| import Mathlib | |
| import OAI.Geometry.Cannon.Modulus.Weights | |
| import OAI.Geometry.Cannon.Modulus.PathCellIndices | |
| import OAI.Geometry.Cannon.Energy.StripCrossing | |
| namespace OAI | |
| section | |
| section | |
| section | |
| noncomputable section | |
| section | |
| section | |
| section | |
| section | |
| section | |
| open Set Topology | |
| open Set Topology Filter Metric unitInterval | |
| open scoped InnerProductSpace | |
| open Set Metric Topology | |
| open Set Topology unitInterval | |
| open scoped BigOperators | |
| open Set unitInterval | |
| open Set MeasureTheory | |
| open Set Filter Metric Topology | |
| open Set Filter Topology | |
| open Set Filter MeasureTheory Topology | |
| open scoped NNReal ENNReal | |
| namespace CannonCharts | |
| open CannonCells CannonDuality CannonAnnular | |
| lemma sphere_pathConnected : PathConnectedSpace SphereTwo := by | |
| apply isPathConnected_iff_pathConnectedSpace.mp | |
| apply isPathConnected_sphere _ _ (by norm_num) | |
| rw [← Module.finrank_eq_rank] | |
| simp | |
| lemma macroFamily_nonempty {X : Type*} [MetricSpace X] [PathConnectedSpace X] | |
| {d : ℝ} (hd : 0 < d) (hdia : d < diam (univ : Set X)) : | |
| (macroFamily (X := X) d).Nonempty := by | |
| have hp : ∃ x y : X, d < dist x y := by | |
| by_contra hn | |
| push Not at hn | |
| exact not_le_of_gt hdia (diam_le_of_forall_dist_le hd.le (fun x _ y _ => hn x y)) | |
| obtain ⟨x,y,hxy⟩ := hp | |
| let p := PathConnectedSpace.somePath x y | |
| refine ⟨range p,p,p.continuous,rfl,?_⟩ | |
| exact hxy.le.trans (dist_le_diam_of_mem (isCompact_range p.continuous).isBounded | |
| ⟨0,p.source⟩ ⟨1,p.target⟩) | |
| theorem sphere_macro_modulus_minimum {X ι : Type*} [MetricSpace X] [Fintype ι] | |
| (φ : X ≃ₜ SphereTwo) (d : ℝ) (hd : 0 < d) (hdia : d < diam (univ : Set X)) | |
| (p : ι → X) (r : ℝ) (hcover : ∀ x, ∃ i, x ∈ ball (p i) (2*r)) : | |
| 0 < modulus (fun i => ball (p i) (2*r)) (macroFamily d) ∧ | |
| ∃ ω : ι → ℝ, MacroscopicAdmissible p r d ω ∧ | |
| (∑ i, (ω i)^2) = modulus (fun i => ball (p i) (2*r)) (macroFamily d) := by | |
| let : PathConnectedSpace SphereTwo := sphere_pathConnected | |
| let : PathConnectedSpace X := φ.symm.surjective.pathConnectedSpace φ.symm.continuous | |
| let C := fun i => ball (p i) (2*r) | |
| obtain ⟨hpos,w,hw,he⟩ := modulus_minimum C hcover (macroFamily_nonempty hd hdia) | |
| (fun A hA => by obtain ⟨p,_,rfl,_⟩ := hA; exact ⟨p 0,⟨0,rfl⟩⟩) | |
| refine ⟨hpos,(fun i => w i),?_,?_⟩ | |
| · have ha := (admissible_iff C (macroFamily d) w).mp hw | |
| refine ⟨ha.1,fun u v q hq => ?_⟩ | |
| have hh := ha.2 (range q) ⟨q,q.continuous,rfl,hq⟩ | |
| have hi : indices C (range q) = pathCellIndices p r q := by | |
| ext i | |
| simp only [mem_indices,pathCellIndices,Finset.mem_filter,Finset.mem_univ,true_and] | |
| constructor | |
| · rintro ⟨x,⟨t,rfl⟩,ht⟩ | |
| exact ⟨t,ht⟩ | |
| · rintro ⟨t,ht⟩ | |
| exact ⟨q t,⟨t,rfl⟩,ht⟩ | |
| rw [hi] at hh | |
| exact hh | |
| · rw [← he,EuclideanSpace.norm_sq_eq] | |
| simp only [Real.norm_eq_abs,sq_abs] | |
| end CannonCharts | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |