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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 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.FrozenChecks
import OAI.Analysis.HardSphere.Conditioning.BathPhase
namespace OAI
section
section
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal
namespace HardSphere
lemma insertFirst_preserving {N : ℕ} (hN : 0<N) (L : Length) :
MeasurePreserving (fun p : Torus L×Configuration (N-1) L ↦ insertFirst hN p.1 p.2)
((volume : Measure (Torus L)).prod volume) volume := by
cases N with
| zero => omega
| succ n =>
unfold insertFirst
have h := (cons_preserving (n:=n) L).comp
(Measure.measurePreserving_swap (μ:=(volume : Measure (Torus L))) (ν:=volume))
convert h using 1 <;> simp [Function.comp_def]
lemma bathFirst_preserving {N : ℕ} (hN : 0<N) (L : Length) :
MeasurePreserving (fun p : Configuration (N-1) L×Torus L ↦ insertFirst hN p.2 p.1)
((volume : Measure (Configuration (N-1) L)).prod volume) volume :=
(insertFirst_preserving hN L).comp (Measure.measurePreserving_swap)
lemma ground_slice_memLp {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) :
MemLp (fun p : Configuration (N-1) L×Torus L ↦ ψ (insertFirst hN p.2 p.1)) 2
((volume : Measure (Configuration (N-1) L)).prod volume) :=
(Lp.memLp ψ).comp_measurePreserving (bathFirst_preserving hN L)
lemma occupation_nonneg {N : ℕ} (hN : 0<N) {L : Length} (ψ : ScalarL2 N L) :
0≤occupation hN ψ := by
unfold occupation
exact mul_nonneg (inv_nonneg.mpr (pow_nonneg L.property.le _))
(integral_nonneg fun _ ↦ sq_nonneg _)
end HardSphere
end
end
section
noncomputable section
open MeasureTheory Set
open scoped ENNReal
namespace HardSphere.Comparison
variable {I C : Type*} [Fintype I] [DecidableEq I] [Fintype C]
{X : I → Type*} [∀i,MeasurableSpace (X i)]
omit [Fintype C] in
lemma frozen_uniformInsertion_on (checks : C → LocalCheck X) (U S : Finset I)
(hSU : S⊆U) (x : ∀i,X i) (μ : ∀i,Measure (X i)) {a : ℝ≥0∞}
(h : UniformInsertionOn checks μ a S) :
UniformInsertionOn (frozenChecks checks U x) μ a S := by
intro i hi s his y
have hin : i∉s∪(Finset.univ\U) := by simp [Finset.mem_union,his,hSU hi]
apply (h i hi (s∪(Finset.univ\U)) hin (freezeOutside U x y)).trans
apply lintegral_mono
intro v
dsimp only
rw [←freezeOutside_update U x y (hSU hi) v]
by_cases hp : freezeOutside U x (Function.update y i v)∈insertionEvent checks (s∪(Finset.univ\U)) i
· rw [Set.indicator_of_mem hp,Set.indicator_of_mem (frozen_insertion checks U s x (Function.update y i v) i hp)]
· rw [Set.indicator_of_notMem hp]
exact bot_le
end HardSphere.Comparison
end
end
end
end OAI
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