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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 | import OAI.Probability.FreeIsing.Estimates.FairSpin
import OAI.Probability.FreeIsing.Estimates.MeasurableConnectivity
import OAI.Probability.FreeIsing.Trees.EdgeLabelCoding
import OAI.Probability.FreeIsing.Trees.FiniteClusterSelector
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.ClusterFactorMap
open EdgeLabelCoding
variable {d : ℕ}
abbrev Input (d : ℕ) :=
(TreeVertex d → ℝ) × ((TreeVertex d × ℕ → ℝ) × (TreeVertex d → ℝ))
def relabelInput (g : TreeAut d) (p : Input d) : Input d :=
(relabel g p.1, ((fun q => p.2.1 (g.toEquiv.symm q.1, q.2)), relabel g p.2.2))
def openAdj (theta : ℝ) (p : Input d) (u v : TreeVertex d) : Prop :=
treeAdj u v ∧ edgeLabel p.1 p.2.1 u v ≤ theta
def connected (theta : ℝ) (p : Input d) : TreeVertex d → TreeVertex d → Prop :=
Relation.ReflTransGen (openAdj theta p)
def factor (theta : ℝ) (p : Input d) : TreeVertex d → IsingSpin :=
FiniteCluster.color (connected theta p) p.1 (fun v => fairSpin (p.2.2 v))
@[simp] lemma factor_apply (theta : ℝ) (p : Input d) (v : TreeVertex d) :
factor theta p v =
fairSpin (p.2.2 (FiniteCluster.selector (connected theta p) p.1 v)) := rfl
lemma openAdj_edge_iff (theta : ℝ) (p : Input d) (e : TreeEdge d) :
openAdj theta p e.1.1 e.1.2 ↔ edgeLabels p.1 p.2.1 e ≤ theta := by
exact and_iff_right e.2.1
lemma measurable_relabelInput (g : TreeAut d) :
Measurable (relabelInput g) := by
unfold relabelInput relabel
fun_prop
lemma openAdj_sub_treeAdj (theta : ℝ) (p : Input d) {u v : TreeVertex d}
(h : openAdj theta p u v) : treeAdj u v := h.1
lemma openAdj_symmetric (theta : ℝ) {p : Input d} (hp : Function.Injective p.1) :
∀ ⦃u v : TreeVertex d⦄, openAdj theta p u v → openAdj theta p v u := by
intro u v huv
refine ⟨adj_symm huv.1, ?_⟩
rw [edgeLabel_swap hp p.2.1 huv.1]
exact huv.2
lemma connected_equivalence (theta : ℝ) {p : Input d}
(hp : Function.Injective p.1) : Equivalence (connected theta p) := by
refine ⟨fun _ => .refl, ?_, fun h h' => h.trans h'⟩
intro u v huv
have hsymm := openAdj_symmetric theta hp
apply Relation.reflTransGen_swap.mp
exact Relation.ReflTransGen.mono (fun _ _ h => hsymm h) _ _ huv
lemma openAdj_relabel (g : TreeAut d) (theta : ℝ) (p : Input d)
(u v : TreeVertex d) :
openAdj theta (relabelInput g p) (g.toEquiv u) (g.toEquiv v) ↔
openAdj theta p u v := by
simp only [openAdj, relabelInput, g.map_adj, edgeLabel_relabel]
lemma connected_relabel (g : TreeAut d) (theta : ℝ) (p : Input d)
(u v : TreeVertex d) :
connected theta (relabelInput g p) (g.toEquiv u) (g.toEquiv v) ↔
connected theta p u v := by
exact reflTransGen_equiv_iff g.toEquiv (openAdj_relabel g theta p) u v
lemma connected_relabel_eq (g : TreeAut d) (theta : ℝ) (p : Input d) :
connected theta (relabelInput g p) =
fun u v => connected theta p (g.toEquiv.symm u) (g.toEquiv.symm v) := by
funext u v
apply propext
simpa only [Equiv.apply_symm_apply] using
connected_relabel g theta p (g.toEquiv.symm u) (g.toEquiv.symm v)
theorem factor_relabel (g : TreeAut d) (theta : ℝ) (p : Input d) :
factor theta (relabelInput g p) = relabel g (factor theta p) := by
funext v
unfold factor
rw [connected_relabel_eq]
have h := FiniteCluster.color_reindex g.toEquiv (connected theta p) p.1
(fun w => fairSpin (p.2.2 w)) (g.toEquiv.symm v)
change FiniteCluster.color
(fun x y => connected theta p (g.toEquiv.symm x) (g.toEquiv.symm y))
(fun x => p.1 (g.toEquiv.symm x))
(fun x => fairSpin (p.2.2 (g.toEquiv.symm x))) v =
FiniteCluster.color (connected theta p) p.1
(fun x => fairSpin (p.2.2 x)) (g.toEquiv.symm v)
simpa only [Equiv.apply_symm_apply, Function.comp_def] using h
theorem factor_eq_of_connected (theta : ℝ) {p : Input d}
(hp : Function.Injective p.1) {u v : TreeVertex d}
(hu : {w | connected theta p u w}.Finite) (huv : connected theta p u v) :
factor theta p u = factor theta p v :=
FiniteCluster.color_eq_of_related _ (connected_equivalence theta hp) huv hu hp
theorem factor_eq_at_minimum (theta : ℝ) (p : Input d) (v a : TreeVertex d)
(ha : FiniteCluster.IsMinimum (connected theta p) p.1 v a) :
factor theta p v = fairSpin (p.2.2 a) := by
unfold factor FiniteCluster.color
rw [FiniteCluster.selector_eq_of_minimum ha]
lemma measurableSet_openAdj (theta : ℝ) (u v : TreeVertex d) :
MeasurableSet {p : Input d | openAdj theta p u v} := by
let : MeasurableSpace (TreeVertex d) := ⊤
let : MeasurableSingletonClass (TreeVertex d) := ⟨fun _ => trivial⟩
have hlabel : Measurable (fun p : Input d => edgeLabel p.1 p.2.1 u v) :=
(measurable_edgeLabel u v).comp
(measurable_fst.prodMk (measurable_fst.comp measurable_snd))
exact (MeasurableSet.const (treeAdj u v)).inter
(measurableSet_le hlabel measurable_const)
lemma measurableSet_connected (theta : ℝ) (u v : TreeVertex d) :
MeasurableSet {p : Input d | connected theta p u v} :=
measurableSet_reflTransGen (measurableSet_openAdj theta) u v
theorem measurable_factor (theta : ℝ) : Measurable (@factor d theta) := by
apply FiniteCluster.measurable_color
(connected theta) (fun p : Input d => p.1)
(fun p : Input d => fun v => fairSpin (p.2.2 v))
· exact measurableSet_connected theta
· intro v
exact (measurable_pi_apply v).comp measurable_fst
· intro v
exact measurable_fairSpin.comp
((measurable_pi_apply v).comp (measurable_snd.comp measurable_snd))
end Problem367.ClusterFactorMap
end
end OAI
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