import Mathlib

namespace OAI

namespace Erdos970

section

open MeasureTheory Set
namespace ErdosContinuousBoundary

theorem continuous_variable_upper_integral
    (F : (ℝ×ℝ) → ℝ → ℝ) (hF : Continuous F.uncurry)
    (c : (ℝ×ℝ) → ℝ) (hc : Continuous c) :
    Continuous (fun p => ∫ x in (1:ℝ)..c p,F p x) := by
  let G : (ℝ×ℝ) → ℝ → ℝ := fun p t => F p ((c p-1)*t+1)
  have hG : Continuous G.uncurry := by
    change Continuous (F.uncurry ∘ (fun q : (ℝ×ℝ)×ℝ => (q.1,(c q.1-1)*q.2+1)))
    apply hF.comp
    exact continuous_fst.prodMk
      (((hc.comp continuous_fst).sub continuous_const).mul continuous_snd |>.add continuous_const)
  have hi : Continuous (fun p => ∫ t in Icc (0:ℝ) 1,G p t) :=
    continuous_parametric_integral_of_continuous hG isCompact_Icc
  have heq (p : ℝ×ℝ) : (c p-1)*(∫ t in Icc (0:ℝ) 1,G p t)=∫ x in (1:ℝ)..c p,F p x := by
    have h := intervalIntegral.smul_integral_comp_mul_add (a:=0) (b:=1) (fun x => F p x) (c p-1) 1
    rw [intervalIntegral.integral_of_le (by norm_num : (0:ℝ)≤1),← integral_Icc_eq_integral_Ioc] at h
    simpa only [smul_eq_mul,mul_zero,zero_add,mul_one,sub_add_cancel,G] using h
  exact ((hc.sub continuous_const).mul hi).congr heq

end ErdosContinuousBoundary

end

end Erdos970

end OAI