| import Mathlib |
| import OAI.MeasureTheory.FalconerCompletion.Measures.AmbientGridCell4 |
|
|
| namespace OAI |
|
|
| /-! Graph energies and pair estimates. -/ |
|
|
| section |
| section |
| open Set MeasureTheory |
| open scoped ComplexConjugate RealInnerProductSpace |
| namespace GraphLiteralSupport |
| namespace Attempt4Near |
| open Set MeasureTheory Metric |
|
|
| def nearGridND {d : ℕ} (h : ℕ) (P Q : Fin d → ℤ) : Prop := |
| dist (ambientGridCenter h P) (ambientGridCenter h Q) ≤ 10*(d+1:ℝ)/(2:ℝ)^h |
|
|
| noncomputable instance nearGridND_decidable {d : ℕ} (h : ℕ) : |
| DecidableRel (@nearGridND d h) := fun _ _ => Classical.propDecidable _ |
|
|
| lemma nearGridND_symm {d h : ℕ} {P Q : Fin d → ℤ} : |
| nearGridND h P Q ↔ nearGridND h Q P := by |
| simp only [nearGridND,dist_comm] |
|
|
| theorem nearGridND_partner_card {d : ℕ} (h : ℕ) (P : Fin d → ℤ) |
| (A : Finset (Fin d → ℤ)) : |
| ((A.filter (nearGridND h P)).card:ℝ) ≤ (20*(d+1:ℝ)+5)^d := by |
| have hh := ambientGridCenter_ball_card h (A.filter (nearGridND h P)) |
| (ambientGridCenter h P) (r := 10*(d+1:ℝ)/(2:ℝ)^h) (by positivity) (by |
| intro Q hQ |
| rw [dist_comm] |
| exact (Finset.mem_filter.mp hQ).2) |
| convert hh using 1 |
| congr 1 |
| field_simp |
| ring |
|
|
| def nearGridPairND {d : ℕ} (h : ℕ) |
| (a b : (Fin d → ℤ) × (Fin d → ℤ)) : Prop := |
| nearGridND h a.1 b.1 ∧ nearGridND h a.2 b.2 |
|
|
| noncomputable instance nearGridPairND_decidable {d : ℕ} (h : ℕ) : |
| DecidableRel (@nearGridPairND d h) := fun _ _ => Classical.propDecidable _ |
|
|
| lemma nearGridPairND_symm {d h : ℕ} {a b : (Fin d → ℤ) × (Fin d → ℤ)} : |
| nearGridPairND h a b ↔ nearGridPairND h b a := by |
| simp only [nearGridPairND,nearGridND_symm] |
|
|
| theorem nearGridPairND_row_card {d : ℕ} (h : ℕ) |
| (A B : Finset (Fin d → ℤ)) (a : (Fin d → ℤ) × (Fin d → ℤ)) : |
| (((A.product B).filter (nearGridPairND h a)).card:ℝ) ≤ |
| (20*(d+1:ℝ)+5)^(2*d) := by |
| have he : (A.product B).filter (nearGridPairND h a) = |
| (A.filter (nearGridND h a.1)).product (B.filter (nearGridND h a.2)) := by |
| ext b |
| simp only [Finset.product_eq_sprod,Finset.mem_filter,Finset.mem_product,nearGridPairND] |
| tauto |
| rw [he,Finset.product_eq_sprod,Finset.card_product,Nat.cast_mul] |
| calc |
| _ ≤ (20*(d+1:ℝ)+5)^d*(20*(d+1:ℝ)+5)^d := mul_le_mul |
| (nearGridND_partner_card h a.1 A) (nearGridND_partner_card h a.2 B) |
| (Nat.cast_nonneg _) (by positivity) |
| _ = _ := by rw [← pow_add,← two_mul] |
|
|
| theorem nearGridPairND_col_card {d : ℕ} (h : ℕ) |
| (A B : Finset (Fin d → ℤ)) (b : (Fin d → ℤ) × (Fin d → ℤ)) : |
| (((A.product B).filter (fun a => nearGridPairND h a b)).card:ℝ) ≤ |
| (20*(d+1:ℝ)+5)^(2*d) := by |
| have he : (A.product B).filter (fun a => nearGridPairND h a b) = |
| (A.product B).filter (nearGridPairND h b) := by |
| ext a |
| simp only [Finset.mem_filter,nearGridPairND_symm] |
| rw [he] |
| exact nearGridPairND_row_card h A B b |
|
|
| end Attempt4Near |
| end GraphLiteralSupport |
| end |
| end |
|
|
| end OAI |
|
|
