import Mathlib

namespace OAI

section
section
noncomputable section
namespace LogConcaveSampling.RMSIntegral
open MeasureTheory

variable {Ω E : Type*} [MeasurableSpace Ω] {ν : Measure Ω}
  [NormedAddCommGroup E]

lemma norm_sub_sq_le_two (x y : E) : ‖x-y‖^2≤2*‖x‖^2+2*‖y‖^2 := by
  have h := pow_le_pow_left₀ (norm_nonneg (x-y)) (norm_sub_le x y) 2
  nlinarith [sq_nonneg (‖x‖-‖y‖)]

lemma sub_sq_bound {f g : Ω → E} (hf : AEStronglyMeasurable f ν)
    (hg : AEStronglyMeasurable g ν)
    (hfi : Integrable (fun z => ‖f z‖^2) ν) (hgi : Integrable (fun z => ‖g z‖^2) ν)
    {A B : ℝ} (hA : (∫z,‖f z‖^2 ∂ν)≤A) (hB : (∫z,‖g z‖^2 ∂ν)≤B) :
    Integrable (fun z => ‖f z-g z‖^2) ν ∧
      (∫z,‖f z-g z‖^2 ∂ν)≤2*A+2*B := by
  have hi : Integrable (fun z => 2*‖f z‖^2+2*‖g z‖^2) ν :=
    (hfi.const_mul 2).add (hgi.const_mul 2)
  have hm : AEStronglyMeasurable (fun z => ‖f z-g z‖^2) ν := (hf.sub hg).norm.pow 2
  have hs : Integrable (fun z => ‖f z-g z‖^2) ν :=
    hi.mono' hm (Filter.Eventually.of_forall (fun z => by
      simpa only [Real.norm_eq_abs,abs_sq] using norm_sub_sq_le_two (f z) (g z)))
  refine ⟨hs,(integral_mono hs hi (fun z => norm_sub_sq_le_two (f z) (g z))).trans ?_⟩
  rw [integral_add (hfi.const_mul 2) (hgi.const_mul 2),integral_const_mul,integral_const_mul]
  linarith
end LogConcaveSampling.RMSIntegral

end

end

end

end OAI