import Mathlib
import Analysis.Section_9_6
import Analysis.Tags.Section_9_6
import Analysis.Section_11_3
import Analysis.Tags.Section_11_3

namespace Chapter11
-- <CONTEXT>
inductive BoundedInterval where
  | Ioo (a b:ℝ) : BoundedInterval
  | Icc (a b:ℝ) : BoundedInterval
  | Ioc (a b:ℝ) : BoundedInterval
  | Ico (a b:ℝ) : BoundedInterval

open BoundedInterval

@[coe]
def BoundedInterval.toSet (I: BoundedInterval) : Set ℝ := match I with
  | Ioo a b => .Ioo a b
  | Icc a b => .Icc a b
  | Ioc a b => .Ioc a b
  | Ico a b => .Ico a b

instance BoundedInterval.inst_coeSet : Coe BoundedInterval (Set ℝ) where
  coe := toSet

instance BoundedInterval.instEmpty : EmptyCollection BoundedInterval where
  emptyCollection := Ioo 0 0

@[simp]
theorem BoundedInterval.coe_empty : ((∅ : BoundedInterval):Set ℝ) = ∅ := by
  sorry

open Classical in
noncomputable instance BoundedInterval.decidableEq : DecidableEq BoundedInterval := instDecidableEqOfLawfulBEq

@[simp]
theorem BoundedInterval.set_Ioo (a b:ℝ) : (Ioo a b : Set ℝ) = .Ioo a b := by
  sorry

@[simp]
theorem BoundedInterval.set_Icc (a b:ℝ) : (Icc a b : Set ℝ) = .Icc a b := by
  sorry

@[simp]
theorem BoundedInterval.set_Ioc (a b:ℝ) : (Ioc a b : Set ℝ) = .Ioc a b := by
  sorry

@[simp]
theorem BoundedInterval.set_Ico (a b:ℝ) : (Ico a b : Set ℝ) = .Ico a b := by
  sorry

abbrev BoundedInterval.a (I: BoundedInterval) : ℝ := match I with
  | Ioo a _ => a
  | Icc a _ => a
  | Ioc a _ => a
  | Ico a _ => a

abbrev BoundedInterval.b (I: BoundedInterval) : ℝ := match I with
  | Ioo _ b => b
  | Icc _ b => b
  | Ioc _ b => b
  | Ico _ b => b

abbrev BoundedInterval.length (I: BoundedInterval) : ℝ := max (I.b - I.a) 0

macro:max atomic("|" noWs) a:term noWs "|ₗ" : term => `(BoundedInterval.length $a)

instance BoundedInterval.instMembership : Membership ℝ BoundedInterval where
  mem I x := x ∈ (I:Set ℝ)

theorem BoundedInterval.mem_iff (I: BoundedInterval) (x:ℝ) :
  x ∈ I ↔ x ∈ (I:Set ℝ) := by
  sorry

instance BoundedInterval.instSubset : HasSubset BoundedInterval where
  Subset I J := ∀ x, x ∈ I → x ∈ J

theorem BoundedInterval.subset_iff (I J: BoundedInterval) :
  I ⊆ J ↔ (I:Set ℝ) ⊆ (J:Set ℝ) := by
  sorry

@[ext]
structure Partition (I: BoundedInterval) where
  intervals : Finset BoundedInterval
  exists_unique (x:ℝ) (hx : x ∈ I) : ∃! J, J ∈ intervals ∧ x ∈ J
  contains (J : BoundedInterval) (hJ : J ∈ intervals) : J ⊆ I

instance Partition.instMembership (I: BoundedInterval) : Membership BoundedInterval (Partition I) where
  mem P J := J ∈ P.intervals

abbrev Constant {X Y:Type} (f: X → Y) : Prop := ∃ c, ∀ x, f x = c

open Classical in
noncomputable abbrev constant_value {X Y:Type} [hY: Nonempty Y] (f:X → Y) : Y :=
  if h: Constant f then h.choose else hY.some

abbrev ConstantOn (f: ℝ → ℝ) (X: Set ℝ) : Prop := Constant (fun x : X ↦ f ↑x)

noncomputable abbrev constant_value_on (f:ℝ → ℝ) (X: Set ℝ) : ℝ := constant_value (fun x : X ↦ f ↑x)

abbrev PiecewiseConstantWith (f:ℝ → ℝ) {I: BoundedInterval} (P: Partition I) : Prop := ∀ J ∈ P, ConstantOn f (J:Set ℝ)

abbrev PiecewiseConstantOn (f:ℝ → ℝ) (I: BoundedInterval) : Prop := ∃ P : Partition I, PiecewiseConstantWith f P

noncomputable abbrev PiecewiseConstantWith.integ (f:ℝ → ℝ) {I: BoundedInterval} (P: Partition I)  :
  ℝ := ∑ J ∈ P.intervals, constant_value_on f (J:Set ℝ) * |J|ₗ

open Classical in
noncomputable abbrev PiecewiseConstantOn.integ (f:ℝ → ℝ) (I: BoundedInterval) :
  ℝ := if h: PiecewiseConstantOn f I then PiecewiseConstantWith.integ f h.choose else 0

abbrev Chapter9.BddAboveOn (f:ℝ → ℝ) (X:Set ℝ) : Prop := ∃ M, ∀ x ∈ X, f x ≤ M

abbrev Chapter9.BddBelowOn (f:ℝ → ℝ) (X:Set ℝ) : Prop := ∃ M, ∀ x ∈ X, -M ≤ f x

abbrev Chapter9.BddOn (f:ℝ → ℝ) (X:Set ℝ) : Prop := ∃ M, ∀ x ∈ X, |f x| ≤ M

open Chapter9

abbrev MajorizesOn (g f:ℝ → ℝ) (I: BoundedInterval) : Prop := ∀ x ∈ (I:Set ℝ), f x ≤ g x

abbrev MinorizesOn (g f:ℝ → ℝ) (I: BoundedInterval) : Prop := ∀ x ∈ (I:Set ℝ), g x ≤ f x

noncomputable abbrev upper_integral (f:ℝ → ℝ) (I: BoundedInterval) : ℝ :=
  sInf ((PiecewiseConstantOn.integ · I) '' {g | MajorizesOn g f I ∧ PiecewiseConstantOn g I})

noncomputable abbrev lower_integral (f:ℝ → ℝ) (I: BoundedInterval) : ℝ :=
  sSup ((PiecewiseConstantOn.integ · I) '' {g | MinorizesOn g f I ∧ PiecewiseConstantOn g I})

noncomputable abbrev integ (f:ℝ → ℝ) (I: BoundedInterval) : ℝ := upper_integral f I

noncomputable abbrev IntegrableOn (f:ℝ → ℝ) (I: BoundedInterval) : Prop :=
  BddOn f I ∧ lower_integral f I = upper_integral f I

theorem IntegrableOn.neg {I: BoundedInterval} {f:ℝ → ℝ} (hf: IntegrableOn f I) :
  IntegrableOn (-f) I ∧ integ (-f) I = -integ f I := by
  sorry

theorem IntegrableOn.mono' {I J: BoundedInterval} (hIJ: J ⊆ I)
  {f: ℝ → ℝ} (h: IntegrableOn f I) : IntegrableOn f J := by
  sorry

theorem IntegrableOn.eq {I J: BoundedInterval} (hIJ: J ⊆ I)
  (ha: J.a = I.a) (hb: J.b = I.b)
  {f: ℝ → ℝ} (h: IntegrableOn f I) : integ f J = integ f I := by
  sorry

lemma nonneg_of_le_const_mul_eps {x C:ℝ} (h: ∀ ε>0, x ≤ C * ε) : x ≤ 0 := by
  sorry

theorem IntegrableOn.max {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I) :
  IntegrableOn (f ⊔ g) I  := by
  sorry

theorem IntegrableOn.abs {I: BoundedInterval} {f:ℝ → ℝ} (hf: IntegrableOn f I) :
  IntegrableOn (abs f) I := by
  sorry

theorem integ_of_mul_nonneg {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I)
  (hf_nonneg: MajorizesOn f 0 I) (hg_nonneg: MajorizesOn g 0 I) :
  IntegrableOn (f * g) I := by
  sorry

theorem integ_of_mul {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I) :
  IntegrableOn (f * g) I := by
  sorry
open BoundedInterval
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem IntegrableOn.split {I: BoundedInterval} {f: ℝ → ℝ} (hf: IntegrableOn f I) (P: Partition I) :
  integ f I = ∑ J ∈ P.intervals, integ f J := by
-- </TARGET_THEOREM>

end Chapter11