| import OAI.Probability.FreeIsing.Model | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367.ClusterAssembly | |
| def openMask {Ω E : Type*} (edges : Finset E) (X : E → Ω → Bool) | |
| (ω : Ω) : edges.powerset := by | |
| classical | |
| exact ⟨edges.filter (fun e => X e ω = true), | |
| Finset.mem_powerset.mpr (Finset.filter_subset _ _)⟩ | |
| theorem measurable_openMask {Ω E : Type*} [MeasurableSpace Ω] | |
| (edges : Finset E) (X : E → Ω → Bool) | |
| (hX : ∀ e ∈ edges, Measurable (X e)) : Measurable (openMask edges X) := by | |
| classical | |
| apply Measurable.subtype_mk | |
| apply measurable_finset_iff.mpr | |
| intro e | |
| by_cases he : e ∈ edges | |
| · have h : Measurable (fun ω => X e ω = true) := (hX e he).eq measurable_const | |
| simpa only [Finset.mem_filter, he, true_and] using h | |
| · simp only [Finset.mem_filter, he, false_and] | |
| exact measurable_const | |
| theorem openMask_fiber {Ω E : Type*} [DecidableEq E] | |
| (edges : Finset E) (X : E → Ω → Bool) | |
| (A : edges.powerset) : | |
| {ω | openMask edges X ω = A} = | |
| ⋂ e ∈ edges, X e ⁻¹' {if e ∈ A.val then true else false} := by | |
| classical | |
| have hsub : A.val ⊆ edges := Finset.mem_powerset.mp A.property | |
| ext ω | |
| simp only [Set.mem_ofPred_eq, Set.mem_iInter, Set.mem_preimage, Set.mem_singleton_iff] | |
| constructor | |
| · intro h e he | |
| have hm := congrArg (fun B : edges.powerset => e ∈ B.val) h | |
| change (e ∈ edges.filter (fun f => X f ω = true)) = (e ∈ A.val) at hm | |
| by_cases ha : e ∈ A.val | |
| · simpa [Finset.mem_filter, he, ha] using hm | |
| · have hn : X e ω ≠ true := by simpa [Finset.mem_filter, he, ha] using hm | |
| cases hx : X e ω <;> simp_all | |
| · intro h | |
| apply Subtype.ext | |
| apply Finset.ext | |
| intro e | |
| change (e ∈ edges.filter (fun f => X f ω = true)) ↔ e ∈ A.val | |
| by_cases he : e ∈ edges | |
| · have hx := h e he | |
| by_cases ha : e ∈ A.val <;> simp_all | |
| · have ha : e ∉ A.val := fun h => he (hsub h) | |
| simp [he, ha] | |
| theorem openMask_atom_measure {Ω E : Type*} [MeasurableSpace Ω] [DecidableEq E] | |
| (μ : Measure Ω) (edges : Finset E) (X : E → Ω → Bool) | |
| (hindep : iIndepFun X μ) (p q : ENNReal) | |
| (htrue : ∀ e ∈ edges, μ {ω | X e ω = true} = p) | |
| (hfalse : ∀ e ∈ edges, μ {ω | X e ω = false} = q) | |
| (A : edges.powerset) : | |
| μ {ω | openMask edges X ω = A} = p ^ A.val.card * q ^ (edges \ A.val).card := by | |
| classical | |
| rw [openMask_fiber] | |
| rw [hindep.measure_inter_preimage_eq_mul edges (fun e _ => measurableSet_singleton _)] | |
| have hprod : (∏ e ∈ edges, μ (X e ⁻¹' {if e ∈ A.val then true else false})) = | |
| ∏ e ∈ edges, if e ∈ A.val then p else q := by | |
| apply Finset.prod_congr rfl | |
| intro e he | |
| have hset (b : Bool) : X e ⁻¹' {b} = {ω | X e ω = b} := by | |
| ext ω | |
| simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_ofPred_eq] | |
| by_cases ha : e ∈ A.val | |
| · rw [ite_eq_left ha, ite_eq_left ha, hset] | |
| exact htrue e he | |
| · rw [ite_eq_right ha, ite_eq_right ha, hset] | |
| exact hfalse e he | |
| rw [hprod, Finset.prod_ite] | |
| have hsub : A.val ⊆ edges := Finset.mem_powerset.mp A.property | |
| have hfilter : edges.filter (fun e => e ∈ A.val) = A.val := by | |
| ext e | |
| simp only [Finset.mem_filter] | |
| exact and_iff_right_of_imp (fun h => hsub h) | |
| rw [hfilter, ← Finset.sdiff_eq_filter] | |
| simp only [Finset.prod_const] | |
| theorem openMask_atom_measure_ofReal {Ω E : Type*} [MeasurableSpace Ω] | |
| [DecidableEq E] (μ : Measure Ω) (edges : Finset E) (X : E → Ω → Bool) | |
| (hindep : iIndepFun X μ) {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta ≤ 1) | |
| (htrue : ∀ e ∈ edges, μ {ω | X e ω = true} = ENNReal.ofReal theta) | |
| (hfalse : ∀ e ∈ edges, μ {ω | X e ω = false} = ENNReal.ofReal (1 - theta)) | |
| (A : edges.powerset) : | |
| μ {ω | openMask edges X ω = A} = | |
| ENNReal.ofReal (theta ^ A.val.card * (1 - theta) ^ (edges \ A.val).card) := by | |
| rw [openMask_atom_measure μ edges X hindep _ _ htrue hfalse A] | |
| rw [ENNReal.ofReal_mul (pow_nonneg h0 _), ENNReal.ofReal_pow h0, | |
| ENNReal.ofReal_pow (sub_nonneg.mpr h1)] | |
| theorem openMask_atom_measure_of_restricted_indep | |
| {Ω E : Type*} [MeasurableSpace Ω] [DecidableEq E] | |
| (μ : Measure Ω) (edges : Finset E) (X : E → Ω → Bool) | |
| (hindep : iIndepFun (fun e : edges => X e.val) μ) (p q : ENNReal) | |
| (htrue : ∀ e ∈ edges, μ {ω | X e ω = true} = p) | |
| (hfalse : ∀ e ∈ edges, μ {ω | X e ω = false} = q) | |
| (A : edges.powerset) : | |
| μ {ω | openMask edges X ω = A} = p ^ A.val.card * q ^ (edges \ A.val).card := by | |
| classical | |
| let sets : E → Set Bool := fun e => {if e ∈ A.val then true else false} | |
| have hi := hindep.measure_inter_preimage_eq_mul Finset.univ | |
| (sets := fun e : edges => sets e.val) (fun _ _ => measurableSet_singleton _) | |
| have hset : (⋂ e : edges, X e.val ⁻¹' sets e.val) = | |
| ⋂ e ∈ edges, X e ⁻¹' sets e := by | |
| ext ω | |
| simp | |
| simp only [Finset.mem_univ, Set.iInter_true] at hi | |
| rw [hset, Finset.prod_coe_sort edges (fun e => μ (X e ⁻¹' sets e))] at hi | |
| rw [openMask_fiber] | |
| change μ (⋂ e ∈ edges, X e ⁻¹' sets e) = _ | |
| rw [hi] | |
| have hprod : (∏ e ∈ edges, μ (X e ⁻¹' sets e)) = | |
| ∏ e ∈ edges, if e ∈ A.val then p else q := by | |
| apply Finset.prod_congr rfl | |
| intro e he | |
| have hset' (b : Bool) : X e ⁻¹' {b} = {ω | X e ω = b} := by | |
| ext ω | |
| simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_ofPred_eq] | |
| dsimp only [sets] | |
| by_cases ha : e ∈ A.val | |
| · rw [ite_eq_left ha, ite_eq_left ha, hset'] | |
| exact htrue e he | |
| · rw [ite_eq_right ha, ite_eq_right ha, hset'] | |
| exact hfalse e he | |
| rw [hprod, Finset.prod_ite] | |
| have hsub : A.val ⊆ edges := Finset.mem_powerset.mp A.property | |
| have hfilter : edges.filter (fun e => e ∈ A.val) = A.val := by | |
| ext e | |
| simp only [Finset.mem_filter] | |
| exact and_iff_right_of_imp (fun h => hsub h) | |
| rw [hfilter, ← Finset.sdiff_eq_filter] | |
| simp only [Finset.prod_const] | |
| theorem openMask_atom_measure_ofReal_restricted | |
| {Ω E : Type*} [MeasurableSpace Ω] [DecidableEq E] | |
| (μ : Measure Ω) (edges : Finset E) (X : E → Ω → Bool) | |
| (hindep : iIndepFun (fun e : edges => X e.val) μ) | |
| {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta ≤ 1) | |
| (htrue : ∀ e ∈ edges, μ {ω | X e ω = true} = ENNReal.ofReal theta) | |
| (hfalse : ∀ e ∈ edges, μ {ω | X e ω = false} = ENNReal.ofReal (1 - theta)) | |
| (A : edges.powerset) : | |
| μ {ω | openMask edges X ω = A} = | |
| ENNReal.ofReal (theta ^ A.val.card * (1 - theta) ^ (edges \ A.val).card) := by | |
| rw [openMask_atom_measure_of_restricted_indep μ edges X hindep _ _ htrue hfalse A] | |
| rw [ENNReal.ofReal_mul (pow_nonneg h0 _), ENNReal.ofReal_pow h0, | |
| ENNReal.ofReal_pow (sub_nonneg.mpr h1)] | |
| end Problem367.ClusterAssembly | |
| end | |
| end OAI | |