| import Mathlib |
| import OAI.Analysis.HardSphere.Conditioning.ReverseConfigurationPath |
| import OAI.Analysis.HardSphere.Bridges.NormalizedPrefix |
| import OAI.Analysis.HardSphere.Geometry.InjectiveNear |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Set Filter ProbabilityTheory |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.PathCutoff |
| open Wiener |
| attribute [local irreducible] torusDistance freeTrajectory |
|
|
| def intervalTime (a b s : Time) : Time := |
| ⟨(1-(s:ℝ))*(a:ℝ)+(s:ℝ)*(b:ℝ), by |
| constructor |
| · exact add_nonneg (mul_nonneg (sub_nonneg.mpr s.property.2) a.property.1) (mul_nonneg s.property.1 b.property.1) |
| · nlinarith [a.property.1,a.property.2,b.property.1,b.property.2, |
| s.property.1,s.property.2,mul_nonneg (sub_nonneg.mpr s.property.2) (sub_nonneg.mpr a.property.2), |
| mul_nonneg s.property.1 (sub_nonneg.mpr b.property.2)]⟩ |
| |
| lemma intervalTime_continuous (a b : Time) : Continuous (intervalTime a b) := by |
| apply Continuous.subtype_mk |
| fun_prop |
|
|
| @[simp] lemma intervalTime_zero (a b : Time) : intervalTime a b 0=a := by |
| apply Subtype.ext |
| simp [intervalTime] |
| @[simp] lemma intervalTime_one (a b : Time) : intervalTime a b 1=b := by |
| apply Subtype.ext |
| simp [intervalTime] |
|
|
| def diameterMap {L : Length} (a b : Time) (ω : ConfigurationPath 1 L) : C(Time×Time,ℝ) := |
| ⟨fun p ↦ torusDistance L (ω (intervalTime a b p.1) 0) (ω (intervalTime a b p.2) 0), by |
| apply (torusDistance_continuous_pair L).comp (f := fun p : Time×Time ↦ (ω (intervalTime a b p.1) 0,ω (intervalTime a b p.2) 0)) |
| exact ((continuous_apply 0).comp (ω.continuous.comp ((intervalTime_continuous a b).comp continuous_fst))).prodMk |
| ((continuous_apply 0).comp (ω.continuous.comp ((intervalTime_continuous a b).comp continuous_snd)))⟩ |
|
|
| def intervalDiameter {L : Length} (a b : Time) (ω : ConfigurationPath 1 L) : ℝ := |
| ‖diameterMap a b ω‖ |
|
|
| lemma diameterMap_continuous (L : Length) (a b : Time) : Continuous (diameterMap (L:=L) a b) := by |
| apply ContinuousMap.continuous_of_continuous_uncurry |
| change Continuous (fun p : ConfigurationPath 1 L × (Time×Time) ↦ |
| torusDistance L (p.1 (intervalTime a b p.2.1) 0) (p.1 (intervalTime a b p.2.2) 0)) |
| apply (torusDistance_continuous_pair L).comp (f := fun p : ConfigurationPath 1 L×(Time×Time) ↦ (p.1 (intervalTime a b p.2.1) 0,p.1 (intervalTime a b p.2.2) 0)) |
| apply Continuous.prodMk |
| · exact (continuous_apply 0).comp (continuous_fst.eval ((intervalTime_continuous a b).comp (continuous_fst.comp continuous_snd))) |
| · exact (continuous_apply 0).comp (continuous_fst.eval ((intervalTime_continuous a b).comp (continuous_snd.comp continuous_snd))) |
|
|
| lemma intervalDiameter_continuous (L : Length) (a b : Time) : |
| Continuous (intervalDiameter (L:=L) a b) := (diameterMap_continuous L a b).norm |
| |
| lemma intervalDiameter_nonneg {L : Length} (a b : Time) (ω : ConfigurationPath 1 L) : |
| 0 ≤ intervalDiameter a b ω := norm_nonneg _ |
|
|
| lemma intervalDistance_le {L : Length} (a b s q : Time) (ω : ConfigurationPath 1 L) : |
| torusDistance L (ω (intervalTime a b s) 0) (ω (intervalTime a b q) 0) ≤ intervalDiameter a b ω := by |
| exact (le_abs_self _).trans (ContinuousMap.norm_coe_le_norm (diameterMap a b ω) (s,q)) |
|
|
| lemma intervalDiameter_le {L : Length} (a b : Time) (ω : ConfigurationPath 1 L) {R : ℝ} (hR : 0 ≤ R) |
| (h : ∀ s q, torusDistance L (ω (intervalTime a b s) 0) (ω (intervalTime a b q) 0) ≤ R) : |
| intervalDiameter a b ω ≤ R := by |
| apply (ContinuousMap.norm_le _ hR).mpr |
| intro p |
| change |torusDistance L _ _| ≤ R |
| rw [abs_of_nonneg (by unfold torusDistance |
| exact h p.1 p.2 |
|
|
| lemma intervalTime_reverse (a b s : Time) : |
| reverseTime (intervalTime a b s)=intervalTime (reverseTime b) (reverseTime a) (reverseTime s) := by |
| apply Subtype.ext |
| change 1-((1-(s:ℝ))*(a:ℝ)+(s:ℝ)*(b:ℝ)) = (1-(1-(s:ℝ)))*(1-(b:ℝ))+(1-(s:ℝ))*(1-(a:ℝ)) |
| ring |
|
|
| lemma intervalDiameter_reverse {L : Length} (a b : Time) (ω : ConfigurationPath 1 L) : |
| intervalDiameter a b (reverseConfigurationPath 1 L ω)= |
| intervalDiameter (reverseTime b) (reverseTime a) ω := by |
| apply le_antisymm |
| · apply intervalDiameter_le _ _ _ (intervalDiameter_nonneg _ _ _) |
| intro s q |
| change torusDistance L (ω (reverseTime (intervalTime a b s)) 0) |
| (ω (reverseTime (intervalTime a b q)) 0) ≤ _ |
| simp only [intervalTime_reverse] |
| exact intervalDistance_le _ _ _ _ _ |
| · have hh : reverseTime (reverseTime a)=a := reverseTime_reverse _ |
| have hh' : reverseTime (reverseTime b)=b := reverseTime_reverse _ |
| apply intervalDiameter_le _ _ _ (intervalDiameter_nonneg _ _ _) |
| intro s q |
| have h := intervalDistance_le a b (reverseTime s) (reverseTime q) (reverseConfigurationPath 1 L ω) |
| change torusDistance L (ω (reverseTime (intervalTime a b (reverseTime s))) 0) |
| (ω (reverseTime (intervalTime a b (reverseTime q))) 0) ≤ _ at h |
| simpa only [intervalTime_reverse,reverseTime_reverse] using h |
| |
| end HardSphere.PathCutoff |
| |
| namespace HardSphere.PathCutoff |
| open Wiener |
| attribute [local irreducible] torusDistance freeTrajectory |
| |
| def meshTime {m : ℕ} (j : Fin m) (s : Time) : Time := |
| ⟨((j:ℝ)+(s:ℝ))/(m:ℝ), by |
| have hm : (0:ℝ) < m := by exact_mod_cast j.pos |
| refine ⟨div_nonneg (add_nonneg (Nat.cast_nonneg _) s.property.1) hm.le,(div_le_one hm).mpr ?_⟩ |
| have hj : (j:ℝ)+1≤(m:ℝ) := by exact_mod_cast j.isLt |
| linarith [s.property.2]⟩ |
| |
| lemma meshTime_continuous {m : ℕ} (j : Fin m) : Continuous (meshTime j) := by |
| apply Continuous.subtype_mk |
| fun_prop |
| |
| def meshSegment {m : ℕ} {L : Length} (j : Fin m) (ω : ConfigurationPath 1 L) : ConfigurationPath 1 L := |
| ⟨fun s ↦ ω (meshTime j s),ω.continuous.comp (meshTime_continuous j)⟩ |
| |
| lemma meshSegment_continuous {m : ℕ} (L : Length) (j : Fin m) : |
| Continuous (meshSegment (L:=L) j) := by |
| apply ContinuousMap.continuous_of_continuous_uncurry |
| change Continuous (fun p : ConfigurationPath 1 L×Time ↦ p.1 (meshTime j p.2)) |
| exact continuous_fst.eval ((meshTime_continuous j).comp continuous_snd) |
| |
| @[simp] lemma meshSegment_one {L : Length} (ω : ConfigurationPath 1 L) : |
| meshSegment (0:Fin 1) ω=ω := by |
| apply ContinuousMap.ext |
| intro s |
| change ω (meshTime (0:Fin 1) s)=ω s |
| congr 1 |
| apply Subtype.ext |
| simp [meshTime] |
| |
| lemma mesh_firstTime {m : ℕ} (hm : 0 < m) (s : Time) : |
| meshTime (0:Fin (m+1)) s=prefixTime (splitTime 1 (m:ℝ) (by norm_num) (by exact_mod_cast hm)) s := by |
| apply Subtype.ext |
| simp only [meshTime,Fin.val_zero,Nat.cast_zero,zero_add,prefixTime,ContinuousMap.coe_mk,splitTime,Nat.cast_add,Nat.cast_one] |
| ring |
| |
| lemma mesh_nextTime {m : ℕ} (hm : 0 < m) (j : Fin m) (s : Time) : |
| meshTime j.succ s=suffixTime (splitTime 1 (m:ℝ) (by norm_num) (by exact_mod_cast hm)) (meshTime j s) := by |
| apply Subtype.ext |
| have hm' : (m:ℝ)≠0 := by exact_mod_cast (Nat.ne_of_gt hm) |
| have hm1 : (1:ℝ)+m≠0 := by positivity |
| simp only [meshTime,Fin.val_succ,suffixTime,ContinuousMap.coe_mk,splitTime,Nat.cast_add,Nat.cast_one] |
| field_simp |
| ring |
|
|
| lemma meshSegment_first {m : ℕ} {L : Length} (hm : 0 < m) (ω : ConfigurationPath 1 L) : |
| meshSegment (0:Fin (m+1)) ω=configurationPrefix 1 L (splitTime 1 (m:ℝ) (by norm_num) (by exact_mod_cast hm)) ω := by |
| ext s i j |
| exact congrArg (fun u : Time ↦ ω u i j) (mesh_firstTime hm s) |
|
|
| lemma meshSegment_next {m : ℕ} {L : Length} (hm : 0 < m) (j : Fin m) (ω : ConfigurationPath 1 L) : |
| meshSegment j.succ ω=meshSegment j (configurationSuffix 1 L (splitTime 1 (m:ℝ) (by norm_num) (by exact_mod_cast hm)) ω) := by |
| ext s i k |
| exact congrArg (fun u : Time ↦ ω u i k) (mesh_nextTime hm j s) |
|
|
| lemma meshTime_reverse {m : ℕ} (j : Fin m) (s : Time) : |
| reverseTime (meshTime j s)=meshTime j.rev (reverseTime s) := by |
| apply Subtype.ext |
| have hm : (m:ℝ)≠0 := by exact_mod_cast (Nat.ne_of_gt j.pos) |
| change 1-((j:ℝ)+(s:ℝ))/(m:ℝ)=((j.rev:ℝ)+(1-(s:ℝ)))/(m:ℝ) |
| have hj : (j.rev:ℝ)=(m:ℝ)-1-(j:ℝ) := by |
| rw [Fin.val_rev,Nat.cast_sub (by omega)] |
| push_cast |
| ring |
| rw [hj] |
| field_simp |
| ring |
|
|
| lemma meshSegment_reverse {m : ℕ} {L : Length} (j : Fin m) (ω : ConfigurationPath 1 L) : |
| meshSegment j (reverseConfigurationPath 1 L ω)=reverseConfigurationPath 1 L (meshSegment j.rev ω) := by |
| ext s i k |
| exact congrArg (fun u : Time ↦ ω u i k) (meshTime_reverse j s) |
|
|
| lemma mesh_sum_reverse {m : ℕ} {L : Length} (F : ConfigurationPath 1 L→ℝ) |
| (hF : ∀ω,F (reverseConfigurationPath 1 L ω)=F ω) (ω : ConfigurationPath 1 L) : |
| ∑ j : Fin m,F (meshSegment j (reverseConfigurationPath 1 L ω))= |
| ∑ j : Fin m,F (meshSegment j ω) := by |
| simp only [meshSegment_reverse,hF] |
| exact Equiv.sum_comp (Fin.revPerm) (fun j : Fin m ↦ F (meshSegment j ω)) |
|
|
| end HardSphere.PathCutoff |
|
|
| end |
|
|
| end OAI |
|
|
