| 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 |
| |
