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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
open scoped RealInnerProductSpace
open Set Filter
open scoped Topology RealInnerProductSpace
open scoped NNReal
open Set Filter
open scoped Topology RealInnerProductSpace NNReal
open MeasureTheory ProbabilityTheory Set Filter
open scoped Topology RealInnerProductSpace
open MeasureTheory Set Filter
open scoped Topology BigOperators
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology ENNReal
namespace GaussianPropeller.PairIntegral
lemma integrableOn_pos_sub_pow (b : ℝ) {n : ℕ} (hn : n ≠ 0) :
IntegrableOn (fun x : ℝ => (max (b-x) 0)^n) (Ici 0) := by
by_cases hb : 0 ≤ b
· have hi : IntegrableOn (fun x : ℝ => (max (b-x) 0)^n) (Icc 0 b) :=
(by fun_prop : Continuous (fun x : ℝ => (max (b-x) 0)^n)).integrableOn_Icc
have hz : IntegrableOn (fun x : ℝ => (max (b-x) 0)^n) (Ioi b) := by
apply integrableOn_zero.congr_fun _ measurableSet_Ioi
intro x hx
have hx' : b ≤ x := le_of_lt hx
simp only [max_eq_right (sub_nonpos.mpr hx'), zero_pow hn]
simpa [Icc_union_Ioi_eq_Ici hb] using hi.union hz
· apply integrableOn_zero.congr_fun _ measurableSet_Ici
intro x hx
have hx' : 0 ≤ x := hx
simp only [max_eq_right (by linarith : b-x ≤ 0), zero_pow hn]
lemma integral_pos_sub_pow (b : ℝ) {n : ℕ} (hn : n ≠ 0) :
∫ x : ℝ in Ici 0, (max (b-x) 0)^n = (max b 0)^(n+1)/(n+1) := by
by_cases hb : 0 ≤ b
· rw [max_eq_left hb, setIntegral_eq_of_subset_of_forall_sdiff_eq_zero
measurableSet_Ici (show Icc 0 b ⊆ Ici 0 from fun _ hx => hx.1)]
· rw [setIntegral_congr_fun measurableSet_Icc (fun x hx =>
congrArg (fun z : ℝ => z^n) (max_eq_left (by linarith [hx.2] : 0 ≤ b-x)))]
rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le hb,
intervalIntegral.integral_comp_sub_left (fun x : ℝ => x^n) b]
simp [integral_pow]
· intro x hx
have hxb : b < x := lt_of_not_ge (fun h => hx.2 ⟨hx.1,h⟩)
simp [max_eq_right (by linarith : b-x ≤ 0), hn]
· rw [max_eq_right (le_of_not_ge hb)]
simp only [zero_pow (Nat.succ_ne_zero n), zero_div]
apply setIntegral_eq_zero_of_forall_eq_zero
intro x hx
have hx' : 0 ≤ x := hx
simp only [max_eq_right (by linarith : b-x ≤ 0), zero_pow hn]
lemma lintegral_pos_sub_pow (b : ℝ) {n : ℕ} (hn : n ≠ 0) :
∫⁻ x : ℝ in Ici 0, ENNReal.ofReal ((max (b-x) 0)^n) =
ENNReal.ofReal ((max b 0)^(n+1)/(n+1)) := by
rw [← ofReal_integral_eq_lintegral_ofReal (integrableOn_pos_sub_pow b hn)
(ae_of_all _ (fun _ => pow_nonneg (le_max_right _ _) _)), integral_pos_sub_pow b hn]
lemma lintegral_triangle_shifted (s : ℝ) :
∫⁻ a : ℝ in Ici 0, ∫⁻ b : ℝ in Ici 0,
ENNReal.ofReal (max (3*s-a-b) 0) = ENNReal.ofReal ((max (3*s) 0)^3/6) := by
have hl (c : ℝ) : ∫⁻ b : ℝ in Ici 0, ENNReal.ofReal (max (c-b) 0) =
ENNReal.ofReal ((max c 0)^2/2) := by
simpa only [pow_one, Nat.cast_one, one_add_one_eq_two] using
lintegral_pos_sub_pow c (by decide : (1:ℕ) ≠ 0)
simp_rw [hl]
simp_rw [div_eq_mul_inv, ENNReal.ofReal_mul (sq_nonneg _)]
rw [lintegral_mul_const _ (by fun_prop), lintegral_pos_sub_pow _ (by decide : (2:ℕ) ≠ 0)]
rw [← ENNReal.ofReal_mul (by positivity)]
congr 1
norm_num
ring
lemma gaussianPDF_std_le (x : ℝ) :
gaussianPDF 0 1 x ≤ ENNReal.ofReal ((Real.sqrt (2*Real.pi))⁻¹) := by
apply ENNReal.ofReal_le_ofReal
simp only [gaussianPDFReal, NNReal.coe_one, sub_zero, mul_one]
have he : Real.exp (-(x^2)/2) ≤ 1 := Real.exp_le_one_iff.mpr (by nlinarith only [sq_nonneg x])
exact (mul_le_mul_of_nonneg_left he (by positivity)).trans_eq (mul_one _)
lemma lintegral_gaussian_std_le (f : ℝ → ℝ≥0∞) (hf : Measurable f) :
∫⁻ x, f x ∂gaussianReal 0 1 ≤
ENNReal.ofReal ((Real.sqrt (2*Real.pi))⁻¹) * ∫⁻ x, f x := by
rw [gaussianReal_of_var_ne_zero _ (by norm_num : (1:NNReal) ≠ 0),
lintegral_withDensity_eq_lintegral_mul _ (measurable_gaussianPDF _ _) hf]
calc
_ ≤ ∫⁻ x, ENNReal.ofReal ((Real.sqrt (2*Real.pi))⁻¹) * f x :=
lintegral_mono (fun x => by
simpa only [mul_comm, Pi.mul_apply] using
mul_le_mul_right (gaussianPDF_std_le x) (f x))
_ = _ := lintegral_const_mul _ hf
lemma lintegral_affine_volume (f : ℝ → ℝ≥0∞) (hf : Measurable f) (b : ℝ)
{a : ℝ} (ha : a ≠ 0) :
∫⁻ x : ℝ, f (a*x+b) = ENNReal.ofReal |a⁻¹| * ∫⁻ u, f u := by
have hm := lintegral_map (μ := (volume : Measure ℝ))
(f := fun u => f (u+b)) (g := fun x => a*x)
(by fun_prop) (by fun_prop)
rw [← hm, Real.map_volume_mul_left ha, lintegral_smul_measure,
lintegral_add_right_eq_self]
rfl
lemma lintegral_affine_gaussian_le (f : ℝ → ℝ≥0∞) (hf : Measurable f) (b : ℝ)
{a : ℝ} (ha : 0 < a) :
∫⁻ x : ℝ, f (a*x+b) ∂gaussianReal 0 1 ≤
ENNReal.ofReal ((Real.sqrt (2*Real.pi)*a)⁻¹) * ∫⁻ u, f u := by
calc
_ ≤ ENNReal.ofReal ((Real.sqrt (2*Real.pi))⁻¹) * ∫⁻ x : ℝ, f (a*x+b) :=
lintegral_gaussian_std_le _ (by fun_prop)
_ = _ := by
rw [lintegral_affine_volume f hf b ha.ne', ← mul_assoc,
← ENNReal.ofReal_mul (by positivity)]
simp only [abs_of_pos (inv_pos.mpr ha), mul_inv_rev, mul_comm]
noncomputable def triangle (s u v : ℝ) : ℝ≥0∞ :=
if u ≤ s then if v ≤ s then ENNReal.ofReal (max (s+u+v) 0) else 0 else 0
lemma measurable_triangle : Measurable (fun p : ℝ × ℝ × ℝ => triangle p.1 p.2.1 p.2.2) := by
unfold triangle
exact Measurable.ite (measurableSet_le (by fun_prop) (by fun_prop))
(Measurable.ite (measurableSet_le (by fun_prop) (by fun_prop)) (by fun_prop)
measurable_const) measurable_const
lemma lintegral_triangle (s : ℝ) :
∫⁻ u : ℝ, ∫⁻ v : ℝ, triangle s u v = ENNReal.ofReal ((max (3*s) 0)^3/6) := by
calc
_ = ∫⁻ a : ℝ, ∫⁻ b : ℝ, triangle s (s-a) (s-b) := by
simp only [lintegral_sub_left_eq_self]
exact (lintegral_sub_left_eq_self (fun u : ℝ => ∫⁻ v : ℝ, triangle s u v) s).symm
_ = ∫⁻ a : ℝ in Ici 0, ∫⁻ b : ℝ in Ici 0,
ENNReal.ofReal (max (3*s-a-b) 0) := by
have hfun (a b : ℝ) : triangle s (s-a) (s-b) =
(Ici (0:ℝ)).indicator (fun a => (Ici (0:ℝ)).indicator
(fun b => ENNReal.ofReal (max (3*s-a-b) 0)) b) a := by
simp only [triangle, Set.indicator, mem_Ici]
simp only [show s-a ≤ s ↔ 0 ≤ a by constructor <;> intro h <;> linarith,
show s-b ≤ s ↔ 0 ≤ b by constructor <;> intro h <;> linarith,
show s+(s-a)+(s-b) = 3*s-a-b by ring]
simp_rw [hfun]
have hint (a : ℝ) :
(∫⁻ b : ℝ, (Ici (0:ℝ)).indicator (fun a => (Ici (0:ℝ)).indicator
(fun b => ENNReal.ofReal (max (3*s-a-b) 0)) b) a) =
(Ici (0:ℝ)).indicator (fun a => ∫⁻ b : ℝ in Ici 0,
ENNReal.ofReal (max (3*s-a-b) 0)) a := by
by_cases ha : 0 ≤ a
· simp only [Set.indicator, mem_Ici, ite_eq_left ha]
simpa only [Set.indicator, mem_Ici] using lintegral_indicator measurableSet_Ici
(fun b : ℝ => ENNReal.ofReal (max (3*s-a-b) 0))
· simp only [Set.indicator, mem_Ici, ite_eq_right ha, lintegral_zero]
simp_rw [hint]
exact lintegral_indicator measurableSet_Ici _
_ = _ := lintegral_triangle_shifted s
lemma lintegral_gaussian_pair_le (f : ℝ × ℝ → ℝ≥0∞) (hf : Measurable f)
(b : ℝ) {a c : ℝ} (ha : 0 < a) (hc : 0 < c) :
∫⁻ x : ℝ, (∫⁻ y : ℝ, f (a*x,b*x+c*y) ∂gaussianReal 0 1) ∂gaussianReal 0 1 ≤
ENNReal.ofReal ((2*Real.pi*a*c)⁻¹) * ∫⁻ u : ℝ, ∫⁻ v : ℝ, f (u,v) := by
let k := ENNReal.ofReal ((Real.sqrt (2*Real.pi)*c)⁻¹)
have hm : Measurable (fun u : ℝ => ∫⁻ v : ℝ, f (u,v)) := hf.lintegral_prod_right'
calc
_ ≤ ∫⁻ x : ℝ, k * (∫⁻ v : ℝ, f (a*x,v)) ∂gaussianReal 0 1 := by
apply lintegral_mono
intro x
simpa only [add_comm, k] using lintegral_affine_gaussian_le
(fun v => f (a*x,v)) (by fun_prop) (b*x) hc
_ = k * (∫⁻ x : ℝ, (∫⁻ v : ℝ, f (a*x,v)) ∂gaussianReal 0 1) :=
lintegral_const_mul k (hm.comp (by fun_prop))
_ ≤ k * (ENNReal.ofReal ((Real.sqrt (2*Real.pi)*a)⁻¹) *
∫⁻ u : ℝ, ∫⁻ v : ℝ, f (u,v)) := by
gcongr
simpa only [add_zero] using lintegral_affine_gaussian_le _ hm 0 ha
_ = _ := by
rw [← mul_assoc, ← ENNReal.ofReal_mul (by positivity)]
congr 2
rw [← mul_inv]
congr 1
calc
_ = (Real.sqrt (2*Real.pi))^2*a*c := by ring
_ = _ := by rw [Real.sq_sqrt (show 0 ≤ 2*Real.pi by positivity)]
lemma integral_pos_cube_gaussian :
∫ x : ℝ, (max x 0)^3 ∂gaussianReal 0 1 = 2/Real.sqrt (2*Real.pi) := by
have hc : ∫ x : ℝ in Ioi 0, x^3*Real.exp (-(1/2:ℝ)*x^2) = 2 := by
have hh := integral_rpow_mul_exp_neg_mul_rpow (p := 2) (q := 3) (b := 1/2)
(by norm_num) (by norm_num) (by norm_num)
norm_num [Real.rpow_natCast, Real.rpow_neg, Real.Gamma_nat_eq_factorial] at hh
simpa only [neg_mul] using hh
rw [integral_gaussianReal_eq_integral_smul (by norm_num : (1:NNReal) ≠ 0)]
have hf : (fun x : ℝ => gaussianPDFReal 0 1 x • (max x 0)^3) =
(Real.sqrt (2*Real.pi))⁻¹ • (fun x : ℝ => (max x 0)^3*Real.exp (-(1/2:ℝ)*x^2)) := by
ext x
simp only [gaussianPDFReal, NNReal.coe_one, sub_zero, mul_one, Pi.smul_apply,
smul_eq_mul]
rw [show -(x^2)/2 = -(1/2:ℝ)*x^2 by ring]
ring
rw [hf]
change (∫ x : ℝ, (Real.sqrt (2*Real.pi))⁻¹ * ((max x 0)^3*Real.exp (-(1/2:ℝ)*x^2))) = _
rw [integral_const_mul]
have hs : ∫ x : ℝ, (max x 0)^3*Real.exp (-(1/2:ℝ)*x^2) = 2 := by
rw [← setIntegral_eq_integral_of_forall_compl_eq_zero (s := Ioi (0:ℝ))]
· rw [setIntegral_congr_fun measurableSet_Ioi (fun x hx => by
rw [max_eq_left (le_of_lt hx)])]
exact hc
· intro x hx
have hx' : x ≤ 0 := le_of_not_gt hx
simp [max_eq_right hx']
rw [hs]
ring
end GaussianPropeller.PairIntegral
namespace GaussianPropeller.PairIntegral
lemma integrable_pos_cube_gaussian :
Integrable (fun x : ℝ => (max x 0)^3) (gaussianReal 0 1) := by
have hi : Integrable (fun x : ℝ => ‖x‖^3) (gaussianReal 0 1) :=
(memLp_id_gaussianReal 3).integrable_norm_pow (by decide : (3:ℕ) ≠ 0)
apply hi.mono' (by fun_prop)
filter_upwards [] with x
rw [Real.norm_eq_abs, abs_of_nonneg (pow_nonneg (le_max_right _ _) _)]
apply pow_le_pow_left₀ (le_max_right _ _)
exact max_le (le_abs_self _) (abs_nonneg _)
lemma lintegral_pos_cube_gaussian :
∫⁻ x : ℝ, ENNReal.ofReal ((max x 0)^3) ∂gaussianReal 0 1 =
ENNReal.ofReal (2/Real.sqrt (2*Real.pi)) := by
rw [← ofReal_integral_eq_lintegral_ofReal integrable_pos_cube_gaussian
(ae_of_all _ (fun _ => pow_nonneg (le_max_right _ _) _)), integral_pos_cube_gaussian]
lemma lintegral_triangle_gaussian {a c : ℝ} (b : ℝ) (ha : 0 < a) (hc : 0 < c) :
∫⁻ s : ℝ, (∫⁻ x : ℝ, (∫⁻ y : ℝ, triangle s (a*x) (b*x+c*y)
∂gaussianReal 0 1) ∂gaussianReal 0 1) ∂gaussianReal 0 1 ≤
ENNReal.ofReal (9/(2*Real.pi*Real.sqrt (2*Real.pi)*a*c)) := by
have hm : Measurable (fun s : ℝ => ENNReal.ofReal ((max (3*s) 0)^3/6)) := by fun_prop
calc
_ ≤ ∫⁻ s : ℝ, ENNReal.ofReal ((2*Real.pi*a*c)⁻¹) *
ENNReal.ofReal ((max (3*s) 0)^3/6) ∂gaussianReal 0 1 := by
apply lintegral_mono
intro s
exact (lintegral_gaussian_pair_le (fun p => triangle s p.1 p.2)
(measurable_triangle.comp (measurable_const.prodMk measurable_id)) b ha hc).trans_eq
(congrArg (ENNReal.ofReal ((2*Real.pi*a*c)⁻¹) * ·) (lintegral_triangle s))
_ = _ := by
have he (s : ℝ) : (max (3*s) 0)^3/6 = (9/2:ℝ)*(max s 0)^3 := by
rw [show max (3*s) 0 = 3*max s 0 by
by_cases hs : 0 ≤ s
· rw [max_eq_left hs, max_eq_left (mul_nonneg (by norm_num) hs)]
· rw [max_eq_right (le_of_not_ge hs), max_eq_right (mul_nonpos_of_nonneg_of_nonpos (by norm_num) (le_of_not_ge hs)), mul_zero]]
ring
rw [lintegral_const_mul _ hm]
simp_rw [he, ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 9/2)]
rw [lintegral_const_mul _ (by fun_prop), lintegral_pos_cube_gaussian,
← ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 9/2),
← ENNReal.ofReal_mul (by positivity)]
congr 1
field_simp
end GaussianPropeller.PairIntegral
namespace GaussianPropeller.PairIntegral
lemma integral_le_of_lintegral_ofReal_le {X : Type uX} [MeasurableSpace X]
{μ : Measure X} {f : X → ℝ} (hf : Integrable f μ) {b : ℝ} (hb : 0 ≤ b)
(h : ∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ ENNReal.ofReal b) : ∫ x, f x ∂μ ≤ b := by
rw [integral_eq_lintegral_pos_part_sub_lintegral_neg_part hf]
have hh := ENNReal.toReal_mono (ENNReal.ofReal_ne_top) h
rw [ENNReal.toReal_ofReal hb] at hh
linarith only [hh, ENNReal.toReal_nonneg (a := ∫⁻ x, ENNReal.ofReal (-f x) ∂μ)]
end GaussianPropeller.PairIntegral
end OAI
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