lean/OAI/Dynamics/StandardMap/ArrayMoments.lean

import OAI.Dynamics.StandardMap.SampleLaws

namespace OAI

open MeasureTheory Set
open scoped ENNReal BigOperators

open MeasureTheory Set Filter
open scoped ENNReal Topology Classical
namespace StandardMapEntropy
noncomputable def meanSquareDeficit (k : ℝ) (n : ℕ) : ℝ :=
  ∫ z,(torusShortfall k z 0 n)^2 ∂area
lemma meanSquareDeficit_bounds (k : ℝ) (hk : 0 ≤ k) (n : ℕ) (hn : 0 < n) :
    0 ≤ meanSquareDeficit k n ∧ meanSquareDeficit k n ≤ meanDeficit k n := by
  refine ⟨integral_nonneg (fun z => sq_nonneg _),?_⟩
  rw [meanDeficit_eq_integral_shortfall k hk n]
  apply integral_mono
  · exact ((continuous_torusShortfall k hk 0 n).pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
  · exact (continuous_torusShortfall k hk 0 n).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
  · intro z
    have h := torusShortfall_mem k hk z 0 n hn
    nlinarith [h.1,h.2]
lemma integrable_arrayShortfall (μ : Measure DistanceArray) [IsFiniteMeasure μ] (s t : DyadicTime) :
    Integrable (arrayShortfall s t) μ :=
  (continuous_arrayShortfall s t).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
lemma integrable_arrayShortfall_sq (μ : Measure DistanceArray) [IsFiniteMeasure μ] (s t : DyadicTime) :
    Integrable (fun d => (arrayShortfall s t d)^2) μ :=
  ((continuous_arrayShortfall s t).pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
lemma integral_shortfall_sq_sample (k : ℝ) (hk : 0 ≤ k) (n : ℕ) (hn : 0 < n)
    (s t : DyadicTime) (a : ℤ) (m : ℕ) (hm : 0 < m)
    (hs : (n:ℝ)*(s:ℝ)=(a:ℝ)) (ht : (n:ℝ)*(t:ℝ)=(a:ℝ)+(m:ℝ)) :
    (∫ d,(arrayShortfall s t d)^2 ∂sampleLaw k hk n hn)=meanSquareDeficit k m := by
  rw [integral_sampleLaw k hk n hn (fun d => (arrayShortfall s t d)^2) ((continuous_arrayShortfall s t).pow 2)]
  simp_rw [shortfall_sample_aligned k hk _ n hn s t a m hm hs ht]
  have he (z : Torus) : torusShortfall k z a m=torusShortfall k (torusIter k a z) 0 m := by
    rw [torusShortfall_shift,zero_add]
  simp_rw [he]
  exact integral_torusIter k a (fun z => (torusShortfall k z 0 m)^2)
lemma integral_J_sample (k : ℝ) (hk : 0 ≤ k) (n : ℕ) (hn : 0 < n)
    (s t : DyadicTime) (a : ℤ) (m : ℕ) (hm : 0 < m)
    (hs : (n:ℝ)*(s:ℝ)=(a:ℝ)) (ht : (n:ℝ)*(t:ℝ)=(a:ℝ)+2*(m:ℝ)) :
    (∫ d,arrayJ s t d ∂sampleLaw k hk n hn)=meanDeficit k (m+m)-meanDeficit k m := by
  have hmid : (n:ℝ)*(dyadicMid s t:ℝ)=(a:ℝ)+(m:ℝ) := by rw [dyadicMid_val]; nlinarith
  have hwhole := integral_shortfall_sample k hk n hn s t a (m+m) (by omega) hs (by simpa only [Nat.cast_add,two_mul] using ht)
  have hl := integral_shortfall_sample k hk n hn s (dyadicMid s t) a m hm hs hmid
  have hr := integral_shortfall_sample k hk n hn (dyadicMid s t) t (a+(m:ℤ)) m hm
    (by simpa only [Int.cast_add,Int.cast_natCast] using hmid) (by push_cast; linarith)
  have hadd : (∫ d,arrayShortfall s (dyadicMid s t) d+arrayShortfall (dyadicMid s t) t d ∂sampleLaw k hk n hn)=
      (∫ d,arrayShortfall s (dyadicMid s t) d ∂sampleLaw k hk n hn)+
        (∫ d,arrayShortfall (dyadicMid s t) t d ∂sampleLaw k hk n hn) := by
    convert! integral_add (integrable_arrayShortfall (sampleLaw k hk n hn) s (dyadicMid s t)) (integrable_arrayShortfall (sampleLaw k hk n hn) (dyadicMid s t) t) using 1
  have hsub : (∫ d,arrayJ s t d ∂sampleLaw k hk n hn)=
      (∫ d,arrayShortfall s t d ∂sampleLaw k hk n hn)-
        (∫ d,(arrayShortfall s (dyadicMid s t) d+arrayShortfall (dyadicMid s t) t d)/2 ∂sampleLaw k hk n hn) := by
    convert! integral_sub (integrable_arrayShortfall (sampleLaw k hk n hn) s t)
      (((integrable_arrayShortfall (sampleLaw k hk n hn) s (dyadicMid s t)).add (integrable_arrayShortfall (sampleLaw k hk n hn) (dyadicMid s t) t)).div_const 2) using 1
  rw [hsub,integral_div,hadd,hwhole,hl,hr]
  ring
lemma scalar_variance_test {a b c : ℝ} (ha : a∈Icc 0 1) (hb : b∈Icc 0 1)
    (hc : c∈Icc 0 1) (hJ : 0 ≤ c-(a+b)/2) :
    (a-b)^2/4 ≤ (a^2+b^2)/2-c^2+2*(c-(a+b)/2) ∧ (a-b)^2 ≤ a+b ∧ c-(a+b)/2 ≤ c := by
  have hab : 0 ≤ a*b := mul_nonneg ha.1 hb.1
  have hca : (a+b)/2 ≤ c := by linarith
  have hbar : 0 ≤ (a+b)/2 := by linarith [ha.1,hb.1]
  have hcc : 0 ≤ (c-(a+b)/2)*(2-c-(a+b)/2) := mul_nonneg hJ (by linarith [hc.2])
  refine ⟨by nlinarith,by nlinarith [ha.1,ha.2,hb.1,hb.2],by linarith⟩
lemma integral_V_sample_bound (k : ℝ) (hk : 0 ≤ k) (n : ℕ) (hn : 0 < n)
    (s t : DyadicTime) (a : ℤ) (m : ℕ) (hm : 0 < m)
    (hs : (n:ℝ)*(s:ℝ)=(a:ℝ)) (ht : (n:ℝ)*(t:ℝ)=(a:ℝ)+2*(m:ℝ)) :
    (∫ d,arrayV s t d ∂sampleLaw k hk n hn)/4 ≤
      meanSquareDeficit k m-meanSquareDeficit k (m+m)+2*(meanDeficit k (m+m)-meanDeficit k m) := by
  let μ := sampleLaw k hk n hn
  have : IsProbabilityMeasure μ := sampleLawProbability k hk n hn
  have hmid : (n:ℝ)*(dyadicMid s t:ℝ)=(a:ℝ)+(m:ℝ) := by rw [dyadicMid_val]; nlinarith
  have hwhole : (n:ℝ)*(t:ℝ)=(a:ℝ)+((m+m:ℕ):ℝ) := by push_cast; linarith
  have hrstart : (n:ℝ)*(dyadicMid s t:ℝ)=((a+(m:ℤ):ℤ):ℝ) := by simpa only [Int.cast_add,Int.cast_natCast] using hmid
  have hrend : (n:ℝ)*(t:ℝ)=((a+(m:ℤ):ℤ):ℝ)+(m:ℝ) := by push_cast; linarith
  have hslt : (s:ℝ) < (t:ℝ) := by
    have hnR : (0:ℝ) < n := by exact_mod_cast hn
    have hmR : (0:ℝ) < m := by exact_mod_cast hm
    nlinarith
  let F : DistanceArray → ℝ := fun d =>
    ((arrayShortfall s (dyadicMid s t) d)^2+(arrayShortfall (dyadicMid s t) t d)^2)/2-
      (arrayShortfall s t d)^2+2*arrayJ s t d
  have hF : Continuous F := by
    exact ((((continuous_arrayShortfall s _).pow 2).add ((continuous_arrayShortfall _ t).pow 2)).div_const 2).sub
      ((continuous_arrayShortfall s t).pow 2) |>.add (continuous_const.mul (continuous_arrayJ s t))
  have hb (z : Torus) : arrayV s t (sampleArray k hk z n hn)/4 ≤ F (sampleArray k hk z n hn) := by
    have ha := torusShortfall_mem k hk z a m hm
    have hb := torusShortfall_mem k hk z (a+(m:ℤ)) m hm
    have hcc := torusShortfall_mem k hk z a (m+m) (by omega)
    rw [← shortfall_sample_aligned k hk z n hn s (dyadicMid s t) a m hm hs hmid] at ha
    rw [← shortfall_sample_aligned k hk z n hn (dyadicMid s t) t (a+(m:ℤ)) m hm hrstart hrend] at hb
    rw [← shortfall_sample_aligned k hk z n hn s t a (m+m) (by omega) hs hwhole] at hcc
    exact (scalar_variance_test ha hb hcc (arrayJ_nonneg s t hslt _)).1
  have hi := integral_mono (μ := area)
    (((continuous_arrayV s t).comp (continuous_sampleArray k hk n hn)).div_const 4 |>.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _))
    ((hF.comp (continuous_sampleArray k hk n hn)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) hb
  simp only [Function.comp_apply] at hi
  rw [integral_div,← integral_sampleLaw k hk n hn _ (continuous_arrayV s t),← integral_sampleLaw k hk n hn F hF] at hi
  have hEval : (∫ d,F d ∂μ)=meanSquareDeficit k m-meanSquareDeficit k (m+m)+2*(meanDeficit k (m+m)-meanDeficit k m) := by
    have h1 : (∫ d,F d ∂μ)=
        (∫ d,((arrayShortfall s (dyadicMid s t) d)^2+(arrayShortfall (dyadicMid s t) t d)^2)/2-(arrayShortfall s t d)^2 ∂μ)+
          (∫ d,2*arrayJ s t d ∂μ) := by
      convert! integral_add
        ((((integrable_arrayShortfall_sq μ s (dyadicMid s t)).add (integrable_arrayShortfall_sq μ (dyadicMid s t) t)).div_const 2).sub (integrable_arrayShortfall_sq μ s t))
        (((continuous_arrayJ s t).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)).const_mul 2) using 1
    have h2 : (∫ d,((arrayShortfall s (dyadicMid s t) d)^2+(arrayShortfall (dyadicMid s t) t d)^2)/2-(arrayShortfall s t d)^2 ∂μ)=
        (∫ d,((arrayShortfall s (dyadicMid s t) d)^2+(arrayShortfall (dyadicMid s t) t d)^2)/2 ∂μ)-
          (∫ d,(arrayShortfall s t d)^2 ∂μ) := by
      convert! integral_sub (((integrable_arrayShortfall_sq μ s (dyadicMid s t)).add (integrable_arrayShortfall_sq μ (dyadicMid s t) t)).div_const 2) (integrable_arrayShortfall_sq μ s t) using 1
    have h3 : (∫ d,(arrayShortfall s (dyadicMid s t) d)^2+(arrayShortfall (dyadicMid s t) t d)^2 ∂μ)=
        (∫ d,(arrayShortfall s (dyadicMid s t) d)^2 ∂μ)+(∫ d,(arrayShortfall (dyadicMid s t) t d)^2 ∂μ) := by
      convert! integral_add (integrable_arrayShortfall_sq μ s (dyadicMid s t)) (integrable_arrayShortfall_sq μ (dyadicMid s t) t) using 1
    rw [h1,h2,integral_div,h3,integral_const_mul]
    change ((∫ d,(arrayShortfall s (dyadicMid s t) d)^2 ∂sampleLaw k hk n hn)+
      (∫ d,(arrayShortfall (dyadicMid s t) t d)^2 ∂sampleLaw k hk n hn))/2-
      (∫ d,(arrayShortfall s t d)^2 ∂sampleLaw k hk n hn)+2*(∫ d,arrayJ s t d ∂sampleLaw k hk n hn)=_
    rw [integral_shortfall_sq_sample k hk n hn s (dyadicMid s t) a m hm hs hmid,
      integral_shortfall_sq_sample k hk n hn (dyadicMid s t) t (a+(m:ℤ)) m hm hrstart hrend,
      integral_shortfall_sq_sample k hk n hn s t a (m+m) (by omega) hs hwhole,
      integral_J_sample k hk n hn s t a m hm hs ht]
    ring
  exact hi.trans_eq hEval
end StandardMapEntropy

end OAI