| 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 |
| |
