| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.ReverseConfigurationPath |
| import OAI.Analysis.HardSphere.Bridges.NormalizedPrefix |
| import OAI.Analysis.HardSphere.Geometry.LargeParentMarked |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.FreeVisits |
| open HardSphere.BathSlice HardSphere.Lattice HardSphere.CellOrbital |
|
|
| def packingBox (c : Fin 3 → ℝ) : Finset Vertex := |
| Finset.Icc (fun j ↦ ⌊c j - 1/2⌋ - 14) (fun j ↦ ⌊c j - 1/2⌋ + 14) |
|
|
| lemma card_packingBox (c : Fin 3 → ℝ) : (packingBox c).card=29^3 := by |
| unfold packingBox |
| rw [Pi.card_Icc] |
| have hc (j : Fin 3) : (Finset.Icc (⌊c j-1/2⌋-14) (⌊c j-1/2⌋+14)).card=29 := by |
| rw [Int.card_Icc] |
| omega |
| simp only [hc,Finset.prod_const,Finset.card_univ,Fintype.card_fin] |
|
|
| lemma site_packing_witness {K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ)) |
| (c : Fin 3 → ℝ) (v : DiscreteTorus K) |
| (hv : ∀ j, dist (siteCenter L v j) ((c j:ℝ):AddCircle L.val)≤13) : |
| ∃ z ∈ packingBox c, torusProjection K z=v := by |
| let k : Vertex := fun j ↦ round (L.val⁻¹*((((v j).val:ℝ)+1/2)-c j)) |
| let z : Vertex := fun j ↦ ((v j).val:ℤ)-k j*(K:ℤ) |
| refine ⟨z,?_,?_⟩ |
| · apply Finset.mem_Icc.mpr |
| have hz (j : Fin 3) : ⌊c j-1/2⌋-14 ≤ z j ∧ z j ≤ ⌊c j-1/2⌋+14 := by |
| have hd := hv j |
| change dist ((((v j).val:ℝ)+1/2:ℝ):AddCircle L.val) ((c j:ℝ):AddCircle L.val)≤13 at hd |
| rw [dist_eq_norm,←AddCircle.coe_sub,AddCircle.norm_eq] at hd |
| have hd' : |(z j:ℝ)-(c j-1/2)|≤13 := by |
| convert hd using 1 |
| dsimp [z,k] |
| push_cast |
| rw [hL] |
| ring_nf |
| have hl := Int.floor_le (c j-1/2) |
| have hu := Int.lt_floor_add_one (c j-1/2) |
| obtain ⟨h₁,h₂⟩ := abs_le.mp hd' |
| constructor <;> (apply (Int.cast_le (R := ℝ)).mp; push_cast; linarith) |
| exact ⟨fun j ↦ (hz j).1,fun j ↦ (hz j).2⟩ |
| · ext j |
| simp [torusProjection,z] |
|
|
| lemma card_near_sites_le {K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ)) |
| (x : Torus L) (A : Finset (DiscreteTorus K)) |
| (hA : ∀ v∈A, ∀ j, dist (siteCenter L v j) (x j)≤13) : A.card≤29^3 := by |
| classical |
| let c : Fin 3 → ℝ := fun j ↦ (AddCircle.equivIco L.val 0 (x j)).val |
| have hc (j : Fin 3) : ((c j:ℝ):AddCircle L.val)=x j := AddCircle.coe_equivIco |
| have hsub : A ⊆ (packingBox c).image (torusProjection K) := by |
| intro v hv |
| obtain ⟨z,hz,he⟩ := site_packing_witness L hL c v (fun j ↦ by rw [hc]; exact hA v hv j) |
| exact Finset.mem_image.mpr ⟨z,hz,he⟩ |
| exact (Finset.card_le_card hsub).trans ((Finset.card_image_le).trans (le_of_eq (card_packingBox c))) |
| end HardSphere.FreeVisits |
|
|
| namespace HardSphere |
| open MeasureTheory Wiener Set |
| attribute [local irreducible] torusDistance |
|
|
| def pathVisitsBall {n : ℕ} (L : Length) (R : ℝ) (v : Torus L) |
| (i : Fin n) (ω : ConfigurationPath n L) : Prop := |
| ∃ q : Time, torusDistance L (ω q i) v ≤ R |
|
|
| lemma pathVisitsBall_isClosed {n : ℕ} (L : Length) (R : ℝ) (v : Torus L) (i : Fin n) : |
| IsClosed {ω : ConfigurationPath n L | pathVisitsBall L R v i ω} := by |
| have hc : Continuous (fun p : ConfigurationPath n L × Time ↦ torusDistance L (p.1 p.2 i) v) := by |
| have he : Continuous (fun p : ConfigurationPath n L × Time ↦ p.1 p.2 i) := |
| (continuous_apply i).comp (continuous_eval : Continuous (fun p : ConfigurationPath n L × Time ↦ p.1 p.2)) |
| exact (continuous_torusDistance L).comp (he.prodMk continuous_const) |
| have hs := isClosedMap_fst_of_compactSpace |
| {p : ConfigurationPath n L × Time | torusDistance L (p.1 p.2 i) v ≤ R} |
| (isClosed_le hc continuous_const) |
| convert hs using 1 |
| ext ω |
| simp only [pathVisitsBall,Set.mem_ofPred_eq,Set.mem_image,Prod.exists] |
| constructor |
| · rintro ⟨q,hq⟩ |
| exact ⟨ω,q,hq,rfl⟩ |
| · rintro ⟨ω',q,hq,rfl⟩ |
| exact ⟨q,hq⟩ |
|
|
| @[simp] lemma pathVisitsBall_reverse {n : ℕ} (L : Length) (R : ℝ) (v : Torus L) |
| (i : Fin n) (ω : ConfigurationPath n L) : |
| pathVisitsBall L R v i (reverseConfigurationPath n L ω) ↔ pathVisitsBall L R v i ω := by |
| constructor |
| · rintro ⟨q,hq⟩ |
| exact ⟨reverseTime q,hq⟩ |
| · rintro ⟨q,hq⟩ |
| refine ⟨reverseTime q,?_⟩ |
| simpa only [reverseConfigurationPath_apply,reverseTime_reverse] using hq |
|
|
| def pathVisitedSites {S : Type*} {n : ℕ} (L : Length) (v : S → Torus L) |
| (A : Finset S) (i : Fin n) (ω : ConfigurationPath n L) : Finset S := by |
| classical |
| exact A.filter (fun a ↦ pathVisitsBall L 12 (v a) i ω) |
|
|
| lemma pathVisitedSites_measurable {S : Type*} {n : ℕ} (L : Length) |
| (v : S → Torus L) (A : Finset S) (i : Fin n) : |
| Measurable (fun ω : ConfigurationPath n L ↦ ((pathVisitedSites L v A i ω).card:ℝ)) := by |
| classical |
| simp only [pathVisitedSites,Finset.card_filter,Nat.cast_sum,Nat.cast_ite,Nat.cast_one,Nat.cast_zero] |
| apply Finset.measurable_sum |
| intro a _ |
| exact Measurable.ite (pathVisitsBall_isClosed L 12 (v a) i).measurableSet |
| measurable_const measurable_const |
|
|
| @[simp] lemma pathVisitedSites_reverse {S : Type*} {n : ℕ} (L : Length) |
| (v : S → Torus L) (A : Finset S) (i : Fin n) (ω : ConfigurationPath n L) : |
| pathVisitedSites L v A i (reverseConfigurationPath n L ω)=pathVisitedSites L v A i ω := by |
| classical |
| simp only [pathVisitedSites,pathVisitsBall_reverse] |
|
|
| end HardSphere |
|
|
| namespace HardSphere.FreeVisits |
| open HardSphere.BathSlice HardSphere.Lattice HardSphere.CellOrbital HardSphere.Wiener |
|
|
| lemma visited_subset_near {n K : ℕ} [NeZero K] (L : Length) (A : Finset (DiscreteTorus K)) |
| (i : Fin n) (ω : ConfigurationPath n L) (x : Torus L) |
| (hω : ∀ (u : Time) (j : Fin 3), dist (ω u i j) (x j)≤1) : |
| ∀ v∈pathVisitedSites L (siteCenter L) A i ω, ∀ j, dist (siteCenter L v j) (x j)≤13 := by |
| classical |
| intro v hv j |
| obtain ⟨_,u,hu⟩ := Finset.mem_filter.mp hv |
| have hc := dist_coord_le_torusDistance L (ω u i) (siteCenter L v) j |
| calc |
| dist (siteCenter L v j) (x j) ≤ dist (siteCenter L v j) (ω u i j)+dist (ω u i j) (x j) := dist_triangle _ _ _ |
| _ ≤ 12+1 := add_le_add (by rw [dist_comm]; exact hc.trans hu) (hω u j) |
| _ = 13 := by norm_num |
|
|
| lemma visited_card_le_of_short {n K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ)) |
| (A : Finset (DiscreteTorus K)) (i : Fin n) (ω : ConfigurationPath n L) (x : Torus L) |
| (hω : ∀ (u : Time) (j : Fin 3), dist (ω u i j) (x j)≤1) : |
| (pathVisitedSites L (siteCenter L) A i ω).card≤29^3 := |
| card_near_sites_le L hL x _ (visited_subset_near L A i ω x hω) |
|
|
| lemma visited_split {S : Type*} [DecidableEq S] {n : ℕ} (L : Length) (v : S → Torus L) |
| (A : Finset S) (i : Fin n) (ω : ConfigurationPath n L) |
| (u : Time) (hu : 0<(u:ℝ)) (hu' : (u:ℝ)<1) : |
| pathVisitedSites L v A i ω = pathVisitedSites L v A i (configurationPrefix n L u ω) ∪ |
| pathVisitedSites L v A i (configurationSuffix n L u ω) := by |
| classical |
| ext a |
| rw [Finset.mem_union] |
| constructor |
| · intro hav |
| obtain ⟨ha,q,hq⟩ := Finset.mem_filter.mp hav |
| by_cases hq' : (q:ℝ)≤u |
| · let z : Time := ⟨(q:ℝ)/(u:ℝ),div_nonneg q.property.1 hu.le,(div_le_one hu).mpr hq'⟩ |
| have hz : prefixTime u z=q := by |
| apply Subtype.ext |
| change (u:ℝ)*((q:ℝ)/(u:ℝ))=(q:ℝ) |
| field_simp |
| exact Or.inl (Finset.mem_filter.mpr ⟨ha,z,by simpa only [configurationPrefix,ContinuousMap.comp_apply,hz] using hq⟩) |
| · let z : Time := ⟨((q:ℝ)-(u:ℝ))/(1-(u:ℝ)), |
| div_nonneg (sub_nonneg.mpr (le_of_not_ge hq')) (sub_pos.mpr hu').le, |
| (div_le_one (sub_pos.mpr hu')).mpr (by linarith [q.property.2])⟩ |
| have hz : suffixTime u z=q := by |
| apply Subtype.ext |
| change (u:ℝ)+(1-(u:ℝ))*(((q:ℝ)-(u:ℝ))/(1-(u:ℝ)))=(q:ℝ) |
| field_simp [ne_of_gt (sub_pos.mpr hu')] |
| ring |
| exact Or.inr (Finset.mem_filter.mpr ⟨ha,z,by simpa only [configurationSuffix,ContinuousMap.comp_apply,hz] using hq⟩) |
| · rintro (hav|hav) |
| · obtain ⟨ha,q,hq⟩ := Finset.mem_filter.mp hav |
| exact Finset.mem_filter.mpr ⟨ha,prefixTime u q,hq⟩ |
| · obtain ⟨ha,q,hq⟩ := Finset.mem_filter.mp hav |
| exact Finset.mem_filter.mpr ⟨ha,suffixTime u q,hq⟩ |
|
|
| lemma visited_card_split_le {S : Type*} {n : ℕ} (L : Length) (v : S → Torus L) |
| (A : Finset S) (i : Fin n) (ω : ConfigurationPath n L) |
| (u : Time) (hu : 0<(u:ℝ)) (hu' : (u:ℝ)<1) : |
| (pathVisitedSites L v A i ω).card ≤ (pathVisitedSites L v A i (configurationPrefix n L u ω)).card + |
| (pathVisitedSites L v A i (configurationSuffix n L u ω)).card := by |
| classical |
| rw [visited_split L v A i ω u hu hu'] |
| exact Finset.card_union_le _ _ |
|
|
| lemma freeTrajectory_coord_dist_le {n : ℕ} (L : Length) (t : ℝ) (w : CoordinatePaths n) |
| (X : Configuration n L) (i : Fin n) (u : Time) (j : Fin 3) : |
| dist (freeTrajectory n L t w X u i j) (X i j) ≤ Real.sqrt (2*t)*‖w (i,j)‖ := by |
| change dist (X i j+((Real.sqrt (2*t)*w (i,j) u:ℝ):AddCircle L.val)) (X i j) ≤ _ |
| simp only [dist_eq_norm,add_sub_cancel_left] |
| calc |
| ‖((Real.sqrt (2*t)*w (i,j) u:ℝ):AddCircle L.val)‖ ≤ ‖Real.sqrt (2*t)*w (i,j) u‖ := |
| QuotientAddGroup.norm_mk_le_norm |
| _ = Real.sqrt (2*t)*|w (i,j) u| := by rw [Real.norm_eq_abs,abs_mul,abs_of_nonneg (Real.sqrt_nonneg _)] |
| _ ≤ _ := mul_le_mul_of_nonneg_left (by simpa only [Real.norm_eq_abs] using ContinuousMap.norm_coe_le_norm (w (i,j)) u) (Real.sqrt_nonneg _) |
|
|
| lemma free_visited_card_le_of_norm {n K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ)) |
| (A : Finset (DiscreteTorus K)) (i : Fin n) (t : ℝ) (w : CoordinatePaths n) |
| (X : Configuration n L) (hw : ∀ j, Real.sqrt (2*t)*‖w (i,j)‖≤1) : |
| (pathVisitedSites L (siteCenter L) A i (freeTrajectory n L t w X)).card≤29^3 := by |
| apply visited_card_le_of_short L hL A i _ (X i) |
| exact fun u j ↦ (freeTrajectory_coord_dist_le L t w X i u j).trans (hw j) |
|
|
| end HardSphere.FreeVisits |
|
|
| end |
|
|
| end OAI |
|
|
