| 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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|
