import Mathlib
import OAI.Probability.SKBarriers.Parisi.CDFPartitionRepresentation
import OAI.Probability.SKBarriers.Gaussian.LipschitzObservable

namespace OAI

section

noncomputable section
open scoped NNReal Topology
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic

theorem scalarWindowDerivative_lipschitz_bound {g : ℝ → ℝ} {L : ℝ≥0}
    (hg : LipschitzWith L g) {h : ℝ} (hh : 0<h) (x : ℝ) :
    |scalarWindowDerivative g h x| ≤ L := by
  have H := hg.dist_le_mul (x+h) x
  simp only [Real.dist_eq,add_sub_cancel_left,abs_of_pos hh] at H
  rw [scalarWindowDerivative,abs_div,abs_of_pos hh,div_le_iff₀ hh]
  exact H

theorem scalarWindow_lipschitz {g : ℝ → ℝ} {L : ℝ≥0}
    (hg : LipschitzWith L g) {h : ℝ} (hh : 0<h) : LipschitzWith L (scalarWindow g h) := by
  apply lipschitzWith_of_nnnorm_deriv_le
    (fun x => (scalarWindow_hasDerivAt hg.continuous h x).differentiableAt)
  intro x
  rw [(scalarWindow_hasDerivAt hg.continuous h x).deriv]
  exact_mod_cast scalarWindowDerivative_lipschitz_bound hg hh x

theorem scalarCDFAverage_eq_chainAverage_lipschitz {f g : ℝ → ℝ}
    (hf : BoundedDerivs f) {K B L : ℝ≥0} (hfK : LipschitzWith K f)
    (hgL : LipschitzWith L g) (hB : ∀ z, |g z| ≤ B)
    (β : ℝ) {α : ℝ → ℝ} (hα : ∀ z, α z∈Icc (0:ℝ) 1) (hmono : Monotone α)
    (l : List (ℝ × ℝ≥0)) (hm : ∀ p∈l, p.1∈Icc (0:ℝ) 1)
    (s : ℝ) (t : ℝ≥0) (ht : t≤1) (hdur : chainDuration l=t)
    (hmodel : TimeChainModels α s l) (x : ℝ) :
    scalarCDFAverage β α s t f g x=scalarTimeChainAverage β l f g x := by
  apply sub_eq_zero.mp
  apply abs_eq_zero.mp
  apply le_antisymm ?_ (abs_nonneg _)
  apply le_of_forall_pos_le_add
  intro ε hε
  simp only [zero_add]
  let h := ε/(2*((L:ℝ)+1))
  have hL : 0<(L:ℝ)+1 := by positivity
  have hh : 0<h := div_pos hε (mul_pos (by norm_num) hL)
  have hsmall : (L:ℝ)*h ≤ ε/2 := by
    calc
      _ ≤ ((L:ℝ)+1)*h := mul_le_mul_of_nonneg_right (by linarith) hh.le
      _ = ε/2 := by dsimp [h]; field_simp
  have hd (z) : |g z-scalarWindow g h z| ≤ (L:ℝ)*h := by
    rw [abs_sub_comm]
    exact scalarWindow_approx hgL hh z
  have H₁ := scalarCDFAverage_observable_distance β hα hmono hf hfK hgL
    (scalarWindow_lipschitz hgL hh) hB (scalarWindow_bound hgL.continuous hB hh)
    (mul_nonneg L.coe_nonneg hh.le) hd s t ht x
  have H₂ := scalarHierarchyAverage_observable_distance hf
    ⟨hgL.continuous,B,B.coe_nonneg,hB⟩
    ⟨(scalarWindow_lipschitz hgL hh).continuous,B,B.coe_nonneg,scalarWindow_bound hgL.continuous hB hh⟩
    (mul_nonneg L.coe_nonneg hh.le) hd l.length (fun i => (l.get i).1)
    (fun i => β*Real.sqrt ((l.get i).2:ℝ)) x
  change |scalarTimeChainAverage β l f g x-scalarTimeChainAverage β l f (scalarWindow g h) x|≤_ at H₂
  have HE := scalarCDFAverage_eq_chainAverage hf
    (scalarWindow_hasDerivAt hgL.continuous h) (scalarWindowDerivative_continuous hgL.continuous h)
    hfK (scalarWindow_bound hgL.continuous hB hh) (scalarWindowDerivative_lipschitz_bound hgL hh)
    β hα hmono l hm s t ht hdur hmodel x
  rw [HE] at H₁
  have H := abs_sub_le (scalarCDFAverage β α s t f g x)
    (scalarTimeChainAverage β l f (scalarWindow g h) x) (scalarTimeChainAverage β l f g x)
  rw [abs_sub_comm (scalarTimeChainAverage β l f (scalarWindow g h) x)] at H
  linarith

theorem scalarCDFAverage_eq_partition_lipschitz {f g : ℝ → ℝ} (hf : BoundedDerivs f)
    {K B L : ℝ≥0} (hK : LipschitzWith K f) (hgL : LipschitzWith L g) (hB : ∀ y, |g y|≤B)
    (β : ℝ) {α : ℝ → ℝ} (hα : ∀ z, α z∈Icc (0:ℝ) 1) (hαm : Monotone α)
    (n : ℕ) (q : Fin (n+1) → ℝ) (hq : Monotone q) (m : Fin n → ℝ)
    (hm : ∀ i, m i∈Icc (0:ℝ) 1)
    (hmodel : ∀ i : Fin n, ∀ z∈Ico (q i.castSucc) (q i.succ), α z=m i)
    (t : ℝ≥0) (ht : t≤1) (hd : (t:ℝ)=q (Fin.last n)-q 0) (x : ℝ) :
    scalarCDFAverage β α (q 0) t f g x=scalarHierarchyAverage n m (timeGridCoefficients n β q) f g x := by
  have hdur : chainDuration (partitionTimeChain n q hq m)=t := by
    apply NNReal.coe_injective
    exact (partitionTimeChain_duration n q hq m).trans hd.symm
  have hml (p) (hp : p∈partitionTimeChain n q hq m) : p.1∈Icc (0:ℝ) 1 := by
    obtain ⟨i,rfl⟩ := List.mem_ofFn.mp hp
    exact hm i
  have H := scalarCDFAverage_eq_chainAverage_lipschitz hf hK hgL hB β hα hαm
    (partitionTimeChain n q hq m) hml (q 0) t ht hdur (partitionTimeChain_models n q hq m hmodel) x
  simp only [partitionTimeChain,scalarTimeChainAverage_ofFn,NNReal.coe_mk] at H
  exact H

end SK.Analytic

end
end

end OAI