| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.PolygonArea | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.PolygonGridComparison | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section PolygonTrackArea | |
| open Classical Set MeasureTheory | |
| namespace PolygonGrid | |
| noncomputable def trackUnion {m : ℕ} (S : Fin m → Set (ℝ×ℝ)) : Set PlaneTracks := | |
| ⋃i, {i.val} ×ˢ S i | |
| @[simp] theorem mem_trackUnion {m : ℕ} (S : Fin m → Set (ℝ×ℝ)) (i : Fin m) (x : ℝ×ℝ) : | |
| (i.val,x)∈trackUnion S ↔ x∈S i := by | |
| constructor | |
| · intro h | |
| obtain ⟨j,hj,hx⟩ := Set.mem_iUnion.mp h | |
| have he : i=j := Fin.ext hj | |
| simpa only [←he] using hx | |
| · intro hx | |
| exact Set.mem_iUnion.mpr ⟨i,rfl,hx⟩ | |
| theorem trackUnion_mono {m : ℕ} {S T : Fin m → Set (ℝ×ℝ)} (h : ∀i,S i⊆T i) : | |
| trackUnion S⊆trackUnion T := by | |
| intro p hp | |
| obtain ⟨i,hi,hp⟩ := Set.mem_iUnion.mp hp | |
| exact Set.mem_iUnion.mpr ⟨i,hi,h i hp⟩ | |
| theorem trackUnion_region {m : ℕ} {S : Fin m → Set (ℝ×ℝ)} | |
| (h : ∀i,S i⊆PolygonArea.square) : trackUnion S⊆region m := by | |
| intro p hp | |
| obtain ⟨i,hi,hp⟩ := Set.mem_iUnion.mp hp | |
| exact ⟨hi ▸ i.isLt,h i hp⟩ | |
| @[simp] theorem volume_trackUnion {m : ℕ} (S : Fin m → Set (ℝ×ℝ)) | |
| (hS : ∀i,MeasurableSet (S i)) : μ (trackUnion S)=∑i,volume (S i) := by | |
| have hdis : Pairwise (fun i j : Fin m => Disjoint ({i.val}×ˢ S i) ({j.val}×ˢ S j)) := by | |
| intro i j hij | |
| apply Set.disjoint_left.mpr | |
| rintro p ⟨hi,-⟩ ⟨hj,-⟩ | |
| exact hij (Fin.ext (hi.symm.trans hj)) | |
| rw [trackUnion,measure_iUnion hdis (fun i => (measurableSet_singleton _).prod (hS i))] | |
| simp only [μ,Measure.prod_prod,Measure.count_singleton,one_mul,tsum_fintype] | |
| rfl | |
| theorem bank_representative {a m : ℕ} (U : Fin m → polygonAlgebra a) : | |
| ∃ S : Set PlaneTracks, S⊆region m ∧ | |
| (∀i : Fin m,∀x : GenericSquare a,(i.val,x.val)∈S ↔ x∈(U i).val) ∧ | |
| μ (closure S)=∑i,PolygonArea.area (U i) ∧ | |
| μ (interior S)=∑i,PolygonArea.area (U i) := by | |
| choose R hR hm hn he using fun i => PolygonArea.representative (U i).property | |
| have hmR : ∀i,MeasurableSet (R i) := hm | |
| have harea (i : Fin m) : volume (R i)=PolygonArea.area (U i) := | |
| (PolygonArea.area_eq_rep (U i) (hR i) (he i)).symm | |
| have hvol : μ (trackUnion R)=∑i,PolygonArea.area (U i) := by | |
| rw [volume_trackUnion R hmR] | |
| exact Finset.sum_congr rfl (fun i _ => harea i) | |
| have hclosure : closure (trackUnion R)⊆trackUnion (fun i => closure (R i)) := by | |
| apply (isClosed_iUnion_of_finite (fun i => isClosed_singleton.prod isClosed_closure)).closure_subset_iff.mpr | |
| exact trackUnion_mono (fun _ => subset_closure) | |
| have hinterior : trackUnion (fun i => interior (R i))⊆interior (trackUnion R) := by | |
| apply interior_maximal (trackUnion_mono (fun _ => interior_subset)) | |
| exact isOpen_iUnion (fun i => (isOpen_discrete _).prod isOpen_interior) | |
| have hvclosure : μ (trackUnion (fun i => closure (R i)))=∑i,PolygonArea.area (U i) := by | |
| rw [volume_trackUnion _ (fun _ => isClosed_closure.measurableSet)] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [measure_congr (PolygonArea.closure_ae_eq (hn i)),harea] | |
| have hvinterior : μ (trackUnion (fun i => interior (R i)))=∑i,PolygonArea.area (U i) := by | |
| rw [volume_trackUnion _ (fun _ => isOpen_interior.measurableSet)] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [measure_congr (PolygonArea.interior_ae_eq (hn i)),harea] | |
| refine ⟨trackUnion R,trackUnion_region hR,?_,?_,?_⟩ | |
| · intro i x | |
| exact (mem_trackUnion R i x.val).trans (he i x).symm | |
| · exact le_antisymm ((measure_mono hclosure).trans_eq hvclosure) | |
| (hvol.symm.trans_le (measure_mono subset_closure)) | |
| · exact le_antisymm ((measure_mono interior_subset).trans_eq hvol) | |
| (hvinterior.symm.trans_le (measure_mono hinterior)) | |
| theorem polygon_grid_comparison {a m : ℕ} (U V : Fin m → polygonAlgebra a) | |
| (h : (∑i,PolygonArea.area (U i)) < ∑i,PolygonArea.area (V i)) : | |
| ∃ (S T : Set PlaneTracks) (δ : ℝ), 0 < δ ∧ | |
| (∀i : Fin m,∀x : GenericSquare a,(i.val,x.val)∈S ↔ x∈(U i).val) ∧ | |
| (∀i : Fin m,∀x : GenericSquare a,(i.val,x.val)∈T ↔ x∈(V i).val) ∧ | |
| ∀ε : ℝ, 0 < ε → ε < δ → | |
| ∃ (N : ℕ) (I J : Finset (Index m N)), Nonempty (I ↪ J) ∧ | |
| S⊆⋃i∈I,cell ε i ∧ ∀j∈J,cell ε j⊆T := by | |
| obtain ⟨S,hS,heS,hcS,hiS⟩ := bank_representative U | |
| obtain ⟨T,hT,heT,hcT,hiT⟩ := bank_representative V | |
| have hh : μ (closure S)<μ (interior T) := by rwa [hcS,hiT] | |
| obtain ⟨δ,hδ,hmatch⟩ := strict_grid_comparison hS hT hh | |
| exact ⟨S,T,δ,hδ,heS,heT,hmatch⟩ | |
| end PolygonGrid | |
| end PolygonTrackArea | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |