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f205175 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 | import Mathlib
import OAI.Probability.CriticalPercolation.Entropy.BinaryParents
namespace OAI
section
section
namespace CriticalPercolation.BinaryLaw
open scoped Classical
open FiniteProduct BondGraph BinaryInterpolation InterpolationForest ExactExposure FiniteEntropy
variable {V E I : Type*} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] [Fintype I]
variable (G : BondGraph V E) (order : List (LazyDFS.Dart V E)) (d : ℕ) (x : ℕ → V)
variable (p : ℝ) (w : E → Prop → ℝ) (s n : ℕ) (h : History E n) (ω : E → Bool)
theorem active_cluster_users_le
(hr : density G order d x p w (seed G w (x 0) (x d) s) n h (configuration ω) ≠ 0)
(idx : I → ℕ) (hinj : Function.Injective idx)
(hidx : ∀ i, idx i ∈ activeNodes G d x n h (configuration ω)) (v : V) :
(clusterUsers G (fun i => (node d x n h (idx i)).1)
(fun i => (node d x n h (nodeParent G d x (configuration ω) n h (idx i))).1) ω v).card ≤ 2 := by
classical
let F := historyForest G d x (configuration ω) n h
let S := activeNodes G d x n h (configuration ω)
let users := clusterUsers G (fun i => (node d x n h (idx i)).1)
(fun i => (node d x n h (F.parent (idx i))).1) ω v
have hb (i : I) : 2 ≤ idx i ∧ idx i<n+2 :=
⟨(Forest.mem_active _).mp (hidx i) |>.2,active_lt G d x n h _ _ (hidx i)⟩
have hcan (i : I) : Canonical G (anchors d x n h) (configuration ω) (idx i) :=
active_canonical G order d x p w _ n h _ hr _ (hidx i)
have hpcan (i : I) : Canonical G (anchors d x n h) (configuration ω) (F.parent (idx i)) :=
nodeParent_canonical G order d x p w _ n h _ hr _ (hb i).1 (hb i).2
have hpbound (i : I) : F.parent (idx i)<n+2 := (F.parent_lt _ (hb i).1).trans (hb i).2
by_cases hu : users.Nonempty
· obtain ⟨i,hi⟩ := hu
have hi' := hi
simp only [users,clusterUsers,Finset.mem_filter,Finset.mem_univ,true_and] at hi'
obtain ⟨j,hj,hjc,hjv⟩ : ∃ j<n+2, Canonical G (anchors d x n h) (configuration ω) j ∧
(G.openGraph (configuration ω)).Reachable (node d x n h j).1 v := by
rcases hi' with hi'|hi'
· exact ⟨idx i,(hb i).2,hcan i,hi'⟩
· exact ⟨F.parent (idx i),hpbound i,hpcan i,hi'⟩
have hm : Set.MapsTo idx (↑users : Set I) (↑(S.filter fun k => k=j ∨ F.parent k=j) : Set ℕ) := by
intro k hk
refine Finset.mem_filter.mpr ⟨hidx k,?_⟩
simp only [users,clusterUsers,Finset.mem_coe,Finset.mem_filter,Finset.mem_univ,true_and] at hk
rcases hk with hk|hk
· exact Or.inl (canonical_node_reachable_eq G d x n h (configuration ω) _ j
(hb k).2 hj (hcan k) hjc (hk.trans hjv.symm))
· exact Or.inr (canonical_node_reachable_eq G d x n h (configuration ω) _ j
(hpbound k) hj (hpcan k) hjc (hk.trans hjv.symm))
have hc : users.card ≤ (S.filter fun k => k=j ∨ F.parent k=j).card :=
Finset.card_le_card_of_injOn idx hm hinj.injOn
have ht := terminal_roots G order d x p w s n h (configuration ω) hr
have hne : F.root (owner G d x n h (configuration ω) (x (interval d n h).1)) ≠
F.root (owner G d x n h (configuration ω) (x (interval d n h).2)) := by
change (historyForest G d x (configuration ω) n h).root _ ≠ _
rw [ht.1,ht.2]
omega
exact hc.trans (F.active_pair_multiplicity hne j)
· have he : users=∅ := Finset.not_nonempty_iff_eq_empty.mp hu
change users.card ≤ 2
rw [he,Finset.card_empty]
omega
abbrev ActiveIndex (n : ℕ) (D : FixedDescriptor n) := {i : Fin n // (D i).2.2 = true}
def activeLeft (d : ℕ) (x : ℕ → V) (n : ℕ) (D : FixedDescriptor n) (i : ActiveIndex n D) : V :=
(descriptorNode (E := E) d x n D (i.val.val+2)).1
def activeRight (d : ℕ) (x : ℕ → V) (n : ℕ) (D : FixedDescriptor n) (i : ActiveIndex n D) : V :=
(descriptorNode (E := E) d x n D (descriptorParent (E := E) d x n D (i.val.val+2))).1
omit [Fintype I] in
lemma fixed_cluster_users_le (D : FixedDescriptor n) (m : Fin n → Option E)
(hz : conditionedMass G order d x p w s n D (ω,m) ≠ 0) (v : V) :
(clusterUsers G (activeLeft (E := E) d x n D) (activeRight (E := E) d x n D) ω v).card ≤ 2 := by
classical
obtain ⟨hD,hr⟩ := conditionedMass_support G order d x p w s n D (ω,m) hz
let h := fromWords (E := E) n (fun j => (D j).1) m
have hi (i : ActiveIndex n D) : i.val.val+2 ∈ activeNodes G d x n h (configuration ω) := by
rw [activeNodes_eq_descriptor G d x n h _ D hD]
exact Finset.mem_image.mpr ⟨i.val,Finset.mem_filter.mpr ⟨Finset.mem_univ _,i.property⟩,rfl⟩
have hinj : Function.Injective (fun i : ActiveIndex n D => i.val.val+2) := by
intro i j he
apply Subtype.ext
apply Fin.ext
dsimp only at he
omega
have he := active_cluster_users_le G order d x p w s n h ω hr _ hinj hi v
have hleft (i : ActiveIndex n D) : (node d x n h (i.val.val+2)).1 = activeLeft (E := E) d x n D i :=
congrArg Prod.fst (node_eq_descriptor G d x n h _ D hD _)
have hright (i : ActiveIndex n D) :
(node d x n h (nodeParent G d x (configuration ω) n h (i.val.val+2))).1 =
activeRight (E := E) d x n D i := by
rw [active_parent_eq_descriptor G order d x p w s n h _ hr D hD _ (hi i)]
exact congrArg Prod.fst (node_eq_descriptor G d x n h _ D hD _)
simp_rw [hleft,hright] at he
exact he
noncomputable section Entropy
local instance pairsClassicalFinDecidable (n : ℕ) : DecidableEq (Fin n) := Classical.decEq _
omit [Fintype I] in
lemma configurationMass_support (D : FixedDescriptor n) (η : E → Bool)
(hη : 0 < pushMass Prod.fst (conditionedMass G order d x p w s n D) η) :
∃ m, 0 < conditionedMass G order d x p w s n D (η,m) := by
by_contra! hn
have hz : pushMass Prod.fst (conditionedMass G order d x p w s n D) η ≤ 0 := by
unfold pushMass
apply Finset.sum_nonpos
intro z _
split_ifs with he
· rcases z with ⟨η',m⟩
dsimp only at he
subst η'
exact hn m
· rfl
linarith
omit [Fintype I] in
theorem conditioned_pair_entropy (D : FixedDescriptor n)
(hp : p ∈ Set.Ioo 0 1) (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p)
(hδ : 0 < prob w (endpointEvent G (x 0) (x d) s))
(hD : 0 < descriptorMass G order d x p w s n D) :
let P := pushMass Prod.fst (conditionedMass G order d x p w s n D)
(∑ i : ActiveIndex n D,
divergence (pushMass (pair G (activeLeft (E := E) d x n D i)
(activeRight (E := E) d x n D i)).bare.run P)
((pair G (activeLeft (E := E) d x n D i) (activeRight (E := E) d x n D i)).bare.reference p)) ≤
16/(2*p*(1-p)) * divergence P (FiniteProduct.density (QueryTree.bitWeight p)) := by
cases Subsingleton.elim (inferInstance : DecidableEq V) (Classical.decEq V)
cases Subsingleton.elim (inferInstance : DecidableEq E) (Classical.decEq E)
dsimp only
let Q := conditionedMass G order d x p w s n D
have hQ : ∀ z, 0 ≤ Q z := conditionedMass_nonneg G order d x p w s n hp h0 h1 hδ D hD
have hQ1 : ∑ z, Q z = 1 := by
have he := conditionedMass_normalized G order d x p w s n D hD
refine Eq.trans ?_ he
apply Finset.sum_congr
· ext z; simp only [Finset.mem_univ]
· intro z _; rfl
apply pair_entropy hp G (activeLeft (E := E) d x n D) (activeRight (E := E) d x n D)
(pushMass Prod.fst Q) (pushMass_nonneg _ _ hQ) (by rw [pushMass_sum,hQ1])
intro η hη v
obtain ⟨m,hm⟩ := configurationMass_support G order d x p w s n D η hη
exact fixed_cluster_users_le G order d x p w s n η D m hm.ne' v
end Entropy
end CriticalPercolation.BinaryLaw
end
end
end OAI
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