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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 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.ExpectMulReal
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
namespace CriticalPercolation.FiniteProduct
open scoped Classical
variable {E : Type*} [Fintype E]
noncomputable def prob (w : E → Prop → ℝ) (P : Set E → Prop) : ℝ :=
expect w (fun ω => if P ω then 1 else 0)
lemma prob_nonneg (w : E → Prop → ℝ) (hp : ∀ e b, 0 ≤ w e b) (P : Set E → Prop) :
0 ≤ prob w P := expect_nonneg hp (fun _ => by split_ifs <;> norm_num)
lemma prob_mono (w : E → Prop → ℝ) (hp : ∀ e b, 0 ≤ w e b)
{P Q : Set E → Prop} (h : ∀ ω, P ω → Q ω) : prob w P ≤ prob w Q := by
apply expect_mono hp
intro ω
by_cases hP : P ω <;> by_cases hQ : Q ω <;> simp [hP, hQ]
exact (hQ (h ω hP)).elim
lemma prob_split (w : E → Prop → ℝ) (P Q : Set E → Prop) :
prob w P = prob w (fun ω => P ω ∧ Q ω) + prob w (fun ω => P ω ∧ ¬Q ω) := by
unfold prob
rw [← expect_add]
congr 1
funext ω
by_cases hp : P ω <;> by_cases hq : Q ω <;> simp [hp, hq]
lemma prob_congr (w : E → Prop → ℝ) {P Q : Set E → Prop}
(h : ∀ ω, P ω ↔ Q ω) : prob w P = prob w Q := by
congr 1
funext ω
exact propext (h ω)
lemma indicator_mask (P Q : Prop) :
(if P then (if Q then (1 : ℝ) else 0) else 0) = if P ∧ Q then 1 else 0 := by
by_cases hP : P <;> by_cases hQ : Q <;> simp [hP, hQ]
lemma indicator_mask_mul (P Q R : Prop) :
(if P then (if Q then (1 : ℝ) else 0)*(if R then 1 else 0) else 0) =
if P ∧ Q ∧ R then 1 else 0 := by
by_cases hP : P <;> by_cases hQ : Q <;> by_cases hR : R <;> simp [hP, hQ, hR]
lemma indicator_antitone {X : Type*} [Preorder X] {P : X → Prop}
(h : ∀ ⦃x y⦄, x ≤ y → P y → P x) :
Antitone (fun x => if P x then (1 : ℝ) else 0) := by
intro x y hxy
by_cases hx : P x <;> by_cases hy : P y <;> simp [hx, hy]
exact (hx (h hxy hy)).elim
lemma indicator_monotone {X : Type*} [Preorder X] {P : X → Prop}
(h : ∀ ⦃x y⦄, x ≤ y → P x → P y) :
Monotone (fun x => if P x then (1 : ℝ) else 0) := by
intro x y hxy
by_cases hx : P x <;> by_cases hy : P y <;> simp [hx, hy]
exact (hy (h hxy hx)).elim
lemma expect_mul_ge_antitone (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(hp : ∀ e b, 0 ≤ w e b) (f g : (E → Prop) → ℝ)
(hf : Antitone f) (hg : Antitone g) :
expect w f * expect w g ≤ expect w (fun ω => f ω*g ω) := by
have hh := expect_mul_ge_real w hw hp (fun ω => -f ω) (fun ω => -g ω) hf.neg hg.neg
have hn (f : (E → Prop) → ℝ) : expect w (fun ω => -f ω) = -expect w f := by
simp [expect, Finset.sum_neg_distrib]
simpa only [hn, neg_mul_neg] using hh
lemma prob_harris_decreasing (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(hp : ∀ e b, 0 ≤ w e b) (P Q : Set E → Prop)
(hP : ∀ ⦃ω ω'⦄, ω ≤ ω' → P ω' → P ω)
(hQ : ∀ ⦃ω ω'⦄, ω ≤ ω' → Q ω' → Q ω) :
prob w P * prob w Q ≤ prob w (fun ω => P ω ∧ Q ω) := by
have hh := expect_mul_ge_antitone w hw hp _ _ (indicator_antitone hP) (indicator_antitone hQ)
change expect w _ * expect w _ ≤ _
refine hh.trans_eq ?_
apply congrArg (expect w)
funext ω
by_cases hP : P ω <;> by_cases hQ : Q ω <;> simp [hP, hQ]
end CriticalPercolation.FiniteProduct
end
end
end OAI
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