| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Groups |
| attribute [local instance] Classical.propDecidable |
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| lemma certified_allocated_joint {W Z V : Type*} [MeasurableSpace W] [MeasurableSpace Z] |
| [Fintype V] [DecidableEq V] (μ : Measure W) [SFinite μ] (κ : Kernel W Z) [IsSFiniteKernel κ] |
| (cert : V→W→Prop) (bad : V→W×Z→Prop) |
| (hcert : ∀v,MeasurableSet {w | cert v w}) (hbad : ∀v,MeasurableSet {p | bad v p}) |
| (p q : ℝ≥0∞) |
| (hjoint : ∀S : Finset V,μ {w | ∀v∈S,¬cert v w}≤p^S.card) |
| (hcond : ∀w,∀T : Finset V,(∀v∈T,cert v w)→κ w {z | ∀v∈T,bad v (w,z)}≤q^T.card) |
| (A : Finset V) : |
| μ.compProd κ {wz | ∀v∈A,¬cert v wz.1 ∨ bad v wz}≤(p+q)^A.card := by |
| let F : Finset V→Set (W×Z) := fun S ↦ {wz | (∀v∈S,¬cert v wz.1) ∧ |
| ∀v∈A\S,cert v wz.1 ∧ bad v wz} |
| have hF (S : Finset V) : MeasurableSet (F S) := by |
| have h₁ : MeasurableSet {p : W×Z | ∀v∈S,¬cert v p.1} := by |
| simp only [Set.ofPred_forall] |
| exact MeasurableSet.iInter (fun v ↦ MeasurableSet.iInter (fun _ ↦ (hcert v).compl.preimage measurable_fst)) |
| have h₂ : MeasurableSet {p : W×Z | ∀v∈A\S,cert v p.1 ∧ bad v p} := by |
| simp only [Set.ofPred_forall] |
| exact MeasurableSet.iInter (fun v ↦ MeasurableSet.iInter (fun _ ↦ ((hcert v).preimage measurable_fst).inter (hbad v))) |
| exact h₁.inter h₂ |
| have hsub : {wz : W×Z | ∀v∈A,¬cert v wz.1 ∨ bad v wz}⊆⋃S∈A.powerset,F S := by |
| intro wz hwz |
| let S := A.filter (fun v ↦ ¬cert v wz.1) |
| refine mem_iUnion.mpr ⟨S,mem_iUnion.mpr ⟨Finset.mem_powerset.mpr (Finset.filter_subset _ _),?_⟩⟩ |
| constructor |
| · exact fun v hv ↦ (Finset.mem_filter.mp hv).2 |
| · intro v hv |
| obtain ⟨hvA,hvS⟩ := Finset.mem_sdiff.mp hv |
| have hc : cert v wz.1 := by |
| by_contra hn |
| exact hvS (Finset.mem_filter.mpr ⟨hvA,hn⟩) |
| exact ⟨hc,(hwz v hvA).resolve_left (not_not.mpr hc)⟩ |
| have hbound (S : Finset V) : μ.compProd κ (F S)≤p^S.card*q^(A\S).card := by |
| rw [Measure.compProd_apply (hF S)] |
| let C : Set W := {w | ∀v∈S,¬cert v w} |
| have hC : MeasurableSet C := by |
| simp only [C,ofPred_forall] |
| exact MeasurableSet.iInter (fun v ↦ MeasurableSet.iInter (fun _ ↦ (hcert v).compl)) |
| calc |
| _ ≤ ∫⁻w,C.indicator (fun _ ↦ q^(A\S).card) w ∂μ := by |
| apply lintegral_mono |
| intro w |
| change κ w {z | (w,z)∈F S} ≤ _ |
| by_cases hc : w∈C |
| · rw [Set.indicator_of_mem hc] |
| by_cases hT : ∀v∈A\S,cert v w |
| · apply (measure_mono (show {z | (w,z)∈F S}⊆{z | ∀v∈A\S,bad v (w,z)} from fun z hz v hv ↦ (hz.2 v hv).2)).trans |
| exact hcond w (A\S) hT |
| · have he : {z | (w,z)∈F S}=∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro z hz |
| exact hT (fun v hv ↦ (hz.2 v hv).1) |
| rw [he,measure_empty] |
| exact bot_le |
| · rw [Set.indicator_of_notMem hc] |
| have he : {z | (w,z)∈F S}=∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro z hz |
| exact hc hz.1 |
| rw [he,measure_empty] |
| _ = q^(A\S).card*μ C := by rw [lintegral_indicator hC]; simp |
| _ ≤ q^(A\S).card*p^S.card := mul_le_mul_right (hjoint S) _ |
| _ = _ := mul_comm _ _ |
| calc |
| _ ≤ μ.compProd κ (⋃S∈A.powerset,F S) := measure_mono hsub |
| _ ≤ ∑S∈A.powerset,μ.compProd κ (F S) := measure_biUnion_finset_le _ _ |
| _ ≤ ∑S∈A.powerset,p^S.card*q^(A\S).card := Finset.sum_le_sum (fun S _ ↦ hbound S) |
| _ = _ := by |
| calc |
| _ = ∑S∈A.powerset,p^S.card*q^(A.card-S.card) := by |
| apply Finset.sum_congr rfl |
| intro S hS |
| rw [Finset.card_sdiff_of_subset (Finset.mem_powerset.mp hS)] |
| _ = _ := Finset.sum_pow_mul_eq_add_pow p q A |
|
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| end HardSphere.Groups |
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| end |
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| noncomputable section |
| open scoped BigOperators |
| namespace HardSphere.Groups |
|
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| def groupCount (ell D : ℝ) (M : ℕ) : ℕ := ⌊ell*D/(4*(M:ℝ))⌋₊ |
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| lemma groupCount_allocation {ell D : ℝ} {M m : ℕ} |
| (hell : 0≤ell) (hD : 0≤D) (hM : 0<M) (hm : ell*D/2 ≤ (m:ℝ)) : |
| M*groupCount ell D M ≤ m := by |
| have hMp : (0:ℝ)<M := by exact_mod_cast hM |
| have hfloor := Nat.floor_le (show 0≤ell*D/(4*(M:ℝ)) by positivity) |
| have hh : (M:ℝ)*(groupCount ell D M:ℝ) ≤ ell*D/4 := by |
| unfold groupCount |
| calc |
| _ ≤ (M:ℝ)*(ell*D/(4*(M:ℝ))) := mul_le_mul_of_nonneg_left hfloor hMp.le |
| _ = ell*D/4 := by field_simp |
| have hm' : ell*D/4≤(m:ℝ) := by nlinarith [mul_nonneg hell hD] |
| have h := hh.trans hm' |
| exact_mod_cast h |
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| lemma groupCount_ratio {ell D : ℝ} {M : ℕ} |
| (hell : 0<ell) (hD : 0<D) (hM : 0<M) |
| (hlarge : 2≤ell*D/(4*(M:ℝ))) : |
| ell/(8*(M:ℝ)) ≤ (groupCount ell D M:ℝ)/D := by |
| have hMp : (0:ℝ)<M := by exact_mod_cast hM |
| have hfloor : ell*D/(4*(M:ℝ)) < (⌊ell*D/(4*(M:ℝ))⌋₊:ℝ)+1 := by |
| have hnonneg := div_nonneg (mul_nonneg hell.le hD.le) |
| (mul_nonneg (by norm_num : (0:ℝ)≤4) hMp.le) |
| exact_mod_cast (Nat.floor_lt hnonneg).mp (Nat.lt_succ_self _) |
| have hh : ell*D/(8*(M:ℝ)) ≤ (groupCount ell D M:ℝ) := by |
| unfold groupCount |
| have he : ell*D/(4*(M:ℝ))=2*(ell*D/(8*(M:ℝ))) := by ring |
| rw [he] at hlarge hfloor |
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| have he' : ⌊2*(ell*D/(8*(M:ℝ)))⌋₊=⌊ell*D/(4*(M:ℝ))⌋₊ := by rw [←he] |
| rw [he'] at hfloor |
| linarith |
| apply (le_div_iff₀ hD).mpr |
| have he : (ell/(8*(M:ℝ)))*D=ell*D/(8*(M:ℝ)) := by ring |
| rw [he] |
| exact hh |
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| end HardSphere.Groups |
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| end |
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| end OAI |
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