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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Entropy.BinaryConfiguration
import OAI.Probability.CriticalPercolation.Entropy.BinaryCosts
namespace OAI
section
section
namespace CriticalPercolation.BinaryLaw
open scoped Classical
open FiniteProduct BondGraph BinaryInterpolation InterpolationForest ExactExposure FiniteEntropy
variable {V E : Type*}
abbrev stepLeft (d : ℕ) (x : ℕ → V) (n : ℕ) (D : FixedDescriptor n) (i : Fin n) : V :=
(descriptorNode (E := E) d x n D (i.val+2)).1
abbrev stepRight (d : ℕ) (x : ℕ → V) (n : ℕ) (D : FixedDescriptor n) (i : Fin n) : V :=
(descriptorNode (E := E) d x n D (descriptorParent (E := E) d x n D (i.val+2))).1
abbrev stepTargets (d : ℕ) (x : ℕ → V) (n : ℕ) (D : FixedDescriptor n) (i : Fin n) : Set V :=
scheduledTargets (E := E) d x n (fun j => (D j).1) i.val
def activeSteps (n : ℕ) (D : FixedDescriptor n) : Finset (Fin n) :=
Finset.univ.filter fun i => (D i).2.2 = true
variable [Fintype V] [Fintype E]
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 pairQuery (i : Fin n) : QueryTree E :=
(pair G (stepLeft (E := E) d x n D i) (stepRight (E := E) d x n D i)).bare
noncomputable def selectedTranscript (i : Fin n) (ω : E → Bool) : E → Bool :=
if (D i).2.2 = true then
representative (pairQuery G d x n D i).run (fun _ => false) ((pairQuery G d x n D i).run ω)
else ω
noncomputable def selectedWeight (i : Fin n) (ω : E → Bool) (m : Option E) : ℝ :=
if (D i).2.2 = true then match m with
| none => 0
| some e => markWeight G order (stepTargets (E := E) d x n D i)
(stepLeft (E := E) d x n D i) (stepRight (E := E) d x n D i) p w ω e
else scheduleWeight G order d x p w n (fun j => (D j).1) i (configuration ω) m
noncomputable def selectedCost (i : Fin n) (ω : E → Bool) (m : Option E) : ℝ :=
if (D i).2.2 = true then match m with
| none => 0
| some e => markCost G order (stepTargets (E := E) d x n D i)
(stepLeft (E := E) d x n D i) (stepRight (E := E) d x n D i) p w ω e
else 0
lemma selected_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 ≤ selectedWeight G order d x p w n D i ω m ∧
0 ≤ selectedCost G order d x p w n D i ω m := by
unfold selectedWeight selectedCost
split_ifs
· cases m with
| none => exact ⟨le_rfl,le_rfl⟩
| some e => exact mark_nonneg G order _ _ _ hp w h0 h1 ω e
· exact ⟨scheduleWeight_nonneg G order d x p w hp h0 h1 n _ i _ m,le_rfl⟩
lemma selected_run (i : Fin n) (ω : E → Bool) (m : Option E) :
selectedWeight G order d x p w n D i (selectedTranscript G d x n D i ω) m =
selectedWeight G order d x p w n D i ω m ∧
selectedCost G order d x p w n D i (selectedTranscript G d x n D i ω) m =
selectedCost G order d x p w n D i ω m := by
unfold selectedWeight selectedCost selectedTranscript
split_ifs with hi
· cases m with
| none => exact ⟨rfl,rfl⟩
| some e =>
exact pair_mark_congr G order _ _ _ (scheduled_parent_mem d x n D i) p w _ ω
(representative_run (pairQuery G d x n D i).run (fun _ => false) ω) e
· exact ⟨rfl,rfl⟩
lemma selectedWeight_le_raw (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) :
selectedWeight G order d x p w n D i ω m ≤
scheduleWeight G order d x p w n (fun j => (D j).1) i (configuration ω) m := by
unfold selectedWeight
split_ifs
· cases m with
| none => exact scheduleWeight_nonneg G order d x p w hp h0 h1 n _ i _ none
| some e =>
unfold scheduleWeight
rw [scheduledAnchor_node]
exact G.markWeight_le_raw order _ _ _ (scheduled_parent_mem d x n D i) hp w h0 h1 ω e
· exact le_rfl
lemma supported_selected_eq (s : ℕ) (ω : E → Bool) (m : Fin n → Option E)
(hz : conditionedMass G order d x p w s n D (ω,m) ≠ 0) (i : Fin n) :
selectedWeight G order d x p w n D i ω (m i) =
scheduleWeight G order d x p w n (fun j => (D j).1) i (configuration ω) (m i) := by
unfold selectedWeight
split_ifs with hi
· obtain ⟨e,he,ha⟩ := supported_active_admissible G order d x p w s n D ω m hz ⟨i,hi⟩
dsimp only at he ha
rw [he]
unfold scheduleWeight
rw [scheduledAnchor_node]
exact G.markWeight_eq_raw_of_admissible order _ _ _ (scheduled_parent_mem d x n D i) p w ω e ha
· rfl
lemma supported_selected_positive (s : ℕ) (hp : p ∈ Set.Ioo 0 1)
(h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p)
(z : (E → Bool) × (Fin n → Option E))
(hz : 0 < conditionedMass G order d x p w s n D z) (i : Fin n) :
0 < selectedWeight G order d x p w n D i (selectedTranscript G d x n D i z.1) (z.2 i) := by
rw [(selected_run G order d x p w n D i z.1 (z.2 i)).1,
supported_selected_eq G order d x p w n D s z.1 z.2 hz.ne' i]
exact lt_of_le_of_ne (scheduleWeight_nonneg G order d x p w hp h0 h1 n _ i _ _)
(Ne.symm (supported_schedule_ne_zero G order d x p w s n D z.1 z.2 hz.ne' i))
lemma selected_moment (horder : LazyDFS.GoodDarts G order) (ho : LazyDFS.Exhaustive G order)
(hp : p ∈ Set.Ioo 0 1) (h0 : ∀ e, w e False = 1-p) (h1 : ∀ e, w e True = p)
(i : Fin n) (hi : i ∈ activeSteps n D) :
(∑ ym : (E → Bool) × Option E,
pushMass (selectedTranscript G d x n D i) (FiniteProduct.density (QueryTree.bitWeight p)) ym.1 *
selectedWeight G order d x p w n D i ym.1 ym.2 *
(1+selectedCost G order d x p w n D i ym.1 ym.2)) ≤ 4 := by
have hi' : (D i).2.2 = true := (Finset.mem_filter.mp hi).2
rw [Fintype.sum_prod_type]
simp_rw [mul_assoc]
simp_rw [← Finset.mul_sum]
rw [sum_pushMass_mul]
simp_rw [(selected_run G order d x p w n D i _ _).1,
(selected_run G order d x p w n D i _ _).2]
simp only [selectedWeight,selectedCost,ite_eq_left hi',Fintype.sum_option,zero_mul,zero_add]
exact marked_pair_bound G order horder ho _ _ _ (scheduled_parent_mem d x n D i) hp w h0 h1
end CriticalPercolation.BinaryLaw
end
end
end OAI
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