import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Topology.Instances.Real

namespace Chapter11

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

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

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

@[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 (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) * (0:ℝ)

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

noncomputable abbrev PiecewiseConstantWith_RS_integ (f:ℝ → ℝ) {I: BoundedInterval} (P: Partition I) (α: ℝ → ℝ)   :
  ℝ := ∑ J ∈ P.intervals, constant_value_on f (J:Set ℝ) * (0:ℝ)

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

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

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

noncomputable abbrev upper_RS_integral (f:ℝ → ℝ) (I: BoundedInterval) (α: ℝ → ℝ): ℝ :=
  sInf ((PiecewiseConstantOn_RS_integ · I α) '' {g | MajorizesOn g f I ∧ PiecewiseConstantOn g I})

noncomputable abbrev RS_integ (f:ℝ → ℝ) (I: BoundedInterval) (α:ℝ → ℝ) : ℝ := upper_RS_integral f I α

open BoundedInterval

-- Lemma 11.10.5 / Exercise 11.10.2-
 theorem PiecewiseConstantOn.RS_integ_of_comp {a b:ℝ} (hab: a < b) {phi f:ℝ → ℝ}
  (hφ_cont: Continuous phi) (hφ_mono: Monotone phi) (hf: PiecewiseConstantOn f (Icc (phi a) (phi b))) :
  PiecewiseConstantOn (f ∘ phi) (Icc a b) ∧ RS_integ (f ∘ phi) (Icc a b) phi =
    integ f (Icc (phi a) (phi b)) := by
end Chapter11