| import Mathlib | |
| import OAI.Geometry.Cannon.Modulus.RadialCellEnergy | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Metric Topology | |
| open scoped BigOperators OnePoint | |
| namespace CannonRadialEnergy | |
| attribute [local instance] Classical.propDecidable | |
| open Set Metric | |
| theorem clipped_radial_energy {ι X : Type*} [Fintype ι] [PseudoMetricSpace X] | |
| (K : ι → Set X) (c : ι → X) (r w : ι → ℝ) (good : ι → Prop) | |
| (hr : ∀ i,0≤r i) (hw : ∀ i,0≤w i) | |
| (p : X) {a b R P : ℝ} (hab : a≤b) (hb : b≤R) (hR : 0≤R) | |
| (hcell : ∀ i,good i → K i⊆closedBall (c i) (3*r i)) | |
| (hbudget : (∑ i,if good i ∧ (K i∩closedBall p R).Nonempty then | |
| (min (r i) R)^2 else 0)≤P) | |
| (D : ℕ) (hbad : (Finset.univ.filter (fun i=>¬good i)).card≤D) | |
| (hadm : ∀ t∈Icc a b,1≤∑ i,if (K i∩sphere p t).Nonempty then w i else 0) : | |
| (b-a)^2≤36*(P+(D:ℝ)*R^2)*∑ i,w i^2 := by | |
| classical | |
| let act (i : ι) := (K i∩closedBall p R).Nonempty | |
| let U (i : ι) := K i∩closedBall p R | |
| let rad (i : ι) := if act i then (if good i then min (r i) R else R) else 0 | |
| let cen (i : ι) := if good i ∧ r i≤R then c i else p | |
| have hrad (i : ι) : 0≤rad i := by | |
| dsimp [rad] | |
| split_ifs | |
| · exact le_min (hr i) hR | |
| · exact hR | |
| · exact le_rfl | |
| have hU (i : ι) : U i⊆closedBall (cen i) (3*rad i) := by | |
| intro x hx | |
| have ha : act i:=⟨x,hx⟩ | |
| dsimp [rad,cen] | |
| rw [ite_eq_left ha] | |
| by_cases hg : good i | |
| · rw [ite_eq_left hg] | |
| by_cases hrR : r i≤R | |
| · rw [ite_eq_left ⟨hg,hrR⟩,min_eq_left hrR] | |
| exact hcell i hg hx.1 | |
| · rw [ite_eq_right (fun h=>hrR h.2),min_eq_right (le_of_not_ge hrR)] | |
| exact (mem_closedBall.mp hx.2).trans (by linarith) | |
| · rw [ite_eq_right hg,ite_eq_right (fun h=>hg h.1)] | |
| exact (mem_closedBall.mp hx.2).trans (by linarith) | |
| have hbud : (∑ i,rad i^2)≤P+(D:ℝ)*R^2 := by | |
| have he (i : ι) : rad i^2 ≤ | |
| (if good i ∧ act i then (min (r i) R)^2 else 0)+ | |
| (if ¬good i then R^2 else 0) := by | |
| by_cases ha : act i <;> by_cases hg : good i <;> simp [rad,ha,hg,sq_nonneg] | |
| calc | |
| _ ≤ ∑ i,((if good i ∧ act i then (min (r i) R)^2 else 0)+ | |
| (if ¬good i then R^2 else 0)) := Finset.sum_le_sum (fun i _=>he i) | |
| _ = (∑ i,if good i ∧ act i then (min (r i) R)^2 else 0)+ | |
| ((Finset.univ.filter (fun i=>¬good i)).card:ℝ)*R^2 := by | |
| rw [Finset.sum_add_distrib] | |
| congr 1 | |
| rw [←Finset.sum_filter] | |
| simp | |
| _ ≤ P+(D:ℝ)*R^2 := add_le_add hbudget | |
| (mul_le_mul_of_nonneg_right (by exact_mod_cast hbad) (sq_nonneg R)) | |
| have hadm' : ∀ t∈Icc a b,1≤∑ i,if (U i∩sphere p t).Nonempty then w i else 0 := by | |
| intro t ht | |
| convert hadm t ht using 1 | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| congr 1 | |
| apply propext | |
| constructor | |
| · rintro ⟨x,⟨hx,-⟩,hxt⟩ | |
| exact ⟨x,hx,hxt⟩ | |
| · rintro ⟨x,hx,hxt⟩ | |
| exact ⟨x,⟨hx,(mem_sphere.mp hxt).le.trans (ht.2.trans hb)⟩,hxt⟩ | |
| have hh:=radial_cell_energy_budget hab (show (0:ℝ)≤3 by norm_num) p cen rad w U | |
| hrad hw hU hbud hadm' | |
| norm_num at hh ⊢ | |
| exact hh | |
| end CannonRadialEnergy | |
| namespace CannonCarrierCharging | |
| open CannonCells CannonDuality | |
| attribute [local instance] Classical.propDecidable | |
| theorem clipped_spherical_carrier_modulus {X ι κ : Type*} [MetricSpace X] | |
| [LocallyPathConnectedSpace X] [Fintype ι] [Fintype κ] | |
| [DecidableEq ι] [DecidableEq κ] | |
| (A : κ → Set X) (K : κ → Set (OnePoint ℂ)) (v : κ → X) (q : ι → X) | |
| (mesh η diam : ℝ) (hη : 0<η) | |
| (hdiam : ∀ d,∀ x∈A d,dist x (v d)≤diam) | |
| (F : Set (Set X)) (hFn : ∀ E∈F,E.Nonempty) | |
| (hcover : ∀ x,∃ i,x∈ball (q i) (2*mesh)) | |
| (c : κ → ℂ) (rad : κ → ℝ) (good : κ → Prop) | |
| (hr : ∀ d,0≤rad d) (p : ℂ) {a b R : ℝ} | |
| (hab : a≤b) (hb : b≤R) (hR : 0≤R) (D : ℕ) | |
| (hcell : ∀ d,good d → ((↑) : ℂ → OnePoint ℂ) ⁻¹' K d⊆closedBall (c d) (3*rad d)) | |
| (hbudget : (∑ d,if good d ∧ ((((↑) : ℂ → OnePoint ℂ) ⁻¹' K d)∩closedBall p R).Nonempty | |
| then (min (rad d) R)^2 else 0)≤(25*(D:ℝ)+1)*R^2) | |
| (hbad : (Finset.univ.filter (fun d=>¬good d)).card≤D) | |
| (hconn : ∀ t∈Icc a b,IsPreconnected | |
| (⋃ (d : {d : κ // (((↑) '' sphere p t : Set (OnePoint ℂ))∩K d).Nonempty}),A d.val)) | |
| (hcross : ∀ t∈Icc a b, | |
| ∃ x∈⋃ (d : {d : κ // (((↑) '' sphere p t : Set (OnePoint ℂ))∩K d).Nonempty}),A d.val, | |
| ∃ y∈⋃ (d : {d : κ // (((↑) '' sphere p t : Set (OnePoint ℂ))∩K d).Nonempty}),A d.val, | |
| ∀ γ : Path x y,range γ∈F) | |
| (kr kc : ℕ) | |
| (hrow : ∀ d,(Finset.univ.filter (fun i=>dist (q i) (v d)<2*mesh+η+diam)).card≤kr) | |
| (hcol : ∀ i,(Finset.univ.filter (fun d=>dist (q i) (v d)<2*mesh+η+diam)).card≤kc) : | |
| (b-a)^2≤36*(26*(D:ℝ)+1)*R^2*(kr:ℝ)*kc*modulus (fun i=>ball (q i) (2*mesh)) F := by | |
| classical | |
| have hF : F.Nonempty := by | |
| obtain ⟨x,hx,y,hy,hxy⟩:=hcross a ⟨le_rfl,hab⟩ | |
| obtain ⟨γ,-⟩:=CannonFiniteCarrier.path_in_open_enlargement (hconn a ⟨le_rfl,hab⟩) | |
| isOpen_thickening (self_subset_thickening hη _) hx hy | |
| exact ⟨range γ,hxy γ⟩ | |
| obtain ⟨_,w,hw,he⟩:=modulus_minimum (fun i=>ball (q i) (2*mesh)) hcover hF hFn | |
| let J : κ → Finset ι:=fun d=>Finset.univ.filter (fun i=>dist (q i) (v d)<2*mesh+η+diam) | |
| let W : κ → ℝ:=fun d=>∑ i∈J d,w i | |
| have hW : ∀ d,0≤W d:=fun d=>Finset.sum_nonneg fun i _=>hw.1 i | |
| have hiff (d : κ) (t : ℝ) : | |
| (((↑) '' sphere p t : Set (OnePoint ℂ))∩K d).Nonempty ↔ | |
| ((((↑) : ℂ → OnePoint ℂ) ⁻¹' K d)∩sphere p t).Nonempty := by | |
| constructor | |
| · rintro ⟨x,⟨z,hz,rfl⟩,hx⟩ | |
| exact ⟨z,hx,hz⟩ | |
| · rintro ⟨z,hz,hzt⟩ | |
| exact ⟨(z:OnePoint ℂ),⟨z,hzt,rfl⟩,hz⟩ | |
| have hadm : ∀ t∈Icc a b,1≤∑ d,if ((((↑) : ℂ → OnePoint ℂ) ⁻¹' K d)∩sphere p t).Nonempty | |
| then W d else 0 := by | |
| intro t ht | |
| obtain ⟨x,hx,y,hy,hxy⟩:=hcross t ht | |
| have hh:=carrier_admissible_charging A K K id v q ((↑) '' sphere p t : Set (OnePoint ℂ)) | |
| mesh η diam hη hdiam (fun _=>Subset.rfl) (hconn t ht) hx hy F hxy w hw | |
| simpa only [W,J,Finset.sum_filter,hiff] using hh | |
| have hrad:=CannonRadialEnergy.clipped_radial_energy | |
| (fun d=>((↑) : ℂ → OnePoint ℂ) ⁻¹' K d) c rad W good hr hW p hab hb hR hcell hbudget D hbad hadm | |
| have heW:=CannonTransfer.overlap_energy J (fun i=>w i) kr kc hrow (by | |
| intro i | |
| simpa only [J,Finset.mem_filter,Finset.mem_univ,true_and] using hcol i) | |
| have hnorm : ∑ i,(w i)^2=‖w‖^2 := by | |
| rw [PiLp.norm_sq_eq_of_L2] | |
| simp only [Real.norm_eq_abs,sq_abs] | |
| calc | |
| _ ≤ 36*((25*(D:ℝ)+1)*R^2+(D:ℝ)*R^2)*∑ d,W d^2 := hrad | |
| _ ≤ 36*((25*(D:ℝ)+1)*R^2+(D:ℝ)*R^2)*((kr:ℝ)*kc*∑ i,(w i)^2) := | |
| mul_le_mul_of_nonneg_left heW (by positivity) | |
| _ = _ := by rw [hnorm,he]; ring | |
| end CannonCarrierCharging | |
| end | |
| end | |
| end OAI | |