| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.MicroCubeCover |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
| noncomputable section |
| open scoped BigOperators Classical |
| noncomputable section |
| open scoped BigOperators Classical |
| namespace HardSphere.Lattice |
|
|
| lemma isBadComponent_iff (B : Set Vertex) (C : Finset Vertex) : |
| IsBadComponent B C ↔ C.Nonempty ∧ (↑C : Set Vertex) ⊆ B ∧ |
| starConnected (↑C) ∧ |
| ∀ c ∈ C, ∀ x, x ∈ B → starStep c x → x ∈ C := by |
| constructor |
| · intro hC |
| obtain ⟨z, hzB, hz⟩ := hC |
| have hzC : z ∈ C := by simpa only [← hz, Finset.mem_coe] using liftedBadComponent_self hzB |
| have hC : IsBadComponent B C := ⟨z, hzB, hz⟩ |
| exact ⟨⟨z, hzC⟩, hC.subset, hC.connected, fun _ hc _ hx hcx ↦ hC.closed hc hx hcx⟩ |
| · rintro ⟨⟨z, hz⟩, hsub, hconn, hclosed⟩ |
| refine ⟨z, hsub hz, ?_⟩ |
| ext w |
| constructor |
| · intro hw |
| exact ⟨hsub hz, hsub hw, withinReach_mono hsub (hconn z hz w hw)⟩ |
| · intro hw |
| exact withinReach_end hz |
| (withinReach_invariant (fun _ hu _ hv huv ↦ hclosed _ hu _ hv huv) hz hw.2.2) |
|
|
| lemma finiteBadComponents_iff (B : Set Vertex) : |
| FiniteBadComponents B ↔ ∀ z ∈ B, ∃ C : Finset Vertex, IsBadComponent B C ∧ z ∈ C := by |
| classical |
| constructor |
| · intro h z hz |
| exact ⟨(h z hz).toFinset, ⟨z, hz, (h z hz).coe_toFinset⟩, |
| (h z hz).mem_toFinset.mpr (liftedBadComponent_self hz)⟩ |
| · intro h z hz |
| obtain ⟨C, hC, hzC⟩ := h z hz |
| apply C.finite_toSet.subset |
| intro w hw |
| exact withinReach_end hzC |
| (withinReach_invariant (fun _ hu _ hv huv ↦ hC.closed hu hv huv) hzC hw.2.2) |
|
|
| end HardSphere.Lattice |
|
|
| namespace HardSphere.BathSlice |
| open HardSphere.Lattice |
|
|
| lemma measurable_component_test (L : Length) (h r : ℝ) (C : Finset Vertex) : |
| Measurable (fun Y : Configuration n L ↦ IsBadComponent (bad h r Y) C) := by |
| simp_rw [isBadComponent_iff] |
| apply Measurable.and measurable_const |
| apply Measurable.and |
| · change Measurable (fun Y : Configuration n L ↦ ∀ z, z ∈ C → z ∈ bad h r Y) |
| exact Measurable.forall fun z ↦ Measurable.imp measurable_const |
| (measurableSet_setOfPred.mp (measurableSet_bad_test L h r z)) |
| · apply Measurable.and measurable_const |
| apply Measurable.forall |
| intro c |
| apply Measurable.imp measurable_const |
| apply Measurable.forall |
| intro x |
| apply Measurable.imp |
| · exact measurableSet_setOfPred.mp (measurableSet_bad_test L h r x) |
| · exact Measurable.imp measurable_const measurable_const |
|
|
| lemma measurable_finiteComponents_test (L : Length) (h r : ℝ) : |
| Measurable (fun Y : Configuration n L ↦ FiniteBadComponents (bad h r Y)) := by |
| simp_rw [finiteBadComponents_iff] |
| apply Measurable.forall |
| intro z |
| apply Measurable.imp |
| · exact measurableSet_setOfPred.mp (measurableSet_bad_test L h r z) |
| · exact Measurable.exists fun C ↦ (measurable_component_test L h r C).and measurable_const |
|
|
| lemma measurable_exceptional (L : Length) (h r : ℝ) : |
| Measurable (fun Y : Configuration n L ↦ exceptional h r Y) := |
| ((isOpen_allowed n L r).measurableSet.mem.not).or |
| (measurable_finiteComponents_test L h r).not |
|
|
| theorem measurableSet_omitted_graph (L : Length) (h r : ℝ) : |
| MeasurableSet := by |
| apply Measurable.setOf |
| apply Measurable.or ((measurable_exceptional L h r).comp measurable_fst) |
| apply Measurable.exists |
| intro C |
| apply Measurable.and ((measurable_component_test L h r C).comp measurable_fst) |
| apply Measurable.exists |
| intro z |
| exact Measurable.and measurable_const |
| (((isCompact_microCube L h z).isClosed.measurableSet.mem).comp measurable_snd) |
|
|
| attribute [local irreducible] omitted |
|
|
| lemma measurableSet_omitted (h r : ℝ) (Y : Configuration n L) : |
| MeasurableSet (omitted h r Y) := by |
| have hm : Measurable (fun x : Torus L ↦ (Y, x)) := |
| (measurable_const : Measurable (fun _ : Torus L ↦ Y)).prodMk measurable_id |
| exact (measurableSet_omitted_graph (n := n) L h r).preimage hm |
|
|
| lemma measurableSet_retained (h r : ℝ) (Y : Configuration n L) : |
| MeasurableSet (retained h r Y) := (measurableSet_omitted h r Y).compl |
|
|
| theorem measurable_omitted_volume (L : Length) (h r : ℝ) : |
| Measurable (fun Y : Configuration n L ↦ (volume : Measure (Torus L)) (omitted h r Y)) := |
| measurable_measure_prodMk_left (measurableSet_omitted_graph L h r) |
|
|
| end HardSphere.BathSlice |
|
|
| namespace HardSphere.Lattice |
|
|
| theorem fill_column_meets (C : Finset Vertex) (hx : x ∈ fill C) |
| (i : Fin 3) : ∃ c ∈ C, ∀ j, j ≠ i → c j = x j := by |
| by_contra hn |
| apply hx |
| apply coordinateRay_exterior C x i 1 (by norm_num) |
| intro n hc |
| apply hn |
| refine ⟨coordinateRay x i 1 n, hc, ?_⟩ |
| intro j hj |
| simp [coordinateRay, hj] |
|
|
| def columnCover (C : Finset Vertex) (i : Fin 3) (lo hi : ℤ) : Finset Vertex := |
| C.biUnion fun c ↦ (Finset.Icc lo hi).image fun k ↦ Function.update c i k |
|
|
| lemma fill_subset_columnCover (C : Finset Vertex) (i : Fin 3) (lo hi : ℤ) |
| (hb : ∀ c ∈ C, lo ≤ c i ∧ c i ≤ hi) : fill C ⊆ ↑(columnCover C i lo hi) := by |
| intro x hx |
| obtain ⟨c, hc, hcol⟩ := fill_column_meets C hx i |
| have hbox := fill_subset_coordinateBox C hx i |
| obtain ⟨cl, hcl, hclx⟩ := hbox.1 |
| obtain ⟨cu, hcu, hxcu⟩ := hbox.2 |
| apply Finset.mem_biUnion.mpr |
| refine ⟨c, hc, Finset.mem_image.mpr ⟨x i, Finset.mem_Icc.mpr |
| ⟨(hb cl hcl).1.trans hclx, hxcu.trans (hb cu hcu).2⟩, ?_⟩⟩ |
| ext j |
| by_cases hj : j = i |
| · subst j; simp |
| · simp [Function.update_of_ne hj, hcol j hj] |
|
|
| lemma card_columnCover_le (C : Finset Vertex) (i : Fin 3) (lo hi : ℤ) : |
| (columnCover C i lo hi).card ≤ C.card * (hi - lo + 1).toNat := by |
| calc |
| _ ≤ ∑ c ∈ C, ((Finset.Icc lo hi).image fun k ↦ Function.update c i k).card := |
| Finset.card_biUnion_le |
| _ ≤ ∑ _c ∈ C, (Finset.Icc lo hi).card := |
| Finset.sum_le_sum fun c _ ↦ Finset.card_image_le |
| _ = _ := by simp [Int.card_Icc, show hi + 1 - lo = hi - lo + 1 by omega] |
|
|
| lemma finite_fill (C : Finset Vertex) : (fill C).Finite := by |
| classical |
| by_cases hc : C.Nonempty |
| · let lo := C.inf' hc (fun c ↦ c 0) |
| let hi := C.sup' hc (fun c ↦ c 0) |
| exact (columnCover C 0 lo hi).finite_toSet.subset |
| (fill_subset_columnCover C 0 lo hi fun c hc ↦ |
| ⟨Finset.inf'_le (fun c : Vertex ↦ c 0) hc, |
| Finset.le_sup' (fun c : Vertex ↦ c 0) hc⟩) |
| · have hC : C = ∅ := Finset.not_nonempty_iff_eq_empty.mp hc |
| subst C |
| have he : fill ∅ = ∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro x hx |
| obtain ⟨c, hc, _⟩ := fill_column_meets ∅ hx 0 |
| exact Finset.notMem_empty c hc |
| rw [he] |
| exact Set.finite_empty |
|
|
| theorem ncard_fill_le_columns (C : Finset Vertex) (i : Fin 3) (lo hi : ℤ) |
| (hb : ∀ c ∈ C, lo ≤ c i ∧ c i ≤ hi) : |
| (fill C).ncard ≤ C.card * (hi - lo + 1).toNat := by |
| calc |
| _ ≤ (↑(columnCover C i lo hi) : Set Vertex).ncard := |
| Set.ncard_le_ncard (fill_subset_columnCover C i lo hi hb) |
| (columnCover C i lo hi).finite_toSet |
| _ = (columnCover C i lo hi).card := by simp |
| _ ≤ _ := card_columnCover_le C i lo hi |
|
|
| end HardSphere.Lattice |
|
|
| end |
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
