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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.PrefixScoreNonneg
import OAI.Probability.CriticalPercolation.Entropy.BinaryWords
import OAI.Probability.CriticalPercolation.Entropy.BinaryMarks
namespace OAI
section
section
namespace CriticalPercolation.BinaryLaw
open scoped Classical
open FiniteProduct BondGraph BinaryInterpolation InterpolationForest FiniteEntropy ExactExposure
variable {V E : Type*}
abbrev FixedDescriptor (n : ℕ) := Fin n → Bool × Bool × Bool
lemma fixedDescriptor_card (n : ℕ) : Fintype.card (FixedDescriptor n) = 8^n := by
simp only [FixedDescriptor,Fintype.card_fun,Fintype.card_prod,Fintype.card_bool,Fintype.card_fin]
noncomputable def descriptor (G : BondGraph V E) (d : ℕ) (x : ℕ → V)
(n : ℕ) (h : History E n) (ω : Set E) : FixedDescriptor n := fun i =>
(sides n h i, decide (marks n h i ≠ none), decide (i.val+2 ∈ activeNodes G d x n h ω))
lemma descriptor_sides (G : BondGraph V E) (d : ℕ) (x : ℕ → V)
(n : ℕ) (h : History E n) (ω : Set E) (D : FixedDescriptor n)
(hD : descriptor G d x n h ω = D) : sides n h = fun i => (D i).1 := by
funext i
exact congrArg (fun D : FixedDescriptor n => (D i).1) hD
lemma sum_history_words [Fintype E] (n : ℕ) (f : History E n → ℝ) :
(∑ h, f h) = ∑ b : Fin n → Bool, ∑ m : Fin n → Option E, f (fromWords n b m) := by
rw [← (wordsEquiv (E := E) n).symm.sum_comp f,Fintype.sum_prod_type]
rfl
section Finite
variable [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
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)
noncomputable def jointMass (z : (E → Bool) × History E n) : ℝ :=
FiniteProduct.density (QueryTree.bitWeight p) z.1 *
density G order d x p w (seed G w (x 0) (x d) s) n z.2 (configuration z.1)
lemma jointMass_nonneg (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))
(z : (E → Bool) × History E n) : 0 ≤ jointMass G order d x p w s n z := by
exact mul_nonneg (FiniteProduct.density_nonneg (fun e b => (QueryTree.bitWeight_pos hp e b).le) _)
(density_nonneg G order d x p w _ hp h0 h1 (seed_nonneg G w _ _ s hδ) n z.2 _)
lemma jointMass_normalized (ho : LazyDFS.Exhaustive G order) (hd : 2 ≤ d)
(hp : p ∈ Set.Ioo 0 1) (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p)
{c : ℝ} (hc : 0<c)
(hconn : ∀ j k, j<k → k≤d → k-j<d →
c ≤ prob w (fun ω => (G.openGraph ω).Reachable (x j) (x k)))
(hδ : 0 < prob w (endpointEvent G (x 0) (x d) s)) :
∑ z, jointMass G order d x p w s n z = 1 := by
have hw : QueryTree.bitWeight p = fun e (b : Bool) => w e (b = true) := by
funext e b
cases b <;> simp [QueryTree.bitWeight,h0,h1]
rw [Fintype.sum_prod_type,Finset.sum_comm]
simp only [jointMass]
have ht := endpoint_density_normalized G order ho hd x hp w h0 h1 hc hconn s hδ n
rw [← ht]
apply Finset.sum_congr rfl
intro h _
have he := expect_configuration w
(fun η : Set E => density G order d x p w (seed G w (x 0) (x d) s) n h η)
rw [← hw] at he
refine Eq.trans ?_ he
unfold expect
apply Finset.sum_congr
· ext f; simp only [Finset.mem_univ]
· intro f _; rfl
noncomputable def descriptorMass (D : FixedDescriptor n) : ℝ :=
pushMass (fun z : (E → Bool) × History E n => descriptor G d x n z.2 (configuration z.1))
(jointMass G order d x p w s n) D
lemma descriptorMass_sum :
(∑ D, descriptorMass G order d x p w s n D) = ∑ z, jointMass G order d x p w s n z :=
pushMass_sum _ _
theorem exists_descriptor (hZ : ∑ z, jointMass G order d x p w s n z = 1) :
∃ D : FixedDescriptor n, (8:ℝ)^(- (n:ℤ)) ≤ descriptorMass G order d x p w s n D := by
obtain ⟨D,hD,hmax⟩ := Finset.exists_max_image (Finset.univ : Finset (FixedDescriptor n))
(descriptorMass G order d x p w s n) Finset.univ_nonempty
have hs : 1 ≤ (8:ℝ)^n * descriptorMass G order d x p w s n D := by
calc
1 = ∑ D', descriptorMass G order d x p w s n D' := (descriptorMass_sum G order d x p w s n).trans hZ |>.symm
_ ≤ ∑ _D' : FixedDescriptor n, descriptorMass G order d x p w s n D :=
Finset.sum_le_sum (fun D' hD' => hmax D' hD')
_ = _ := by simp only [Finset.sum_const,Finset.card_univ,fixedDescriptor_card,nsmul_eq_mul,Nat.cast_pow,Nat.cast_ofNat]
refine ⟨D,?_⟩
rw [zpow_neg, zpow_natCast]
rw [← one_div]
exact (div_le_iff₀ (pow_pos (by norm_num : (0:ℝ)<8) n)).mpr (by simpa only [mul_comm] using hs)
noncomputable def fixedMass (D : FixedDescriptor n) (z : (E → Bool) × (Fin n → Option E)) : ℝ :=
let h := fromWords n (fun i => (D i).1) z.2
if descriptor G d x n h (configuration z.1) = D then
jointMass G order d x p w s n (z.1,h) else 0
lemma fixedMass_sum (D : FixedDescriptor n) :
(∑ z, fixedMass G order d x p w s n D z) = descriptorMass G order d x p w s n D := by
unfold descriptorMass pushMass
conv_rhs => rw [Fintype.sum_prod_type]
simp_rw [sum_history_words]
rw [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro ω _
symm
apply (Finset.sum_eq_single (fun i => (D i).1) ?_ ?_).trans
· apply Finset.sum_congr rfl
intro m _
unfold fixedMass
dsimp only
split_ifs <;> rfl
· intro b _ hb
apply Finset.sum_eq_zero
intro m _
have hne : descriptor G d x n (fromWords n b m) (configuration ω) ≠ D := by
intro he
have hs := descriptor_sides G d x n _ _ D he
rw [sides_fromWords] at hs
exact hb hs
rw [ite_eq_right hne]
· simp
noncomputable def conditionedMass (D : FixedDescriptor n)
(z : (E → Bool) × (Fin n → Option E)) : ℝ :=
fixedMass G order d x p w s n D z / descriptorMass G order d x p w s n D
lemma conditionedMass_normalized (D : FixedDescriptor n)
(hD : 0 < descriptorMass G order d x p w s n D) :
∑ z, conditionedMass G order d x p w s n D z = 1 := by
simp only [conditionedMass,← Finset.sum_div,fixedMass_sum,div_self hD.ne']
lemma conditionedMass_nonneg (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))
(D : FixedDescriptor n) (hD : 0 < descriptorMass G order d x p w s n D)
(z : (E → Bool) × (Fin n → Option E)) : 0 ≤ conditionedMass G order d x p w s n D z := by
apply div_nonneg _ hD.le
unfold fixedMass
dsimp only
split_ifs
· exact jointMass_nonneg G order d x p w s n hp h0 h1 hδ _
· rfl
lemma conditionedMass_support (D : FixedDescriptor n)
(z : (E → Bool) × (Fin n → Option E))
(hz : conditionedMass G order d x p w s n D z ≠ 0) :
let h := fromWords n (fun i => (D i).1) z.2
descriptor G d x n h (configuration z.1) = D ∧
density G order d x p w (seed G w (x 0) (x d) s) n h (configuration z.1) ≠ 0 := by
have hf := (div_ne_zero_iff.mp hz).1
unfold fixedMass at hf
dsimp only at hf ⊢
split_ifs at hf with ht
· exact ⟨ht,(mul_ne_zero_iff.mp hf).2⟩
· exact (hf rfl).elim
end Finite
end CriticalPercolation.BinaryLaw
end
end
end OAI
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