import Mathlib
import Analysis.Section_8_4
import Analysis.Tags.Section_8_4

namespace Chapter8

-- <CONTEXT>

def PartialOrder.mk {X:Type} [LE X]
  (hrefl: ∀ x:X, x ≤ x)
  (hantisymm: ∀ x y:X, x ≤ y → y ≤ x → x = y)
  (htrans: ∀ x y z:X, x ≤ y → y ≤ z → x ≤ z) : PartialOrder X :=
{
  le := (· ≤ ·)
  le_refl := hrefl
  le_antisymm := hantisymm
  le_trans := htrans
}

def IsTotal (X:Type) [PartialOrder X] : Prop := ∀ x y:X, x ≤ y ∨ y ≤ x

open Classical in
noncomputable def LinearOrder.mk {X:Type} [PartialOrder X]
  (htotal: IsTotal X) : LinearOrder X :=
{
   le_total := htotal
   toDecidableLE := decRel LE.le
}

noncomputable def LinearOrder.subtype {X:Type} [LinearOrder X] (A: Set X) : LinearOrder A :=
LinearOrder.mk (by
  sorry
  )

theorem IsTotal.subtype {X:Type} [PartialOrder X] {A: Set X} (hA: IsTotal X) : IsTotal A := by
  sorry

theorem IsTotal.subset {X:Type} [PartialOrder X] {A B: Set X} (hA: IsTotal A) (hAB: B ⊆ A) : IsTotal B := by
  sorry

abbrev X_8_5_4 : Set (Set ℕ) := { {1,2}, {2}, {2,3}, {2,3,4}, {5} }

theorem IsMax.iff {X:Type} [PartialOrder X] (x:X) :
  IsMax x ↔ ¬ ∃ y, x < y := by
  sorry

theorem IsMin.iff {X:Type} [PartialOrder X] (x:X) :
  IsMin x ↔ ¬ ∃ y, x > y := by
  sorry

theorem WellFoundedLT.iff (X:Type) [LinearOrder X] :
  WellFoundedLT X ↔ ∀ A:Set X, A.Nonempty → ∃ x:A, IsMin x := by
  sorry

theorem WellFoundedLT.iff' {X:Type} [PartialOrder X] (h: IsTotal X) :
  WellFoundedLT X ↔ ∀ A:Set X, A.Nonempty → ∃ x:A, IsMin x := by
  sorry

theorem IsMin.ofFinite {X:Type} [LinearOrder X] [Finite X] [Nonempty X] : ∃ x:X, IsMin x := by
  sorry

theorem WellFoundedLT.subset {X:Type} [PartialOrder X] {A B: Set X} (hA: IsTotal A) [hwell: WellFoundedLT A] (hAB: B ⊆ A) : WellFoundedLT B := by
  sorry

abbrev IsUpperBound {X:Type} [PartialOrder X] (A:Set X) (x:X) : Prop :=
  ∀ y ∈ A, y ≤ x

theorem IsUpperBound.iff {X:Type} [PartialOrder X] (A:Set X) (x:X) :
  IsUpperBound A x ↔ x ∈ upperBounds A := by
  sorry

abbrev IsStrictUpperBound {X:Type} [PartialOrder X] (A:Set X) (x:X) : Prop :=
  IsUpperBound A x ∧ x ∉ A

theorem IsStrictUpperBound.iff {X:Type} [PartialOrder X] (A:Set X) (x:X) :
  IsStrictUpperBound A x ↔ ∀ y ∈ A, y < x := by
  sorry

theorem IsStrictUpperBound.iff' {X:Type} [PartialOrder X] (A:Set X) (x:X) :
  IsStrictUpperBound A x ↔ x ∈ upperBounds A \ A := by
  sorry

theorem IsMin.iff_lowerbound {X:Type} [PartialOrder X] {Y: Set X} (hY: IsTotal Y) (x₀ : X) : (∃ hx₀ : x₀ ∈ Y, IsMin (⟨ x₀, hx₀ ⟩:Y)) ↔ x₀ ∈ Y ∧ ∀ x ∈ Y, x₀ ≤ x := by
  sorry

theorem IsMin.iff_lowerbound' {X:Type} [PartialOrder X] {Y: Set X} (hY: IsTotal Y) : (∃ x₀ : Y, IsMin x₀) ↔ ∃ x₀, x₀ ∈ Y ∧ ∀ x ∈ Y, x₀ ≤ x := by
  sorry

theorem WellFoundedLT.partialOrder {X:Type} [PartialOrder X] (x₀ : X) : ∃ Y : Set X, IsTotal Y ∧ WellFoundedLT Y ∧ (∃ hx₀ : x₀ ∈ Y, IsMin (⟨ x₀, hx₀ ⟩: Y)) ∧ ¬ ∃ x, IsStrictUpperBound Y x := by
  sorry

def empty_set_partial_order [h₀: LE Empty] : Decidable (∃ h : PartialOrder Empty, h.le = h₀.le) := by
  sorry

def empty_set_linear_order [h₀: LE Empty] : Decidable (∃ h : LinearOrder Empty, h.le = h₀.le) := by
  sorry

def empty_set_well_order [h₀: LT Empty]: Decidable (Nonempty (WellFoundedLT Empty)) := by
  sorry

def Ex_8_5_5_b : Decidable (∀ (X Y:Type) (h: LinearOrder Y) (f:X → Y), ∃ h₀: LinearOrder X, h₀.le = (fun x y ↦ f x < f y ∨ x = y)) := by
  sorry

abbrev OrderIdeals (X: Type) [PartialOrder X] : Set (Set X) := .Iic '' (.univ : Set X)

def OrderIdeals.iso {X: Type} [PartialOrder X] : X ≃o OrderIdeals X := {
  toFun x := ⟨ .Iic x, by simp ⟩
  invFun := by sorry
  left_inv := by sorry
  right_inv := by sorry
  map_rel_iff' := by sorry
  }

def Lex' (α : Type) := α

instance Lex'.partialOrder {X Y: Type} [PartialOrder X] [PartialOrder Y] : PartialOrder (Lex' (X × Y)) := {
  le := fun ⟨ x, y ⟩ ⟨ x', y' ⟩ ↦ (x < x') ∨ (x = x' ∧ y ≤ y')
  le_refl := by sorry
  le_antisymm := by sorry
  le_trans := by sorry
}

instance Lex'.linearOrder {X Y:Type} [LinearOrder X] [LinearOrder Y] : LinearOrder (Lex' (X × Y)) := by
  sorry

instance Lex'.WellFoundedLT {X Y:Type} [LinearOrder X] [WellFoundedLT X] [LinearOrder Y] [WellFoundedLT Y]:
  WellFoundedLT (Lex' (X × Y)) := by
  sorry

-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Exercise_8_5_5 {X Y:Type} [PartialOrder Y] (f:X → Y) :
  ∃ h₀: PartialOrder X, h₀.le = (fun x y ↦ f x < f y ∨ x = y) := by
-- </TARGET_THEOREM>

end Chapter8