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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.SpinGlass.FiniteProduct
import OAI.Probability.SKBarriers.SpinGlass.FiniteIndicators
namespace OAI
section
noncomputable section
open scoped BigOperators
open Classical
namespace SK.Analytic.FiniteLaw
variable {X Y I J : Type*} [Fintype X] [Fintype Y] [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J]
theorem expect_expect_comm (P : FiniteLaw X) (Q : FiniteLaw Y) (f : X → Y → ℝ) :
P.expect (fun x => Q.expect (f x))=Q.expect (fun y => P.expect (fun x => f x y)) := by
unfold expect
simp only [Finset.mul_sum]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro y _
apply Finset.sum_congr rfl
intro x _
ring
theorem iid_expect_sum (P : FiniteLaw X) (f : (I ⊕ J → X) → ℝ) :
(P.iid (I ⊕ J)).expect f=(P.iid I).expect (fun z => (P.iid J).expect (fun w => f (Sum.elim z w))) := by
classical
unfold expect iid pi
calc
_ = ∑ z : (I → X) × (J → X), (∏ i, P.weight (z.1 i))*(∏ j, P.weight (z.2 j))*f (Sum.elim z.1 z.2) := by
apply Fintype.sum_equiv (Equiv.sumArrowEquivProdArrow I J X)
intro z
simp only [Equiv.sumArrowEquivProdArrow_apply_fst,Equiv.sumArrowEquivProdArrow_apply_snd,Fintype.prod_sum_type]
congr 1
congr 1
funext i
cases i <;> rfl
_ = _ := by simp only [Fintype.sum_prod_type,Finset.mul_sum,mul_assoc]
theorem iid_miss_bound (P : FiniteLaw X) (A : X → Prop) (K : ℕ) {a : ℝ}
(_ha : 0 ≤ a) (hA : a ≤ P.prob A) :
(P.iid (Fin K)).prob (fun z => ∀ i, ¬A (z i)) ≤ Real.exp (-a*K) := by
have H := P.iid_predictable_bound (fun _ _ x => ¬A x) (fun _ => 1-P.prob A)
(fun _ => sub_nonneg.mpr (P.prob_le_one A)) (fun _ _ => (P.prob_not A).le) K
have H' : (P.iid (Fin K)).prob (fun z => ∀ i, ¬A (z i)) ≤ (1-P.prob A)^K := by
simpa using H
refine H'.trans ?_
have hbase : 1-P.prob A ≤ Real.exp (-a) := by
have H := Real.add_one_le_exp (-a)
linarith only [H,hA]
calc
(1-P.prob A)^K ≤ (Real.exp (-a))^K := pow_le_pow_left₀ (sub_nonneg.mpr (P.prob_le_one A)) hbase K
_ = Real.exp (-a*K) := by rw [← Real.exp_nat_mul]; congr 1; ring
theorem iid_cell_miss_bound (P : FiniteLaw X) (S : X → Prop) (R : X → X → Prop)
(K : ℕ) {a δ : ℝ} (ha : 0 ≤ a)
(hcov : P.prob (fun x => S x ∧ P.prob (fun y => S y ∧ R x y)<a) ≤ δ) :
(P.iid (Fin K)).expect (fun z => P.prob (fun x => S x ∧ ∀ i, ¬(S (z i) ∧ R x (z i)))) ≤
δ+Real.exp (-a*K) := by
classical
unfold prob
rw [expect_expect_comm]
calc
_ ≤ P.expect (fun x => (if S x ∧ P.prob (fun y => S y ∧ R x y)<a then 1 else 0)+Real.exp (-a*K)) := by
apply expect_mono
intro x
by_cases hb : S x ∧ P.prob (fun y => S y ∧ R x y)<a
· rw [ite_eq_left hb]
exact (prob_le_one _ _).trans (by linarith only [Real.exp_pos (-a*K)])
· rw [ite_eq_right hb,zero_add]
by_cases hx : S x
· have hmass : a ≤ P.prob (fun y => S y ∧ R x y) := le_of_not_gt (fun h => hb ⟨hx,h⟩)
simp only [hx,true_and]
rw [expect_indicator_eq_prob]
exact P.iid_miss_bound (fun y => S y ∧ R x y) K ha hmass
· simp only [hx,false_and,ite_false,expect_zero,Real.exp_nonneg]
_ = P.prob (fun x => S x ∧ P.prob (fun y => S y ∧ R x y)<a)+Real.exp (-a*K) := by
rw [expect_add,expect_const,expect_indicator_eq_prob]
_ ≤ _ := by linarith only [hcov]
theorem exists_le_expect (P : FiniteLaw X) (f : X → ℝ) : ∃ x, f x ≤ P.expect f := by
classical
by_contra! H
have he : P.expect f < P.expect f := by
calc
P.expect f = ∑ x, P.weight x*P.expect f := by simp only [← Finset.sum_mul,P.sum_one,one_mul]
_ < ∑ x, P.weight x*f x := by
apply Finset.sum_lt_sum
· intro x _
exact mul_le_mul_of_nonneg_left (H x).le (P.nonneg x)
· have hp : ∃ x, 0 < P.weight x := by
by_contra! hp
have hz : ∑ x, P.weight x=0 := Finset.sum_eq_zero (fun x _ => le_antisymm (hp x) (P.nonneg x))
rw [P.sum_one] at hz
norm_num at hz
obtain ⟨x,hx⟩ := hp
exact ⟨x,Finset.mem_univ x,mul_lt_mul_of_pos_left (H x) hx⟩
_ = P.expect f := rfl
exact lt_irrefl _ he
end SK.Analytic.FiniteLaw
end
end
end OAI
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