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openAI-math-dataset / lean /OAI /Probability /GaussianPropeller /Polar.lean
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import OAI.Probability.GaussianPropeller.SpanReduction
namespace OAI
universe uι
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
open MeasureTheory ProbabilityTheory Set Filter
open scoped RealInnerProductSpace Topology ENNReal
open Metric
namespace GaussianPropeller.Polar
open Translation
variable {ι : Type uι} [Fintype ι] [Nonempty ι]
local notation "E" => EuclideanSpace ℝ ι
local notation "S" => sphere (0 : E) 1
local notation "V" => (volume : Measure E)
lemma integral_volume_polar (f : E → ℝ) :
∫ x, f x = ∫ p : S × Ioi (0 : ℝ), f (p.2.1 • p.1.1)
∂(V).toSphere.prod (Measure.volumeIoiPow (Fintype.card ι - 1)) := by
have hh := (V).measurePreserving_homeomorphUnitSphereProd.integral_comp
(homeomorphUnitSphereProd E).measurableEmbedding
(fun p => f ((homeomorphUnitSphereProd E).symm p).val)
simp only [Homeomorph.symm_apply_apply, homeomorphUnitSphereProd_symm_apply_coe,
finrank_euclideanSpace] at hh
rw [integral_subtype_comap (measurableSet_singleton (0:E)).compl,
restrict_compl_singleton] at hh
exact hh
lemma integral_stdGaussian_polar (f : E → ℝ) :
∫ x, f x ∂stdGaussian E = (Real.sqrt (2*Real.pi))⁻¹ ^ Fintype.card ι *
∫ p : S × Ioi (0 : ℝ), Real.exp (-p.2.1^2/2) * f (p.2.1 • p.1.1)
∂(V).toSphere.prod (Measure.volumeIoiPow (Fintype.card ι - 1)) := by
rw [stdGaussian_density]
simp_rw [ENNReal.ofReal]
rw [integral_withDensity_eq_integral_smul measurable_stdDensity.real_toNNReal]
have heq : (fun x : E => (stdDensity x).toNNReal • f x) =
fun x => stdDensity x * f x := by
funext x
simp only [NNReal.smul_def, Real.coe_toNNReal _ (stdDensity_pos x).le, smul_eq_mul]
rw [heq, integral_volume_polar, ← integral_const_mul]
apply integral_congr_ae
filter_upwards [] with p
rw [stdDensity_eq]
have hn : ‖p.2.1 • p.1.1‖ = p.2.1 := by
rw [norm_smul, Real.norm_eq_abs, abs_of_pos p.2.2,
mem_sphere_zero_iff_norm.mp p.1.2, mul_one]
rw [hn]
ring
lemma integral_volumeIoiPow (d : ℕ) (g : ℝ → ℝ) :
(∫ r : Ioi (0:ℝ), g r.1 ∂Measure.volumeIoiPow d) =
∫ r in Ioi (0:ℝ), r^d * g r := by
rw [Measure.volumeIoiPow]
simp_rw [ENNReal.ofReal]
rw [integral_withDensity_eq_integral_smul
(measurable_subtype_coe.pow_const d).real_toNNReal]
have heq : (fun r : Ioi (0:ℝ) => (r.1^d).toNNReal • g r.1) =
fun r => r.1^d * g r.1 := by
funext r
simp only [NNReal.smul_def, Real.coe_toNNReal _ (pow_nonneg r.2.le d), smul_eq_mul]
rw [heq]
exact integral_subtype_comap measurableSet_Ioi (fun r : ℝ => r^d * g r)
lemma integral_homogeneous (f : E → ℝ) (q : ℕ)
(hf : ∀ (r : ℝ), 0 < r → ∀ x, f (r • x) = r^q * f x) :
∫ x, f x ∂stdGaussian E =
((Real.sqrt (2*Real.pi))⁻¹ ^ Fintype.card ι *
∫ r in Ioi (0:ℝ), r^(Fintype.card ι - 1 + q) * Real.exp (-r^2/2)) *
∫ θ : S, f θ.1 ∂(V).toSphere := by
rw [integral_stdGaussian_polar]
have heq : (fun p : S × Ioi (0:ℝ) => Real.exp (-p.2.1^2/2) * f (p.2.1 • p.1.1)) =
fun p => f p.1.1 * (p.2.1^q * Real.exp (-p.2.1^2/2)) := by
funext p
rw [hf _ p.2.2]
ring
rw [heq, integral_prod_mul (fun θ : S => f θ.1)
(fun r : Ioi (0:ℝ) => r.1^q * Real.exp (-r.1^2/2)), integral_volumeIoiPow (Fintype.card ι - 1)
(fun r : ℝ => r^q * Real.exp (-r^2/2))]
have heq' : (fun r : ℝ => r^(Fintype.card ι-1) * (r^q * Real.exp (-r^2/2))) =
fun r => r^(Fintype.card ι-1+q) * Real.exp (-r^2/2) := by
funext r
rw [pow_add]
ring
rw [heq']
ring
end GaussianPropeller.Polar
namespace GaussianPropeller.Polar
open Real
noncomputable def radial (n : ℕ) (b : ℝ) : ℝ := ∫ r in Ioi (0:ℝ), r^n * exp (-b*r^2)
lemma radial_eq (n : ℕ) {b : ℝ} (hb : 0<b) :
radial n b = b ^ (-((n:ℝ)+1)/2) * (1/2) * Gamma (((n:ℝ)+1)/2) := by
have hh := integral_rpow_mul_exp_neg_mul_rpow (p:=2) (q:=n) (b:=b)
(by norm_num) (by linarith [Nat.cast_nonneg (α:=ℝ) n]) hb
simpa only [Real.rpow_natCast, Real.rpow_two, neg_mul, radial] using hh
lemma radial_integrable (n : ℕ) {b : ℝ} (hb : 0<b) :
IntegrableOn (fun r : ℝ => r^n*exp (-b*r^2)) (Ioi 0) := by
have h := integrableOn_rpow_mul_exp_neg_mul_sq hb (s:=(n:ℝ)) (by linarith [Nat.cast_nonneg (α:=ℝ) n])
simpa only [Real.rpow_natCast] using h
lemma radial_pos (n : ℕ) {b : ℝ} (hb : 0<b) : 0<radial n b := by
rw [radial_eq n hb]
exact mul_pos (mul_pos (rpow_pos_of_pos hb _) (by norm_num))
(Gamma_pos_of_pos (by positivity))
lemma radial_scale (n : ℕ) {b : ℝ} (hb : 0<b) :
radial n b = (sqrt (2*b))⁻¹ ^ (n+1) * radial n (1/2) := by
have hp : 0<sqrt (2*b) := by positivity
have hs := sq_sqrt (show 0≤2*b by positivity)
have hh := integral_comp_mul_left_Ioi (fun r : ℝ => r^n*exp (-(1/2:ℝ)*r^2)) 0 hp
rw [mul_zero] at hh
have heq : (fun r : ℝ => (sqrt (2*b)*r)^n * exp (-(1/2:ℝ)*(sqrt (2*b)*r)^2)) =
fun r => sqrt (2*b)^n * (r^n*exp (-b*r^2)) := by
funext r
rw [mul_pow, show -(1/2:ℝ)*(sqrt (2*b)*r)^2 = -b*r^2 by rw [mul_pow,hs]; ring]
ring
change (∫ r in Ioi (0:ℝ), (sqrt (2*b)*r)^n * exp (-(1/2:ℝ)*(sqrt (2*b)*r)^2)) = _ at hh
rw [heq, integral_const_mul] at hh
change sqrt (2*b)^n*radial n b = (sqrt (2*b))⁻¹*radial n (1/2) at hh
apply (mul_left_cancel₀ (pow_ne_zero n hp.ne'))
rw [hh, pow_succ, mul_assoc, ← mul_assoc (sqrt (2*b)^n)]
simp only [inv_pow, mul_inv_cancel₀ (pow_ne_zero n hp.ne'), one_mul]
lemma radial_half_zero : radial 0 (1/2) = sqrt (2*Real.pi)/2 := by
simpa [radial, sqrt_mul (by norm_num : (0:ℝ)≤2), mul_comm] using integral_gaussian_Ioi (1/2:ℝ)
lemma radial_half_one : radial 1 (1/2) = 1 := by
rw [radial_eq 1 (by norm_num)]
norm_num
lemma radial_half_two : radial 2 (1/2) = sqrt (2*Real.pi)/2 := by
rw [radial_eq 2 (by norm_num)]
have hg := Gamma_nat_add_one_add_half 0
norm_num at hg ⊢
rw [hg]
rw [show (-((3:ℝ)/2))=-(1+(1/2:ℝ)) by ring, rpow_neg (by norm_num : (0:ℝ)≤1/2),
rpow_add (by norm_num : (0:ℝ)<1/2), rpow_one, ← sqrt_eq_rpow]
have h1 := sq_sqrt (by norm_num : (0:ℝ)≤1/2)
have h2 := sq_sqrt (by norm_num : (0:ℝ)≤2)
have hp1 := sqrt_pos.mpr (by norm_num : (0:ℝ)<1/2)
have hp2 := sqrt_pos.mpr (by norm_num : (0:ℝ)<2)
have hh : sqrt (1/2:ℝ)*sqrt 2=1 := by rw [← sqrt_mul (by norm_num : (0:ℝ)≤1/2)]; norm_num
field_simp
nlinarith only [hh]
lemma radial_half_three : radial 3 (1/2) = 2 := by
rw [radial_eq 3 (by norm_num)]
norm_num [Gamma_nat_eq_factorial, rpow_neg, rpow_natCast]
lemma radial_half_four : radial 4 (1/2) = 3*sqrt (2*Real.pi)/2 := by
have h2 := radial_half_two
rw [radial_eq 2 (by norm_num)] at h2
rw [radial_eq 4 (by norm_num)]
have hg := Gamma_add_one (s:=(3/2:ℝ)) (by norm_num)
norm_num at hg h2 ⊢
rw [hg]
have he : (1/2:ℝ)^(-((5:ℝ)/2)) = 2*(1/2:ℝ)^(-((3:ℝ)/2)) := by
rw [show (-((5:ℝ)/2))=-(1:ℝ)+(-((3:ℝ)/2)) by ring, rpow_add (by norm_num : (0:ℝ)<1/2)]
norm_num
rw [he]
nlinarith only [h2]
variable {ι : Type uι} [Fintype ι] [Nonempty ι]
local notation "E" => EuclideanSpace ℝ ι
local notation "S" => Metric.sphere (0:E) 1
lemma integral_radial (f : ℝ → ℝ) :
∫ x : E, f ‖x‖ ∂stdGaussian E =
((sqrt (2*Real.pi))⁻¹ ^ Fintype.card ι * ∫ _ : S, (1:ℝ) ∂(volume : Measure E).toSphere) *
∫ r in Ioi (0:ℝ), r^(Fintype.card ι-1)*exp (-r^2/2)*f r := by
rw [integral_stdGaussian_polar]
have heq : (fun p : S × Ioi (0:ℝ) => exp (-p.2.1^2/2)*f ‖p.2.1•p.1.1‖) =
fun p => (1:ℝ)*(exp (-p.2.1^2/2)*f p.2.1) := by
funext p
simp [norm_smul, Real.norm_eq_abs, abs_of_pos (show 0<p.2.1 from p.2.2),
mem_sphere_zero_iff_norm.mp p.1.2]
rw [heq, integral_prod_mul (fun θ : S => (1:ℝ))
(fun r : Ioi (0:ℝ) => exp (-r.1^2/2)*f r.1), integral_volumeIoiPow (Fintype.card ι-1) (fun r => exp (-r^2/2)*f r)]
simp_rw [mul_assoc]
lemma integral_radial_normalized (f : ℝ → ℝ) :
∫ x : E, f ‖x‖ ∂stdGaussian E =
(∫ r in Ioi (0:ℝ), r^(Fintype.card ι-1)*exp (-r^2/2)*f r) /
radial (Fintype.card ι-1) (1/2) := by
have h1 := integral_radial (ι:=ι) (fun _ => (1:ℝ))
simp only [integral_const, probReal_univ, smul_eq_mul, mul_one] at h1
have hi : (∫ r in Ioi (0:ℝ), r^(Fintype.card ι-1)*exp (-r^2/2)) = radial (Fintype.card ι-1) (1/2) := by
unfold radial
congr 1; funext r; congr 2; ring
rw [hi] at h1
rw [integral_radial]
rw [← one_div_mul_eq_div]
congr 1
apply (eq_div_iff (radial_pos _ (by norm_num)).ne').mpr
simpa using h1.symm
lemma integral_norm_pow_exp (q : ℕ) {a : ℝ} (_ha : 0≤a) :
∫ x : E, ‖x‖^q*exp (-a*‖x‖^2) ∂stdGaussian E =
radial (Fintype.card ι-1+q) (a+1/2)/radial (Fintype.card ι-1) (1/2) := by
rw [integral_radial_normalized (fun r => r^q*exp (-a*r^2))]
congr 1
unfold radial
apply setIntegral_congr_fun measurableSet_Ioi
intro r hr
dsimp only
rw [pow_add]
rw [show r^(Fintype.card ι-1)*exp (-r^2/2)*(r^q*exp (-a*r^2)) =
r^(Fintype.card ι-1)*r^q*(exp (-r^2/2)*exp (-a*r^2)) by ring, ← exp_add]
congr 2
ring
lemma integral_norm (hd : Fintype.card ι = 3) :
∫ x : E, ‖x‖ ∂stdGaussian E = 4/sqrt (2*Real.pi) := by
have h := integral_norm_pow_exp (ι:=ι) 1 (a:=0) (by norm_num)
convert h using 1 <;> simp only [hd, zero_add, neg_zero, zero_mul, exp_zero, mul_one, pow_one,
show (3-1+1:ℕ)=3 by decide, show (3-1:ℕ)=2 by decide,
radial_half_three,radial_half_two]
ring
end GaussianPropeller.Polar
end OAI