| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.BadComponent |
| import OAI.Analysis.HardSphere.Conditioning.Phase |
| import OAI.Analysis.HardSphere.Conditioning.PhaseTransfer |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory MeasureTheory.Measure Set Filter |
| open scoped ENNReal NNReal Topology BigOperators ComplexConjugate |
| namespace HardSphere |
|
|
| lemma cons_preserving {n : ℕ} (L : Length) : |
| MeasurePreserving (fun p : Configuration n L × Torus L ↦ Fin.cons (α:=fun _ ↦ Torus L) p.2 p.1) |
| (volume.prod volume) volume := by |
| have h := (volume_preserving_piFinSuccAbove (fun _ : Fin (n+1) ↦ Torus L) 0).symm |
| have hh := h.comp (measurePreserving_swap (μ:= (volume : Measure (Configuration n L))) (ν:=volume)) |
| convert hh using 1 |
| funext p i |
| simp [MeasurableEquiv.piFinSuccAbove] |
|
|
| lemma allowed_cons {n : ℕ} {L : Length} {r : ℝ} {Y : Configuration n L} {x : Torus L} |
| (hY : Y∈allowed n L r) (hx : x∈BathSlice.slice r Y) : |
| Fin.cons x Y∈allowed (n+1) L r := by |
| rw [allowed_iff_ne] at hY ⊢ |
| intro i j hij |
| induction i using Fin.cases with |
| | zero => |
| induction j using Fin.cases with |
| | zero => exact (hij rfl).elim |
| | succ j => |
| simpa using (lt_of_not_ge (fun h ↦ hx ⟨j,h⟩)) |
| | succ i => |
| induction j using Fin.cases with |
| | zero => |
| simpa [torusDistance_comm] using (lt_of_not_ge (fun h ↦ hx ⟨i,h⟩)) |
| | succ j => exact hY i j (fun h ↦ hij (congrArg Fin.succ h)) |
|
|
| lemma IsGroundVector.phaseRepresentative_eq_of_joined {n : ℕ} {L : Length} {r : ℝ} |
| {u : Jet n L} (hu : IsGroundVector n L r u) {x y : Configuration n L} |
| (h : JoinedIn (allowed n L r) x y) : |
| groundPhaseRepresentative u x=groundPhaseRepresentative u y := by |
| obtain ⟨p,hp⟩ := h |
| apply hu.phaseRepresentative_eq_on_connected (S:=Set.range p) |
| (isPreconnected_range p.continuous) (by rintro z ⟨t,rfl⟩; exact hp t) |
| · exact ⟨0,p.source⟩ |
| · exact ⟨1,p.target⟩ |
|
|
| lemma IsGroundVector.retained_phaseRepresentative {n : ℕ} {L : Length} {h r : ℝ} |
| {u : Jet (n+1) L} (hu : IsGroundVector (n+1) L r u) (hh : 0<h) |
| {Y : Configuration n L} {x y : Torus L} |
| (hx : x∈BathSlice.retained h r Y) (hy : y∈BathSlice.retained h r Y) : |
| groundPhaseRepresentative u (Fin.cons x Y)=groundPhaseRepresentative u (Fin.cons y Y) := by |
| have hY : Y∈allowed n L r := by |
| by_contra hn |
| exact hx (Or.inl (Or.inl hn)) |
| have hj := BathSlice.retained_joined hh hx hy |
| have hc : Continuous (fun z : Torus L ↦ Fin.cons (α:=fun _ ↦ Torus L) z Y) := by |
| apply continuous_pi |
| intro i |
| induction i using Fin.cases <;> simp only [Fin.cons_zero,Fin.cons_succ] <;> fun_prop |
| apply hu.phaseRepresentative_eq_of_joined |
| exact (hj.map hc.continuousOn).mono (by rintro _ ⟨z,hz,rfl⟩; exact allowed_cons hY hz) |
|
|
| lemma commonPhase_of_ae_phase {α : Type*} [MeasurableSpace α] (μ : Measure α) {f : α → ℂ} |
| {U : Set α} {c : ℂ} (h : ∀ᵐ x ∂μ.restrict U, complexPhase (f x)=c) : |
| Phase.HasCommonPhase f U μ := by |
| by_cases hc : ‖c‖=1 |
| · refine ⟨c,hc,?_⟩ |
| filter_upwards [h] with x hx |
| rw [←hx,complexPhase_mul_norm] |
| · refine ⟨1,norm_one,?_⟩ |
| filter_upwards [h] with x hx |
| have hz : f x=0 := by |
| by_contra hn |
| exact hc (hx ▸ norm_complexPhase hn) |
| simp [hz] |
|
|
| lemma IsGroundVector.ae_retained_commonPhase {n : ℕ} {L : Length} {h r : ℝ} |
| {u : Jet (n+1) L} (hu : IsGroundVector (n+1) L r u) (hh : 0<h) : |
| ∀ᵐ Y : Configuration n L ∂volume, |
| Phase.HasCommonPhase (fun x ↦ u.1 (Fin.cons x Y)) (BathSlice.retained h r Y) volume := by |
| have ha : ∀ᵐ X ∂volume, X∈allowed (n+1) L r → |
| complexPhase (u.1 X)=groundPhaseRepresentative u X := |
| (ae_restrict_iff' (isOpen_allowed (n+1) L r).measurableSet).mp hu.ae_phaseRepresentative |
| have hb := ae_ae_of_ae_prod ((cons_preserving (n:=n) L).quasiMeasurePreserving.ae ha) |
| filter_upwards [hb] with Y hY |
| by_cases he : (BathSlice.retained h r Y).Nonempty |
| · obtain ⟨x,hx⟩ := he |
| apply commonPhase_of_ae_phase volume (c:=groundPhaseRepresentative u (Fin.cons x Y)) |
| filter_upwards [ae_restrict_of_ae hY,ae_restrict_mem (BathSlice.measurableSet_retained h r Y)] with y hy hym |
| have hYa : Y∈allowed n L r := by |
| by_contra hn |
| exact hx (Or.inl (Or.inl hn)) |
| have hys := (BathSlice.retained_joined hh hym hym).source_mem |
| exact (hy (allowed_cons hYa hys)).trans (hu.retained_phaseRepresentative hh hym hx) |
| · rw [not_nonempty_iff_eq_empty.mp he] |
| exact ⟨1,norm_one,by simp⟩ |
|
|
| end HardSphere |
|
|
| end |
| end |
|
|
| end |
|
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| end OAI |
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|
