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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 | import Mathlib
import OAI.Probability.SKBarriers.SpinGlass.FiniteLaw
import OAI.Probability.SKBarriers.SpinGlass.FiniteSampling
namespace OAI
section
noncomputable section
open scoped BigOperators
open Classical
namespace SK.Analytic
namespace FiniteLaw
variable {X I A : Type*} [Fintype X] [Fintype I] [Fintype A] [DecidableEq I] [DecidableEq A]
theorem iid_random_cell_miss (P : FiniteLaw X) (K : ℕ) (u : A → I) (v : Fin K → I)
(he : Function.Injective (Sum.elim u v))
(C : (A → X) → X → Prop) (R : X → X → Prop) {a δ : ℝ} (ha : 0≤a)
(hcov : ∀ w, P.prob (fun x => C w x ∧ P.prob (fun y => C w y ∧ R x y)<a) ≤ δ) :
(P.iid I).expect (fun z => P.prob (fun x => C (z ∘ u) x ∧
∀ k, ¬(C (z ∘ u) (z (v k)) ∧ R x (z (v k))))) ≤ δ+Real.exp (-a*K) := by
classical
let F (w : A ⊕ Fin K → X) : ℝ := P.prob (fun x => C (w ∘ Sum.inl) x ∧
∀ k, ¬(C (w ∘ Sum.inl) (w (Sum.inr k)) ∧ R x (w (Sum.inr k))))
have H := P.iid_expect_injective (Sum.elim u v) he F
change (P.iid I).expect (fun z => F (z ∘ Sum.elim u v)) ≤ _
rw [H,P.iid_expect_sum F]
calc
_ ≤ (P.iid A).expect (fun _ => δ+Real.exp (-a*K)) := by
apply expect_mono
intro w
exact P.iid_cell_miss_bound (C w) R K ha (hcov w)
_ = _ := expect_const _ _
end FiniteLaw
abbrev BankIndex {p : ℕ} (K : Fin p → ℕ) := (j : Fin p) × Fin (K j)
abbrev BankPrior {p : ℕ} (pred : Fin p → Finset (Fin p)) (j : Fin p) := {i : Fin p // i∈pred j}
abbrev BankTuple {p : ℕ} (K : Fin p → ℕ) (pred : Fin p → Finset (Fin p)) (j : Fin p) :=
(i : BankPrior pred j) → Fin (K i.val)
def bankJointIndex {p : ℕ} (K : Fin p → ℕ) (pred : Fin p → Finset (Fin p)) (j : Fin p)
(T : BankTuple K pred j) : BankPrior pred j ⊕ Fin (K j) → BankIndex K :=
Sum.elim (fun i => ⟨i.val,T i⟩) (fun a => ⟨j,a⟩)
theorem bankJointIndex_injective {p : ℕ} (K : Fin p → ℕ) (pred : Fin p → Finset (Fin p))
(j : Fin p) (hj : j∉pred j) (T : BankTuple K pred j) :
Function.Injective (bankJointIndex K pred j T) := by
intro a b H
cases a with
| inl i =>
cases b with
| inl k =>
have he : i.val=k.val := congrArg Sigma.fst H
exact congrArg Sum.inl (Subtype.ext he)
| inr k =>
have he : i.val=j := congrArg Sigma.fst H
apply False.elim; apply hj; simpa only [he] using i.property
| inr i =>
cases b with
| inl k =>
have he : j=k.val := congrArg Sigma.fst H
apply False.elim; apply hj; simpa only [← he] using k.property
| inr k =>
have he : HEq i k := (Sigma.mk.inj_iff.mp H).2
exact congrArg Sum.inr (eq_of_heq he)
def bankCell {n p : ℕ} (K : Fin p → ℕ) (pred : Fin p → Finset (Fin p)) (t : ℝ)
(z : BankIndex K → Config n) (j : Fin p) (T : BankTuple K pred j) (x : Config n) : Prop :=
∀ i : BankPrior pred j, |overlap (z ⟨i.val,T i⟩) x|≤3*t
def bankCoverageBad {n p : ℕ} (K : Fin p → ℕ) (pred : Fin p → Finset (Fin p)) (t q : ℝ)
(z : BankIndex K → Config n) (x : Config n) : Prop :=
∃ j : Fin p, ∃ T : BankTuple K pred j, bankCell K pred t z j T x ∧
∀ a : Fin (K j), ¬(bankCell K pred t z j T (z ⟨j,a⟩) ∧ q≤|overlap x (z ⟨j,a⟩)|)
theorem expected_bank_coverage_bad {n p : ℕ} (P : FiniteLaw (Config n)) (K : Fin p → ℕ)
(pred : Fin p → Finset (Fin p)) (hpred : ∀ j, j∉pred j) (t q : ℝ) (a δ : Fin p → ℝ)
(ha : ∀ j,0≤a j)
(hcov : ∀ j, ∀ w : BankPrior pred j → Config n,
P.prob (fun x => (∀i,|overlap (w i) x|≤3*t) ∧
P.prob (fun y => (∀i,|overlap (w i) y|≤3*t) ∧ q≤|overlap x y|)<a j)≤δ j) :
(P.iid (BankIndex K)).expect (fun z => P.prob (bankCoverageBad K pred t q z)) ≤
∑ j, Fintype.card (BankTuple K pred j)*(δ j+Real.exp (-(a j)*(K j))) := by
classical
calc
_ ≤ (P.iid (BankIndex K)).expect (fun z => ∑ j : Fin p, ∑ T : BankTuple K pred j,
P.prob (fun x => bankCell K pred t z j T x ∧
∀ a : Fin (K j), ¬(bankCell K pred t z j T (z ⟨j,a⟩) ∧ q≤|overlap x (z ⟨j,a⟩)|))) := by
apply FiniteLaw.expect_mono
intro z
refine (P.prob_exists _).trans ?_
apply Finset.sum_le_sum
intro j _
exact P.prob_exists _
_ = ∑ j : Fin p, ∑ T : BankTuple K pred j, (P.iid (BankIndex K)).expect (fun z =>
P.prob (fun x => bankCell K pred t z j T x ∧
∀ a : Fin (K j), ¬(bankCell K pred t z j T (z ⟨j,a⟩) ∧ q≤|overlap x (z ⟨j,a⟩)|))) := by
rw [FiniteLaw.expect_sum]
apply Finset.sum_congr rfl
intro j _
exact FiniteLaw.expect_sum _ _
_ ≤ ∑ j : Fin p, ∑ _T : BankTuple K pred j, (δ j+Real.exp (-(a j)*(K j))) := by
apply Finset.sum_le_sum
intro j _
apply Finset.sum_le_sum
intro T _
exact P.iid_random_cell_miss (I := BankIndex K) (A := BankPrior pred j) (K j) (fun i : BankPrior pred j => (⟨i.val,T i⟩ : BankIndex K)) (fun b => (⟨j,b⟩ : BankIndex K))
(bankJointIndex_injective K pred j (hpred j) T) (fun w x => ∀i,|overlap (w i) x|≤3*t)
(fun x y => q≤|overlap x y|) (ha j) (hcov j)
_ = _ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
end SK.Analytic
end
end
end OAI
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