/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne

! This file was ported from Lean 3 source module data.set.intervals.order_iso
! leanprover-community/mathlib commit c3291da49cfa65f0d43b094750541c0731edc932
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Data.Set.Intervals.Basic
import Mathbin.Order.Hom.Set

/-!
# Lemmas about images of intervals under order isomorphisms.

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

variable {α β : Type _}

open Set

namespace OrderIso

section Preorder

variable [Preorder α] [Preorder β]

/- warning: order_iso.preimage_Iic -> OrderIso.preimage_Iic is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align order_iso.preimage_Iic OrderIso.preimage_Iicₓ'. -/
@[simp]
theorem preimage_Iic (e : α ≃o β) (b : β) : e ⁻¹' Iic b = Iic (e.symm b) :=
  by
  ext x
  simp [← e.le_iff_le]
#align order_iso.preimage_Iic OrderIso.preimage_Iic

/- warning: order_iso.preimage_Ici -> OrderIso.preimage_Ici is a dubious translation:
lean 3 declaration is
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but is expected to have type
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Case conversion may be inaccurate. Consider using '#align order_iso.preimage_Ici OrderIso.preimage_Iciₓ'. -/
@[simp]
theorem preimage_Ici (e : α ≃o β) (b : β) : e ⁻¹' Ici b = Ici (e.symm b) :=
  by
  ext x
  simp [← e.le_iff_le]
#align order_iso.preimage_Ici OrderIso.preimage_Ici

/- warning: order_iso.preimage_Iio -> OrderIso.preimage_Iio is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align order_iso.preimage_Iio OrderIso.preimage_Iioₓ'. -/
@[simp]
theorem preimage_Iio (e : α ≃o β) (b : β) : e ⁻¹' Iio b = Iio (e.symm b) :=
  by
  ext x
  simp [← e.lt_iff_lt]
#align order_iso.preimage_Iio OrderIso.preimage_Iio

/- warning: order_iso.preimage_Ioi -> OrderIso.preimage_Ioi is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.preimage_Ioi OrderIso.preimage_Ioiₓ'. -/
@[simp]
theorem preimage_Ioi (e : α ≃o β) (b : β) : e ⁻¹' Ioi b = Ioi (e.symm b) :=
  by
  ext x
  simp [← e.lt_iff_lt]
#align order_iso.preimage_Ioi OrderIso.preimage_Ioi

/- warning: order_iso.preimage_Icc -> OrderIso.preimage_Icc is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.preimage_Icc OrderIso.preimage_Iccₓ'. -/
@[simp]
theorem preimage_Icc (e : α ≃o β) (a b : β) : e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b) := by
  simp [← Ici_inter_Iic]
#align order_iso.preimage_Icc OrderIso.preimage_Icc

/- warning: order_iso.preimage_Ico -> OrderIso.preimage_Ico is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.preimage_Ico OrderIso.preimage_Icoₓ'. -/
@[simp]
theorem preimage_Ico (e : α ≃o β) (a b : β) : e ⁻¹' Ico a b = Ico (e.symm a) (e.symm b) := by
  simp [← Ici_inter_Iio]
#align order_iso.preimage_Ico OrderIso.preimage_Ico

/- warning: order_iso.preimage_Ioc -> OrderIso.preimage_Ioc is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.preimage_Ioc OrderIso.preimage_Iocₓ'. -/
@[simp]
theorem preimage_Ioc (e : α ≃o β) (a b : β) : e ⁻¹' Ioc a b = Ioc (e.symm a) (e.symm b) := by
  simp [← Ioi_inter_Iic]
#align order_iso.preimage_Ioc OrderIso.preimage_Ioc

/- warning: order_iso.preimage_Ioo -> OrderIso.preimage_Ioo is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.preimage_Ioo OrderIso.preimage_Iooₓ'. -/
@[simp]
theorem preimage_Ioo (e : α ≃o β) (a b : β) : e ⁻¹' Ioo a b = Ioo (e.symm a) (e.symm b) := by
  simp [← Ioi_inter_Iio]
#align order_iso.preimage_Ioo OrderIso.preimage_Ioo

/- warning: order_iso.image_Iic -> OrderIso.image_Iic is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align order_iso.image_Iic OrderIso.image_Iicₓ'. -/
@[simp]
theorem image_Iic (e : α ≃o β) (a : α) : e '' Iic a = Iic (e a) := by
  rw [e.image_eq_preimage, e.symm.preimage_Iic, e.symm_symm]
#align order_iso.image_Iic OrderIso.image_Iic

/- warning: order_iso.image_Ici -> OrderIso.image_Ici is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.image_Ici OrderIso.image_Iciₓ'. -/
@[simp]
theorem image_Ici (e : α ≃o β) (a : α) : e '' Ici a = Ici (e a) :=
  e.dual.image_Iic a
#align order_iso.image_Ici OrderIso.image_Ici

/- warning: order_iso.image_Iio -> OrderIso.image_Iio is a dubious translation:
lean 3 declaration is
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Case conversion may be inaccurate. Consider using '#align order_iso.image_Iio OrderIso.image_Iioₓ'. -/
@[simp]
theorem image_Iio (e : α ≃o β) (a : α) : e '' Iio a = Iio (e a) := by
  rw [e.image_eq_preimage, e.symm.preimage_Iio, e.symm_symm]
#align order_iso.image_Iio OrderIso.image_Iio

/- warning: order_iso.image_Ioi -> OrderIso.image_Ioi is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.image_Ioi OrderIso.image_Ioiₓ'. -/
@[simp]
theorem image_Ioi (e : α ≃o β) (a : α) : e '' Ioi a = Ioi (e a) :=
  e.dual.image_Iio a
#align order_iso.image_Ioi OrderIso.image_Ioi

/- warning: order_iso.image_Ioo -> OrderIso.image_Ioo is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.image_Ioo OrderIso.image_Iooₓ'. -/
@[simp]
theorem image_Ioo (e : α ≃o β) (a b : α) : e '' Ioo a b = Ioo (e a) (e b) := by
  rw [e.image_eq_preimage, e.symm.preimage_Ioo, e.symm_symm]
#align order_iso.image_Ioo OrderIso.image_Ioo

/- warning: order_iso.image_Ioc -> OrderIso.image_Ioc is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.image_Ioc OrderIso.image_Iocₓ'. -/
@[simp]
theorem image_Ioc (e : α ≃o β) (a b : α) : e '' Ioc a b = Ioc (e a) (e b) := by
  rw [e.image_eq_preimage, e.symm.preimage_Ioc, e.symm_symm]
#align order_iso.image_Ioc OrderIso.image_Ioc

/- warning: order_iso.image_Ico -> OrderIso.image_Ico is a dubious translation:
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Case conversion may be inaccurate. Consider using '#align order_iso.image_Ico OrderIso.image_Icoₓ'. -/
@[simp]
theorem image_Ico (e : α ≃o β) (a b : α) : e '' Ico a b = Ico (e a) (e b) := by
  rw [e.image_eq_preimage, e.symm.preimage_Ico, e.symm_symm]
#align order_iso.image_Ico OrderIso.image_Ico

/- warning: order_iso.image_Icc -> OrderIso.image_Icc is a dubious translation:
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but is expected to have type
  forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : Preorder.{u2} α] [_inst_2 : Preorder.{u1} β] (e : OrderIso.{u2, u1} α β (Preorder.toLE.{u2} α _inst_1) (Preorder.toLE.{u1} β _inst_2)) (a : α) (b : α), Eq.{succ u1} (Set.{u1} β) (Set.image.{u2, u1} α β (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} α β) α β (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} α β)) (RelEmbedding.toEmbedding.{u2, u1} α β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} 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 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} α β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} 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 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Icc.{u2} α _inst_1 a b)) (Set.Icc.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) a) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} α β) α β (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} α β)) (RelEmbedding.toEmbedding.{u2, u1} α β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} 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 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} α β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} 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 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} α β) α (fun (_x : α) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : α) => β) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} α β) α β (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} α β)) (RelEmbedding.toEmbedding.{u2, u1} α β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} 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 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} α β (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : α) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : α) => LE.le.{u2} α (Preorder.toLE.{u2} 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 : β) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : β) => LE.le.{u1} β (Preorder.toLE.{u1} β _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) b))
Case conversion may be inaccurate. Consider using '#align order_iso.image_Icc OrderIso.image_Iccₓ'. -/
@[simp]
theorem image_Icc (e : α ≃o β) (a b : α) : e '' Icc a b = Icc (e a) (e b) := by
  rw [e.image_eq_preimage, e.symm.preimage_Icc, e.symm_symm]
#align order_iso.image_Icc OrderIso.image_Icc

end Preorder

/- warning: order_iso.Iic_top -> OrderIso.IicTop is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : OrderTop.{u1} α (Preorder.toLE.{u1} α _inst_1)], OrderIso.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.Iic.{u1} α _inst_1 (Top.top.{u1} α (OrderTop.toHasTop.{u1} α (Preorder.toLE.{u1} α _inst_1) _inst_2)))) α (Subtype.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1) (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.Iic.{u1} α _inst_1 (Top.top.{u1} α (OrderTop.toHasTop.{u1} α (Preorder.toLE.{u1} α _inst_1) _inst_2))))) (Preorder.toLE.{u1} α _inst_1)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : OrderTop.{u1} α (Preorder.toLE.{u1} α _inst_1)], OrderIso.{u1, u1} (Set.Elem.{u1} α (Set.Iic.{u1} α _inst_1 (Top.top.{u1} α (OrderTop.toTop.{u1} α (Preorder.toLE.{u1} α _inst_1) _inst_2)))) α (Subtype.le.{u1} α (Preorder.toLE.{u1} α _inst_1) (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (Set.Iic.{u1} α _inst_1 (Top.top.{u1} α (OrderTop.toTop.{u1} α (Preorder.toLE.{u1} α _inst_1) _inst_2))))) (Preorder.toLE.{u1} α _inst_1)
Case conversion may be inaccurate. Consider using '#align order_iso.Iic_top OrderIso.IicTopₓ'. -/
/-- Order isomorphism between `Iic (⊤ : α)` and `α` when `α` has a top element -/
def IicTop [Preorder α] [OrderTop α] : Set.Iic (⊤ : α) ≃o α :=
  { @Equiv.subtypeUnivEquiv α (Set.Iic (⊤ : α)) fun x => le_top with
    map_rel_iff' := fun x y => by rfl }
#align order_iso.Iic_top OrderIso.IicTop

/- warning: order_iso.Ici_bot -> OrderIso.IciBot is a dubious translation:
lean 3 declaration is
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : OrderBot.{u1} α (Preorder.toLE.{u1} α _inst_1)], OrderIso.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} α) Type.{u1} (Set.hasCoeToSort.{u1} α) (Set.Ici.{u1} α _inst_1 (Bot.bot.{u1} α (OrderBot.toHasBot.{u1} α (Preorder.toLE.{u1} α _inst_1) _inst_2)))) α (Subtype.hasLe.{u1} α (Preorder.toLE.{u1} α _inst_1) (fun (x : α) => Membership.Mem.{u1, u1} α (Set.{u1} α) (Set.hasMem.{u1} α) x (Set.Ici.{u1} α _inst_1 (Bot.bot.{u1} α (OrderBot.toHasBot.{u1} α (Preorder.toLE.{u1} α _inst_1) _inst_2))))) (Preorder.toLE.{u1} α _inst_1)
but is expected to have type
  forall {α : Type.{u1}} [_inst_1 : Preorder.{u1} α] [_inst_2 : OrderBot.{u1} α (Preorder.toLE.{u1} α _inst_1)], OrderIso.{u1, u1} (Set.Elem.{u1} α (Set.Ici.{u1} α _inst_1 (Bot.bot.{u1} α (OrderBot.toBot.{u1} α (Preorder.toLE.{u1} α _inst_1) _inst_2)))) α (Subtype.le.{u1} α (Preorder.toLE.{u1} α _inst_1) (fun (x : α) => Membership.mem.{u1, u1} α (Set.{u1} α) (Set.instMembershipSet.{u1} α) x (Set.Ici.{u1} α _inst_1 (Bot.bot.{u1} α (OrderBot.toBot.{u1} α (Preorder.toLE.{u1} α _inst_1) _inst_2))))) (Preorder.toLE.{u1} α _inst_1)
Case conversion may be inaccurate. Consider using '#align order_iso.Ici_bot OrderIso.IciBotₓ'. -/
/-- Order isomorphism between `Ici (⊥ : α)` and `α` when `α` has a bottom element -/
def IciBot [Preorder α] [OrderBot α] : Set.Ici (⊥ : α) ≃o α :=
  { @Equiv.subtypeUnivEquiv α (Set.Ici (⊥ : α)) fun x => bot_le with
    map_rel_iff' := fun x y => by rfl }
#align order_iso.Ici_bot OrderIso.IciBot

end OrderIso