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0dd3af4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | 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
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