import Mathlib
import Analysis.Section_9_6
import Analysis.Tags.Section_9_6
import Analysis.Section_11_2
import Analysis.Tags.Section_11_2

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

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

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)

@[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

open BigOperators

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

noncomputable abbrev PiecewiseConstantOn.integ' {f:ℝ → ℝ} {I: BoundedInterval} (_:PiecewiseConstantOn f I) := integ f I

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

open BoundedInterval 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

theorem MinorizesOn.iff (g f:ℝ → ℝ) (I: BoundedInterval) : MinorizesOn g f I ↔ MajorizesOn f g I := by
  sorry

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})

theorem upper_integral_congr {f g:ℝ → ℝ} {I: BoundedInterval} (h: Set.EqOn f g I) :
  upper_integral f I = upper_integral g I := by
  sorry

theorem lower_integral_congr {f g:ℝ → ℝ} {I: BoundedInterval} (h: Set.EqOn f g I) :
  lower_integral f I = lower_integral g I := by
  sorry

lemma integral_bound_upper_of_bounded {f:ℝ → ℝ} {M:ℝ} {I: BoundedInterval} (h: ∀ x ∈ (I:Set ℝ), |f x| ≤ M) : M * |I|ₗ ∈ (PiecewiseConstantOn.integ · I) '' {g | MajorizesOn g f I ∧ PiecewiseConstantOn g I} := by
  sorry

lemma integral_bound_lower_of_bounded {f:ℝ → ℝ} {M:ℝ} {I: BoundedInterval} (h: ∀ x ∈ (I:Set ℝ), |f x| ≤ M) : -M * |I|ₗ ∈ (PiecewiseConstantOn.integ · I) '' {g | MinorizesOn g f I ∧ PiecewiseConstantOn g I} := by
  sorry

lemma integral_bound_upper_nonempty {f:ℝ → ℝ} {I: BoundedInterval} (h: BddOn f I) : ((PiecewiseConstantOn.integ · I) '' {g | MajorizesOn g f I ∧ PiecewiseConstantOn g I}).Nonempty := by
  sorry

lemma integral_bound_lower_nonempty {f:ℝ → ℝ} {I: BoundedInterval} (h: BddOn f I) : ((PiecewiseConstantOn.integ · I) '' {g | MinorizesOn g f I ∧ PiecewiseConstantOn g I}).Nonempty := by
  sorry

lemma integral_bound_lower_le_upper {f:ℝ → ℝ} {I: BoundedInterval} {a b:ℝ}
  (ha: a ∈ (PiecewiseConstantOn.integ · I) '' {g | MajorizesOn g f I ∧ PiecewiseConstantOn g I})
  (hb: b ∈ (PiecewiseConstantOn.integ · I) '' {g | MinorizesOn g f I ∧ PiecewiseConstantOn g I})
  : b ≤ a := by
  sorry

lemma integral_bound_below {f:ℝ → ℝ} {I: BoundedInterval} (h: BddOn f I) :
  BddBelow ((PiecewiseConstantOn.integ · I) '' {g | MajorizesOn g f I ∧ PiecewiseConstantOn g I}) := by
  sorry

lemma integral_bound_above {f:ℝ → ℝ} {I: BoundedInterval} (h: BddOn f I) :
  BddAbove ((PiecewiseConstantOn.integ · I) '' {g | MinorizesOn g f I ∧ PiecewiseConstantOn g I}) := by
  sorry

lemma le_lower_integral {f:ℝ → ℝ} {I: BoundedInterval} {M:ℝ} (h: ∀ x ∈ (I:Set ℝ), |f x| ≤ M) :
  -M * |I|ₗ ≤ lower_integral f I := by
  sorry

lemma lower_integral_le_upper {f:ℝ → ℝ} {I: BoundedInterval} (h: BddOn f I) :
  lower_integral f I ≤ upper_integral f I := by
  sorry

lemma upper_integral_le {f:ℝ → ℝ} {I: BoundedInterval} {M:ℝ} (h: ∀ x ∈ (I:Set ℝ), |f x| ≤ M) :
  upper_integral f I ≤ M * |I|ₗ := by
  sorry

lemma upper_integral_le_integ {f g:ℝ → ℝ} {I: BoundedInterval} (hf: BddOn f I)
  (hfg: MajorizesOn g f I) (hg: PiecewiseConstantOn g I) :
  upper_integral f I ≤ hg.integ' := by
  sorry

lemma integ_le_lower_integral {f h:ℝ → ℝ} {I: BoundedInterval} (hf: BddOn f I)
  (hfh: MinorizesOn h f I) (hg: PiecewiseConstantOn h I) :
  hg.integ' ≤ lower_integral f I := by
  sorry

lemma lt_of_gt_upper_integral {f:ℝ → ℝ} {I: BoundedInterval} (hf: BddOn f I)
  {X:ℝ} (hX: upper_integral f I < X ) :
  ∃ g, MajorizesOn g f I ∧ PiecewiseConstantOn g I ∧ PiecewiseConstantOn.integ g I < X := by
  sorry

lemma gt_of_lt_lower_integral {f:ℝ → ℝ} {I: BoundedInterval} (hf: BddOn f I)
  {X:ℝ} (hX: X < lower_integral f I) :
  ∃ h, MinorizesOn h f I ∧ PiecewiseConstantOn h I ∧ X < PiecewiseConstantOn.integ h I := by
  sorry

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

theorem integ_congr {f g:ℝ → ℝ} {I: BoundedInterval} (h: Set.EqOn f g I) :
  integ f I = integ g I := by
  sorry

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

theorem integ_on_subsingleton {f:ℝ → ℝ} {I: BoundedInterval} (hI: |I|ₗ = 0) :
  IntegrableOn f I ∧ integ f I = 0 := by
  sorry

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

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

theorem lower_riemann_sum_ge {f h: ℝ → ℝ} {I:BoundedInterval} (P: Partition I)
  (hf: BddOn f I) (hfh: MinorizesOn h f I) (hg: PiecewiseConstantOn h I) :
  integ h I ≤ lower_riemann_sum f P := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem upper_integ_le_upper_sum {f:ℝ → ℝ} {I:BoundedInterval} (hf: BddOn f I)
  (P: Partition I): upper_integral f I ≤ upper_riemann_sum f P := by
-- </TARGET_THEOREM>

end Chapter11