| import OAI.MathematicalPhysics.RapidForcing.Scales |
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| namespace OAI |
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| open scoped BigOperators ENNReal Topology |
| open Set MeasureTheory |
| namespace RapidForcing |
| namespace ScaleData |
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| lemma weighted_geometric_summable (J : ℕ) : |
| Summable (fun n : ℕ => ((n : ℝ) + 3) ^ J * (1 / 2 : ℝ) ^ n) := by |
| have h := (summable_nat_add_iff 3).mpr |
| (summable_pow_mul_geometric_of_norm_lt_one J (r := (1 / 2 : ℝ)) (by norm_num)) |
| apply (h.mul_left (8 : ℝ)).congr |
| intro n |
| simp only [Nat.cast_add, Nat.cast_ofNat, pow_add] |
| norm_num |
| ring |
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| variable (d : ScaleData) |
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| lemma scaled_b_le_delta (n p : ℕ) (h : p + 1 ≤ n + 10) : |
| d.b n / d.δ n ^ p ≤ d.δ n := by |
| apply (div_le_iff₀ (pow_pos (d.delta_pos n) p)).mpr |
| calc |
| d.b n ≤ d.δ n ^ (n + 10) := d.b_le_delta_pow n |
| _ ≤ d.δ n ^ (p + 1) := pow_le_pow_of_le_one (d.delta_nonneg n) (d.delta_le_one n) h |
| _ = d.δ n * d.δ n ^ p := pow_succ' _ _ |
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|
| lemma scaled_b_sq_le_delta (n p : ℕ) (h : p + 1 ≤ 2 * (n + 10)) : |
| d.b n ^ 2 / d.δ n ^ p ≤ d.δ n := by |
| apply (div_le_iff₀ (pow_pos (d.delta_pos n) p)).mpr |
| calc |
| d.b n ^ 2 ≤ (d.δ n ^ (n + 10)) ^ 2 := |
| pow_le_pow_left₀ (d.b_pos n).le (d.b_le_delta_pow n) 2 |
| _ = d.δ n ^ (2 * (n + 10)) := by rw [← pow_mul, Nat.mul_comm (n + 10)] |
| _ ≤ d.δ n ^ (p + 1) := pow_le_pow_of_le_one (d.delta_nonneg n) (d.delta_le_one n) h |
| _ = d.δ n * d.δ n ^ p := pow_succ' _ _ |
|
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| lemma residual_coefficient_le {n p : ℕ} (h : p ≤ n) : |
| d.b n / d.δ n ^ (p + 2) + d.b n ^ 2 / d.δ n ^ (p + 1) ≤ 2 * d.δ n := by |
| have h₁ := d.scaled_b_le_delta n (p + 2) (by omega) |
| have h₂ := d.scaled_b_sq_le_delta n (p + 1) (by omega) |
| linarith |
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| lemma velocity_coefficient_summable (J p : ℕ) : |
| Summable (fun n : ℕ => ((n : ℝ) + 3) ^ J * (d.b n / d.δ n ^ p)) := by |
| apply (summable_nat_add_iff p).mp |
| refine Summable.of_nonneg_of_le (fun n => ?_) (fun n => ?_) |
| ((summable_nat_add_iff p).mpr (weighted_geometric_summable J)) |
| · exact mul_nonneg (by positivity) |
| (div_nonneg (d.b_pos _).le (pow_nonneg (d.delta_nonneg _) _)) |
| · apply mul_le_mul_of_nonneg_left _ (by positivity) |
| exact (d.scaled_b_le_delta _ _ (by omega)).trans (d.delta_le_geometric _) |
|
|
| lemma residual_coefficient_summable (J p : ℕ) : |
| Summable (fun n : ℕ => ((n : ℝ) + 3) ^ J * |
| (d.b n / d.δ n ^ (p + 2) + d.b n ^ 2 / d.δ n ^ (p + 1))) := by |
| apply (summable_nat_add_iff p).mp |
| refine Summable.of_nonneg_of_le (fun n => ?_) (fun n => ?_) |
| ((summable_nat_add_iff p).mpr ((weighted_geometric_summable J).mul_left 2)) |
| · have hb := d.b_pos (n + p) |
| have hδ := d.delta_pos (n + p) |
| positivity |
| · calc |
| _ ≤ (((n + p : ℕ) : ℝ) + 3) ^ J * (2 * d.δ (n + p)) := |
| mul_le_mul_of_nonneg_left (d.residual_coefficient_le (by omega)) (by positivity) |
| _ ≤ (((n + p : ℕ) : ℝ) + 3) ^ J * (2 * (1 / 2 : ℝ) ^ (n + p)) := |
| mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_left (d.delta_le_geometric _) (by norm_num)) (by positivity) |
| _ = _ := by ring |
|
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| theorem velocity_coefficient_bound (J p : ℕ) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, |
| ((n : ℝ) + 3) ^ J * (d.b n / d.δ n ^ p) ≤ C := by |
| let a := fun n : ℕ => ((n : ℝ) + 3) ^ J * (d.b n / d.δ n ^ p) |
| have ha : ∀ n, 0 ≤ a n := fun n => |
| mul_nonneg (by positivity) (div_nonneg (d.b_pos _).le (pow_nonneg (d.delta_nonneg _) _)) |
| refine ⟨∑' n : ℕ, a n, tsum_nonneg ha, ?_⟩ |
| intro n |
| exact (d.velocity_coefficient_summable J p).le_tsum n (fun j _ => ha j) |
|
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| theorem residual_coefficient_bound (J p : ℕ) : |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, |
| ((n : ℝ) + 3) ^ J * |
| (d.b n / d.δ n ^ (p + 2) + d.b n ^ 2 / d.δ n ^ (p + 1)) ≤ C := by |
| let a := fun n : ℕ => ((n : ℝ) + 3) ^ J * |
| (d.b n / d.δ n ^ (p + 2) + d.b n ^ 2 / d.δ n ^ (p + 1)) |
| have ha : ∀ n, 0 ≤ a n := by |
| intro n |
| have hb := d.b_pos n |
| have hδ := d.delta_pos n |
| dsimp [a] |
| positivity |
| refine ⟨∑' n : ℕ, a n, tsum_nonneg ha, ?_⟩ |
| intro n |
| exact (d.residual_coefficient_summable J p).le_tsum n (fun j _ => ha j) |
|
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| end ScaleData |
| end RapidForcing |
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| end OAI |
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