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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 | import Mathlib
import OAI.Analysis.HardSphere.Geometry.AuxiliaryChecks
namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace HardSphere.Comparison
variable {I C : Type*} [Fintype I] [DecidableEq I] [Fintype C]
{X : I → Type*} [∀i,MeasurableSpace (X i)]
def freezeOutside (U : Finset I) (x y : ∀i,X i) : ∀i,X i :=
fun i ↦ if i∈U then y i else x i
omit [Fintype I] in
lemma freezeOutside_measurable (U : Finset I) (x : ∀i,X i) : Measurable (freezeOutside U x) := by
apply Measurable.of_eval
intro i
by_cases hi : i∈U
· simpa only [freezeOutside,ite_eq_left hi] using (measurable_pi_apply i)
· simp only [freezeOutside,ite_eq_right hi]
exact measurable_const
omit [Fintype I] [∀i,MeasurableSpace (X i)] in
lemma freezeOutside_update (U : Finset I) (x y : ∀i,X i) {i : I} (hi : i∈U) (v : X i) :
freezeOutside U x (Function.update y i v)=Function.update (freezeOutside U x y) i v := by
funext j
by_cases hj : j=i
· subst j;simp [freezeOutside,hi]
· simp [freezeOutside,Function.update_of_ne hj]
def FrozenCheckIndex (checks : C → LocalCheck X) (U : Finset I) :=
{c : C // ((checks c).support∩U).Nonempty}
instance (checks : C → LocalCheck X) (U : Finset I) : Fintype (FrozenCheckIndex checks U) :=
by
classical
unfold FrozenCheckIndex
infer_instance
def frozenChecks (checks : C → LocalCheck X) (U : Finset I) (x : ∀i,X i) :
FrozenCheckIndex checks U → LocalCheck X := fun c ↦ {
support := (checks c.val).support∩U
nonempty := c.property
arity := (Finset.card_le_card Finset.inter_subset_left).trans (checks c.val).arity
pass := fun y ↦ (checks c.val).pass (freezeOutside U x y)
measurable_pass := (checks c.val).measurable_pass.preimage (freezeOutside_measurable U x)
depends := by
intro y z hyz
apply (checks c.val).depends
intro i hi
by_cases hu : i∈U
· simp only [freezeOutside,ite_eq_left hu]
exact hyz i (Finset.mem_inter.mpr ⟨hi,hu⟩)
· simp only [freezeOutside,ite_eq_right hu]
}
omit [Fintype C]
lemma frozen_support_subset (checks : C → LocalCheck X) (U t : Finset I) (c : C) :
(checks c).support ⊆ t∪(Finset.univ \ U) ↔ (checks c).support∩U ⊆t := by
constructor
· intro h i hi
rcases Finset.mem_union.mp (h (Finset.mem_inter.mp hi).1) with ht|ho
· exact ht
· exact False.elim ((Finset.mem_sdiff.mp ho).2 (Finset.mem_inter.mp hi).2)
· intro h i hi
by_cases hu : i∈U
· exact Finset.mem_union_left _ (h (Finset.mem_inter.mpr ⟨hi,hu⟩))
· exact Finset.mem_union_right _ (Finset.mem_sdiff.mpr ⟨Finset.mem_univ _,hu⟩)
lemma frozen_retained (checks : C → LocalCheck X) (U t : Finset I) (x y : ∀i,X i)
(hx : x∈retainedEvent checks (Finset.univ \ U)) :
y∈retainedEvent (frozenChecks checks U x) t ↔
freezeOutside U x y∈retainedEvent checks (t∪(Finset.univ \ U)) := by
constructor
· intro hy c hc
by_cases he : ((checks c).support∩U).Nonempty
· exact hy ⟨c,he⟩ ((frozen_support_subset checks U t c).mp hc)
· have hout : (checks c).support⊆Finset.univ \ U := by
intro i hi
refine Finset.mem_sdiff.mpr ⟨Finset.mem_univ _,?_⟩
intro hu
exact he ⟨i,Finset.mem_inter.mpr ⟨hi,hu⟩⟩
apply ((checks c).depends x (freezeOutside U x y) ?_).mp (hx c hout)
intro i hi
simp only [freezeOutside,ite_eq_right (Finset.mem_sdiff.mp (hout hi)).2]
· intro hy c hc
exact hy c.val ((frozen_support_subset checks U t c.val).mpr hc)
lemma frozen_checkWeight (checks : C → LocalCheck X) (U t : Finset I) (x y : ∀i,X i)
(hx : x∈retainedEvent checks (Finset.univ \ U)) :
checkWeight (frozenChecks checks U x) t y=
checkWeight checks (t∪(Finset.univ \ U)) (freezeOutside U x y) := by
classical
simp only [checkWeight,Set.indicator_apply,frozen_retained checks U t x y hx]
lemma frozen_insertion (checks : C → LocalCheck X) (U s : Finset I) (x y : ∀i,X i) (i : I) :
freezeOutside U x y∈insertionEvent checks (s∪(Finset.univ \ U)) i →
y∈insertionEvent (frozenChecks checks U x) s i := by
intro hy c hc hcs
apply hy c.val
· rw [←Finset.insert_union]
exact (frozen_support_subset checks U (insert i s) c.val).mpr hc
· intro hs
exact hcs ((frozen_support_subset checks U s c.val).mp hs)
lemma frozen_uniformInsertion (checks : C → LocalCheck X) (U : Finset I) (x : ∀i,X i)
(μ : ∀i,Measure (X i)) {a : ℝ≥0∞}
(h : UniformInsertionOn checks μ a U) :
UniformInsertionOn (frozenChecks checks U x) μ a U := by
intro i hi s his y
have hin : i∉s∪(Finset.univ \ U) := by simp [Finset.mem_union,his,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 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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