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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Entropy.BinaryTranscript
import OAI.Probability.CriticalPercolation.Criticality.ReverseReference
import OAI.Probability.CriticalPercolation.Entropy.MarkedBits
namespace OAI
section
section
noncomputable section
namespace CriticalPercolation.BinaryLaw
open scoped Classical
open FiniteProduct BondGraph BinaryInterpolation ExactExposure FiniteEntropy MarkedBits SearchTree
variable {V E : Type*} [Fintype V] [Fintype E]
local instance kernelsFinDec (n : ℕ) : DecidableEq (Fin n) := Classical.decEq _
variable (G : BondGraph V E) (order : List (LazyDFS.Dart V E)) (d : ℕ) (x : ℕ → V)
variable (p : ℝ) (w : E → Prop → ℝ) (n : ℕ) (D : FixedDescriptor n)
noncomputable def pivotGuess (i : Fin n) (ω : E → Bool) (e : E) : ℝ :=
(LazyDFS.search G order (stepLeft (E := E) d x n D i) (stepTargets (E := E) d x n D i)).reverseGuess
p (fun σ => ∃ b ∈ stepTargets (E := E) d x n D i,
(G.openGraph σ).Reachable (stepLeft (E := E) d x n D i) b) w (configuration ω) 0 e
lemma pivotGuess_subprobability (hp : p ∈ Set.Ioo 0 1)
(h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) (i : Fin n) (ω : E → Bool) :
(∀ e, 0 ≤ pivotGuess G order d x p w n D i ω e) ∧
(∑ e, pivotGuess G order d x p w n D i ω e) ≤ 1 := by
apply SearchTree.reverseGuess_subprobability _ ∅ (LazyDFS.compile_valid G order _ _ _ _) hp
· rintro a b hab ⟨v,hv,hr⟩
exact ⟨v,hv,hr.mono (G.openGraph_mono hab)⟩
· exact homogeneous_normalized w p h0 h1
· exact fun e b => (homogeneous_positive w hp h0 h1 e b).le
· exact fun e _ => ⟨h0 e,h1 e⟩
noncomputable def inverseKernel (j : ℕ) (ω : E → Bool) (m : Option E) : ℝ :=
if hj : j < n then
if (⟨j,hj⟩ : Fin n) ∈ activeSteps n D then
match m with | none => 0 | some e => pivotGuess G order d x p w n D ⟨j,hj⟩ ω e
else if m = none then 1 else 0
else if m = none then 1 else 0
noncomputable def inverseUpdate (j : ℕ) (ω : E → Bool) (m : Option E) : E → Bool :=
if hj : j < n then
if (⟨j,hj⟩ : Fin n) ∈ activeSteps n D then
match m with | none => ω | some e => Function.update ω e false
else ω
else ω
lemma inverseKernel_nonneg (hp : p ∈ Set.Ioo 0 1)
(h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p)
(j : ℕ) (ω : E → Bool) (m : Option E) : 0 ≤ inverseKernel G order d x p w n D j ω m := by
unfold inverseKernel
split_ifs with hj hi hm hm
· cases m with
| none => exact le_rfl
| some e => exact (pivotGuess_subprobability G order d x p w n D hp h0 h1 _ ω).1 e
· exact zero_le_one
· exact le_rfl
· exact zero_le_one
· exact le_rfl
lemma inverseKernel_sum (hp : p ∈ Set.Ioo 0 1)
(h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p)
(j : ℕ) (ω : E → Bool) : (∑ m, inverseKernel G order d x p w n D j ω m) ≤ 1 := by
unfold inverseKernel
split_ifs with hj hi
· rw [Fintype.sum_option]
simp only [zero_add]
exact (pivotGuess_subprobability G order d x p w n D hp h0 h1 _ ω).2
· simp
· simp
noncomputable def contactTotal (i : Fin n) (ω : E → Bool) : ℝ :=
∑ m, scheduleWeight G order d x p w n (fun j => (D j).1) i (configuration ω) m
noncomputable def inactiveKernel (i : Fin n) (ω : E → Bool) (m : Option E) : ℝ :=
if i ∈ activeSteps n D then if m=none then 1 else 0
else if contactTotal G order d x p w n D i ω = 0 then if m=none then 1 else 0
else scheduleWeight G order d x p w n (fun j => (D j).1) i (configuration ω) m /
contactTotal G order d x p w n D i ω
lemma contactTotal_nonneg (hp : p ∈ Set.Ioo 0 1)
(h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) (i : Fin n) (ω : E → Bool) :
0 ≤ contactTotal G order d x p w n D i ω :=
Finset.sum_nonneg fun m _ => scheduleWeight_nonneg G order d x p w hp h0 h1 n _ i _ m
lemma inactiveKernel_nonneg (hp : p ∈ Set.Ioo 0 1)
(h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p)
(i : Fin n) (ω : E → Bool) (m : Option E) : 0 ≤ inactiveKernel G order d x p w n D i ω m := by
unfold inactiveKernel
split_ifs
· exact zero_le_one
· exact le_rfl
· exact zero_le_one
· exact le_rfl
· exact div_nonneg (scheduleWeight_nonneg G order d x p w hp h0 h1 n _ i _ m)
(contactTotal_nonneg G order d x p w n D hp h0 h1 i ω)
lemma inactiveKernel_sum (i : Fin n) (ω : E → Bool) :
(∑ m, inactiveKernel G order d x p w n D i ω m) = 1 := by
unfold inactiveKernel
split_ifs with hi ht
· simp
· simp
· rw [← Finset.sum_div]
exact div_self ht
noncomputable def inverseReference : ((E → Bool) × (Fin n → Option E) × (Fin n → Option E)) → ℝ :=
ReverseReference.joint (FiniteProduct.density (QueryTree.bitWeight p))
(inverseKernel G order d x p w n D) (inverseUpdate (E := E) n D)
(inactiveKernel G order d x p w n D) n
lemma inverseReference_nonneg (hp : p ∈ Set.Ioo 0 1)
(h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) (z) :
0 ≤ inverseReference G order d x p w n D z :=
ReverseReference.joint_nonneg _ (FiniteProduct.density_nonneg (QueryTree.bitWeight_nonneg ⟨hp.1.le,hp.2.le⟩))
_ _ (inverseKernel_nonneg G order d x p w n D hp h0 h1)
_ (inactiveKernel_nonneg G order d x p w n D hp h0 h1) n z
lemma inverseReference_marginal (hp : p ∈ Set.Ioo 0 1)
(h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p) (ω : E → Bool) :
pushMass Prod.fst (inverseReference G order d x p w n D) ω ≤
FiniteProduct.density (QueryTree.bitWeight p) ω := by
have he := ReverseReference.joint_marginal_le
(FiniteProduct.density (QueryTree.bitWeight p) : (E → Bool) → ℝ)
(FiniteProduct.density_nonneg (QueryTree.bitWeight_nonneg ⟨hp.1.le,hp.2.le⟩))
(inverseKernel G order d x p w n D) (inverseUpdate (E := E) n D)
(inverseKernel_nonneg G order d x p w n D hp h0 h1)
(inverseKernel_sum G order d x p w n D hp h0 h1)
(inactiveKernel G order d x p w n D) (inactiveKernel_sum G order d x p w n D) n ω
refine Eq.trans_le ?_ he
unfold pushMass
apply Finset.sum_congr
· ext z
simp only [Finset.mem_univ]
· intro z _
rfl
end CriticalPercolation.BinaryLaw
end
end
end
end OAI
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