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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Probability.GaussianPropeller.Mills
namespace OAI
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
namespace GaussianPropeller.Taylor
noncomputable def E (n : ℕ) (x : ℝ) : ℝ :=
∑ k ∈ Finset.range (n+1), (-1:ℝ)^k*x^k/(Nat.factorial k)
noncomputable def J (n : ℕ) (x : ℝ) : ℝ :=
∑ k ∈ Finset.range (n+1), (-1:ℝ)^k*x^(2*k+1)/((2:ℝ)^k*(Nat.factorial k)*(2*k+1))
lemma E_zero (n : ℕ) : E n 0 = 1 := by
unfold E
rw [Finset.sum_eq_single 0]
· norm_num
· intro k hk hk0
simp [zero_pow hk0]
· simp
lemma exp_remainder {x : ℝ} (hx : 0 < x) (n : ℕ) :
∃ c : ℝ, Real.exp (-x)-E n x =
(-1:ℝ)^(n+1)*Real.exp (-c)*x^(n+1)/(Nat.factorial (n+1)) := by
obtain ⟨c,hc,h⟩ := taylor_mean_remainder_lagrange_iteratedDeriv
(f := fun t : ℝ => Real.exp ((-1)*t)) (x₀ := 0) (x := x) (n := n) hx.ne
(by fun_prop)
refine ⟨c, ?_⟩
have he : taylorWithinEval (fun t : ℝ => Real.exp ((-1)*t)) n (uIcc 0 x) 0 x = E n x := by
rw [taylor_within_apply]
apply Finset.sum_congr rfl
intro k hk
rw [iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_uIcc hx.ne)
(by fun_prop) (by simp), iteratedDeriv_exp_const_mul]
simp only [mul_zero, Real.exp_zero, mul_one, sub_zero, smul_eq_mul]
ring
rw [he, iteratedDeriv_exp_const_mul] at h
simpa only [neg_one_mul, sub_zero, mul_assoc] using h
lemma exp_le_E_even (n : ℕ) {x : ℝ} (hx : 0 ≤ x) : Real.exp (-x) ≤ E (2*n) x := by
obtain rfl | hp := hx.eq_or_lt
· simp [E_zero]
obtain ⟨c,h⟩ := exp_remainder hp (2*n)
have hn : (-1:ℝ)^(2*n+1) = -1 := by rw [pow_add, pow_mul]; norm_num
rw [hn] at h
have : (-1:ℝ)*Real.exp (-c)*x^(2*n+1)/(Nat.factorial (2*n+1)) ≤ 0 := by
apply div_nonpos_of_nonpos_of_nonneg
· exact mul_nonpos_of_nonpos_of_nonneg (by linarith [Real.exp_pos (-c)]) (pow_nonneg hx _)
· positivity
linarith only [h,this]
lemma E_odd_le_exp (n : ℕ) {x : ℝ} (hx : 0 ≤ x) : E (2*n+1) x ≤ Real.exp (-x) := by
obtain rfl | hp := hx.eq_or_lt
· simp [E_zero]
obtain ⟨c,h⟩ := exp_remainder hp (2*n+1)
have hn : (-1:ℝ)^(2*n+1+1) = 1 := by rw [show 2*n+1+1=2*(n+1) by omega, pow_mul]; norm_num
rw [hn] at h
have : 0 ≤ (1:ℝ)*Real.exp (-c)*x^(2*n+1+1)/(Nat.factorial (2*n+1+1)) := by positivity
linarith only [h,this]
lemma hasDerivAt_J (n : ℕ) (x : ℝ) : HasDerivAt (J n) (E n (x^2/2)) x := by
unfold J E
convert! HasDerivAt.sum (u := Finset.range (n+1)) (fun k hk =>
(((hasDerivAt_pow (2*k+1) x).const_mul ((-1:ℝ)^k)).div_const
((2:ℝ)^k*(Nat.factorial k)*(2*k+1)))) using 1
· funext y
simp only [Finset.sum_apply]
· apply Finset.sum_congr rfl
intro k hk
have hfact : (Nat.factorial k : ℝ) ≠ 0 := by positivity
have hk' : (2*(k:ℝ)+1) ≠ 0 := by positivity
simp only [Nat.add_sub_cancel, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat, Nat.cast_one]
rw [div_pow, pow_mul]
field_simp
lemma J_zero (n : ℕ) : J n 0 = 0 := by simp [J]
lemma integral_E (n : ℕ) (x : ℝ) :
∫ t in (0:ℝ)..x, E n (t^2/2) = J n x := by
have h := intervalIntegral.integral_eq_sub_of_hasDerivAt
(fun t _ => hasDerivAt_J n t)
(show IntervalIntegrable (fun t : ℝ => E n (t^2/2)) volume 0 x from
(by unfold E; exact (by fun_prop : Continuous _).intervalIntegrable _ _))
simpa only [J_zero, sub_zero] using h
lemma integral_exp_bounds (n : ℕ) {x : ℝ} (hx : 0 ≤ x) :
J (2*n+1) x ≤ (∫ t in (0:ℝ)..x, Real.exp (-t^2/2)) ∧
(∫ t in (0:ℝ)..x, Real.exp (-t^2/2)) ≤ J (2*n) x := by
have hi (k : ℕ) : IntervalIntegrable (fun t : ℝ => E k (t^2/2)) volume 0 x := by
unfold E; exact (by fun_prop : Continuous _).intervalIntegrable _ _
have he : IntervalIntegrable (fun t : ℝ => Real.exp (-t^2/2)) volume 0 x := (by fun_prop : Continuous _).intervalIntegrable _ _
constructor
· rw [← integral_E]
apply intervalIntegral.integral_mono_on hx (hi _) he
intro t ht
simpa only [neg_div] using E_odd_le_exp n (show (0:ℝ) ≤ t^2/2 by positivity)
· rw [← integral_E]
apply intervalIntegral.integral_mono_on hx he (hi _)
intro t ht
simpa only [neg_div] using exp_le_E_even n (show (0:ℝ) ≤ t^2/2 by positivity)
end GaussianPropeller.Taylor
end OAI
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