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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 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Delta
import OAI.Probability.CriticalPercolation.Exploration.QueryTree
import OAI.Probability.CriticalPercolation.Modular.PushMass
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open scoped Classical
open Real
open Set Real
open scoped Topology
open scoped BigOperators
open Set
namespace CriticalPercolation.QueryTree
open FiniteProduct FiniteEntropy
variable {E : Type*} [Fintype E]
noncomputable def bitWeight (p : ℝ) (_ : E) (b : Bool) : ℝ := if b then p else 1-p
omit [Fintype E] in
lemma bitWeight_sum (p : ℝ) (e : E) : ∑ b, bitWeight p e b = 1 := by
simp [bitWeight]
omit [Fintype E] in
lemma bitWeight_nonneg {p : ℝ} (hp : p ∈ Set.Icc 0 1) (e : E) (b : Bool) :
0 ≤ bitWeight p e b := by cases b <;> simp [bitWeight, hp.1, hp.2]
lemma expect_indicator (p : ℝ) {Y : Type*} (f : (E → Bool) → Y) (y : Y) :
expect (bitWeight p) (fun ω => if f ω = y then 1 else 0) =
pushMass f (density (bitWeight p)) y := by
classical
simp only [expect, pushMass, mul_ite, mul_one, mul_zero]
lemma reference_eq_product_push (p : ℝ) (t : QueryTree E)
(used : Finset E) (hv : t.Valid used) :
pushMass t.run (density (bitWeight p)) = t.reference p := by
classical
induction t generalizing used with
| stop =>
funext x
have : x = PUnit.unit := Subsingleton.elim _ _
subst x
simp only [pushMass, run, reference]
exact sum_density _ (bitWeight_sum p)
| ask e left right hl hr =>
have hi (b : Bool) : DependsOn (fun ω : E → Bool => if ω e = b then (1:ℝ) else 0)
({e} : Set E) := by
intro ω ω' hω
dsimp only
rw [hω e (Set.mem_singleton e)]
have hc (u : QueryTree E) (hu : u.Valid (insert e used)) (x : u.Leaf) :
DependsOn (fun ω => if u.run ω = x then (1:ℝ) else 0) ({e} : Set E)ᶜ := by
intro ω ω' hω
have heq : u.run ω = u.run ω' := run_dependsOn u _ hu (fun j hj =>
hω j (by intro hje; have hje : j = e := hje; subst j
exact hj (Finset.mem_insert_self _ _)))
dsimp only
rw [heq]
funext x
cases x with
| inl x =>
have heq : (fun ω => if (ask e left right).run ω = Sum.inl x then (1:ℝ) else 0) =
(fun ω => (if ω e = false then (1:ℝ) else 0) *
(if left.run ω = x then 1 else 0)) := by
funext ω
cases he : ω e <;> simp [run, he]
have hm : pushMass (ask e left right).run (density (bitWeight p)) (Sum.inl x) =
expect (bitWeight p) (fun ω => if (ask e left right).run ω = Sum.inl x then 1 else 0) := by
simp only [pushMass, expect, mul_ite, mul_one, mul_zero]
apply Finset.sum_congr rfl
intro ω _
split_ifs <;> simp_all
rw [hm, heq, expect_indep _ (bitWeight_sum p) _ (hi false)
(hc left hv.2.1 x)]
have hev : expect (bitWeight p) (fun ω => if ω e = false then 1 else 0) =
bitWeight p e false := by
convert expect_eval_indicator (bitWeight p) (bitWeight_sum p) e false using 1
congr 1
funext ω
split_ifs <;> simp_all
rw [hev]
have hm' : expect (bitWeight p) (fun ω => if left.run ω = x then 1 else 0) =
pushMass left.run (density (bitWeight p)) x := by
simp only [pushMass, expect, mul_ite, mul_one, mul_zero]
rw [hm', hl _ hv.2.1]
rfl
| inr x =>
have heq : (fun ω => if (ask e left right).run ω = Sum.inr x then (1:ℝ) else 0) =
(fun ω => (if ω e = true then (1:ℝ) else 0) *
(if right.run ω = x then 1 else 0)) := by
funext ω
cases he : ω e <;> simp [run, he]
have hm : pushMass (ask e left right).run (density (bitWeight p)) (Sum.inr x) =
expect (bitWeight p) (fun ω => if (ask e left right).run ω = Sum.inr x then 1 else 0) := by
simp only [pushMass, expect, mul_ite, mul_one, mul_zero]
apply Finset.sum_congr rfl
intro ω _
split_ifs <;> simp_all
rw [hm, heq, expect_indep _ (bitWeight_sum p) _ (hi true)
(hc right hv.2.2 x)]
have hev : expect (bitWeight p) (fun ω => if ω e = true then 1 else 0) =
bitWeight p e true := by
convert expect_eval_indicator (bitWeight p) (bitWeight_sum p) e true using 1
congr 1
funext ω
split_ifs <;> simp_all
rw [hev]
have hm' : expect (bitWeight p) (fun ω => if right.run ω = x then 1 else 0) =
pushMass right.run (density (bitWeight p)) x := by
simp only [pushMass, expect, mul_ite, mul_one, mul_zero]
rw [hm', hr _ hv.2.2]
rfl
end CriticalPercolation.QueryTree
end
end
end OAI
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