| import Mathlib |
| import OAI.Geometry.Cannon.PlanarGraphs.TraceCard |
| import OAI.Geometry.Cannon.PlaneTopology.VertexNearCard |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonCarrierCharging |
| open Set Metric unitInterval CannonEmbeddedArc CannonEmbeddedArc.Arc CannonIncidence |
|
|
| attribute [local instance] Classical.propDecidable |
| variable {X : Type*} [MetricSpace X] |
|
|
| theorem uniform_original_charging_counts (Q : ℕ) (C L : ℝ) |
| (hpack : CannonPacking.PolynomialPacking (X:=X) Q C) (hC : 0≤C) (hL : 1≤L) |
| (N : ℕ) : |
| ∃ kr kc : ℕ, ∀ (r M : ℝ),0<r → 1≤M → (L+4)*r≤1 → |
| ∀ T : Finset X,(∀ x∈T,∀ y∈T,x≠y → r≤dist x y) → |
| (∀ x : X,∃ y∈T,dist x y≤r) → |
| ∀ {ι : Type*} [Fintype ι],∀ H : ι → Arc ℂ,∀ e : ℂ → X, |
| (∀ x : X,Nat.card {i : ι | ∃ t:I,dist (e (H i t)) x<2*M*r}≤N) → |
| (∀ x : X,(T.filter (fun y=>dist x y<(L+3)*r)).card≤kr) ∧ |
| (∀ x : X,Nat.card {v : ℂ | Vertex H v ∧ dist (e v) x<(L+3)*r}≤kc) := by |
| classical |
| obtain ⟨kr,hkr⟩:=CannonPacking.exists_overlap_bound Q C (L+3) 1 hpack hC (by linarith) zero_lt_one |
| obtain ⟨B,hB⟩:=CannonIncidence.uniform_larger_trace Q C (L+3) hpack hC (by linarith) |
| refine ⟨kr,2*(B*N),?_⟩ |
| intro r M hr hM hscale T hsep hcov ι _ H e hN |
| constructor |
| · intro x |
| apply (Finset.card_le_card (show T.filter (fun y=>dist x y<(L+3)*r)⊆ |
| T.filter (fun y=>dist x y≤(L+3)*r) from fun y hy=> |
| Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hy).1,(Finset.mem_filter.mp hy).2.le⟩)).trans |
| apply hkr id T x r hr |
| · rw [max_eq_left (by linarith : (1:ℝ)≤L+3)] |
| nlinarith |
| · simpa only [id_eq,one_mul] using hsep |
| · intro x |
| have htrace (center : X) (radius : ℝ) : |
| traceCard (fun i=>range (fun t:I=>e (H i t))) (ball center radius)= |
| Nat.card {i : ι | ∃ t:I,dist (e (H i t)) center<radius} := by |
| have heq : {i : ι | (range (fun t:I=>e (H i t))∩ball center radius).Nonempty}= |
| {i : ι | ∃ t:I,dist (e (H i t)) center<radius} := by |
| ext index |
| constructor |
| · rintro ⟨point,⟨parameter,rfl⟩,hpoint⟩ |
| exact ⟨parameter,hpoint⟩ |
| · rintro ⟨parameter,hparameter⟩ |
| exact ⟨e (H index parameter),mem_range_self parameter,hparameter⟩ |
| simp only [traceCard,←heq,Nat.card_eq_fintype_card,Fintype.card_subtype,Set.mem_ofPred_eq] |
| have hb:=hB r M hr hM (by linarith) T hsep hcov (fun i=>range (fun t:I=>e (H i t))) N |
| (by |
| intro center |
| rw [htrace] |
| exact hN center) x |
| have hb' : Nat.card {i : ι | ∃ t:I,dist (e (H i t)) x<(L+3)*r}≤B*N := by |
| simpa only [htrace] using hb |
| exact (vertex_near_card_le H e x ((L+3)*r)).trans (Nat.mul_le_mul_left 2 hb') |
| end CannonCarrierCharging |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
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|
