| import OAI.Probability.FreeIsing.Estimates.ExponentialTaylor |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Filter |
| open scoped BigOperators Topology NNReal |
|
|
| namespace Problem367.ExponentialTaylor |
|
|
| theorem norm_gaussian_recurrence_error_le_sum |
| (z : ℕ → ℂ) (dt e : ℕ → ℝ) (n : ℕ) {c : ℝ} |
| (hc : 0 ≤ c) (hdt : ∀ k < n, 0 ≤ dt k) |
| (he : ∀ k < n, |
| ‖z (k + 1) - Complex.exp ((-c * dt k : ℝ) : ℂ) * z k‖ ≤ e k) : |
| ‖z n - Complex.exp (((-c * ∑ k ∈ Finset.range n, dt k : ℝ) : ℂ)) * z 0‖ ≤ |
| ∑ k ∈ Finset.range n, e k := by |
| have h := norm_recurrence_error_le_sum z |
| (fun k => Complex.exp ((-c * dt k : ℝ) : ℂ)) e n |
| (fun k hk => ?_) he |
| · simpa only [← Complex.exp_sum, ← Complex.ofReal_sum, ← Finset.mul_sum] using h |
| · rw [Complex.norm_exp_ofReal] |
| exact Real.exp_le_one_iff.mpr (mul_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr hc) (hdt k hk)) |
|
|
| theorem eq_gaussian_mul_of_recurrence_errors_tendsto_zero |
| {ι : Type*} {l : Filter ι} [NeBot l] |
| (z : ι → ℕ → ℂ) (dt e : ι → ℕ → ℝ) (N : ι → ℕ) |
| {c T : ℝ} {z₀ zT : ℂ} |
| (hc : 0 ≤ c) |
| (hdt : ∀ i k, k < N i → 0 ≤ dt i k) |
| (htotal : ∀ i, ∑ k ∈ Finset.range (N i), dt i k = T) |
| (hstart : ∀ i, z i 0 = z₀) (hend : ∀ i, z i (N i) = zT) |
| (he : ∀ i k, k < N i → |
| ‖z i (k + 1) - Complex.exp ((-c * dt i k : ℝ) : ℂ) * z i k‖ ≤ e i k) |
| (hlim : Tendsto (fun i => ∑ k ∈ Finset.range (N i), e i k) l (𝓝 0)) : |
| zT = Complex.exp ((-c * T : ℝ) : ℂ) * z₀ := by |
| apply sub_eq_zero.mp |
| apply norm_eq_zero.mp |
| apply le_antisymm _ (norm_nonneg _) |
| apply ge_of_tendsto hlim |
| exact Eventually.of_forall fun i => by |
| simpa only [htotal i, hstart i, hend i] using |
| norm_gaussian_recurrence_error_le_sum (z i) (dt i) (e i) (N i) hc (hdt i) (he i) |
|
|
| theorem sum_mul_sqrt_le_total_mul_sqrt_mesh |
| (dt : ℕ → ℝ) (n : ℕ) {mesh : ℝ} |
| (hdt : ∀ k < n, 0 ≤ dt k) (hmesh : ∀ k < n, dt k ≤ mesh) : |
| (∑ k ∈ Finset.range n, dt k * Real.sqrt (dt k)) ≤ |
| (∑ k ∈ Finset.range n, dt k) * Real.sqrt mesh := by |
| rw [Finset.sum_mul] |
| apply Finset.sum_le_sum |
| intro k hk |
| exact mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt (hmesh k (Finset.mem_range.mp hk))) |
| (hdt k (Finset.mem_range.mp hk)) |
|
|
| theorem sum_sq_le_total_mul_mesh |
| (dt : ℕ → ℝ) (n : ℕ) {mesh : ℝ} |
| (hdt : ∀ k < n, 0 ≤ dt k) (hmesh : ∀ k < n, dt k ≤ mesh) : |
| (∑ k ∈ Finset.range n, (dt k) ^ 2) ≤ |
| (∑ k ∈ Finset.range n, dt k) * mesh := by |
| rw [Finset.sum_mul] |
| apply Finset.sum_le_sum |
| intro k hk |
| simpa only [pow_two] using mul_le_mul_of_nonneg_left |
| (hmesh k (Finset.mem_range.mp hk)) (hdt k (Finset.mem_range.mp hk)) |
|
|
| theorem sum_cube_le_total_mul_mesh_sq |
| (dt : ℕ → ℝ) (n : ℕ) {mesh : ℝ} |
| (hdt : ∀ k < n, 0 ≤ dt k) (hmesh : ∀ k < n, dt k ≤ mesh) : |
| (∑ k ∈ Finset.range n, (dt k) ^ 3) ≤ |
| (∑ k ∈ Finset.range n, dt k) * mesh ^ 2 := by |
| rw [Finset.sum_mul] |
| apply Finset.sum_le_sum |
| intro k hk |
| have hk' := Finset.mem_range.mp hk |
| have hsq : (dt k) ^ 2 ≤ mesh ^ 2 := |
| pow_le_pow_left₀ (hdt k hk') (hmesh k hk') 2 |
| convert mul_le_mul_of_nonneg_left hsq (hdt k hk') using 1; ring |
|
|
| theorem tendsto_sum_brownian_drift_errors |
| {ι : Type*} {l : Filter ι} |
| (dt : ι → ℕ → ℝ) (N : ι → ℕ) (mesh : ι → ℝ) |
| {T A B C : ℝ} (hA : 0 ≤ A) (hB : 0 ≤ B) (hC : 0 ≤ C) |
| (hdt : ∀ i k, k < N i → 0 ≤ dt i k) |
| (hmesh : ∀ i k, k < N i → dt i k ≤ mesh i) |
| (htotal : ∀ i, ∑ k ∈ Finset.range (N i), dt i k = T) |
| (hlim : Tendsto mesh l (𝓝 0)) : |
| Tendsto (fun i => ∑ k ∈ Finset.range (N i), |
| (A * (dt i k * Real.sqrt (dt i k)) + B * (dt i k) ^ 2 + C * (dt i k) ^ 3)) |
| l (𝓝 0) := by |
| have hupper (i) : (∑ k ∈ Finset.range (N i), |
| (A * (dt i k * Real.sqrt (dt i k)) + B * (dt i k) ^ 2 + C * (dt i k) ^ 3)) ≤ |
| A * (T * Real.sqrt (mesh i)) + B * (T * mesh i) + C * (T * (mesh i) ^ 2) := by |
| simp only [Finset.sum_add_distrib, ← Finset.mul_sum] |
| apply add_le_add |
| · apply add_le_add |
| · exact mul_le_mul_of_nonneg_left |
| (by simpa only [htotal i] using (sum_mul_sqrt_le_total_mul_sqrt_mesh |
| (dt i) (N i) (hdt i) (hmesh i))) hA |
| · exact mul_le_mul_of_nonneg_left |
| (by simpa only [htotal i] using (sum_sq_le_total_mul_mesh |
| (dt i) (N i) (hdt i) (hmesh i))) hB |
| · exact mul_le_mul_of_nonneg_left |
| (by simpa only [htotal i] using (sum_cube_le_total_mul_mesh_sq |
| (dt i) (N i) (hdt i) (hmesh i))) hC |
| have hright : Tendsto (fun i => A * (T * Real.sqrt (mesh i)) + |
| B * (T * mesh i) + C * (T * (mesh i) ^ 2)) l (𝓝 0) := by |
| simpa only [Function.comp_def, Real.sqrt_zero, mul_zero, zero_pow (by norm_num : 2 ≠ 0), add_zero] using |
| (((Real.continuous_sqrt.tendsto 0).comp hlim).const_mul T |>.const_mul A).add |
| ((hlim.const_mul T).const_mul B) |>.add (((hlim.pow 2).const_mul T).const_mul C) |
| apply squeeze_zero (fun i => ?_) hupper hright |
| apply Finset.sum_nonneg |
| intro k hk |
| have := hdt i k (Finset.mem_range.mp hk) |
| positivity |
|
|
| theorem eq_gaussian_mul_of_local_error |
| (z : ℝ → ℂ) {c T A B C : ℝ} |
| (hc : 0 ≤ c) (hT : 0 ≤ T) |
| (hA : 0 ≤ A) (hB : 0 ≤ B) (hC : 0 ≤ C) |
| (hstep : ∀ s t : ℝ, 0 ≤ s → s ≤ t → t ≤ T → |
| ‖z t - Complex.exp ((-c * (t - s) : ℝ) : ℂ) * z s‖ ≤ |
| A * ((t - s) * Real.sqrt (t - s)) + B * (t - s) ^ 2 + C * (t - s) ^ 3) : |
| z T = Complex.exp ((-c * T : ℝ) : ℂ) * z 0 := by |
| let mesh : ℕ → ℝ := fun n => T / ((n : ℝ) + 1) |
| let grid : ℕ → ℕ → ℝ := fun n k => (k : ℝ) * mesh n |
| let e : ℕ → ℕ → ℝ := fun n _ => |
| A * (mesh n * Real.sqrt (mesh n)) + B * (mesh n) ^ 2 + C * (mesh n) ^ 3 |
| have hm (n : ℕ) : 0 ≤ mesh n := by dsimp [mesh]; positivity |
| have htotal (n : ℕ) : (∑ _k ∈ Finset.range (n + 1), mesh n) = T := by |
| simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul, Nat.cast_add, Nat.cast_one] |
| dsimp [mesh] |
| field_simp |
| have hmeshlim : Tendsto mesh atTop (𝓝 0) := by |
| simpa only [mesh, div_eq_mul_inv, one_div, one_mul, mul_zero] using |
| (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul T |
| have herrlim : Tendsto (fun n => ∑ k ∈ Finset.range (n + 1), e n k) atTop (𝓝 0) := |
| tendsto_sum_brownian_drift_errors (fun n _ => mesh n) (fun n => n + 1) mesh |
| hA hB hC (fun n _ _ => hm n) (fun _ _ _ => le_rfl) htotal hmeshlim |
| apply eq_gaussian_mul_of_recurrence_errors_tendsto_zero |
| (fun n k => z (grid n k)) (fun n _ => mesh n) e (fun n => n + 1) hc |
| (fun n _ _ => hm n) htotal (fun n => by simp [grid]) |
| (fun n => ?_) (fun n k hk => ?_) herrlim |
| · congr 1 |
| dsimp [grid, mesh] |
| rw [Nat.cast_add, Nat.cast_one] |
| field_simp |
| · have hkn : (k : ℝ) + 1 ≤ (n : ℝ) + 1 := by exact_mod_cast (Nat.succ_le_of_lt hk) |
| have hstart : 0 ≤ grid n k := mul_nonneg (Nat.cast_nonneg k) (hm n) |
| have hdiff : grid n (k + 1) - grid n k = mesh n := by |
| simp only [grid, Nat.cast_add, Nat.cast_one] |
| ring |
| have horder : grid n k ≤ grid n (k + 1) := by linarith [hm n] |
| have hend : grid n (k + 1) ≤ T := by |
| calc |
| grid n (k + 1) = ((k : ℝ) + 1) * mesh n := by simp [grid] |
| _ ≤ ((n : ℝ) + 1) * mesh n := mul_le_mul_of_nonneg_right hkn (hm n) |
| _ = T := by dsimp [mesh]; field_simp |
| simpa only [hdiff, e] using hstep (grid n k) (grid n (k + 1)) hstart horder hend |
|
|
| theorem eq_gaussian_mul_of_local_error_nnreal |
| (z : ℝ≥0 → ℂ) (T : ℝ≥0) {c A B C : ℝ} |
| (hc : 0 ≤ c) (hA : 0 ≤ A) (hB : 0 ≤ B) (hC : 0 ≤ C) |
| (hstep : ∀ s t : ℝ≥0, s ≤ t → t ≤ T → |
| ‖z t - Complex.exp ((-c * ((t : ℝ) - (s : ℝ)) : ℝ) : ℂ) * z s‖ ≤ |
| A * (((t : ℝ) - (s : ℝ)) * Real.sqrt ((t : ℝ) - (s : ℝ))) + |
| B * ((t : ℝ) - (s : ℝ)) ^ 2 + C * ((t : ℝ) - (s : ℝ)) ^ 3) : |
| z T = Complex.exp ((-c * (T : ℝ) : ℝ) : ℂ) * z 0 := by |
| have h := eq_gaussian_mul_of_local_error (fun r => z r.toNNReal) hc T.coe_nonneg |
| hA hB hC (fun s t hs hst ht => ?_) |
| · simpa only [Real.toNNReal_coe, Real.toNNReal_zero] using h |
| · have ht0 : 0 ≤ t := hs.trans hst |
| have hst' : s.toNNReal ≤ t.toNNReal := Real.toNNReal_mono hst |
| have ht' : t.toNNReal ≤ T := by |
| simpa only [Real.toNNReal_coe] using Real.toNNReal_mono ht |
| simpa only [Real.coe_toNNReal _ hs, Real.coe_toNNReal _ ht0] using |
| hstep s.toNNReal t.toNNReal hst' ht' |
|
|
| end Problem367.ExponentialTaylor |
|
|
| end |
|
|
| end OAI |
|
|
