| import OAI.Probability.FreeIsing.Estimates.FiniteBondMask |
| import OAI.Probability.FreeIsing.Factors.Transfer |
| import OAI.Probability.FreeIsing.Model |
| import OAI.Probability.FreeIsing.Trees.FiniteClusterLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
|
|
| namespace Problem367.ClusterAssembly |
|
|
| theorem lintegral_finite_statistic |
| {Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E] |
| [Fintype E] [MeasurableSingletonClass E] |
| (μ : Measure Ω) (A : Ω → E) (hA : Measurable A) |
| (weight : E → ENNReal) |
| (hlaw : ∀ e, μ {ω | A ω = e} = weight e) (f : E → ENNReal) : |
| (∫⁻ ω, f (A ω) ∂μ) = ∑ e, weight e * f e := by |
| rw [← lintegral_map (measurable_of_finite f) hA, lintegral_fintype] |
| apply Finset.sum_congr rfl |
| intro e he |
| rw [Measure.map_apply hA (measurableSet_singleton e)] |
| change f e * μ {ω | A ω = e} = weight e * f e |
| rw [hlaw, mul_comm] |
|
|
| theorem map_prod_apply_eq_finite_mixture |
| {Ω C E S : Type*} [MeasurableSpace Ω] [MeasurableSpace C] |
| [MeasurableSpace E] [MeasurableSpace S] |
| [Fintype E] [MeasurableSingletonClass E] |
| (μ : Measure Ω) (ν : Measure C) [SFinite ν] |
| (A : Ω → E) (hA : Measurable A) |
| (F : Ω × C → S) (hF : Measurable F) |
| (s : Set S) (hs : MeasurableSet s) |
| (weight color : E → ENNReal) |
| (hlaw : ∀ e, μ {ω | A ω = e} = weight e) |
| (hcolor : ∀ᵐ ω ∂μ, ν {c | F (ω, c) ∈ s} = color (A ω)) : |
| Measure.map F (μ.prod ν) s = ∑ e, weight e * color e := by |
| rw [Measure.map_apply hF hs, Measure.prod_apply (hs.preimage hF)] |
| calc |
| (∫⁻ ω, ν (Prod.mk ω ⁻¹' (F ⁻¹' s)) ∂μ) = ∫⁻ ω, color (A ω) ∂μ := by |
| apply lintegral_congr_ae |
| exact hcolor |
| _ = ∑ e, weight e * color e := |
| lintegral_finite_statistic μ A hA weight hlaw color |
|
|
| theorem map_compProd_snd_apply_eq_finite_mixture |
| {Ω E S : Type*} [MeasurableSpace Ω] [MeasurableSpace E] |
| [MeasurableSpace S] [Fintype E] [MeasurableSingletonClass E] |
| (μ : Measure Ω) [SFinite μ] (κ : Kernel Ω S) [IsSFiniteKernel κ] |
| (A : Ω → E) (hA : Measurable A) |
| (s : Set S) (hs : MeasurableSet s) |
| (weight color : E → ENNReal) |
| (hlaw : ∀ e, μ {ω | A ω = e} = weight e) |
| (hcolor : ∀ᵐ ω ∂μ, κ ω s = color (A ω)) : |
| Measure.map Prod.snd (μ ⊗ₘ κ) s = ∑ e, weight e * color e := by |
| rw [Measure.map_apply measurable_snd hs, |
| Measure.compProd_apply (hs.preimage measurable_snd)] |
| calc |
| (∫⁻ ω, κ ω (Prod.mk ω ⁻¹' (Prod.snd ⁻¹' s)) ∂μ) = |
| ∫⁻ ω, color (A ω) ∂μ := by |
| apply lintegral_congr_ae |
| exact hcolor |
| _ = ∑ e, weight e * color e := |
| lintegral_finite_statistic μ A hA weight hlaw color |
|
|
| theorem cluster_mixture_sum {d : ℕ} {theta : ℝ} |
| (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) : |
| (∑ A : (inducedEdges D).powerset, |
| ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) A.val) * |
| ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) A.val eta)) = |
| freeCylinderWeight theta D eta := by |
| classical |
| rw [Finset.sum_coe_sort (inducedEdges D).powerset (fun A => |
| ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) A) * |
| ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) A eta))] |
| rw [← FiniteCluster.marginal_eq_freeCylinderWeight theta D eta] |
| rw [ENNReal.ofReal_sum_of_nonneg (fun A _ => |
| mul_nonneg (FiniteCluster.bondWeight_nonneg h0 h1 _ _) |
| (FiniteCluster.colorWeight_nonneg _ _ _))] |
| apply Finset.sum_congr rfl |
| intro A hA |
| exact (ENNReal.ofReal_mul (FiniteCluster.bondWeight_nonneg h0 h1 _ _)).symm |
|
|
| lemma measurableSet_spinCylinder {d : ℕ} (D : Finset (TreeVertex d)) |
| (eta : TreeVertex d → IsingSpin) : |
| MeasurableSet {x : TreeVertex d → IsingSpin | ∀ v ∈ D, x v = eta v} := by |
| classical |
| simp only [Set.ofPred_forall] |
| apply MeasurableSet.iInter |
| intro v |
| apply MeasurableSet.iInter |
| intro hv |
| convert (show MeasurableSet ({eta v} : Set IsingSpin) by trivial).preimage |
| (measurable_pi_apply v : Measurable (fun x : TreeVertex d → IsingSpin => x v)) using 1 |
| ext x |
| simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_ofPred_eq] |
|
|
| theorem hasFreeIsingLaw_map_prod_of_cluster_conditionals |
| {d : ℕ} {Ω C : Type*} [MeasurableSpace Ω] [MeasurableSpace C] |
| (μ : Measure Ω) (ν : Measure C) [SFinite ν] |
| {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (F : Ω × C → TreeVertex d → IsingSpin) (hF : Measurable F) |
| (A : (D : Finset (TreeVertex d)) → Ω → (inducedEdges D).powerset) |
| (hA : ∀ D, Measurable (A D)) |
| (hmask : ∀ D, ConnectedFinset D → ∀ a : (inducedEdges D).powerset, |
| μ {ω | A D ω = a} = |
| ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) a.val)) |
| (hcolor : ∀ D, ConnectedFinset D → ∀ eta : TreeVertex d → IsingSpin, |
| ∀ᵐ ω ∂μ, ν {c | ∀ v ∈ D, F (ω, c) v = eta v} = |
| ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) (A D ω).val eta)) : |
| HasFreeIsingLaw theta (Measure.map F (μ.prod ν)) := by |
| intro D hD eta |
| rw [map_prod_apply_eq_finite_mixture μ ν (A D) (hA D) F hF _ |
| (measurableSet_spinCylinder D eta) |
| (fun a => ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) a.val)) |
| (fun a => ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) a.val eta)) |
| (hmask D hD) (hcolor D hD eta)] |
| exact cluster_mixture_sum h0 h1 D eta |
|
|
| theorem hasFreeIsingLaw_compProd_of_cluster_conditionals |
| {d : ℕ} {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [SFinite μ] |
| (κ : Kernel Ω (TreeVertex d → IsingSpin)) [IsSFiniteKernel κ] |
| {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (A : (D : Finset (TreeVertex d)) → Ω → (inducedEdges D).powerset) |
| (hA : ∀ D, Measurable (A D)) |
| (hmask : ∀ D, ConnectedFinset D → ∀ a : (inducedEdges D).powerset, |
| μ {ω | A D ω = a} = |
| ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) a.val)) |
| (hcolor : ∀ D, ConnectedFinset D → ∀ eta : TreeVertex d → IsingSpin, |
| ∀ᵐ ω ∂μ, κ ω {x | ∀ v ∈ D, x v = eta v} = |
| ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) (A D ω).val eta)) : |
| HasFreeIsingLaw theta (Measure.map Prod.snd (μ ⊗ₘ κ)) := by |
| intro D hD eta |
| rw [map_compProd_snd_apply_eq_finite_mixture μ κ (A D) (hA D) _ |
| (measurableSet_spinCylinder D eta) |
| (fun a => ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) a.val)) |
| (fun a => ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) a.val eta)) |
| (hmask D hD) (hcolor D hD eta)] |
| exact cluster_mixture_sum h0 h1 D eta |
|
|
| theorem hasFreeIsingLaw_map_prod_of_independent_bonds |
| {d : ℕ} {Ω C : Type*} [MeasurableSpace Ω] [MeasurableSpace C] |
| (μ : Measure Ω) (ν : Measure C) [SFinite ν] |
| {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (X : (TreeVertex d × TreeVertex d) → Ω → Bool) |
| (hX : ∀ e, Measurable (X e)) |
| (hindep : ∀ D : Finset (TreeVertex d), |
| iIndepFun (fun e : inducedEdges D => X e.val) μ) |
| (htrue : ∀ D : Finset (TreeVertex d), ∀ e ∈ inducedEdges D, |
| μ {ω | X e ω = true} = ENNReal.ofReal theta) |
| (hfalse : ∀ D : Finset (TreeVertex d), ∀ e ∈ inducedEdges D, |
| μ {ω | X e ω = false} = ENNReal.ofReal (1 - theta)) |
| (F : Ω × C → TreeVertex d → IsingSpin) (hF : Measurable F) |
| (hcolor : ∀ D, ConnectedFinset D → ∀ eta : TreeVertex d → IsingSpin, |
| ∀ᵐ ω ∂μ, ν {c | ∀ v ∈ D, F (ω, c) v = eta v} = |
| ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) |
| (openMask (inducedEdges D) X ω).val eta)) : |
| HasFreeIsingLaw theta (Measure.map F (μ.prod ν)) := by |
| classical |
| apply hasFreeIsingLaw_map_prod_of_cluster_conditionals μ ν h0 h1 F hF |
| (fun D => openMask (inducedEdges D) X) |
| (fun D => measurable_openMask _ _ (fun e _ => hX e)) _ hcolor |
| intro D hD A |
| exact openMask_atom_measure_ofReal_restricted μ (inducedEdges D) X |
| (hindep D) h0 h1 (htrue D) (hfalse D) A |
|
|
| end Problem367.ClusterAssembly |
|
|
| end |
|
|
| end OAI |
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|
