| import OAI.Probability.GaussianPropeller.GaussianCoordinates |
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| namespace OAI |
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| universe uX |
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| 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 |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped RealInnerProductSpace Topology ENNReal |
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| namespace GaussianPropeller.PairIntegral |
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|
| 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] |
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|
| 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] |
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|
| 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 |
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|
| 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 |
| |
