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

namespace Chapter11

/-- Bounded intervals, from Section 11.1 -/ 
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]
lemma BoundedInterval.coe_empty : ((∅ : BoundedInterval):Set ℝ) = ∅ := by
  simp [toSet]

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

@[simp]
lemma BoundedInterval.set_Ioo (a b:ℝ) : (Ioo a b : Set ℝ) = .Ioo a b := by rfl
@[simp]
lemma BoundedInterval.set_Icc (a b:ℝ) : (Icc a b : Set ℝ) = .Icc a b := by rfl
@[simp]
lemma BoundedInterval.set_Ioc (a b:ℝ) : (Ioc a b : Set ℝ) = .Ioc a b := by rfl
@[simp]
lemma BoundedInterval.set_Ico (a b:ℝ) : (Ico a b : Set ℝ) = .Ico a b := by rfl

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

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

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

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

noncomputable abbrev right_lim (f: ℝ → ℝ) (x₀:ℝ) : ℝ :=
  lim ((nhdsWithin x₀ (.Ioi x₀)).map f)

noncomputable abbrev left_lim (f: ℝ → ℝ) (x₀:ℝ) : ℝ :=
  lim ((nhdsWithin x₀ (.Iio x₀)).map f)

noncomputable abbrev α_length (α: ℝ → ℝ) (I: BoundedInterval) : ℝ := match I with
| Icc a b => if a ≤ b then (right_lim α b) - (left_lim α a) else 0
| Ico a b => if a ≤ b then (left_lim α b) - (left_lim α a) else 0
| Ioc a b => if a ≤ b then (right_lim α b) - (right_lim α a) else 0
| Ioo a b => if a < b then (left_lim α b) - (right_lim α a) else 0

notation3:max α"["I"]ₗ" => α_length α I

@[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 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.RS_integ (f:ℝ → ℝ) {I: BoundedInterval} (P: Partition I) (α: ℝ → ℝ)   :
  ℝ := ∑ J ∈ P.intervals, constant_value_on f (J:Set ℝ) * α[J]ₗ

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

-- We open the namespace so the unqualified `RS_integ` in the target statement resolves correctly.
open Chapter11.PiecewiseConstantOn

open Classical in
-- Theorem 11.8.8 (g) (Laws of RS integration) / Exercise 11.8.8
theorem PiecewiseConstantOn.RS_integ_of_extend {I J: BoundedInterval} (hIJ: I ⊆ J)
  {f: ℝ → ℝ} (h: PiecewiseConstantOn f I) {α:ℝ → ℝ} (hα: Monotone α):
  RS_integ (fun x ↦ if x ∈ I then f x else 0) J α = RS_integ f I α := by
end Chapter11