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9655094 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 | import OAI.Analysis.C0Absorption.Molecules
namespace OAI
namespace C0Absorption
open scoped BigOperators NNReal ENNReal
noncomputable section
open Finset
section SelectedNorm
variable {S : Type*} [MetricSpace S]
def triple (n : ℕ) : ℕ × ℕ × ℕ :=
((Nat.unpair n).1, Nat.unpair (Nat.unpair n).2)
theorem triple_surjective : Function.Surjective triple := by
rintro ⟨h, C, k⟩
exact ⟨Nat.pair h (Nat.pair C k), by simp [triple]⟩
def listConstant (n : ℕ) : ℝ≥0 := (triple n).2.1 + 1
theorem listConstant_pos (n : ℕ) : 0 < listConstant n := by
dsimp [listConstant]
positivity
def listExponent (n : ℕ) : ℝ≥0∞ := 1 + ((triple n).2.2 + 1 : ℝ≥0∞)⁻¹
instance (n : ℕ) : Fact (1 ≤ listExponent n) := ⟨by simp [listExponent]⟩
theorem listExponent_ne_top (n : ℕ) : listExponent n ≠ ∞ := by
simp [listExponent]
theorem listExponent_toReal (n : ℕ) :
(listExponent n).toReal = 1 + 1 / ((triple n).2.2 + 1 : ℝ) := by
simp [listExponent, ENNReal.toReal_add, one_div]
def alpha (n : ℕ) : ℝ := (1 / 2 : ℝ) ^ (n + 1) / (listConstant n : ℝ) ^ 2
theorem alpha_pos (n : ℕ) : 0 < alpha n := by
have hC : 0 < (listConstant n : ℝ) := by exact_mod_cast listConstant_pos n
dsimp [alpha]
positivity
def squareWeight (n : ℕ) : ℝ≥0 :=
⟨Real.sqrt ((1 / 2 : ℝ) ^ (n + 1)) / (listConstant n : ℝ), by positivity⟩
theorem squareWeight_sq (n : ℕ) : (squareWeight n : ℝ) ^ 2 = alpha n := by
change (Real.sqrt ((1 / 2 : ℝ) ^ (n + 1)) / (listConstant n : ℝ)) ^ 2 =
(1 / 2 : ℝ) ^ (n + 1) / (listConstant n : ℝ) ^ 2
rw [div_pow, Real.sq_sqrt (by positivity)]
theorem squareWeight_pos (n : ℕ) : 0 < (squareWeight n : ℝ) := by
have h := alpha_pos n
rw [← squareWeight_sq] at h
have hn := (squareWeight n).coe_nonneg
nlinarith
def selectedCoordinate (L : ℕ → Set (S → ℝ)) (n : ℕ) : Seminorm ℝ (Molecule S) :=
squareWeight n • sigma (L (triple n).1) (listConstant n) (listExponent n)
theorem selectedCoordinate_apply (L : ℕ → Set (S → ℝ)) (n : ℕ) (m : Molecule S) :
selectedCoordinate L n m = (squareWeight n : ℝ) *
sigma (L (triple n).1) (listConstant n) (listExponent n) m := rfl
theorem weighted_bound (n : ℕ) (a b : ℝ) (h : a ≤ (listConstant n : ℝ) * b) :
(squareWeight n : ℝ) * a ≤ Real.sqrt ((1 / 2 : ℝ) ^ (n + 1)) * b := by
have hC : (listConstant n : ℝ) ≠ 0 := ne_of_gt (by exact_mod_cast listConstant_pos n)
have h' := mul_le_mul_of_nonneg_left h (squareWeight n).coe_nonneg
calc
(squareWeight n : ℝ) * a ≤
(Real.sqrt ((1 / 2 : ℝ) ^ (n + 1)) / (listConstant n : ℝ)) *
((listConstant n : ℝ) * b) := h'
_ = Real.sqrt ((1 / 2 : ℝ) ^ (n + 1)) * b := by field_simp
theorem selectedCoordinate_bound (L : ℕ → Set (S → ℝ)) (o : S) (n : ℕ) (m : Molecule S) :
selectedCoordinate L n m ≤ Real.sqrt ((1 / 2 : ℝ) ^ (n + 1)) *
∑ s ∈ m.val.support, |m.val s| * dist s o := by
exact weighted_bound n _ _
(sigma_le (L (triple n).1) (listConstant n) (listExponent n) o m)
theorem geometric_summable : Summable (fun n : ℕ => (1 / 2 : ℝ) ^ (n + 1)) := by
simpa only [pow_succ] using
(summable_geometric_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2)
(by norm_num : (1 / 2 : ℝ) < 1)).mul_right (1 / 2)
theorem geometric_tsum : ∑' n : ℕ, (1 / 2 : ℝ) ^ (n + 1) = 1 := by
simp_rw [pow_succ]
rw [tsum_mul_right, tsum_geometric_of_lt_one (by norm_num) (by norm_num)]
norm_num
theorem selectedCoordinate_summable (L : ℕ → Set (S → ℝ)) (o : S) (m : Molecule S) :
Summable (fun n => (selectedCoordinate L n m) ^ 2) := by
classical
let b := ∑ s ∈ m.val.support, |m.val s| * dist s o
have hb : 0 ≤ b := Finset.sum_nonneg (fun _ _ => mul_nonneg (abs_nonneg _) dist_nonneg)
apply Summable.of_nonneg_of_le (fun n => sq_nonneg _) _ (geometric_summable.mul_right (b ^ 2))
intro n
calc
(selectedCoordinate L n m) ^ 2 ≤ (Real.sqrt ((1 / 2 : ℝ) ^ (n + 1)) * b) ^ 2 := by
apply pow_le_pow_left₀ (apply_nonneg _ _) (selectedCoordinate_bound L o n m)
_ = (1 / 2 : ℝ) ^ (n + 1) * b ^ 2 := by
rw [mul_pow, Real.sq_sqrt (by positivity)]
def selectedNorm (L : ℕ → Set (S → ℝ)) (o : S) : Seminorm ℝ (Molecule S) :=
squareSeminorm (selectedCoordinate L) (selectedCoordinate_summable L o)
theorem selectedNorm_sq (L : ℕ → Set (S → ℝ)) (o : S) (m : Molecule S) :
(selectedNorm L o m) ^ 2 = ∑' n, alpha n *
(sigma (L (triple n).1) (listConstant n) (listExponent n) m) ^ 2 := by
rw [selectedNorm, squareSeminorm_sq]
apply tsum_congr
intro n
rw [selectedCoordinate_apply, mul_pow, squareWeight_sq]
theorem selectedNorm_molecule_le (L : ℕ → Set (S → ℝ)) (o s t : S) :
selectedNorm L o (molecule s t) ≤ dist s t := by
have hh : ∀ n, (selectedCoordinate L n (molecule s t)) ^ 2 ≤
(1 / 2 : ℝ) ^ (n + 1) * (dist s t) ^ 2 := by
intro n
have h := weighted_bound n _ _
(sigma_molecule_le (L (triple n).1) (listConstant n) (listExponent n) s t)
calc
(selectedCoordinate L n (molecule s t)) ^ 2
≤ (Real.sqrt ((1 / 2 : ℝ) ^ (n + 1)) * dist s t) ^ 2 :=
pow_le_pow_left₀ (apply_nonneg _ _) h 2
_ = (1 / 2 : ℝ) ^ (n + 1) * dist s t ^ 2 := by
rw [mul_pow, Real.sq_sqrt (by positivity)]
have hsq : (selectedNorm L o (molecule s t)) ^ 2 ≤ dist s t ^ 2 := by
rw [selectedNorm, squareSeminorm_sq]
calc
∑' n, (selectedCoordinate L n (molecule s t)) ^ 2
≤ ∑' n, (1 / 2 : ℝ) ^ (n + 1) * dist s t ^ 2 :=
Summable.tsum_le_tsum hh (selectedCoordinate_summable L o _)
(geometric_summable.mul_right _)
_ = dist s t ^ 2 := by rw [tsum_mul_right, geometric_tsum, one_mul]
nlinarith [apply_nonneg (selectedNorm L o) (molecule s t), dist_nonneg (x := s) (y := t)]
end SelectedNorm
end
end C0Absorption
end OAI
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