import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.BigOperators.Basic

open scoped BigOperators

/-!
Self-contained slice for Chapter11 Section 11.4 target theorem.
-/

namespace Chapter11

/-! Section 11.1 essentials: Bounded intervals, coercions, length, partitions -/

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

/-- Using ||ₗ subscript here to not override || -/
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

/-! Section 11.2 essentials: Constant/ConstantOn and piecewise-constant integrals -/

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

/-! Section 9.6 essential: boundedness on sets -/
namespace Chapter9
abbrev BddOn (f : ℝ → ℝ) (X : Set ℝ) : Prop := ∃ M, ∀ x ∈ X, |f x| ≤ M
end Chapter9

/-! Section 11.3 essentials: order relations, upper/lower integrals and IntegrableOn -/

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

open Chapter9

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 11.4.1(f) / Exercise 11.4.1 -/ 
lemma IntegrableOn.const (c : ℝ) (I : BoundedInterval) :
  IntegrableOn (fun _ ↦ c) I ∧ integ (fun _ ↦ c) I = c * |I|ₗ := by
  sorry

/-- Theorem 11.4.1(f) / Exercise 11.4.1 -/
theorem IntegrableOn.const' {I: BoundedInterval} {f:ℝ → ℝ} (hf: ConstantOn f I) :
  IntegrableOn f I ∧ integ f I = (constant_value_on f I) * |I|ₗ := by
end Chapter11