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caec230 | 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 | import OAI.MathematicalPhysics.RapidForcing.Scales
namespace OAI
open scoped BigOperators ENNReal Topology
open Set MeasureTheory
namespace RapidForcing
namespace ScaleData
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
variable (d : ScaleData)
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' _ _
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' _ _
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
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
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)
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)
end ScaleData
end RapidForcing
end OAI
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