Analysis
11.4. Basic properties of the Riemann integral

Imports

import Mathlib.Tactic
import Analysis.Section_9_6
import Analysis.Section_11_3

Analysis I, Section 11.4: Basic properties of the Riemann integral

I have attempted to make the translation as faithful a paraphrasing as possible of the original text. When there is a choice between a more idiomatic Lean solution and a more faithful translation, I have generally chosen the latter. In particular, there will be places where the Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided doing so.

Main constructions and results of this section:

Basic properties of the Riemann integral.

namespace Chapter11open Chapter9

Theorem 11.4.1(a) / Exercise 11.4.1

theorem declaration uses `sorry`IntegrableOn.add {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I) :
  IntegrableOn (f + g) I ∧ integ (f + g) I = integ f I + integ g I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f + g) I ∧ integ (f + g) I = integ f I + integ g I
  sorryAll goals completed! 🐙

Theorem 11.4.1(b) / Exercise 11.4.1

theorem declaration uses `sorry`IntegrableOn.smul {I: BoundedInterval} (c:ℝ) {f:ℝ → ℝ} (hf: IntegrableOn f I) :
  IntegrableOn (c • f) I ∧ integ (c • f) I = c * integ f I := byI:BoundedIntervalc:ℝf:ℝ → ℝhf:IntegrableOn f I⊢ IntegrableOn (c • f) I ∧ integ (c • f) I = c * integ f I
  sorryAll goals completed! 🐙

theorem IntegrableOn.neg {I: BoundedInterval} {f:ℝ → ℝ} (hf: IntegrableOn f I) :
  IntegrableOn (-f) I ∧ integ (-f) I = -integ f I := byI:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f I⊢ IntegrableOn (-f) I ∧ integ (-f) I = -integ f I have := IntegrableOn.smul (-1) hfI:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f Ithis:IntegrableOn (-1 • f) I ∧ integ (-1 • f) I = -1 * integ f I⊢ IntegrableOn (-f) I ∧ integ (-f) I = -integ f I; aesopAll goals completed! 🐙

Theorem 11.4.1(c) / Exercise 11.4.1

theorem declaration uses `sorry`IntegrableOn.sub {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I) :
  IntegrableOn (f - g) I ∧ integ (f - g) I = integ f I - integ g I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f - g) I ∧ integ (f - g) I = integ f I - integ g I
  sorryAll goals completed! 🐙

Theorem 11.4.1(d) / Exercise 11.4.1

theorem declaration uses `sorry`IntegrableOn.nonneg {I: BoundedInterval} {f:ℝ → ℝ} (hf: IntegrableOn f I) (hf_nonneg: ∀ x ∈ I, 0 ≤ f x) :
  0 ≤ integ f I := byI:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f Ihf_nonneg:∀ x ∈ I, 0 ≤ f x⊢ 0 ≤ integ f I
  sorryAll goals completed! 🐙

Theorem 11.4.1(e) / Exercise 11.4.1

theorem declaration uses `sorry`IntegrableOn.mono {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I)
  (h: MajorizesOn g f I) :
  integ f I ≤ integ g I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ih:MajorizesOn g f I⊢ integ f I ≤ integ g I
  sorryAll goals completed! 🐙

Theorem 11.4.1(f) / Exercise 11.4.1

theorem declaration uses `sorry`IntegrableOn.const (c:ℝ) (I: BoundedInterval) :
  IntegrableOn (fun _ ↦ c) I ∧ integ (fun _ ↦ c) I = c * |I|ₗ := byc:ℝI:BoundedInterval⊢ IntegrableOn (fun x ↦ c) I ∧ integ (fun x ↦ c) I = c * I.length
  sorryAll goals completed! 🐙

Theorem 11.4.1(f) / Exercise 11.4.1

theorem declaration uses `sorry`IntegrableOn.const' {I: BoundedInterval} {f:ℝ → ℝ} (hf: ConstantOn f I) :
  IntegrableOn f I ∧ integ f I = (constant_value_on f I) * |I|ₗ := byI:BoundedIntervalf:ℝ → ℝhf:ConstantOn f ↑I⊢ IntegrableOn f I ∧ integ f I = constant_value_on f ↑I * I.length
  sorryAll goals completed! 🐙

Theorem 11.4.1 (g) / Exercise 11.4.1

open Classical in

theorem declaration uses `sorry`IntegrableOn.of_extend {I J: BoundedInterval} (hIJ: I ⊆ J)
  {f: ℝ → ℝ} (h: IntegrableOn f I) :
  IntegrableOn (fun x ↦ if x ∈ I then f x else 0) J := byI:BoundedIntervalJ:BoundedIntervalhIJ:I ⊆ Jf:ℝ → ℝh:IntegrableOn f I⊢ IntegrableOn (fun x ↦ if x ∈ I then f x else 0) J
  sorryAll goals completed! 🐙

Theorem 11.4.1 (g) / Exercise 11.4.1

open Classical in

theorem declaration uses `sorry`IntegrableOn.of_extend' {I J: BoundedInterval} (hIJ: I ⊆ J)
  {f: ℝ → ℝ} (h: IntegrableOn f I) :
  integ (fun x ↦ if x ∈ I then f x else 0) J = integ f I := byI:BoundedIntervalJ:BoundedIntervalhIJ:I ⊆ Jf:ℝ → ℝh:IntegrableOn f I⊢ integ (fun x ↦ if x ∈ I then f x else 0) J = integ f I
  sorryAll goals completed! 🐙

Theorem 11.4.1 (h) (Laws of integration) / Exercise 11.4.1

theorem declaration uses `sorry`IntegrableOn.join {I J K: BoundedInterval} (hIJK: K.joins I J)
  {f: ℝ → ℝ} (h: IntegrableOn f K) :
  IntegrableOn f I ∧ IntegrableOn f J ∧ integ f K = integ f I + integ f J := byI:BoundedIntervalJ:BoundedIntervalK:BoundedIntervalhIJK:K.joins I Jf:ℝ → ℝh:IntegrableOn f K⊢ IntegrableOn f I ∧ IntegrableOn f J ∧ integ f K = integ f I + integ f J
  sorryAll goals completed! 🐙

A variant of Theorem 11.4.1(h) that will be useful in later sections.

theorem declaration uses `sorry`IntegrableOn.mono' {I J: BoundedInterval} (hIJ: J ⊆ I)
  {f: ℝ → ℝ} (h: IntegrableOn f I) : IntegrableOn f J := byI:BoundedIntervalJ:BoundedIntervalhIJ:J ⊆ If:ℝ → ℝh:IntegrableOn f I⊢ IntegrableOn f J
  sorryAll goals completed! 🐙

A further variant of Theorem 11.4.1(h) that will be useful in later sections.

theorem declaration uses `sorry`IntegrableOn.eq {I J: BoundedInterval} (hIJ: J ⊆ I)
  (ha: J.a = I.a) (hb: J.b = I.b)
  {f: ℝ → ℝ} (h: IntegrableOn f I) : integ f J = integ f I := byI:BoundedIntervalJ:BoundedIntervalhIJ:J ⊆ Iha:J.a = I.ahb:J.b = I.bf:ℝ → ℝh:IntegrableOn f I⊢ integ f J = integ f I
  sorryAll goals completed! 🐙

A handy little lemma for "epsilon of room" type arguments

lemma nonneg_of_le_const_mul_eps {x C:ℝ} (h: ∀ ε>0, x ≤ C * ε) : x ≤ 0 := byx:ℝC:ℝh:∀ ε > 0, x ≤ C * ε⊢ x ≤ 0
  by_cases hC: C > 0posx:ℝC:ℝh:∀ ε > 0, x ≤ C * εhC:C > 0⊢ x ≤ 0negx:ℝC:ℝh:∀ ε > 0, x ≤ C * εhC:¬C > 0⊢ x ≤ 0
  .posx:ℝC:ℝh:∀ ε > 0, x ≤ C * εhC:C > 0⊢ x ≤ 0 by_contra!posx:ℝC:ℝh:∀ ε > 0, x ≤ C * εhC:C > 0this:0 < x⊢ False
    specialize h (x/(2*C)) (byx:ℝC:ℝh:∀ ε > 0, x ≤ C * εhC:C > 0this:0 < x⊢ x / (2 * C) > 0 positivityAll goals completed! 🐙); convert_to x ≤ x/2 at hh.e'_4x:ℝC:ℝhC:C > 0this:0 < xh:x ≤ C * (x / (2 * C))⊢ C * (x / (2 * C)) = x / 2posx:ℝC:ℝhC:C > 0this:0 < xh:x ≤ x / 2⊢ False; grindposx:ℝC:ℝhC:C > 0this:0 < xh:x ≤ x / 2⊢ False
    linarithAll goals completed! 🐙
  specialize h 1 ?_negx:ℝC:ℝh:∀ ε > 0, x ≤ C * εhC:¬C > 0⊢ 1 > 0negx:ℝC:ℝhC:¬C > 0h:x ≤ C * 1⊢ x ≤ 0 <;>negx:ℝC:ℝh:∀ ε > 0, x ≤ C * εhC:¬C > 0⊢ 1 > 0negx:ℝC:ℝhC:¬C > 0h:x ≤ C * 1⊢ x ≤ 0 grindAll goals completed! 🐙

Theorem 11.4.3 (Max and min preserve integrability)

theorem declaration uses `sorry`IntegrableOn.max {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I) :
  IntegrableOn (f ⊔ g) I  := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I
  -- This proof is written to follow the structure of the original text.
  unfold IntegrableOn at hf hgI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g I⊢ IntegrableOn (f ⊔ g) I
  have hmax_bound : BddOn (f ⊔ g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I
    choose M hM using hf.1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IM:ℝhM:∀ x ∈ ↑I, |f x| ≤ M⊢ BddOn (f ⊔ g) ↑I; choose M' hM' using hg.1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IM:ℝhM:∀ x ∈ ↑I, |f x| ≤ MM':ℝhM':∀ x ∈ ↑I, |g x| ≤ M'⊢ BddOn (f ⊔ g) ↑I
    use M ⊔ M'hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IM:ℝhM:∀ x ∈ ↑I, |f x| ≤ MM':ℝhM':∀ x ∈ ↑I, |g x| ≤ M'⊢ ∀ x ∈ ↑I, |(f ⊔ g) x| ≤ Max.max M M'; peel hM with x hx hMh.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IM:ℝhM✝:∀ x ∈ ↑I, |f x| ≤ MM':ℝhM':∀ x ∈ ↑I, |g x| ≤ M'x:ℝhx:x ∈ ↑IhM:|f x| ≤ M⊢ |(f ⊔ g) x| ≤ Max.max M M'; specialize hM' _ hxh.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IM:ℝhM✝:∀ x ∈ ↑I, |f x| ≤ MM':ℝx:ℝhx:x ∈ ↑IhM:|f x| ≤ MhM':|g x| ≤ M'⊢ |(f ⊔ g) x| ≤ Max.max M M'
    simp only [Pi.sup_apply]h.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IM:ℝhM✝:∀ x ∈ ↑I, |f x| ≤ MM':ℝx:ℝhx:x ∈ ↑IhM:|f x| ≤ MhM':|g x| ≤ M'⊢ |Max.max (f x) (g x)| ≤ Max.max M M'
    exact abs_max_le_max_abs_abs.trans (sup_le_sup hM hM')All goals completed! 🐙
  have lower_le_upper : 0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I linarith [lower_integral_le_upper hmax_bound]All goals completed! 🐙
  have (ε:ℝ) (hε: 0 < ε) : upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4*ε := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I
    choose f' hf'min hf'const hf'int using gt_of_lt_lower_integral hf.1 (show integ f I - ε < lower_integral f I
    byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I grindAll goals completed! 🐙)
    choose g' hg'min hg'const hg'int using gt_of_lt_lower_integral hg.1 (show integ g I - ε < lower_integral g I byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I grindAll goals completed! 🐙)
    choose f'' hf''max hf''const hf''int using lt_of_gt_upper_integral hf.1 (show upper_integral f I < integ f I + ε byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I grindAll goals completed! 🐙)
    choose g'' hg''max hg''const hg''int using lt_of_gt_upper_integral hg.1 (show upper_integral g I < integ g I + ε byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I grindAll goals completed! 🐙)
    set h := (f'' - f') + (g'' - g')I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hf'_integ := integ_of_piecewise_const hf'constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hg'_integ := integ_of_piecewise_const hg'constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hf''_integ := integ_of_piecewise_const hf''constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hg''_integ := integ_of_piecewise_const hg''constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hf''f'_integ := hf''_integ.1.sub hf'_integ.1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' I⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hg''g'_integ := hg''_integ.1.sub hg'_integ.1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' I⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hh_IntegrableOn.eq := hf''f'_integ.1.add hg''g'_integ.1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') I⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hinteg_le : integ h I ≤ 4 * ε := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I linarithAll goals completed! 🐙
    have hf''g''_const := hf''const.max hg''constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') I⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hf''g''_maj : MajorizesOn (f'' ⊔ g'') (f ⊔ g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I
      sorryAll goals completed! 🐙
    have hf'g'_const := hf'const.max hg'constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') I⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hf'g'_maj : MinorizesOn (f' ⊔ g') (f ⊔ g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I
      sorryAll goals completed! 🐙
    have hff'g''_ge := upper_integral_le_integ hmax_bound hf''g''_maj hf''g''_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hf'g'_le := integ_le_lower_integral hmax_bound hf'g'_maj hf'g'_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) I⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have : MinorizesOn (f'' ⊔ g'') (f' ⊔ g' + h) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊔ g) I
      peel hf'min with x hx hf'minh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f x⊢ (f'' ⊔ g'') x ≤ (f' ⊔ g' + h) x; specialize hg'min _ hxh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g x⊢ (f'' ⊔ g'') x ≤ (f' ⊔ g' + h) x; specialize hf''max _ hxh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf''max:f x ≤ f'' x⊢ (f'' ⊔ g'') x ≤ (f' ⊔ g' + h) x; specialize hg''max _ hxh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf''max:f x ≤ f'' xhg''max:g x ≤ g'' x⊢ (f'' ⊔ g'') x ≤ (f' ⊔ g' + h) x
      simp [h]h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf''max:f x ≤ f'' xhg''max:g x ≤ g'' x⊢ f'' x ≤ Max.max (f' x) (g' x) + (f'' x - f' x + (g'' x - g' x)) ∧
  g'' x ≤ Max.max (f' x) (g' x) + (f'' x - f' x + (g'' x - g' x)); split_andsh.h.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf''max:f x ≤ f'' xhg''max:g x ≤ g'' x⊢ f'' x ≤ Max.max (f' x) (g' x) + (f'' x - f' x + (g'' x - g' x))h.h.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf''max:f x ≤ f'' xhg''max:g x ≤ g'' x⊢ g'' x ≤ Max.max (f' x) (g' x) + (f'' x - f' x + (g'' x - g' x)) <;>h.h.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf''max:f x ≤ f'' xhg''max:g x ≤ g'' x⊢ f'' x ≤ Max.max (f' x) (g' x) + (f'' x - f' x + (g'' x - g' x))h.h.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf''max:f x ≤ f'' xhg''max:g x ≤ g'' x⊢ g'' x ≤ Max.max (f' x) (g' x) + (f'' x - f' x + (g'' x - g' x)) linarith [le_max_left (f' x) (g' x), le_max_right (f' x) (g' x)]All goals completed! 🐙
    have hf'g'_integ := integ_of_piecewise_const hf'g'_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ithis:MinorizesOn (f'' ⊔ g'') (f' ⊔ g' + h) Ihf'g'_integ:IntegrableOn (f' ⊔ g') I ∧ integ (f' ⊔ g') I = hf'g'_const.integ'⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hf''g''_integ := integ_of_piecewise_const hf''g''_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ithis:MinorizesOn (f'' ⊔ g'') (f' ⊔ g' + h) Ihf'g'_integ:IntegrableOn (f' ⊔ g') I ∧ integ (f' ⊔ g') I = hf'g'_const.integ'hf''g''_integ:IntegrableOn (f'' ⊔ g'') I ∧ integ (f'' ⊔ g'') I = hf''g''_const.integ'⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    have hf'g'h_integ := hf'g'_integ.1.add hh_IntegrableOn.eq.1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ithis:MinorizesOn (f'' ⊔ g'') (f' ⊔ g' + h) Ihf'g'_integ:IntegrableOn (f' ⊔ g') I ∧ integ (f' ⊔ g') I = hf'g'_const.integ'hf''g''_integ:IntegrableOn (f'' ⊔ g'') I ∧ integ (f'' ⊔ g'') I = hf''g''_const.integ'hf'g'h_integ:IntegrableOn (f' ⊔ g' + (f'' - f' + (g'' - g'))) I ∧
  integ (f' ⊔ g' + (f'' - f' + (g'' - g'))) I = integ (f' ⊔ g') I + integ (f'' - f' + (g'' - g')) I⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    rw [MinorizesOn.iff] at thisI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''max:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εg'':ℝ → ℝhg''max:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εh:ℝ → ℝ := f'' - f' + (g'' - g')hf'_integ:IntegrableOn f' I ∧ integ f' I = hf'const.integ'hg'_integ:IntegrableOn g' I ∧ integ g' I = hg'const.integ'hf''_integ:IntegrableOn f'' I ∧ integ f'' I = hf''const.integ'hg''_integ:IntegrableOn g'' I ∧ integ g'' I = hg''const.integ'hf''f'_integ:IntegrableOn (f'' - f') I ∧ integ (f'' - f') I = integ f'' I - integ f' Ihg''g'_integ:IntegrableOn (g'' - g') I ∧ integ (g'' - g') I = integ g'' I - integ g' Ihh_IntegrableOn.eq:IntegrableOn (f'' - f' + (g'' - g')) I ∧ integ (f'' - f' + (g'' - g')) I = integ (f'' - f') I + integ (g'' - g') Ihinteg_le:integ h I ≤ 4 * εhf''g''_const:PiecewiseConstantOn (f'' ⊔ g'') Ihf''g''_maj:MajorizesOn (f'' ⊔ g'') (f ⊔ g) Ihf'g'_const:PiecewiseConstantOn (f' ⊔ g') Ihf'g'_maj:MinorizesOn (f' ⊔ g') (f ⊔ g) Ihff'g''_ge:upper_integral (f ⊔ g) I ≤ hf''g''_const.integ'hf'g'_le:hf'g'_const.integ' ≤ lower_integral (f ⊔ g) Ithis:MajorizesOn (f' ⊔ g' + h) (f'' ⊔ g'') Ihf'g'_integ:IntegrableOn (f' ⊔ g') I ∧ integ (f' ⊔ g') I = hf'g'_const.integ'hf''g''_integ:IntegrableOn (f'' ⊔ g'') I ∧ integ (f'' ⊔ g'') I = hf''g''_const.integ'hf'g'h_integ:IntegrableOn (f' ⊔ g' + (f'' - f' + (g'' - g'))) I ∧
  integ (f' ⊔ g' + (f'' - f' + (g'' - g'))) I = integ (f' ⊔ g') I + integ (f'' - f' + (g'' - g')) I⊢ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε
    linarith [hf''g''_integ.1.mono hf'g'h_integ.1 this]All goals completed! 🐙
  exact ⟨ hmax_bound, byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihmax_bound:BddOn (f ⊔ g) ↑Ilower_le_upper:0 ≤ upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) Ithis:∀ (ε : ℝ), 0 < ε → upper_integral (f ⊔ g) I - lower_integral (f ⊔ g) I ≤ 4 * ε⊢ lower_integral (f ⊔ g) I = upper_integral (f ⊔ g) I linarith [nonneg_of_le_const_mul_eps this]All goals completed! 🐙 ⟩

Theorem 11.4.5 / Exercise 11.4.3. The objective here is to create a shorter proof than the one above.

theorem declaration uses `sorry`IntegrableOn.min {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I) :
  IntegrableOn (f ⊓ g) I  := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f ⊓ g) I
  sorryAll goals completed! 🐙

Corollary 11.4.4

theorem IntegrableOn.abs {I: BoundedInterval} {f:ℝ → ℝ} (hf: IntegrableOn f I) :
  IntegrableOn (abs f) I := byI:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f I⊢ IntegrableOn |f| I
  have := (IntegrableOn.const 0 I).1I:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f Ithis:IntegrableOn (fun x ↦ 0) I⊢ IntegrableOn |f| I
  convert ((hf.max this).sub (hf.min this)).1 using 1h.e'_1I:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f Ithis:IntegrableOn (fun x ↦ 0) I⊢ |f| = (f ⊔ fun x ↦ 0) - f ⊓ fun x ↦ 0
  ext xh.e'_1.hI:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f Ithis:IntegrableOn (fun x ↦ 0) Ix:ℝ⊢ |f| x = ((f ⊔ fun x ↦ 0) - f ⊓ fun x ↦ 0) x; obtain h | h := (show f x ≤ 0 ∨ f x ≥ 0 byI:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f I⊢ IntegrableOn |f| I grindAll goals completed! 🐙) <;>h.e'_1.h.inlI:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f Ithis:IntegrableOn (fun x ↦ 0) Ix:ℝh:f x ≤ 0⊢ |f| x = ((f ⊔ fun x ↦ 0) - f ⊓ fun x ↦ 0) xh.e'_1.h.inrI:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f Ithis:IntegrableOn (fun x ↦ 0) Ix:ℝh:f x ≥ 0⊢ |f| x = ((f ⊔ fun x ↦ 0) - f ⊓ fun x ↦ 0) x simp [h]All goals completed! 🐙

Theorem 11.4.5 (Products preserve Riemann integrability). It is convenient to first establish the non-negative case.

theorem integ_of_mul_nonneg {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I)
  (hf_nonneg: MajorizesOn f 0 I) (hg_nonneg: MajorizesOn g 0 I) :
  IntegrableOn (f * g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
  -- This proof is written to follow the structure of the original text.
  by_cases hI : (I:Set ℝ).NonemptyposI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).Nonempty⊢ IntegrableOn (f * g) InegI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:¬(↑I).Nonempty⊢ IntegrableOn (f * g) I
  swapnegI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:¬(↑I).Nonempty⊢ IntegrableOn (f * g) IposI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).Nonempty⊢ IntegrableOn (f * g) I
  .negI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:¬(↑I).Nonempty⊢ IntegrableOn (f * g) I apply (integ_on_subsingleton _).1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:¬(↑I).Nonempty⊢ I.length = 0
    rw [←BoundedInterval.length_of_subsingleton]I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:¬(↑I).Nonempty⊢ Subsingleton ↑↑I
    simp_all [Set.not_nonempty_iff_eq_empty]All goals completed! 🐙
  unfold IntegrableOn at hf hgposI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).Nonempty⊢ IntegrableOn (f * g) I
  choose M₁ hM₁ using hf.1posI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁⊢ IntegrableOn (f * g) I
  choose M₂ hM₂ using hg.1posI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂⊢ IntegrableOn (f * g) I
  have hM₁pos : 0 ≤ M₁ := (abs_nonneg _).trans (hM₁ hI.some hI.some_mem)posI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁⊢ IntegrableOn (f * g) I
  have hM₂pos : 0 ≤ M₂ := (abs_nonneg _).trans (hM₂ hI.some hI.some_mem)posI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂⊢ IntegrableOn (f * g) I
  have hmul_bound : BddOn (f * g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
    use M₁ * M₂hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂⊢ ∀ x ∈ ↑I, |(f * g) x| ≤ M₁ * M₂; peel hM₁ with x hx hM₁h.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁✝:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂x:ℝhx:x ∈ ↑IhM₁:|f x| ≤ M₁⊢ |(f * g) x| ≤ M₁ * M₂; specialize hM₂ _ hxh.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁✝:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂x:ℝhx:x ∈ ↑IhM₁:|f x| ≤ M₁hM₂:|g x| ≤ M₂⊢ |(f * g) x| ≤ M₁ * M₂
    simp [abs_mul]h.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁✝:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂x:ℝhx:x ∈ ↑IhM₁:|f x| ≤ M₁hM₂:|g x| ≤ M₂⊢ |f x| * |g x| ≤ M₁ * M₂; apply mul_le_mul hM₁ hM₂h.h.h.c0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁✝:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂x:ℝhx:x ∈ ↑IhM₁:|f x| ≤ M₁hM₂:|g x| ≤ M₂⊢ 0 ≤ |g x|h.h.h.b0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁✝:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂x:ℝhx:x ∈ ↑IhM₁:|f x| ≤ M₁hM₂:|g x| ≤ M₂⊢ 0 ≤ M₁ <;>h.h.h.c0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁✝:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂x:ℝhx:x ∈ ↑IhM₁:|f x| ≤ M₁hM₂:|g x| ≤ M₂⊢ 0 ≤ |g x|h.h.h.b0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁✝:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂x:ℝhx:x ∈ ↑IhM₁:|f x| ≤ M₁hM₂:|g x| ≤ M₂⊢ 0 ≤ M₁ positivityAll goals completed! 🐙
  have lower_le_upper : 0 ≤ upper_integral (f * g) I - lower_integral (f * g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
    linarith [lower_integral_le_upper hmul_bound]All goals completed! 🐙
  have (ε:ℝ) (hε: 0 < ε) : upper_integral (f * g) I - lower_integral (f * g) I ≤ 2*(M₁+M₂)*ε := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
    have : ∃ f', MinorizesOn f' f I ∧ PiecewiseConstantOn f' I ∧ integ f I - ε < PiecewiseConstantOn.integ f' I ∧ MajorizesOn f' 0 I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
      choose f' hf'min hf'const hf'int using gt_of_lt_lower_integral hf.1 (show integ f I - ε < lower_integral f I byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I linarithAll goals completed! 🐙)
      use max f' 0hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' I⊢ MinorizesOn (f' ⊔ 0) f I ∧
  PiecewiseConstantOn (f' ⊔ 0) I ∧ integ f I - ε < PiecewiseConstantOn.integ (f' ⊔ 0) I ∧ MajorizesOn (f' ⊔ 0) 0 I
      have hzero := (ConstantOn.of_const' 0 I).piecewiseConstantOnhI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ MinorizesOn (f' ⊔ 0) f I ∧
  PiecewiseConstantOn (f' ⊔ 0) I ∧ integ f I - ε < PiecewiseConstantOn.integ (f' ⊔ 0) I ∧ MajorizesOn (f' ⊔ 0) 0 I
      split_andsh.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ MinorizesOn (f' ⊔ 0) f Ih.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ PiecewiseConstantOn (f' ⊔ 0) Ih.refine_2.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ integ f I - ε < PiecewiseConstantOn.integ (f' ⊔ 0) Ih.refine_2.refine_2.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ MajorizesOn (f' ⊔ 0) 0 I
      .h.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ MinorizesOn (f' ⊔ 0) f I peel hf_nonneg with x hx _h.refine_1.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) Ix:ℝhx:x ∈ ↑Ithis:0 x ≤ f x⊢ (f' ⊔ 0) x ≤ f x; specialize hf'min _ hxh.refine_1.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) Ix:ℝhx:x ∈ ↑Ithis:0 x ≤ f xhf'min:f' x ≤ f x⊢ (f' ⊔ 0) x ≤ f x; aesopAll goals completed! 🐙
      .h.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ PiecewiseConstantOn (f' ⊔ 0) I exact hf'const.max hzeroAll goals completed! 🐙
      .h.refine_2.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ integ f I - ε < PiecewiseConstantOn.integ (f' ⊔ 0) I apply lt_of_lt_of_le hf'int (hf'const.integ_mono _ (hf'const.max hzero))I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ ∀ x ∈ I, f' x ≤ (f' ⊔ fun x ↦ 0) x; simpAll goals completed! 🐙
      intro _h.refine_2.refine_2.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihzero:PiecewiseConstantOn (fun x ↦ 0) Ix✝:ℝ⊢ x✝ ∈ ↑I → 0 x✝ ≤ (f' ⊔ 0) x✝; simpAll goals completed! 🐙
    choose f' hf'min hf'const hf'int hf'_nonneg using thisI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have : ∃ g', MinorizesOn g' g I ∧ PiecewiseConstantOn g' I ∧ integ g I - ε < PiecewiseConstantOn.integ g' I ∧ MajorizesOn g' 0 I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
      obtain ⟨ g', hg'min, hg'const, hg'int ⟩ := gt_of_lt_lower_integral hg.1 (show integ g I - ε < lower_integral g I byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I linarithAll goals completed! 🐙)
      use max g' 0hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' I⊢ MinorizesOn (g' ⊔ 0) g I ∧
  PiecewiseConstantOn (g' ⊔ 0) I ∧ integ g I - ε < PiecewiseConstantOn.integ (g' ⊔ 0) I ∧ MajorizesOn (g' ⊔ 0) 0 I
      have hzero := (ConstantOn.of_const' 0 I).piecewiseConstantOnhI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ MinorizesOn (g' ⊔ 0) g I ∧
  PiecewiseConstantOn (g' ⊔ 0) I ∧ integ g I - ε < PiecewiseConstantOn.integ (g' ⊔ 0) I ∧ MajorizesOn (g' ⊔ 0) 0 I
      split_andsh.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ MinorizesOn (g' ⊔ 0) g Ih.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ PiecewiseConstantOn (g' ⊔ 0) Ih.refine_2.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ integ g I - ε < PiecewiseConstantOn.integ (g' ⊔ 0) Ih.refine_2.refine_2.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ MajorizesOn (g' ⊔ 0) 0 I
      .h.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ MinorizesOn (g' ⊔ 0) g I peel hg_nonneg with x hx _h.refine_1.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) Ix:ℝhx:x ∈ ↑Ithis:0 x ≤ g x⊢ (g' ⊔ 0) x ≤ g x; specialize hg'min _ hxh.refine_1.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) Ix:ℝhx:x ∈ ↑Ithis:0 x ≤ g xhg'min:g' x ≤ g x⊢ (g' ⊔ 0) x ≤ g x; aesopAll goals completed! 🐙
      .h.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ PiecewiseConstantOn (g' ⊔ 0) I exact hg'const.max hzeroAll goals completed! 🐙
      .h.refine_2.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ integ g I - ε < PiecewiseConstantOn.integ (g' ⊔ 0) I apply lt_of_lt_of_le hg'int (hg'const.integ_mono _ (hg'const.max hzero))I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) I⊢ ∀ x ∈ I, g' x ≤ (g' ⊔ fun x ↦ 0) x; simpAll goals completed! 🐙
      intro _h.refine_2.refine_2.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihzero:PiecewiseConstantOn (fun x ↦ 0) Ix✝:ℝ⊢ x✝ ∈ ↑I → 0 x✝ ≤ (g' ⊔ 0) x✝; simpAll goals completed! 🐙
    choose g' hg'min hg'const hg'int hg'_nonneg using thisI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have : ∃ f'', MajorizesOn f'' f I ∧ PiecewiseConstantOn f'' I ∧ PiecewiseConstantOn.integ f'' I < integ f I + ε ∧ MinorizesOn f'' (fun _ ↦ M₁) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
      obtain ⟨ f'', hf''maj, hf''const, hf''int ⟩ := lt_of_gt_upper_integral hf.1 (show upper_integral f I < integ f I + ε  byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I linarithAll goals completed! 🐙)
      use min f'' (fun _ ↦ M₁)hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + ε⊢ MajorizesOn (f'' ⊓ fun x ↦ M₁) f I ∧
  PiecewiseConstantOn (f'' ⊓ fun x ↦ M₁) I ∧
    PiecewiseConstantOn.integ (f'' ⊓ fun x ↦ M₁) I < integ f I + ε ∧ MinorizesOn (f'' ⊓ fun x ↦ M₁) (fun x ↦ M₁) I
      have hM₁_piece := (ConstantOn.of_const' M₁ I).piecewiseConstantOnhI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) I⊢ MajorizesOn (f'' ⊓ fun x ↦ M₁) f I ∧
  PiecewiseConstantOn (f'' ⊓ fun x ↦ M₁) I ∧
    PiecewiseConstantOn.integ (f'' ⊓ fun x ↦ M₁) I < integ f I + ε ∧ MinorizesOn (f'' ⊓ fun x ↦ M₁) (fun x ↦ M₁) I
      split_andsh.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) I⊢ MajorizesOn (f'' ⊓ fun x ↦ M₁) f Ih.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) I⊢ PiecewiseConstantOn (f'' ⊓ fun x ↦ M₁) Ih.refine_2.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) I⊢ PiecewiseConstantOn.integ (f'' ⊓ fun x ↦ M₁) I < integ f I + εh.refine_2.refine_2.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) I⊢ MinorizesOn (f'' ⊓ fun x ↦ M₁) (fun x ↦ M₁) I
      .h.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) I⊢ MajorizesOn (f'' ⊓ fun x ↦ M₁) f I peel hM₁ with x hx hM₁h.refine_1.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁✝:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) Ix:ℝhx:x ∈ ↑IhM₁:|f x| ≤ M₁⊢ f x ≤ (f'' ⊓ fun x ↦ M₁) x; rw [abs_le'] at hM₁h.refine_1.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁✝:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) Ix:ℝhx:x ∈ ↑IhM₁:f x ≤ M₁ ∧ -f x ≤ M₁⊢ f x ≤ (f'' ⊓ fun x ↦ M₁) x
        simp [hf''maj _ hx, hM₁.1]All goals completed! 🐙
      .h.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) I⊢ PiecewiseConstantOn (f'' ⊓ fun x ↦ M₁) I exact hf''const.min hM₁_pieceAll goals completed! 🐙
      .h.refine_2.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) I⊢ PiecewiseConstantOn.integ (f'' ⊓ fun x ↦ M₁) I < integ f I + ε apply lt_of_le_of_lt ((hf''const.min hM₁_piece).integ_mono _ hf''const) hf''intI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) I⊢ ∀ x ∈ I, (f'' ⊓ fun x ↦ M₁) x ≤ f'' x
        simpAll goals completed! 🐙
      intro _h.refine_2.refine_2.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhM₁_piece:PiecewiseConstantOn (fun x ↦ M₁) Ix✝:ℝ⊢ x✝ ∈ ↑I → (f'' ⊓ fun x ↦ M₁) x✝ ≤ (fun x ↦ M₁) x✝; simpAll goals completed! 🐙
    choose f'' hf''maj hf''const hf''int hf''bound using thisI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have : ∃ g'', MajorizesOn g'' g I ∧ PiecewiseConstantOn g'' I ∧ PiecewiseConstantOn.integ g'' I < integ g I + ε ∧ MinorizesOn g'' (fun _ ↦ M₂) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
      obtain ⟨ g'', hg''maj, hg''const, hg''int ⟩ := lt_of_gt_upper_integral hg.1 (show upper_integral g I < integ g I + ε byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I linarithAll goals completed! 🐙)
      use min g'' (fun _ ↦ M₂)hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + ε⊢ MajorizesOn (g'' ⊓ fun x ↦ M₂) g I ∧
  PiecewiseConstantOn (g'' ⊓ fun x ↦ M₂) I ∧
    PiecewiseConstantOn.integ (g'' ⊓ fun x ↦ M₂) I < integ g I + ε ∧ MinorizesOn (g'' ⊓ fun x ↦ M₂) (fun x ↦ M₂) I
      have hM₂_piece := (ConstantOn.of_const' M₂ I).piecewiseConstantOnhI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) I⊢ MajorizesOn (g'' ⊓ fun x ↦ M₂) g I ∧
  PiecewiseConstantOn (g'' ⊓ fun x ↦ M₂) I ∧
    PiecewiseConstantOn.integ (g'' ⊓ fun x ↦ M₂) I < integ g I + ε ∧ MinorizesOn (g'' ⊓ fun x ↦ M₂) (fun x ↦ M₂) I
      split_andsh.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) I⊢ MajorizesOn (g'' ⊓ fun x ↦ M₂) g Ih.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) I⊢ PiecewiseConstantOn (g'' ⊓ fun x ↦ M₂) Ih.refine_2.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) I⊢ PiecewiseConstantOn.integ (g'' ⊓ fun x ↦ M₂) I < integ g I + εh.refine_2.refine_2.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) I⊢ MinorizesOn (g'' ⊓ fun x ↦ M₂) (fun x ↦ M₂) I
      .h.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) I⊢ MajorizesOn (g'' ⊓ fun x ↦ M₂) g I peel hM₂ with x hx hM₂h.refine_1.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂✝:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) Ix:ℝhx:x ∈ ↑IhM₂:|g x| ≤ M₂⊢ g x ≤ (g'' ⊓ fun x ↦ M₂) x; rw [abs_le'] at hM₂h.refine_1.h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂✝:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) Ix:ℝhx:x ∈ ↑IhM₂:g x ≤ M₂ ∧ -g x ≤ M₂⊢ g x ≤ (g'' ⊓ fun x ↦ M₂) x
        simp [hg''maj _ hx, hM₂.1]All goals completed! 🐙
      .h.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) I⊢ PiecewiseConstantOn (g'' ⊓ fun x ↦ M₂) I exact hg''const.min hM₂_pieceAll goals completed! 🐙
      .h.refine_2.refine_2.refine_1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) I⊢ PiecewiseConstantOn.integ (g'' ⊓ fun x ↦ M₂) I < integ g I + ε apply lt_of_le_of_lt ((hg''const.min hM₂_piece).integ_mono _ hg''const) hg''intI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) I⊢ ∀ x ∈ I, (g'' ⊓ fun x ↦ M₂) x ≤ g'' x
        simpAll goals completed! 🐙
      intro _ _h.refine_2.refine_2.refine_2I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhM₂_piece:PiecewiseConstantOn (fun x ↦ M₂) Ix✝:ℝa✝:x✝ ∈ ↑I⊢ (g'' ⊓ fun x ↦ M₂) x✝ ≤ (fun x ↦ M₂) x✝; simpAll goals completed! 🐙
    choose g'' hg''maj hg''const hg''int hg''bound using thisI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hf'g'_const := hf'const.mul hg'constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hf'g'_maj : MinorizesOn (f' * g') (f * g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
      peel hf'min with x hx hf'minh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f x⊢ (f' * g') x ≤ (f * g) x; specialize hg'min _ hxh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g x⊢ (f' * g') x ≤ (f * g) x;
      specialize hf'_nonneg _ hxh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf'_nonneg:0 x ≤ f' x⊢ (f' * g') x ≤ (f * g) x; specialize hg'_nonneg _ hxh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf'_nonneg:0 x ≤ f' xhg'_nonneg:0 x ≤ g' x⊢ (f' * g') x ≤ (f * g) x
      simp at *h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf'_nonneg:0 ≤ f' xhg'_nonneg:0 ≤ g' xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ f' x * g' x ≤ f x * g x; apply mul_le_mul hf'min hg'minh.h.c0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf'_nonneg:0 ≤ f' xhg'_nonneg:0 ≤ g' xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ 0 ≤ g' xh.h.b0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf'_nonneg:0 ≤ f' xhg'_nonneg:0 ≤ g' xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ 0 ≤ f x <;>h.h.c0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf'_nonneg:0 ≤ f' xhg'_nonneg:0 ≤ g' xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ 0 ≤ g' xh.h.b0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min✝:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ig':ℝ → ℝhg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ix:ℝhx:x ∈ ↑Ihf'min:f' x ≤ f xhg'min:g' x ≤ g xhf'_nonneg:0 ≤ f' xhg'_nonneg:0 ≤ g' xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ 0 ≤ f x grindAll goals completed! 🐙
    have hf''g''_const := hf''const.mul hg''constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hf''g''_maj : MajorizesOn (f'' * g'') (f * g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
      peel hf''maj with x hx hf''majh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj✝:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ix:ℝhx:x ∈ ↑Ihf''maj:f x ≤ f'' x⊢ (f * g) x ≤ (f'' * g'') x; specialize hg''maj _ hxh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj✝:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ix:ℝhx:x ∈ ↑Ihf''maj:f x ≤ f'' xhg''maj:g x ≤ g'' x⊢ (f * g) x ≤ (f'' * g'') x
      specialize hg_nonneg _ hxh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj✝:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ix:ℝhx:x ∈ ↑Ihf''maj:f x ≤ f'' xhg''maj:g x ≤ g'' xhg_nonneg:0 x ≤ g x⊢ (f * g) x ≤ (f'' * g'') x; specialize hf_nonneg _ hxh.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj✝:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ix:ℝhx:x ∈ ↑Ihf''maj:f x ≤ f'' xhg''maj:g x ≤ g'' xhg_nonneg:0 x ≤ g xhf_nonneg:0 x ≤ f x⊢ (f * g) x ≤ (f'' * g'') x
      simp at *h.hI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj✝:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ix:ℝhx:x ∈ ↑Ihf''maj:f x ≤ f'' xhg''maj:g x ≤ g'' xhg_nonneg:0 ≤ g xhf_nonneg:0 ≤ f xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ f x * g x ≤ f'' x * g'' x; apply mul_le_mul hf''maj hg''majh.h.c0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj✝:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ix:ℝhx:x ∈ ↑Ihf''maj:f x ≤ f'' xhg''maj:g x ≤ g'' xhg_nonneg:0 ≤ g xhf_nonneg:0 ≤ f xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ 0 ≤ g xh.h.b0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj✝:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ix:ℝhx:x ∈ ↑Ihf''maj:f x ≤ f'' xhg''maj:g x ≤ g'' xhg_nonneg:0 ≤ g xhf_nonneg:0 ≤ f xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ 0 ≤ f'' x <;>h.h.c0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj✝:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ix:ℝhx:x ∈ ↑Ihf''maj:f x ≤ f'' xhg''maj:g x ≤ g'' xhg_nonneg:0 ≤ g xhf_nonneg:0 ≤ f xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ 0 ≤ g xh.h.b0I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj✝:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ix:ℝhx:x ∈ ↑Ihf''maj:f x ≤ f'' xhg''maj:g x ≤ g'' xhg_nonneg:0 ≤ g xhf_nonneg:0 ≤ f xlower_le_upper:lower_integral (f * g) I ≤ upper_integral (f * g) I⊢ 0 ≤ f'' x grindAll goals completed! 🐙
    have hupper_le := upper_integral_le_integ hmul_bound hf''g''_maj hf''g''_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hlower_ge := integ_le_lower_integral hmul_bound hf'g'_maj hf'g'_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hh_const := hf''g''_const.sub hf'g'_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hh_integ := hf''g''_const.integ_sub hf'g'_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hhmin : MinorizesOn (f'' * g'' - f' * g') (M₁ • (g''-g') + M₂ • (f''-f')) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 I⊢ IntegrableOn (f * g) I
      intro x hxI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ (f'' * g'' - f' * g') x ≤ (M₁ • (g'' - g') + M₂ • (f'' - f')) x
      simp only [Pi.sub_apply, Pi.mul_apply, Pi.add_apply, Pi.smul_apply, smul_eq_mul]I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ f'' x * g'' x - f' x * g' x ≤ M₁ * (g'' x - g' x) + M₂ * (f'' x - f' x)
      calc
        _ = (f'' x) * (g'' x - g' x) + (g' x) * (f'' x - f' x) := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ f'' x * g'' x - f' x * g' x = f'' x * (g'' x - g' x) + g' x * (f'' x - f' x) ringAll goals completed! 🐙
        _ ≤ _ := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ f'' x * (g'' x - g' x) + g' x * (f'' x - f' x) ≤ M₁ * (g'' x - g' x) + M₂ * (f'' x - f' x) gcongrh₁.haI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ 0 ≤ g'' x - g' xh₁.hbcI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ f'' x ≤ M₁h₂.haI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ 0 ≤ f'' x - f' xh₂.hbcI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ g' x ≤ M₂ <;>h₁.haI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ 0 ≤ g'' x - g' xh₁.hbcI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ f'' x ≤ M₁h₂.haI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ 0 ≤ f'' x - f' xh₂.hbcI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ix:ℝhx:x ∈ ↑I⊢ g' x ≤ M₂ grindAll goals completed! 🐙
    have hg''g'_const := hg''const.sub hg'constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hg''g'_integ := hg''const.integ_sub hg'constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hM₁g''g'_const := hg''g'_const.smul M₁I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hM₁g''g_integ := hg''g'_const.integ_smul M₁I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hf''f'_const := hf''const.sub hf'constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hf''f_integ := hf''const.integ_sub hf'constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hM₂f''f'_const := hf''f'_const.smul M₂I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hM₂f''f_integ := hf''f'_const.integ_smul M₂I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hsum_const := hM₁g''g'_const.add hM₂f''f'_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hsum_integ := hM₁g''g'_const.integ_add hM₂f''f'_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihsum_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I =
  PiecewiseConstantOn.integ (M₁ • (g'' - g')) I + PiecewiseConstantOn.integ (M₂ • (f'' - f')) I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    have hsum_bound := hh_const.integ_mono hhmin hsum_constI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihsum_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I =
  PiecewiseConstantOn.integ (M₁ • (g'' - g')) I + PiecewiseConstantOn.integ (M₂ • (f'' - f')) Ihsum_bound:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I ≤ PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε
    calc
      _ ≤ M₁ * PiecewiseConstantOn.integ (g'' - g') I + M₂ * PiecewiseConstantOn.integ (f'' - f') I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihsum_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I =
  PiecewiseConstantOn.integ (M₁ • (g'' - g')) I + PiecewiseConstantOn.integ (M₂ • (f'' - f')) Ihsum_bound:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I ≤ PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I⊢ upper_integral (f * g) I - lower_integral (f * g) I ≤
  M₁ * PiecewiseConstantOn.integ (g'' - g') I + M₂ * PiecewiseConstantOn.integ (f'' - f') I linarithAll goals completed! 🐙
      _ ≤ M₁ * (2*ε) + M₂ * (2*ε) := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihsum_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I =
  PiecewiseConstantOn.integ (M₁ • (g'' - g')) I + PiecewiseConstantOn.integ (M₂ • (f'' - f')) Ihsum_bound:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I ≤ PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I⊢ M₁ * PiecewiseConstantOn.integ (g'' - g') I + M₂ * PiecewiseConstantOn.integ (f'' - f') I ≤ M₁ * (2 * ε) + M₂ * (2 * ε) gcongrh₁.hbcI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihsum_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I =
  PiecewiseConstantOn.integ (M₁ • (g'' - g')) I + PiecewiseConstantOn.integ (M₂ • (f'' - f')) Ihsum_bound:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I ≤ PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I⊢ PiecewiseConstantOn.integ (g'' - g') I ≤ 2 * εh₂.hbcI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihsum_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I =
  PiecewiseConstantOn.integ (M₁ • (g'' - g')) I + PiecewiseConstantOn.integ (M₂ • (f'' - f')) Ihsum_bound:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I ≤ PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I⊢ PiecewiseConstantOn.integ (f'' - f') I ≤ 2 * ε <;>h₁.hbcI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihsum_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I =
  PiecewiseConstantOn.integ (M₁ • (g'' - g')) I + PiecewiseConstantOn.integ (M₂ • (f'' - f')) Ihsum_bound:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I ≤ PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I⊢ PiecewiseConstantOn.integ (g'' - g') I ≤ 2 * εh₂.hbcI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihsum_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I =
  PiecewiseConstantOn.integ (M₁ • (g'' - g')) I + PiecewiseConstantOn.integ (M₂ • (f'' - f')) Ihsum_bound:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I ≤ PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I⊢ PiecewiseConstantOn.integ (f'' - f') I ≤ 2 * ε linarithAll goals completed! 🐙
      _ = _ := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Iε:ℝhε:0 < εf':ℝ → ℝhf'min:MinorizesOn f' f Ihf'const:PiecewiseConstantOn f' Ihf'int:integ f I - ε < PiecewiseConstantOn.integ f' Ihf'_nonneg:MajorizesOn f' 0 Ig':ℝ → ℝhg'min:MinorizesOn g' g Ihg'const:PiecewiseConstantOn g' Ihg'int:integ g I - ε < PiecewiseConstantOn.integ g' Ihg'_nonneg:MajorizesOn g' 0 If'':ℝ → ℝhf''maj:MajorizesOn f'' f Ihf''const:PiecewiseConstantOn f'' Ihf''int:PiecewiseConstantOn.integ f'' I < integ f I + εhf''bound:MinorizesOn f'' (fun x ↦ M₁) Ig'':ℝ → ℝhg''maj:MajorizesOn g'' g Ihg''const:PiecewiseConstantOn g'' Ihg''int:PiecewiseConstantOn.integ g'' I < integ g I + εhg''bound:MinorizesOn g'' (fun x ↦ M₂) Ihf'g'_const:PiecewiseConstantOn (f' * g') Ihf'g'_maj:MinorizesOn (f' * g') (f * g) Ihf''g''_const:PiecewiseConstantOn (f'' * g'') Ihf''g''_maj:MajorizesOn (f'' * g'') (f * g) Ihupper_le:upper_integral (f * g) I ≤ hf''g''_const.integ'hlower_ge:hf'g'_const.integ' ≤ lower_integral (f * g) Ihh_const:PiecewiseConstantOn (f'' * g'' - f' * g') Ihh_integ:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I =
  PiecewiseConstantOn.integ (f'' * g'') I - PiecewiseConstantOn.integ (f' * g') Ihhmin:MinorizesOn (f'' * g'' - f' * g') (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihg''g'_const:PiecewiseConstantOn (g'' - g') Ihg''g'_integ:PiecewiseConstantOn.integ (g'' - g') I = PiecewiseConstantOn.integ g'' I - PiecewiseConstantOn.integ g' IhM₁g''g'_const:PiecewiseConstantOn (M₁ • (g'' - g')) IhM₁g''g_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g')) I = M₁ * PiecewiseConstantOn.integ (g'' - g') Ihf''f'_const:PiecewiseConstantOn (f'' - f') Ihf''f_integ:PiecewiseConstantOn.integ (f'' - f') I = PiecewiseConstantOn.integ f'' I - PiecewiseConstantOn.integ f' IhM₂f''f'_const:PiecewiseConstantOn (M₂ • (f'' - f')) IhM₂f''f_integ:PiecewiseConstantOn.integ (M₂ • (f'' - f')) I = M₂ * PiecewiseConstantOn.integ (f'' - f') Ihsum_const:PiecewiseConstantOn (M₁ • (g'' - g') + M₂ • (f'' - f')) Ihsum_integ:PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I =
  PiecewiseConstantOn.integ (M₁ • (g'' - g')) I + PiecewiseConstantOn.integ (M₂ • (f'' - f')) Ihsum_bound:PiecewiseConstantOn.integ (f'' * g'' - f' * g') I ≤ PiecewiseConstantOn.integ (M₁ • (g'' - g') + M₂ • (f'' - f')) I⊢ M₁ * (2 * ε) + M₂ * (2 * ε) = 2 * (M₁ + M₂) * ε ringAll goals completed! 🐙
  exact ⟨ hmul_bound, byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:BddOn f ↑I ∧ lower_integral f I = upper_integral f Ihg:BddOn g ↑I ∧ lower_integral g I = upper_integral g Ihf_nonneg:MajorizesOn f 0 Ihg_nonneg:MajorizesOn g 0 IhI:(↑I).NonemptyM₁:ℝhM₁:∀ x ∈ ↑I, |f x| ≤ M₁M₂:ℝhM₂:∀ x ∈ ↑I, |g x| ≤ M₂hM₁pos:0 ≤ M₁hM₂pos:0 ≤ M₂hmul_bound:BddOn (f * g) ↑Ilower_le_upper:0 ≤ upper_integral (f * g) I - lower_integral (f * g) Ithis:∀ (ε : ℝ), 0 < ε → upper_integral (f * g) I - lower_integral (f * g) I ≤ 2 * (M₁ + M₂) * ε⊢ lower_integral (f * g) I = upper_integral (f * g) I linarith [nonneg_of_le_const_mul_eps this]All goals completed! 🐙 ⟩

theorem integ_of_mul {I: BoundedInterval} {f g:ℝ → ℝ} (hf: IntegrableOn f I) (hg: IntegrableOn g I) :
  IntegrableOn (f * g) I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f * g) I
  -- This proof is written to follow the structure of the original text.
  set fplus := max f (fun _ ↦ 0)I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0⊢ IntegrableOn (f * g) I
  set fminus := -min f (fun _ ↦ 0)I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)⊢ IntegrableOn (f * g) I
  set gplus := max g (fun _ ↦ 0)I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0⊢ IntegrableOn (f * g) I
  set gminus := -min g (fun _ ↦ 0)I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)⊢ IntegrableOn (f * g) I
  have := (IntegrableOn.const 0 I).1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) I⊢ IntegrableOn (f * g) I
  observe hfplus_integ : IntegrableOn fplus II:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) I⊢ IntegrableOn (f * g) I
  observe hgplus_integ : IntegrableOn gplus II:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) I⊢ IntegrableOn (f * g) I
  have hfminus_integ : IntegrableOn fminus I := (hf.min this).neg.1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus I⊢ IntegrableOn (f * g) I
  have hgminus_integ : IntegrableOn gminus I := (hg.min this).neg.1I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus I⊢ IntegrableOn (f * g) I
  have hfplus_nonneg : MajorizesOn fplus 0 I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f * g) I intro _I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ix✝:ℝ⊢ x✝ ∈ ↑I → 0 x✝ ≤ fplus x✝; simp [fplus]All goals completed! 🐙
  have hfminus_nonneg : MajorizesOn fminus 0 I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f * g) I intro _I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ihfplus_nonneg:MajorizesOn fplus 0 Ix✝:ℝ⊢ x✝ ∈ ↑I → 0 x✝ ≤ fminus x✝; simp [fminus]All goals completed! 🐙
  have hgplus_nonneg : MajorizesOn gplus 0 I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f * g) I intro _I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ihfplus_nonneg:MajorizesOn fplus 0 Ihfminus_nonneg:MajorizesOn fminus 0 Ix✝:ℝ⊢ x✝ ∈ ↑I → 0 x✝ ≤ gplus x✝; simp [gplus]All goals completed! 🐙
  have hgminus_nonneg : MajorizesOn gminus 0 I := byI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g I⊢ IntegrableOn (f * g) I intro _I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ihfplus_nonneg:MajorizesOn fplus 0 Ihfminus_nonneg:MajorizesOn fminus 0 Ihgplus_nonneg:MajorizesOn gplus 0 Ix✝:ℝ⊢ x✝ ∈ ↑I → 0 x✝ ≤ gminus x✝; simp [gminus]All goals completed! 🐙
  have hfplusgplus := integ_of_mul_nonneg hfplus_integ hgplus_integ hfplus_nonneg hgplus_nonnegI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ihfplus_nonneg:MajorizesOn fplus 0 Ihfminus_nonneg:MajorizesOn fminus 0 Ihgplus_nonneg:MajorizesOn gplus 0 Ihgminus_nonneg:MajorizesOn gminus 0 Ihfplusgplus:IntegrableOn ((f ⊔ fun x ↦ 0) * (g ⊔ fun x ↦ 0)) I⊢ IntegrableOn (f * g) I
  have hfplusgminus := integ_of_mul_nonneg hfplus_integ hgminus_integ hfplus_nonneg hgminus_nonnegI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ihfplus_nonneg:MajorizesOn fplus 0 Ihfminus_nonneg:MajorizesOn fminus 0 Ihgplus_nonneg:MajorizesOn gplus 0 Ihgminus_nonneg:MajorizesOn gminus 0 Ihfplusgplus:IntegrableOn ((f ⊔ fun x ↦ 0) * (g ⊔ fun x ↦ 0)) Ihfplusgminus:IntegrableOn ((f ⊔ fun x ↦ 0) * gminus) I⊢ IntegrableOn (f * g) I
  have hfminusgplus := integ_of_mul_nonneg hfminus_integ hgplus_integ hfminus_nonneg hgplus_nonnegI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ihfplus_nonneg:MajorizesOn fplus 0 Ihfminus_nonneg:MajorizesOn fminus 0 Ihgplus_nonneg:MajorizesOn gplus 0 Ihgminus_nonneg:MajorizesOn gminus 0 Ihfplusgplus:IntegrableOn ((f ⊔ fun x ↦ 0) * (g ⊔ fun x ↦ 0)) Ihfplusgminus:IntegrableOn ((f ⊔ fun x ↦ 0) * gminus) Ihfminusgplus:IntegrableOn (fminus * (g ⊔ fun x ↦ 0)) I⊢ IntegrableOn (f * g) I
  have hfminusgminus := integ_of_mul_nonneg hfminus_integ hgminus_integ hfminus_nonneg hgminus_nonnegI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ihfplus_nonneg:MajorizesOn fplus 0 Ihfminus_nonneg:MajorizesOn fminus 0 Ihgplus_nonneg:MajorizesOn gplus 0 Ihgminus_nonneg:MajorizesOn gminus 0 Ihfplusgplus:IntegrableOn ((f ⊔ fun x ↦ 0) * (g ⊔ fun x ↦ 0)) Ihfplusgminus:IntegrableOn ((f ⊔ fun x ↦ 0) * gminus) Ihfminusgplus:IntegrableOn (fminus * (g ⊔ fun x ↦ 0)) Ihfminusgminus:IntegrableOn (fminus * gminus) I⊢ IntegrableOn (f * g) I
  rw [show f = fplus - fminus by ext; simp [fplus, fminus],
      show g = gplus - gminus by ext; simp [gplus, gminus]]I:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ihfplus_nonneg:MajorizesOn fplus 0 Ihfminus_nonneg:MajorizesOn fminus 0 Ihgplus_nonneg:MajorizesOn gplus 0 Ihgminus_nonneg:MajorizesOn gminus 0 Ihfplusgplus:IntegrableOn ((f ⊔ fun x ↦ 0) * (g ⊔ fun x ↦ 0)) Ihfplusgminus:IntegrableOn ((f ⊔ fun x ↦ 0) * gminus) Ihfminusgplus:IntegrableOn (fminus * (g ⊔ fun x ↦ 0)) Ihfminusgminus:IntegrableOn (fminus * gminus) I⊢ IntegrableOn ((fplus - fminus) * (gplus - gminus)) I
  ring_nfI:BoundedIntervalf:ℝ → ℝg:ℝ → ℝhf:IntegrableOn f Ihg:IntegrableOn g Ifplus:ℝ → ℝ := f ⊔ fun x ↦ 0fminus:ℝ → ℝ := -(f ⊓ fun x ↦ 0)gplus:ℝ → ℝ := g ⊔ fun x ↦ 0gminus:ℝ → ℝ := -(g ⊓ fun x ↦ 0)this:IntegrableOn (fun x ↦ 0) Ihfplus_integ:IntegrableOn (f ⊔ fun x ↦ 0) Ihgplus_integ:IntegrableOn (g ⊔ fun x ↦ 0) Ihfminus_integ:IntegrableOn fminus Ihgminus_integ:IntegrableOn gminus Ihfplus_nonneg:MajorizesOn fplus 0 Ihfminus_nonneg:MajorizesOn fminus 0 Ihgplus_nonneg:MajorizesOn gplus 0 Ihgminus_nonneg:MajorizesOn gminus 0 Ihfplusgplus:IntegrableOn ((f ⊔ fun x ↦ 0) * (g ⊔ fun x ↦ 0)) Ihfplusgminus:IntegrableOn ((f ⊔ fun x ↦ 0) * gminus) Ihfminusgplus:IntegrableOn (fminus * (g ⊔ fun x ↦ 0)) Ihfminusgminus:IntegrableOn (fminus * gminus) I⊢ IntegrableOn (fplus * gplus + (-(fplus * gminus) - fminus * gplus) + fminus * gminus) I
  exact ((hfplusgplus.add (hfplusgminus.neg.1.sub hfminusgplus).1).1.add hfminusgminus).1All goals completed! 🐙

open BoundedInterval

Exercise 11.4.2

theorem declaration uses `sorry`IntegrableOn.split {I: BoundedInterval} {f: ℝ → ℝ} (hf: IntegrableOn f I) (P: Partition I) :
  integ f I = ∑ J ∈ P.intervals, integ f J := byI:BoundedIntervalf:ℝ → ℝhf:IntegrableOn f IP:Partition I⊢ integ f I = ∑ J ∈ P.intervals, integ f J
    sorryAll goals completed! 🐙

end Chapter11