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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | 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
end OAI
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