lean/OAI/Probability/GaussianPropeller/Quantile.lean

import OAI.Probability.GaussianPropeller.HeatConvolution

namespace OAI

universe uX

open MeasureTheory ProbabilityTheory
open scoped ENNReal
open scoped RealInnerProductSpace
open scoped RealInnerProductSpace
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal RealInnerProductSpace
open Filter
open scoped Topology

open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology

namespace GaussianPropeller.Quantile

noncomputable def φ (x : ℝ) : ℝ := (Real.sqrt (2*Real.pi))⁻¹ * Real.exp (-x^2/2)
noncomputable def Φ (x : ℝ) : ℝ := ∫ y in Iic x, φ y

lemma φ_pos (x : ℝ) : 0 < φ x := by unfold φ; positivity
lemma φ_neg (x : ℝ) : φ (-x) = φ x := by simp [φ]
lemma integrable_φ : Integrable φ := by
  unfold φ
  have h := integrable_exp_neg_mul_sq (by norm_num : (0:ℝ) < 1/2)
  simpa only [φ, show ∀ x : ℝ, -x^2/2 = -(1/2:ℝ)*x^2 by intro x; ring] using
    h.const_mul ((Real.sqrt (2*Real.pi))⁻¹)
lemma integral_φ : ∫ x, φ x = 1 := by
  unfold φ
  rw [integral_const_mul]
  simp_rw [show ∀ x : ℝ, -x^2/2 = -(1/2:ℝ)*x^2 by intro x; ring]
  rw [integral_gaussian]
  have hp : Real.pi/(1/2:ℝ) = 2*Real.pi := by ring
  rw [hp, inv_mul_cancel₀ (by positivity)]
lemma hasDerivAt_φ (x : ℝ) : HasDerivAt φ (-x*φ x) x := by
  unfold φ
  convert ((((hasDerivAt_pow 2 x).neg.div_const 2).exp).const_mul
    ((Real.sqrt (2*Real.pi))⁻¹)) using 1
  first | rfl | (dsimp; ring)
lemma continuous_φ : Continuous φ := by exact continuous_iff_continuousAt.mpr fun x =>
  (hasDerivAt_φ x).continuousAt
lemma hasDerivAt_Φ (x : ℝ) : HasDerivAt Φ (φ x) x := by
  have heq : Φ = fun a => Φ 0 + ∫ t in (0:ℝ)..a, φ t := by
    funext a
    have hi := intervalIntegral.integral_Iic_sub_Iic
      (a := 0) (b := a) integrable_φ.integrableOn integrable_φ.integrableOn
    change Φ a-Φ 0 = _ at hi
    linarith only [hi]
  rw [heq]
  exact (intervalIntegral.integral_hasDerivAt_right integrable_φ.intervalIntegrable
    continuous_φ.stronglyMeasurable.stronglyMeasurableAtFilter continuous_φ.continuousAt).const_add _
lemma continuous_Φ : Continuous Φ := continuous_iff_continuousAt.mpr fun x =>
  (hasDerivAt_Φ x).continuousAt
lemma strictMono_Φ : StrictMono Φ := strictMono_of_deriv_pos (fun x => by
    rw [(hasDerivAt_Φ x).deriv]; exact φ_pos x)
lemma Φ_pos (x : ℝ) : 0 < Φ x := by
  apply (setIntegral_pos_iff_support_of_nonneg_ae (ae_of_all _ (fun x => (φ_pos x).le))
    integrable_φ.integrableOn).mpr
  simp [Function.support, ne_of_gt (φ_pos _)]
lemma Φ_eq_one_sub_tail (x : ℝ) : Φ x = 1 - ∫ y in Ioi x, φ y := by
  have h := integral_add_compl measurableSet_Iic integrable_φ (s := Iic x)
  rw [compl_Iic, integral_φ] at h
  unfold Φ
  linarith only [h]
lemma Φ_neg (x : ℝ) : Φ (-x) = 1-Φ x := by
  have h := integral_comp_neg_Iic (-x) φ
  simp only [φ_neg, neg_neg] at h
  change Φ (-x) = (∫ y in Ioi x, φ y) at h
  rw [h, Φ_eq_one_sub_tail]
  ring
lemma Φ_lt_one (x : ℝ) : Φ x < 1 := by
  have h := Φ_pos (-x)
  rw [Φ_neg] at h
  linarith only [h]
lemma Φ_mem (x : ℝ) : Φ x ∈ Ioo (0:ℝ) 1 := ⟨Φ_pos x, Φ_lt_one x⟩
lemma Φ_zero : Φ 0 = 1/2 := by
  have h := Φ_neg 0
  simp only [neg_zero] at h
  linarith only [h]
lemma tendsto_Φ_atTop : Tendsto Φ atTop (𝓝 1) := by
  have ht := tendsto_integral_Ioi_zero (f := φ) (μ := volume) tendsto_id
  convert ht.const_sub 1 using 1
  · exact funext Φ_eq_one_sub_tail
  · norm_num
lemma tendsto_Φ_atBot : Tendsto Φ atBot (𝓝 0) := by
  have ht := tendsto_Φ_atTop.comp tendsto_neg_atBot_atTop
  have hh := ht.const_sub 1
  simpa only [Function.comp_def, Φ_neg, sub_sub_cancel, sub_self] using hh
lemma range_Φ : range Φ = Ioo (0:ℝ) 1 := by
  apply Subset.antisymm
  · rintro _ ⟨x,rfl⟩; exact Φ_mem x
  · have hh := isPreconnected_univ.intermediate_value_Ioo
      (l₁ := atBot) (l₂ := atTop) (by simp) (by simp) continuous_Φ.continuousOn
      tendsto_Φ_atBot tendsto_Φ_atTop
    simpa only [image_univ] using hh

noncomputable def cdfIso : ℝ ≃o Ioo (0:ℝ) 1 :=
  (strictMono_Φ.orderIso Φ).trans (Set.orderIsoOfEq _ _ range_Φ)
lemma cdfIso_coe (x : ℝ) : (cdfIso x : ℝ) = Φ x := rfl
noncomputable def q (p : ℝ) : ℝ :=
  if h : p ∈ Ioo (0:ℝ) 1 then cdfIso.symm ⟨p,h⟩ else 0
lemma q_of_mem {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) : q p = cdfIso.symm ⟨p,hp⟩ := by
  simp [q, hp]
lemma Φ_q {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) : Φ (q p) = p := by
  rw [q_of_mem hp, ← cdfIso_coe]
  exact congrArg Subtype.val (cdfIso.apply_symm_apply ⟨p,hp⟩)
lemma q_Φ (x : ℝ) : q (Φ x) = x := by
  rw [q_of_mem (Φ_mem x)]
  exact cdfIso.symm_apply_apply x
lemma continuousOn_q : ContinuousOn q (Ioo (0:ℝ) 1) := by
  rw [continuousOn_iff_continuous_domRestrict]
  change Continuous (fun p : Ioo (0:ℝ) 1 => q p)
  have heq : (fun p : Ioo (0:ℝ) 1 => q p) = cdfIso.symm := by
    funext p
    exact q_of_mem p.2
  rw [heq]
  exact cdfIso.symm.continuous
lemma continuousAt_q {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) : ContinuousAt q p :=
  continuousOn_q.continuousAt (isOpen_Ioo.mem_nhds hp)
lemma hasDerivAt_q {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) :
    HasDerivAt q (φ (q p))⁻¹ p := by
  apply HasDerivAt.of_local_left_inverse (continuousAt_q hp) (hasDerivAt_Φ (q p))
    (ne_of_gt (φ_pos _))
  filter_upwards [isOpen_Ioo.mem_nhds hp] with y hy
  exact Φ_q hy
lemma strictMonoOn_q : StrictMonoOn q (Ioo (0:ℝ) 1) := by
  intro a ha b hb hab
  rw [q_of_mem ha, q_of_mem hb]
  exact cdfIso.symm.strictMono hab

noncomputable def I (p : ℝ) : ℝ := φ (q p)
lemma I_pos (p : ℝ) : 0 < I p := φ_pos _
lemma I_Φ (x : ℝ) : I (Φ x) = φ x := by simp [I, q_Φ]
lemma hasDerivAt_I {p : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) : HasDerivAt I (-q p) p := by
  convert (hasDerivAt_φ (q p)).comp p (hasDerivAt_q hp) using 1 <;>
    first | rfl | field_simp [ne_of_gt (φ_pos (q p))]
lemma antitoneOn_I_deriv : AntitoneOn (fun p => -q p) (Ioo (0:ℝ) 1) := by
  intro a ha b hb hab
  exact neg_le_neg (strictMonoOn_q.monotoneOn ha hb hab)
lemma concaveOn_I : ConcaveOn ℝ (Ioo (0:ℝ) 1) I := by
  have hanti : AntitoneOn (deriv I) (Ioo (0:ℝ) 1) := by
    intro a ha b hb hab
    rw [(hasDerivAt_I ha).deriv, (hasDerivAt_I hb).deriv]
    exact antitoneOn_I_deriv ha hb hab
  apply (hanti.mono interior_subset).concaveOn_of_deriv (convex_Ioo 0 1)
  · exact fun p hp => (hasDerivAt_I hp).continuousAt.continuousWithinAt
  · exact fun p hp => (hasDerivAt_I (interior_subset hp)).differentiableAt.differentiableWithinAt

lemma concave_le_tangent {S : Set ℝ} {f : ℝ → ℝ} (hf : ConcaveOn ℝ S f)
    {x y d : ℝ} (hx : x ∈ S) (hy : y ∈ S) (hd : HasDerivAt f d x) :
    f y ≤ f x+d*(y-x) := by
  rcases lt_trichotomy x y with h | rfl | h
  · have hh := hf.slope_le_of_hasDerivAt hx hy h hd
    rw [slope_def_field] at hh
    have hh' := (div_le_iff₀ (sub_pos.mpr h)).mp hh
    linarith
  · simp
  · have hh := hf.le_slope_of_hasDerivAt hy hx h hd
    rw [slope_def_field] at hh
    have hh' := (le_div_iff₀ (sub_pos.mpr h)).mp hh
    nlinarith only [hh']

lemma I_le_tangent {p r : ℝ} (hp : p ∈ Ioo (0:ℝ) 1) (hr : r ∈ Ioo (0:ℝ) 1) :
    I r ≤ I p-q p*(r-p) := by
  simpa only [neg_mul, sub_eq_add_neg] using concave_le_tangent concaveOn_I hp hr (hasDerivAt_I hp)

lemma φ_le_zero (x : ℝ) : φ x ≤ φ 0 := by
  unfold φ
  simp only [zero_pow (by norm_num : 2 ≠ 0), neg_zero, zero_div, Real.exp_zero, mul_one]
  exact mul_le_of_le_one_right (by positivity) (Real.exp_le_one_iff.mpr (by nlinarith [sq_nonneg x]))

section Integration
variable {X : Type uX} [MeasurableSpace X] {μ : Measure X} [IsProbabilityMeasure μ]

lemma integrable_Φ_comp {f : X → ℝ} (hf : AEStronglyMeasurable f μ) :
    Integrable (fun x => Φ (f x)) μ := by
  apply Integrable.mono' (integrable_const (1:ℝ)) (continuous_Φ.comp_aestronglyMeasurable hf)
  filter_upwards [] with x
  rw [Real.norm_eq_abs, abs_of_pos (Φ_pos _)]
  exact (Φ_lt_one _).le
lemma integrable_φ_comp {f : X → ℝ} (hf : AEStronglyMeasurable f μ) :
    Integrable (fun x => φ (f x)) μ := by
  apply Integrable.mono' (integrable_const (φ 0)) (continuous_φ.comp_aestronglyMeasurable hf)
  filter_upwards [] with x
  rw [Real.norm_eq_abs, abs_of_pos (φ_pos _)]
  exact φ_le_zero _
lemma integral_Φ_mem {f : X → ℝ} (hf : AEStronglyMeasurable f μ) :
    (∫ x, Φ (f x) ∂μ) ∈ Ioo (0:ℝ) 1 := by
  have hpos {g : X → ℝ} (hg : Integrable g μ) (hh : ∀ x, 0 < g x) : 0 < ∫ x, g x ∂μ := by
    apply (integral_pos_iff_support_of_nonneg (fun x => (hh x).le) hg).mpr
    have heq : Function.support g = univ := by ext x; simp [Function.mem_support, ne_of_gt (hh x)]
    simp [heq]
  refine ⟨hpos (integrable_Φ_comp hf) (fun x => Φ_pos _), ?_⟩
  have hh := hpos ((integrable_const (1:ℝ)).sub (integrable_Φ_comp hf))
    (fun x => sub_pos.mpr (Φ_lt_one (f x)))
  simp only [Pi.sub_apply] at hh
  rw [integral_sub (integrable_const (1:ℝ)) (integrable_Φ_comp hf), integral_const] at hh
  simp only [probReal_univ, smul_eq_mul, one_mul] at hh
  linarith only [hh]

lemma integral_φ_le_I {f : X → ℝ} (hf : AEStronglyMeasurable f μ) :
    (∫ x, φ (f x) ∂μ) ≤ I (∫ x, Φ (f x) ∂μ) := by
  let p := ∫ x, Φ (f x) ∂μ
  have hp : p ∈ Ioo (0:ℝ) 1 := integral_Φ_mem hf
  have hi := integral_mono (integrable_φ_comp hf)
    ((integrable_const (I p)).sub (((integrable_Φ_comp hf).sub (integrable_const p)).const_mul (q p)))
    (fun x => by simpa only [I_Φ, Pi.sub_apply] using I_le_tangent hp (Φ_mem (f x)))
  simp only [Pi.sub_apply] at hi
  rw [integral_sub (integrable_const (I p)), integral_const, integral_const_mul,
    integral_sub (integrable_Φ_comp hf) (integrable_const p), integral_const] at hi
  · simpa only [probReal_univ, smul_eq_mul, one_mul, show (∫ x, Φ (f x) ∂μ) = p from rfl,
      sub_self, mul_zero, sub_zero] using hi
  · exact ((integrable_Φ_comp hf).sub (integrable_const p)).const_mul (q p)

lemma hasDerivAt_integral_shift {f : X → ℝ} (hf : AEStronglyMeasurable f μ) (a : ℝ) :
    HasDerivAt (fun s => ∫ x, Φ (f x+s) ∂μ) (∫ x, φ (f x+a) ∂μ) a := by
  apply (hasDerivAt_integral_of_dominated_loc_of_deriv_le (s := univ) (bound := fun _ => φ 0)
    (F' := fun s x => φ (f x+s)) (by simp) ?_ (integrable_Φ_comp (hf.add_const a))
    (continuous_φ.comp_aestronglyMeasurable (hf.add_const a)) ?_ (integrable_const (φ 0)) ?_).2
  · exact Filter.Eventually.of_forall fun s => continuous_Φ.comp_aestronglyMeasurable (hf.add_const s)
  · filter_upwards [] with x
    intro s _
    rw [Real.norm_eq_abs, abs_of_pos (φ_pos _)]
    exact φ_le_zero _
  · filter_upwards [] with x
    intro s _
    simpa only [mul_one, Function.comp_def, id_eq] using (hasDerivAt_Φ (f x+s)).comp s ((hasDerivAt_id s).const_add (f x))

lemma hasDerivAt_quantile_shift {f : X → ℝ} (hf : AEStronglyMeasurable f μ) (a : ℝ) :
    HasDerivAt (fun s => q (∫ x, Φ (f x+s) ∂μ))
      ((∫ x, φ (f x+a) ∂μ)/I (∫ x, Φ (f x+a) ∂μ)) a := by
  convert (hasDerivAt_q (integral_Φ_mem (hf.add_const a))).comp a
    (hasDerivAt_integral_shift hf a) using 1 <;> first | rfl | (simp only [I]; ring)

lemma quantile_shift_le {f : X → ℝ} (hf : AEStronglyMeasurable f μ) {a : ℝ} (ha : 0 ≤ a) :
    q (∫ x, Φ (f x+a) ∂μ) ≤ q (∫ x, Φ (f x) ∂μ)+a := by
  have hd (s : ℝ) := (hasDerivAt_quantile_shift hf s).sub (hasDerivAt_id s)
  have hm : Antitone (fun s => q (∫ x, Φ (f x+s) ∂μ)-s) := antitone_of_hasDerivAt_nonpos hd
    (fun s => by
      have hh := (div_le_one (I_pos _)).mpr (integral_φ_le_I (hf.add_const s))
      exact sub_nonpos.mpr hh)
  have hh := hm ha
  simp only [add_zero, sub_zero] at hh
  linarith only [hh]

end Integration
end GaussianPropeller.Quantile

end OAI