import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Finset.Basic
import Mathlib.Algebra.BigOperators.Basic

/-!
Self-contained slice for Chapter 11.4 target theorem.
-/

namespace Chapter11

/-- Bounded intervals, as in 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

/-- There is a technical issue in that this coercion is not injective: the empty set is represented by multiple bounded intervals.  This causes some of the statements in this section to be a little uglier than necessary.-/
@[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

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

/-- Constant maps and their values (Section 11.2). -/
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)

/-- Piecewise constant functions (Section 11.2). -/
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

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

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

/-- Shorthand used elsewhere in the chapter. -/
noncomputable abbrev integ' {f:ℝ → ℝ} {I: BoundedInterval} (_:PiecewiseConstantOn f I) := integ f I
end PiecewiseConstantOn

/-- Majorization/minorization (Section 11.3). -/
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

/-- Upper and lower integrals (Section 11.3). -/
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})

/-- The Riemann integral is the upper integral (junk value convention) (Section 11.3). -/
noncomputable abbrev integ (f:ℝ → ℝ) (I: BoundedInterval) : ℝ := upper_integral f I

end Chapter11

namespace Chapter9
/-- Boundedness on a set (from Section 9, repeated in Section 11.4). -/
abbrev BddOn (f : ℝ → ℝ) (X : Set ℝ) : Prop :=
  ∃ M, ∀ x ∈ X, |f x| ≤ M
end Chapter9

namespace Chapter11
open Chapter9

/-- Integrable functions (bounded with coinciding lower/upper integrals). -/
noncomputable abbrev IntegrableOn (f:ℝ → ℝ) (I: BoundedInterval) : Prop :=
  BddOn f I ∧ lower_integral f I = upper_integral f I

/-- Theorem 11.4.1(d) / Exercise 11.4.1 -/
theorem IntegrableOn.nonneg {I: BoundedInterval} {f:ℝ → ℝ} (hf: IntegrableOn f I) (hf_nonneg: ∀ x ∈ I, 0 ≤ f x) :
  0 ≤ integ f I := by
end Chapter11