import Mathlib
import Analysis.Section_11_1
import Analysis.Tags.Section_11_1

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

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

@[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

theorem Constant.eq {X Y:Type} {f: X → Y} [Nonempty Y] (h: Constant f) (x:X) :
  f x = constant_value f := by sorry

theorem Constant.of_const {X Y:Type} {f:X → Y} {c:Y} (h: ∀ x, f x = c) :
  Constant f := by sorry

theorem Constant.const_eq {X Y:Type} {f:X → Y} [hX: Nonempty X] [Nonempty Y] {c:Y} (h: ∀ x, f x = c) :
  constant_value f = c := by sorry

theorem Constant.of_subsingleton {X Y:Type} [hs: Subsingleton X] [hY: Nonempty Y] {f:X → Y} :
  Constant f := by sorry

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)

theorem ConstantOn.eq {f: ℝ → ℝ} {X: Set ℝ} (h: ConstantOn f X) {x:ℝ} (hx: x ∈ X) :
  f x = constant_value_on f X := by sorry

theorem ConstantOn.of_const {f:ℝ → ℝ} {X: Set ℝ} {c:ℝ} (h: ∀ x ∈ X, f x = c) :
  ConstantOn f X := by sorry

theorem ConstantOn.of_const' (c:ℝ) (X:Set ℝ): ConstantOn (fun _ ↦ c) X := by sorry

theorem ConstantOn.const_eq {f:ℝ → ℝ} {X: Set ℝ} (hX: X.Nonempty) {c:ℝ} (h: ∀ x ∈ X, f x = c) :
  constant_value_on f X = c := by sorry

theorem ConstantOn.congr {f g: ℝ → ℝ} {X: Set ℝ} (h: ∀ x ∈ X, f x = g x) : ConstantOn f X ↔ ConstantOn g X := by sorry

theorem ConstantOn.congr' {f g: ℝ → ℝ} {X: Set ℝ} (hf: ConstantOn f X) (h: ∀ x ∈ X, f x = g x) : ConstantOn g X := by sorry

theorem ConstantOn.of_subsingleton {f: ℝ → ℝ} {X: Set ℝ} [Subsingleton X] :
  ConstantOn f X := by sorry

theorem constant_value_on_congr {f g: ℝ → ℝ} {X: Set ℝ} (h: ∀ x ∈ X, f x = g x) :
  constant_value_on f X = constant_value_on g X := by sorry

abbrev PiecewiseConstantWith (f:ℝ → ℝ) {I: BoundedInterval} (P: Partition I) : Prop := ∀ J ∈ P, ConstantOn f (J:Set ℝ)

theorem PiecewiseConstantWith.def (f:ℝ → ℝ) {I: BoundedInterval} {P: Partition I} :
  PiecewiseConstantWith f P ↔ ∀ J ∈ P, ∃ c, ∀ x ∈ J, f x = c := by sorry

theorem PiecewiseConstantWith.congr {f g:ℝ → ℝ} {I: BoundedInterval} {P: Partition I}
  (h: ∀ x ∈ (I:Set ℝ), f x = g x) :
  PiecewiseConstantWith f P ↔ PiecewiseConstantWith g P := by sorry

abbrev PiecewiseConstantOn (f:ℝ → ℝ) (I: BoundedInterval) : Prop := ∃ P : Partition I, PiecewiseConstantWith f P

theorem PiecewiseConstantOn.def (f:ℝ → ℝ) (I: BoundedInterval):
  PiecewiseConstantOn f I ↔ ∃ P : Partition I, ∀ J ∈ P, ConstantOn f (J:Set ℝ) := by sorry

theorem PiecewiseConstantOn.congr {f g: ℝ → ℝ} {I: BoundedInterval} (h: ∀ x ∈ (I:Set ℝ), f x = g x) :
  PiecewiseConstantOn f I ↔ PiecewiseConstantOn g I := by sorry

theorem PiecewiseConstantOn.congr' {f g: ℝ → ℝ} {I: BoundedInterval} (hf: PiecewiseConstantOn f I) (h: ∀ x ∈ (I:Set ℝ), f x = g x) : PiecewiseConstantOn g I := by sorry

noncomputable abbrev f_11_2_4 : ℝ → ℝ := fun x ↦
  if x < 1 then 0 else
    if x < 3 then 7 else
      if x = 3 then 4 else
        if x < 6 then 5 else
          if x = 6 then 2 else
            0

theorem ConstantOn.piecewiseConstantOn {f:ℝ → ℝ} {I: BoundedInterval} (h: ConstantOn f (I:Set ℝ)) :
  PiecewiseConstantOn f I := by sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem PiecewiseConstantOn.max {f g: ℝ → ℝ} {I: BoundedInterval}
  (hf: PiecewiseConstantOn f I) (hg: PiecewiseConstantOn g I) : PiecewiseConstantOn (max f g) I := by
-- </TARGET_THEOREM>

end Chapter11