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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 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.ProbOr
import OAI.Probability.CriticalPercolation.ProductMeasure.Potential
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
open scoped Classical
namespace CriticalPercolation.FiniteProduct
variable {E A : Type*} [Fintype E] [Fintype A]
lemma expect_eval_indicator (w : E → A → ℝ) (hw : ∀ e, ∑ a, w e a = 1)
(e : E) (b : A) : expect w (fun ω => if ω e = b then 1 else 0) = w e b := by
classical
have ht (ω : E → A) : density w ω * (if ω e = b then 1 else 0) =
∏ j, w j (ω j) * (if j = e then if ω j = b then 1 else 0 else 1) := by
rw [Finset.prod_mul_distrib]
simp [density]
simp only [expect, ht]
rw [← Fintype.prod_sum (fun j a => w j a * (if j = e then if a = b then 1 else 0 else 1))]
have hs (j : E) : (∑ a, w j a * (if j = e then if a = b then 1 else 0 else 1)) =
if j = e then w e b else 1 := by
split_ifs with he
· subst j; simp
· simp [hw]
simp only [hs]
simp
end CriticalPercolation.FiniteProduct
namespace CriticalPercolation.FiniteProduct
open scoped Classical
variable {E A : Type*} [Fintype E] [Fintype A]
lemma expect_coordinate_mask (w : E → A → ℝ) (hw : ∀ e, ∑ a, w e a = 1)
(e : E) (b : A) (f : (E → A) → ℝ) :
expect w (fun ω => if ω e = b then f ω else 0) =
w e b * expect w (fun ω => f (Function.update ω e b)) := by
have hid : (fun ω => if ω e = b then f ω else 0) =
(fun ω => (if ω e = b then (1:ℝ) else 0)*f (Function.update ω e b)) := by
funext ω
split_ifs with h
· have hu : Function.update ω e b = ω := by rw [← h]; exact Function.update_eq_self _ _
simp [hu]
· simp
rw [hid]
have hI : DependsOn (fun ω : E → A => if ω e = b then (1:ℝ) else 0) ({e} : Set E) := by
intro ω ω' heq
dsimp only
rw [heq e (Set.mem_singleton e)]
have hF : DependsOn (fun ω => f (Function.update ω e b)) ({e} : Set E)ᶜ := by
intro ω ω' heq
change f (Function.update ω e b) = f (Function.update ω' e b)
congr 1
funext j
by_cases hj : j = e
· subst j; simp
· simp only [Function.update_of_ne hj]
exact heq j hj
rw [expect_indep w hw _ hI hF]
congr 1
exact expect_eval_indicator w hw e b
end CriticalPercolation.FiniteProduct
namespace CriticalPercolation.FiniteProduct
open scoped Classical
variable {E : Type*} [Fintype E]
lemma expect_pin_eq_force (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(e : E) (b : Prop) (hb : w e b ≠ 0) (f : (E → Prop) → ℝ) :
expect (pin w e b) f = expect w (fun ω => f (Function.update ω e b)) := by
apply mul_left_cancel₀ hb
exact (expect_mask_pin w e b f).symm.trans (expect_coordinate_mask w hw e b f)
lemma expect_pin_transport (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(e : E) (a b : Prop) (ha : w e a ≠ 0) (hb : w e b ≠ 0) (f : (E → Prop) → ℝ) :
expect (pin w e b) f = expect (pin w e a) (fun ω => f (Function.update ω e b)) := by
rw [expect_pin_eq_force w hw e b hb,expect_pin_eq_force w hw e a ha]
simp only [Function.update_idem]
lemma prob_congr_support (w : E → Prop → ℝ) (P Q : Set E → Prop)
(h : ∀ ω, density w ω ≠ 0 → (P ω ↔ Q ω)) : prob w P = prob w Q := by
apply expect_congr_support
intro ω hω
rw [h ω hω]
lemma prob_pin_transport (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(e : E) (a b : Prop) (ha : w e a ≠ 0) (hb : w e b ≠ 0) (P : Set E → Prop) :
prob (pin w e b) P = prob (pin w e a) (fun ω => P (Function.update ω e b)) := by
exact expect_pin_transport w hw e a b ha hb (fun ω => if P ω then 1 else 0)
noncomputable def delta (w : E → Prop → ℝ) (F : Set E → Prop) (e : E) : ℝ :=
prob (pin w e True) F - prob (pin w e False) F
def openPivot (F : Set E → Prop) (e : E) (ω : Set E) : Prop :=
ω e ∧ F ω ∧ ¬F (Function.update ω e False)
omit [Fintype E] in
lemma openPivot_force_true (F : Set E → Prop) (e : E) (ω : Set E) :
openPivot F e (Function.update ω e True) ↔
F (Function.update ω e True) ∧ ¬F (Function.update ω e False) := by
unfold openPivot
erw [Function.update_self,Function.update_idem]
simp only [true_and]
lemma delta_eq_pivotal (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(e : E) (he0 : w e False ≠ 0) (he1 : w e True ≠ 0)
(F : Set E → Prop) (hF : ∀ s t, s ⊆ t → F s → F t) :
delta w F e = prob (pin w e False)
(fun ω => F (Function.update ω e True) ∧ ¬F ω) := by
rw [delta,prob_pin_transport w hw e False True he0 he1]
have hh := prob_diff (pin w e False)
(fun ω => F (Function.update ω e True)) F
have he : prob (pin w e False) (fun ω => F (Function.update ω e True) ∧ F ω) =
prob (pin w e False) F := by
apply prob_congr
intro ω
constructor
· exact And.right
· intro h
refine ⟨hF _ _ ?_ h,h⟩
intro j hj
change Function.update (ω : Set E) e True j
by_cases hje : j = e
· subst j; erw [Function.update_self]; trivial
· erw [Function.update_of_ne hje]; exact hj
erw [he] at hh
exact hh.symm
end CriticalPercolation.FiniteProduct
end
end
end OAI
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