import Mathlib.MeasureTheory.Group.Integral
import OAI.NumberTheory.Jacobsthal.Estimates.InvariantFiniteness

namespace OAI

namespace Erdos970

section

open Set MeasureTheory
namespace Erdos970Dependency.DeficitIntegralValues
open NumberTheoryLean.LinearSieveFunctions NumberTheoryLean.BuchstabBridge
open NumberTheoryLean.NormalizedDeficits NumberTheoryLean.DeficitFutureIntegrals

theorem shift_indicator (f : ℝ → ℝ) (a d : ℝ) :
    (fun s ↦ (Ioi a).indicator f (s - d)) = (Ioi (a + d)).indicator (fun s ↦ f (s - d)) := by
  funext s
  by_cases hs : a + d < s
  · rw [indicator_of_mem (show s ∈ Ioi (a + d) from hs),
      indicator_of_mem (show s - d ∈ Ioi a by change a < s - d; linarith)]
  · rw [indicator_of_notMem (show s ∉ Ioi (a + d) from hs),
      indicator_of_notMem (show s - d ∉ Ioi a by change ¬a < s - d; linarith)]

theorem integrableOn_shift_Ioi {f : ℝ → ℝ} {a : ℝ} (hf : IntegrableOn f (Ioi a)) (d : ℝ) :
    IntegrableOn (fun s ↦ f (s - d)) (Ioi (a + d)) := by
  have h := ((integrable_indicator_iff measurableSet_Ioi).mpr hf).comp_sub_right d
  rw [shift_indicator] at h
  exact (integrable_indicator_iff measurableSet_Ioi).mp h

theorem integral_shift_Ioi (f : ℝ → ℝ) (a d : ℝ) :
    (∫ s in Ioi (a + d), f (s - d)) = ∫ s in Ioi a, f s := by
  have h := integral_sub_right_eq_self (μ := (volume : Measure ℝ)) ((Ioi a).indicator f) d
  rw [shift_indicator, integral_indicator measurableSet_Ioi, integral_indicator measurableSet_Ioi] at h
  exact h

theorem upper_integral_one : IntegrableOn upperDeficit (Ioi (1 : ℝ)) ∧
    (∫ s in Ioi (1 : ℝ), upperDeficit s) = 2 := by
  have h := lower_future_integral (s := 2) (by norm_num)
  norm_num only [show (2 : ℝ) - 1 = 1 by norm_num] at h
  refine ⟨h.1, ?_⟩
  rw [← h.2]
  norm_num [lowerDeficit, f_initial sieveA (s := 2) (by norm_num)]

theorem lower_integral_zero : IntegrableOn lowerDeficit (Ioi (0 : ℝ)) ∧
    (∫ s in Ioi (0 : ℝ), lowerDeficit s) = sieveA - 1 := by
  have h := upper_future_integral (s := 1) (by norm_num)
  norm_num only [show (1 : ℝ) - 1 = 0 by norm_num, one_mul] at h
  refine ⟨h.1, ?_⟩
  rw [← h.2]
  norm_num [upperDeficit, F_initial sieveA (s := 1) (by norm_num) (by norm_num)]

theorem lower_integral_two : IntegrableOn lowerDeficit (Ioi (2 : ℝ)) ∧
    (∫ s in Ioi (2 : ℝ), lowerDeficit s) = sieveA - 3 := by
  have h := upper_future_integral (s := 3) (by norm_num)
  norm_num only [show (3 : ℝ) - 1 = 2 by norm_num] at h
  refine ⟨h.1, ?_⟩
  rw [← h.2]
  simp only [upperDeficit, F_initial sieveA (s := 3) (by norm_num) (by norm_num)]
  ring

theorem even_majorant_integral :
    IntegrableOn (fun s ↦ lowerDeficit s + upperDeficit (s - 1)) (Ici (2 : ℝ)) ∧
      (∫ s in Ici (2 : ℝ), lowerDeficit s + upperDeficit (s - 1)) = sieveA - 1 := by
  have ha := lower_integral_two.1
  have hc : IntegrableOn (fun s ↦ upperDeficit (s - 1)) (Ioi (2 : ℝ)) := by
    simpa only [show (1 : ℝ) + 1 = 2 by norm_num] using integrableOn_shift_Ioi upper_integral_one.1 1
  refine ⟨Iff.mpr integrableOn_Ici_iff_integrableOn_Ioi (ha.add hc), ?_⟩
  rw [integral_Ici_eq_integral_Ioi, integral_add ha hc, lower_integral_two.2]
  have h := integral_shift_Ioi upperDeficit 1 1
  norm_num only [show (1 : ℝ) + 1 = 2 by norm_num] at h
  rw [h, upper_integral_one.2]
  ring

theorem odd_majorant_integral :
    IntegrableOn (fun s ↦ upperDeficit s + lowerDeficit (s - 1)) (Ici (1 : ℝ)) ∧
      (∫ s in Ici (1 : ℝ), upperDeficit s + lowerDeficit (s - 1)) = sieveA + 1 := by
  have hc := upper_integral_one.1
  have ha : IntegrableOn (fun s ↦ lowerDeficit (s - 1)) (Ioi (1 : ℝ)) := by
    simpa only [zero_add] using integrableOn_shift_Ioi lower_integral_zero.1 1
  refine ⟨Iff.mpr integrableOn_Ici_iff_integrableOn_Ioi (hc.add ha), ?_⟩
  rw [integral_Ici_eq_integral_Ioi, integral_add hc ha, upper_integral_one.2]
  have h := integral_shift_Ioi lowerDeficit 0 1
  rw [zero_add] at h
  rw [h, lower_integral_zero.2]
  ring

end Erdos970Dependency.DeficitIntegralValues

end

end Erdos970

end OAI