ↔ a < b :=
  Iff.rfl
#align to_antisymmetrization_lt_to_antisymmetrization_iff toAntisymmetrization_lt_toAntisymmetrization_iff

/- warning: of_antisymmetrization_le_of_antisymmetrization_iff -> ofAntisymmetrization_le_ofAntisymmetrization_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)} {b : Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)}, Iff (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) (ofAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) a) (ofAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) b)) (LE.le.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1))) a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1466 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1468 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1466 x._@.Mathlib.Order.Antisymmetrization._hyg.1468) (instIsPreorderLeToLE.{u1} α _inst_1)} {b : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1483 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1485 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1483 x._@.Mathlib.Order.Antisymmetrization._hyg.1485) (instIsPreorderLeToLE.{u1} α _inst_1)}, Iff (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) (ofAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1505 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1507 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1505 x._@.Mathlib.Order.Antisymmetrization._hyg.1507) (instIsPreorderLeToLE.{u1} α _inst_1) a) (ofAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1522 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1524 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1522 x._@.Mathlib.Order.Antisymmetrization._hyg.1524) (instIsPreorderLeToLE.{u1} α _inst_1) b)) (LE.le.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1466 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1468 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1466 x._@.Mathlib.Order.Antisymmetrization._hyg.1468) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1466 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1468 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1466 x._@.Mathlib.Order.Antisymmetrization._hyg.1468) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1466 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1468 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1466 x._@.Mathlib.Order.Antisymmetrization._hyg.1468) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1))) a b)
Case conversion may be inaccurate. Consider using '#align of_antisymmetrization_le_of_antisymmetrization_iff ofAntisymmetrization_le_ofAntisymmetrization_iffₓ'. -/
@[simp]
theorem ofAntisymmetrization_le_ofAntisymmetrization_iff {a b : Antisymmetrization α (· ≤ ·)} :
    ofAntisymmetrization (· ≤ ·) a ≤ ofAntisymmetrization (· ≤ ·) b ↔ a ≤ b :=
  RelEmbedding.map_rel_iff (Quotient.out'RelEmbedding _)
#align of_antisymmetrization_le_of_antisymmetrization_iff ofAntisymmetrization_le_ofAntisymmetrization_iff

/- warning: of_antisymmetrization_lt_of_antisymmetrization_iff -> ofAntisymmetrization_lt_ofAntisymmetrization_iff is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)} {b : Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)}, Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1) (ofAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) a) (ofAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) b)) (LT.lt.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1))) a b)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] {a : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1590 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1592 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1590 x._@.Mathlib.Order.Antisymmetrization._hyg.1592) (instIsPreorderLeToLE.{u1} α _inst_1)} {b : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1607 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1609 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1607 x._@.Mathlib.Order.Antisymmetrization._hyg.1609) (instIsPreorderLeToLE.{u1} α _inst_1)}, Iff (LT.lt.{u1} α (Preorder.toLT.{u1} α _inst_1) (ofAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1629 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1631 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1629 x._@.Mathlib.Order.Antisymmetrization._hyg.1631) (instIsPreorderLeToLE.{u1} α _inst_1) a) (ofAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1646 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1648 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1646 x._@.Mathlib.Order.Antisymmetrization._hyg.1648) (instIsPreorderLeToLE.{u1} α _inst_1) b)) (LT.lt.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1590 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1592 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1590 x._@.Mathlib.Order.Antisymmetrization._hyg.1592) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLT.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1590 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1592 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1590 x._@.Mathlib.Order.Antisymmetrization._hyg.1592) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1590 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1592 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1590 x._@.Mathlib.Order.Antisymmetrization._hyg.1592) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1))) a b)
Case conversion may be inaccurate. Consider using '#align of_antisymmetrization_lt_of_antisymmetrization_iff ofAntisymmetrization_lt_ofAntisymmetrization_iffₓ'. -/
@[simp]
theorem ofAntisymmetrization_lt_ofAntisymmetrization_iff {a b : Antisymmetrization α (· ≤ ·)} :
    ofAntisymmetrization (· ≤ ·) a < ofAntisymmetrization (· ≤ ·) b ↔ a < b :=
  (Quotient.out'RelEmbedding _).map_rel_iff
#align of_antisymmetrization_lt_of_antisymmetrization_iff ofAntisymmetrization_lt_ofAntisymmetrization_iff

/- warning: to_antisymmetrization_mono -> toAntisymmetrization_mono is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α], Monotone.{u1, u1} α (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) _inst_1 (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α], Monotone.{u1, u1} α (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1717 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1719 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1717 x._@.Mathlib.Order.Antisymmetrization._hyg.1719) (instIsPreorderLeToLE.{u1} α _inst_1)) _inst_1 (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1717 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1719 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1717 x._@.Mathlib.Order.Antisymmetrization._hyg.1719) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1)) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1717 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1719 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1717 x._@.Mathlib.Order.Antisymmetrization._hyg.1719) (instIsPreorderLeToLE.{u1} α _inst_1))
Case conversion may be inaccurate. Consider using '#align to_antisymmetrization_mono toAntisymmetrization_monoₓ'. -/
@[mono]
theorem toAntisymmetrization_mono : Monotone (@toAntisymmetrization α (· ≤ ·) _) := fun a b => id
#align to_antisymmetrization_mono toAntisymmetrization_mono

/-- `to_antisymmetrization` as an order homomorphism. -/
@[simps]
def OrderHom.toAntisymmetrization : α →o Antisymmetrization α (· ≤ ·) :=
  ⟨toAntisymmetrization (· ≤ ·), fun a b => id⟩
#align order_hom.to_antisymmetrization OrderHom.toAntisymmetrization

private theorem lift_fun_antisymm_rel (f : α →o β) :
    ((AntisymmRel.setoid α (· ≤ ·)).R ⇒ (AntisymmRel.setoid β (· ≤ ·)).R) f f := fun a b h =>
  ⟨f.mono h.1, f.mono h.2⟩
#align lift_fun_antisymm_rel lift_fun_antisymm_rel

/- warning: order_hom.antisymmetrization -> OrderHom.antisymmetrization is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Preorder.{u1} α] [_inst_2 : Preorder.{u2} β], (OrderHom.{u1, u2} α β _inst_1 _inst_2) -> (OrderHom.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2)))
but is expected to have type
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Preorder.{u1} α] [_inst_2 : Preorder.{u2} β], (OrderHom.{u1, u2} α β _inst_1 _inst_2) -> (OrderHom.{u1, u2} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u1} α _inst_1)) (Antisymmetrization.{u2} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u2} β _inst_2)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u2} β _inst_2)))
Case conversion may be inaccurate. Consider using '#align order_hom.antisymmetrization OrderHom.antisymmetrizationₓ'. -/
/-- Turns an order homomorphism from `α` to `β` into one from `antisymmetrization α` to
`antisymmetrization β`. `antisymmetrization` is actually a functor. See `Preorder_to_PartialOrder`.
-/
protected def OrderHom.antisymmetrization (f : α →o β) :
    Antisymmetrization α (· ≤ ·) →o Antisymmetrization β (· ≤ ·) :=
  ⟨Quotient.map' f <| liftFun_antisymm_rel f, fun a b => Quotient.inductionOn₂' a b <| f.mono⟩
#align order_hom.antisymmetrization OrderHom.antisymmetrization

/- warning: order_hom.coe_antisymmetrization -> OrderHom.coe_antisymmetrization is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Preorder.{u1} α] [_inst_2 : Preorder.{u2} β] (f : OrderHom.{u1, u2} α β _inst_1 _inst_2), Eq.{max (succ u1) (succ u2)} ((Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) -> (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderHom.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2))) (fun (_x : OrderHom.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2))) => (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) -> (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2))) (OrderHom.hasCoeToFun.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2))) (OrderHom.antisymmetrization.{u1, u2} α β _inst_1 _inst_2 f)) (Quotient.map'.{succ u1, succ u2} α β (AntisymmRel.setoid.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (AntisymmRel.setoid.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderHom.{u1, u2} α β _inst_1 _inst_2) (fun (_x : OrderHom.{u1, u2} α β _inst_1 _inst_2) => α -> β) (OrderHom.hasCoeToFun.{u1, u2} α β _inst_1 _inst_2) f) (_Private.3495585149.liftFun_antisymm_rel.{u1, u2} α β _inst_1 _inst_2 f))
but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Preorder.{u2} α] [_inst_2 : Preorder.{u1} β] (f : OrderHom.{u2, u1} α β _inst_1 _inst_2), Eq.{max (succ u2) (succ u1)} ((Antisymmetrization.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u2} α _inst_1)) -> (Antisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2))) (OrderHom.toFun.{u2, u1} (Antisymmetrization.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u2} α _inst_1)) (Antisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u2} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u2} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} β _inst_2)) (OrderHom.antisymmetrization.{u2, u1} α β _inst_1 _inst_2 f)) (Quotient.map'.{succ u2, succ u1} α β (AntisymmRel.setoid.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1767 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1769 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1767 x._@.Mathlib.Order.Antisymmetrization._hyg.1769) (instIsPreorderLeToLE.{u2} α _inst_1)) (AntisymmRel.setoid.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1787 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1789 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1787 x._@.Mathlib.Order.Antisymmetrization._hyg.1789) (instIsPreorderLeToLE.{u1} β _inst_2)) (OrderHom.toFun.{u2, u1} α β _inst_1 _inst_2 f) (_private.Mathlib.Order.Antisymmetrization.0.lift_fun_antisymmRel.{u1, u2} α β _inst_1 _inst_2 f))
Case conversion may be inaccurate. Consider using '#align order_hom.coe_antisymmetrization OrderHom.coe_antisymmetrizationₓ'. -/
@[simp]
theorem OrderHom.coe_antisymmetrization (f : α →o β) :
    ⇑f.Antisymmetrization = Quotient.map' f (liftFun_antisymm_rel f) :=
  rfl
#align order_hom.coe_antisymmetrization OrderHom.coe_antisymmetrization

/- warning: order_hom.antisymmetrization_apply -> OrderHom.antisymmetrization_apply is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Preorder.{u1} α] [_inst_2 : Preorder.{u2} β] (f : OrderHom.{u1, u2} α β _inst_1 _inst_2) (a : Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)), Eq.{succ u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderHom.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2))) (fun (_x : OrderHom.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2))) => (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) -> (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2))) (OrderHom.hasCoeToFun.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2))) (OrderHom.antisymmetrization.{u1, u2} α β _inst_1 _inst_2 f) a) (Quotient.map'.{succ u1, succ u2} α β (AntisymmRel.setoid.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (AntisymmRel.setoid.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderHom.{u1, u2} α β _inst_1 _inst_2) (fun (_x : OrderHom.{u1, u2} α β _inst_1 _inst_2) => α -> β) (OrderHom.hasCoeToFun.{u1, u2} α β _inst_1 _inst_2) f) (_Private.3495585149.liftFun_antisymm_rel.{u1, u2} α β _inst_1 _inst_2 f) a)
but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Preorder.{u2} α] [_inst_2 : Preorder.{u1} β] (f : OrderHom.{u2, u1} α β _inst_1 _inst_2) (a : Antisymmetrization.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1954 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1956 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1954 x._@.Mathlib.Order.Antisymmetrization._hyg.1956) (instIsPreorderLeToLE.{u2} α _inst_1)), Eq.{succ u1} (Antisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2)) (OrderHom.toFun.{u2, u1} (Antisymmetrization.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u2} α _inst_1)) (Antisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u2} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u2} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} β _inst_2)) (OrderHom.antisymmetrization.{u2, u1} α β _inst_1 _inst_2 f) a) (Quotient.map'.{succ u2, succ u1} α β (AntisymmRel.setoid.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1767 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1769 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1767 x._@.Mathlib.Order.Antisymmetrization._hyg.1769) (instIsPreorderLeToLE.{u2} α _inst_1)) (AntisymmRel.setoid.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1787 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1789 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1787 x._@.Mathlib.Order.Antisymmetrization._hyg.1789) (instIsPreorderLeToLE.{u1} β _inst_2)) (OrderHom.toFun.{u2, u1} α β _inst_1 _inst_2 f) (_private.Mathlib.Order.Antisymmetrization.0.lift_fun_antisymmRel.{u1, u2} α β _inst_1 _inst_2 f) a)
Case conversion may be inaccurate. Consider using '#align order_hom.antisymmetrization_apply OrderHom.antisymmetrization_applyₓ'. -/
@[simp]
theorem OrderHom.antisymmetrization_apply (f : α →o β) (a : Antisymmetrization α (· ≤ ·)) :
    f.Antisymmetrization a = Quotient.map' f (liftFun_antisymm_rel f) a :=
  rfl
#align order_hom.antisymmetrization_apply OrderHom.antisymmetrization_apply

/- warning: order_hom.antisymmetrization_apply_mk -> OrderHom.antisymmetrization_apply_mk is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : Preorder.{u1} α] [_inst_2 : Preorder.{u2} β] (f : OrderHom.{u1, u2} α β _inst_1 _inst_2) (a : α), Eq.{succ u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderHom.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2))) (fun (_x : OrderHom.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2))) => (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) -> (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2))) (OrderHom.hasCoeToFun.{u1, u2} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2)) (Antisymmetrization.partialOrder.{u2} β _inst_2))) (OrderHom.antisymmetrization.{u1, u2} α β _inst_1 _inst_2 f) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) a)) (toAntisymmetrization.{u2} β (LE.le.{u2} β (Preorder.toLE.{u2} β _inst_2)) (LE.le.isPreorder.{u2} β _inst_2) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderHom.{u1, u2} α β _inst_1 _inst_2) (fun (_x : OrderHom.{u1, u2} α β _inst_1 _inst_2) => α -> β) (OrderHom.hasCoeToFun.{u1, u2} α β _inst_1 _inst_2) f a))
but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Preorder.{u2} α] [_inst_2 : Preorder.{u1} β] (f : OrderHom.{u2, u1} α β _inst_1 _inst_2) (a : α), Eq.{succ u1} (Antisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2)) (OrderHom.toFun.{u2, u1} (Antisymmetrization.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u2} α _inst_1)) (Antisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2)) (PartialOrder.toPreorder.{u2} (Antisymmetrization.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u2} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u2} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} β _inst_2)) (OrderHom.antisymmetrization.{u2, u1} α β _inst_1 _inst_2 f) (toAntisymmetrization.{u2} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1843 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.1845 : α) => LE.le.{u2} α (Preorder.toLE.{u2} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.1843 x._@.Mathlib.Order.Antisymmetrization._hyg.1845) (instIsPreorderLeToLE.{u2} α _inst_1) a)) (toAntisymmetrization.{u1} β (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.1860 : β) (x._@.Mathlib.Order.Antisymmetrization._hyg.1862 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Antisymmetrization._hyg.1860 x._@.Mathlib.Order.Antisymmetrization._hyg.1862) (instIsPreorderLeToLE.{u1} β _inst_2) (OrderHom.toFun.{u2, u1} α β _inst_1 _inst_2 f a))
Case conversion may be inaccurate. Consider using '#align order_hom.antisymmetrization_apply_mk OrderHom.antisymmetrization_apply_mkₓ'. -/
@[simp]
theorem OrderHom.antisymmetrization_apply_mk (f : α →o β) (a : α) :
    f.Antisymmetrization (toAntisymmetrization _ a) = toAntisymmetrization _ (f a) :=
  Quotient.map'_mk'' f (liftFun_antisymm_rel f) _
#align order_hom.antisymmetrization_apply_mk OrderHom.antisymmetrization_apply_mk

variable (α)

/- warning: order_embedding.of_antisymmetrization -> OrderEmbedding.ofAntisymmetrization is a dubious translation:
lean 3 declaration is
  forall (α : Type.{u1}) [_inst_1 : Preorder.{u1} α], OrderEmbedding.{u1, u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) α (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1))) (Preorder.toLE.{u1} α _inst_1)
but is expected to have type
  forall (α : Type.{u1}) [_inst_1 : Preorder.{u1} α], OrderEmbedding.{u1, u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2074 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2076 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2074 x._@.Mathlib.Order.Antisymmetrization._hyg.2076) (instIsPreorderLeToLE.{u1} α _inst_1)) α (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2074 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2076 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2074 x._@.Mathlib.Order.Antisymmetrization._hyg.2076) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2074 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2076 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2074 x._@.Mathlib.Order.Antisymmetrization._hyg.2076) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1))) (Preorder.toLE.{u1} α _inst_1)
Case conversion may be inaccurate. Consider using '#align order_embedding.of_antisymmetrization OrderEmbedding.ofAntisymmetrizationₓ'. -/
/-- `of_antisymmetrization` as an order embedding. -/
@[simps]
noncomputable def OrderEmbedding.ofAntisymmetrization : Antisymmetrization α (· ≤ ·) ↪o α :=
  { Quotient.out'RelEmbedding _ with toFun := ofAntisymmetrization _ }
#align order_embedding.of_antisymmetrization OrderEmbedding.ofAntisymmetrization

/- warning: order_iso.dual_antisymmetrization -> OrderIso.dualAntisymmetrization is a dubious translation:
lean 3 declaration is
  forall (α : Type.{u1}) [_inst_1 : Preorder.{u1} α], OrderIso.{u1, u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (OrderDual.hasLe.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))))
but is expected to have type
  forall (α : Type.{u1}) [_inst_1 : Preorder.{u1} α], OrderIso.{u1, u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (OrderDual.instLEOrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1)))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))))
Case conversion may be inaccurate. Consider using '#align order_iso.dual_antisymmetrization OrderIso.dualAntisymmetrizationₓ'. -/
/-- `antisymmetrization` and `order_dual` commute. -/
def OrderIso.dualAntisymmetrization :
    (Antisymmetrization α (· ≤ ·))ᵒᵈ ≃o Antisymmetrization αᵒᵈ (· ≤ ·)
    where
  toFun := Quotient.map' id fun _ _ => And.symm
  invFun := Quotient.map' id fun _ _ => And.symm
  left_inv a := Quotient.inductionOn' a fun a => by simp_rw [Quotient.map'_mk'', id]
  right_inv a := Quotient.inductionOn' a fun a => by simp_rw [Quotient.map'_mk'', id]
  map_rel_iff' a b := Quotient.inductionOn₂' a b fun a b => Iff.rfl
#align order_iso.dual_antisymmetrization OrderIso.dualAntisymmetrization

/- warning: order_iso.dual_antisymmetrization_apply -> OrderIso.dualAntisymmetrization_apply is a dubious translation:
lean 3 declaration is
  forall (α : Type.{u1}) [_inst_1 : Preorder.{u1} α] (a : α), Eq.{succ u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (OrderDual.hasLe.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))))) (fun (_x : RelIso.{u1, u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (LE.le.{u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (OrderDual.hasLe.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1))))) (LE.le.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))))) => (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) -> (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1))) (RelIso.hasCoeToFun.{u1, u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (LE.le.{u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (OrderDual.hasLe.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1))))) (LE.le.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))))) (OrderIso.dualAntisymmetrization.{u1} α _inst_1) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)))) (fun (_x : Equiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)))) => (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) -> (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)))) (Equiv.hasCoeToFun.{succ u1, succ u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)))) (OrderDual.toDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) a))) (toAntisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (OrderDual.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (OrderDual.{u1} α)) => α -> (OrderDual.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (OrderDual.{u1} α)) (OrderDual.toDual.{u1} α) a))
but is expected to have type
  forall (α : Type.{u1}) [_inst_1 : Preorder.{u1} α] (a : α), Eq.{succ u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) => Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)))) (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (fun (a : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) => OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) a) (Equiv.instFunLikeEquiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)))) (OrderDual.toDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1) a))) (FunLike.coe.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (fun (_x : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) => Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) _x) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (Function.instEmbeddingLikeEmbedding.{succ u1, succ u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))))) (RelEmbedding.toEmbedding.{u1, u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) => LE.le.{u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (OrderDual.instLEOrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) => LE.le.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) => LE.le.{u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (OrderDual.instLEOrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) => LE.le.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.dualAntisymmetrization.{u1} α _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)))) (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (fun (_x : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) => OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)))) (OrderDual.toDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1) a))) (toAntisymmetrization.{u1} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => OrderDual.{u1} α) a) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (OrderDual.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => OrderDual.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (OrderDual.{u1} α)) (OrderDual.toDual.{u1} α) a))
Case conversion may be inaccurate. Consider using '#align order_iso.dual_antisymmetrization_apply OrderIso.dualAntisymmetrization_applyₓ'. -/
@[simp]
theorem OrderIso.dualAntisymmetrization_apply (a : α) :
    OrderIso.dualAntisymmetrization _ (toDual <| toAntisymmetrization _ a) =
      toAntisymmetrization _ (toDual a) :=
  rfl
#align order_iso.dual_antisymmetrization_apply OrderIso.dualAntisymmetrization_apply

/- warning: order_iso.dual_antisymmetrization_symm_apply -> OrderIso.dualAntisymmetrization_symm_apply is a dubious translation:
lean 3 declaration is
  forall (α : Type.{u1}) [_inst_1 : Preorder.{u1} α] (a : α), Eq.{succ u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (coeFn.{succ u1, succ u1} (OrderIso.{u1, u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))) (OrderDual.hasLe.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1))))) (fun (_x : RelIso.{u1, u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (LE.le.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))))) (LE.le.{u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (OrderDual.hasLe.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)))))) => (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) -> (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)))) (RelIso.hasCoeToFun.{u1, u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (LE.le.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))))) (LE.le.{u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (OrderDual.hasLe.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)))))) (OrderIso.symm.{u1, u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (OrderDual.hasLe.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} α _inst_1)))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1)) (Antisymmetrization.partialOrder.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))) (OrderIso.dualAntisymmetrization.{u1} α _inst_1)) (toAntisymmetrization.{u1} (OrderDual.{u1} α) (LE.le.{u1} (OrderDual.{u1} α) (OrderDual.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1))) (OrderIso.dualAntisymmetrization._proof_1.{u1} α _inst_1) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} α (OrderDual.{u1} α)) (fun (_x : Equiv.{succ u1, succ u1} α (OrderDual.{u1} α)) => α -> (OrderDual.{u1} α)) (Equiv.hasCoeToFun.{succ u1, succ u1} α (OrderDual.{u1} α)) (OrderDual.toDual.{u1} α) a))) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)))) (fun (_x : Equiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)))) => (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) -> (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)))) (Equiv.hasCoeToFun.{succ u1, succ u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1)))) (OrderDual.toDual.{u1} (Antisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1))) (toAntisymmetrization.{u1} α (LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1)) (LE.le.isPreorder.{u1} α _inst_1) a))
but is expected to have type
  forall (α : Type.{u1}) [_inst_1 : Preorder.{u1} α] (a : α), Eq.{succ u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) => OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (toAntisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (OrderDual.{u1} α)) α (fun (a : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => OrderDual.{u1} α) a) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (OrderDual.{u1} α)) (OrderDual.toDual.{u1} α) a))) (FunLike.coe.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (fun (_x : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) => OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) _x) (EmbeddingLike.toFunLike.{succ u1, succ u1, succ u1} (Function.Embedding.{succ u1, succ u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (Function.instEmbeddingLikeEmbedding.{succ u1, succ u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))))) (RelEmbedding.toEmbedding.{u1, u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) => LE.le.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) => LE.le.{u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (OrderDual.instLEOrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) => LE.le.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) => LE.le.{u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (OrderDual.instLEOrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1)))) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u1, u1} (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (OrderDual.instLEOrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (Preorder.toLE.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} α _inst_1)))) (Preorder.toLE.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (PartialOrder.toPreorder.{u1} (Antisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1))) (instPartialOrderAntisymmetrizationLeToLEInstIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)))) (OrderIso.dualAntisymmetrization.{u1} α _inst_1)))) (toAntisymmetrization.{u1} (OrderDual.{u1} α) (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2167 : OrderDual.{u1} α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2169 : OrderDual.{u1} α) => LE.le.{u1} (OrderDual.{u1} α) (OrderDual.instLEOrderDual.{u1} α (Preorder.toLE.{u1} α _inst_1)) x._@.Mathlib.Order.Antisymmetrization._hyg.2167 x._@.Mathlib.Order.Antisymmetrization._hyg.2169) (instIsPreorderLeToLE.{u1} (OrderDual.{u1} α) (OrderDual.preorder.{u1} α _inst_1)) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} α (OrderDual.{u1} α)) α (fun (_x : α) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : α) => OrderDual.{u1} α) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} α (OrderDual.{u1} α)) (OrderDual.toDual.{u1} α) a))) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)))) (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (fun (_x : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) => OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)) (OrderDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1)))) (OrderDual.toDual.{u1} (Antisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1))) (toAntisymmetrization.{u1} α (fun (x._@.Mathlib.Order.Antisymmetrization._hyg.2147 : α) (x._@.Mathlib.Order.Antisymmetrization._hyg.2149 : α) => LE.le.{u1} α (Preorder.toLE.{u1} α _inst_1) x._@.Mathlib.Order.Antisymmetrization._hyg.2147 x._@.Mathlib.Order.Antisymmetrization._hyg.2149) (instIsPreorderLeToLE.{u1} α _inst_1) a))
Case conversion may be inaccurate. Consider using '#align order_iso.dual_antisymmetrization_symm_apply OrderIso.dualAntisymmetrization_symm_applyₓ'. -/
@[simp]
theorem OrderIso.dualAntisymmetrization_symm_apply (a : α) :
    (OrderIso.dualAntisymmetrization _).symm (toAntisymmetrization _ <| toDual a) =
      toDual (toAntisymmetrization _ a) :=
  rfl
#align order_iso.dual_antisymmetrization_symm_apply OrderIso.dualAntisymmetrization_symm_apply

end Preorder