| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section PolygonGrid |
| open Classical Set MeasureTheory Metric |
| namespace PolygonGrid |
|
|
| abbrev PlaneTracks := ℕ × ℝ × ℝ |
| abbrev Index (m N : ℕ) := Fin m × Fin N × Fin N |
| noncomputable def μ : Measure PlaneTracks := Measure.count.prod ((volume : Measure ℝ).prod volume) |
| noncomputable def interval (ε : ℝ) (i : ℕ) : Set ℝ := Ico (ε*i) (ε*(i+1)) |
| noncomputable def cell {m N : ℕ} (ε : ℝ) (i : Index m N) : Set PlaneTracks := |
| {i.1.val} ×ˢ interval ε i.2.1.val ×ˢ interval ε i.2.2.val |
| noncomputable def region (m : ℕ) : Set PlaneTracks := |
| {i : ℕ | i < m} ×ˢ Ico (0:ℝ) 1 ×ˢ Ico (0:ℝ) 1 |
|
|
| @[simp] theorem measurable_cell {m N : ℕ} (ε : ℝ) (i : Index m N) : MeasurableSet (cell ε i) := |
| (measurableSet_singleton _).prod (measurableSet_Ico.prod measurableSet_Ico) |
|
|
| @[simp] theorem volume_cell {m N : ℕ} (ε : ℝ) (i : Index m N) : |
| μ (cell ε i)=ENNReal.ofReal ε * ENNReal.ofReal ε := by |
| simp only [μ,cell,interval,Measure.prod_prod,Measure.count_singleton,one_mul,Real.volume_Ico] |
| congr 2 <;> ring |
|
|
| theorem cells_disjoint {m N : ℕ} {ε : ℝ} (hε : 0 < ε) : |
| Pairwise (fun i j : Index m N => Disjoint (cell ε i) (cell ε j)) := by |
| intro i j hij |
| apply Set.disjoint_left.mpr |
| intro p hi hj |
| have ht : i.1=j.1 := Fin.ext (hi.1.symm.trans hj.1) |
| have coord {i j : ℕ} {x : ℝ} (hi : x∈interval ε i) (hj : x∈interval ε j) : i=j := by |
| rcases lt_trichotomy i j with h|h|h |
| · have h' : (i:ℝ)+1 ≤ j := by exact_mod_cast h |
| have := mul_le_mul_of_nonneg_left h' hε.le |
| change ε*i ≤ x ∧ x < ε*(i+1) at hi |
| change ε*j ≤ x ∧ x < ε*(j+1) at hj |
| linarith |
| · exact h |
| · have h' : (j:ℝ)+1 ≤ i := by exact_mod_cast h |
| have := mul_le_mul_of_nonneg_left h' hε.le |
| change ε*i ≤ x ∧ x < ε*(i+1) at hi |
| change ε*j ≤ x ∧ x < ε*(j+1) at hj |
| linarith |
| exact hij (Prod.ext ht (Prod.ext (Fin.ext (coord hi.2.1 hj.2.1)) (Fin.ext (coord hi.2.2 hj.2.2)))) |
|
|
| theorem cell_dist {m N : ℕ} {ε : ℝ} (hε : 0 < ε) {i : Index m N} {p q : PlaneTracks} |
| (hp : p∈cell ε i) (hq : q∈cell ε i) : dist p q < ε := by |
| have coord {i : ℕ} {x y : ℝ} (hx : x∈interval ε i) (hy : y∈interval ε i) : dist x y < ε := by |
| rw [Real.dist_eq,abs_lt] |
| change ε*i ≤ x ∧ x < ε*(i+1) at hx |
| change ε*i ≤ y ∧ y < ε*(i+1) at hy |
| constructor <;> nlinarith |
| have ht : p.1=q.1 := hp.1.trans hq.1.symm |
| rw [Prod.dist_eq,Prod.dist_eq,ht,dist_self,max_lt_iff,max_lt_iff] |
| exact ⟨hε,coord hp.2.1 hq.2.1,coord hp.2.2 hq.2.2⟩ |
|
|
| theorem region_covered {m N : ℕ} {ε : ℝ} (hε : 0 < ε) (hN : 1/ε < (N:ℝ)) : |
| region m⊆⋃i : Index m N,cell ε i := by |
| rintro ⟨t,x,y⟩ ⟨ht,hx,hy⟩ |
| have coord {x : ℝ} (hx : x∈Ico (0:ℝ) 1) : ∃i : Fin N,x∈interval ε i.val := by |
| have hnon : 0 ≤ x/ε := div_nonneg hx.1 hε.le |
| have hlt : x/ε < (N:ℝ) := (div_lt_div_of_pos_right hx.2 hε).trans hN |
| have hi : ⌊x/ε⌋₊ < N := by |
| exact_mod_cast lt_of_le_of_lt (Nat.floor_le hnon) hlt |
| refine ⟨⟨⌊x/ε⌋₊,hi⟩,?_,?_⟩ |
| · have h := (le_div_iff₀ hε).mp (Nat.floor_le hnon) |
| nlinarith |
| · have h := (div_lt_iff₀ hε).mp (Nat.lt_floor_add_one (x/ε)) |
| nlinarith |
| obtain ⟨i,hi⟩ := coord hx |
| obtain ⟨j,hj⟩ := coord hy |
| exact Set.mem_iUnion.mpr ⟨(⟨t,ht⟩,i,j),rfl,hi,hj⟩ |
|
|
| theorem volume_cells {m N : ℕ} {ε : ℝ} (hε : 0 < ε) (I : Finset (Index m N)) : |
| μ (⋃i∈I,cell ε i)=I.card*(ENNReal.ofReal ε * ENNReal.ofReal ε) := by |
| rw [measure_biUnion_finset (fun i hi j hj hij => cells_disjoint hε hij) |
| (fun i _ => measurable_cell ε i)] |
| simp only [volume_cell,Finset.sum_const, nsmul_eq_mul] |
|
|
| theorem matching_of_volume {m N : ℕ} {ε : ℝ} (hε : 0 < ε) |
| (I J : Finset (Index m N)) |
| (hμ : μ (⋃i∈I,cell ε i) < μ (⋃j∈J,cell ε j)) : Nonempty (I ↪ J) := by |
| have hc : I.card ≤ J.card := by |
| by_contra hn |
| have hnat : J.card ≤ I.card := by omega |
| have hcast : (J.card : ENNReal) ≤ I.card := by exact_mod_cast hnat |
| rw [volume_cells hε,volume_cells hε] at hμ |
| exact not_lt_of_ge (mul_le_mul hcast le_rfl (by positivity) (by positivity)) hμ |
| have hf : Fintype.card I ≤ Fintype.card J := by simpa using hc |
| exact Function.Embedding.nonempty_of_card_le hf |
|
|
| end PolygonGrid |
| end PolygonGrid |
|
|
| end SimpleAmenable |
| end |
| end |
|
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| end OAI |
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