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a2ee19a | 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 | import OAI.Probability.SATComputability.ReplacementEstimate
namespace OAI
namespace FixedClauseThreshold.Computability
open DilutedSpinGlass
open scoped BigOperators Classical
def testKills {n k : ℕ} (U : Finset (DeletionCandidate n))
(c : Fin k → SignedLiteral n) : Prop :=
∀ x ∈ U, ∀ l, literalFalse x (c l)
def batchKills {n k d : ℕ} (U : Finset (DeletionCandidate n))
(cs : Fin d → Fin k → SignedLiteral n) : Prop :=
∀ x ∈ U, ∃ j, ∀ l, literalFalse x (cs j l)
noncomputable def batchKillProbability {n : ℕ} [NeZero n]
(U : Finset (DeletionCandidate n)) (k d : ℕ) : ℝ :=
(FiniteLaw.uniform : FiniteLaw (Fin d → Fin k → SignedLiteral n)).expect
(fun cs => if batchKills U cs then 1 else 0)
theorem batchKillProbability_ge {n : ℕ} [NeZero n]
(U : Finset (DeletionCandidate n)) (k d : ℕ) :
1 - (1 - oneTestKill U k)^d ≤ batchKillProbability U k d := by
let f : (Fin k → SignedLiteral n) → ℝ := fun c => if ¬testKills U c then 1 else 0
have hf : (FiniteLaw.uniform : FiniteLaw (Fin k → SignedLiteral n)).expect f =
1 - oneTestKill U k := by
have he (c : Fin k → SignedLiteral n) :
f c = 1 - (if testKills U c then 1 else 0) := by
dsimp [f]
by_cases h : testKills U c <;> simp [h]
rw [show f = (fun c => 1 - (if testKills U c then 1 else 0)) from funext he]
rw [FiniteLaw.expect_sub, FiniteLaw.expect_const]
simp only [oneTestKill, testKills]
congr 1
apply FiniteLaw.expect_congr
intro c
split_ifs with h
· exact ite_eq_left h
· exact ite_eq_right h
have hp : (FiniteLaw.uniform : FiniteLaw (Fin d → Fin k → SignedLiteral n)).expect
(fun cs => ∏ j, f (cs j)) = (1 - oneTestKill U k)^d := by
rw [uniformFiniteLaw_pi]
calc
_ = ∏ _ : Fin d, (FiniteLaw.uniform : FiniteLaw (Fin k → SignedLiteral n)).expect f :=
FiniteLaw.expect_pi_product (fun _ => FiniteLaw.uniform) (fun _ => f)
_ = _ := by simp [hf]
have hpoint (cs : Fin d → Fin k → SignedLiteral n) :
1 - ∏ j, f (cs j) ≤ if batchKills U cs then 1 else 0 := by
dsimp [f]
rw [Fintype.prod_boole]
by_cases h : ∀ j, ¬testKills U (cs j)
· rw [ite_eq_left h, sub_self]
split <;> norm_num
· obtain ⟨j, hj⟩ := not_forall.mp h
have hj' := not_not.mp hj
have hb : batchKills U cs := fun x hx => ⟨j, hj' x hx⟩
rw [ite_eq_right h, ite_eq_left hb]
norm_num
have hmean := (FiniteLaw.uniform : FiniteLaw (Fin d → Fin k → SignedLiteral n)).expect_mono hpoint
rw [FiniteLaw.expect_sub, FiniteLaw.expect_const, hp] at hmean
exact hmean
theorem relaxed_one_replacement {n : ℕ} [NeZero n]
{U : Finset (DeletionCandidate n)} (hU : U.Nonempty) {d : ℕ} (hd : 0 < d) :
oneTestKill U 2 - 1 / (d : ℝ)^2 ≤ batchKillProbability U 3 d := by
have hn := frozenFraction_nonneg U
have hh := frozenFraction_le_half hU
have hr := two_to_three_replacement hn hh hd
have hb := batchKillProbability_ge U 3 d
rw [oneTestKill_eq] at hb
rw [oneTestKill_eq]
linarith
end FixedClauseThreshold.Computability
end OAI
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