| import OAI.Probability.GaussianPropeller.Mills |
|
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| namespace OAI |
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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 |
|
|
| 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 |
| |
