File size: 2,907 Bytes
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 | import OAI.Probability.SATComputability.ClauseProcess
namespace OAI
namespace FixedClauseThreshold.Computability
open DilutedSpinGlass _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal Classical
local instance gridBlockCandidateMeasurable (n : ℕ) :
MeasurableSpace (DeletionCandidate n) := ⊤
local instance gridBlockCandidateSingleton (n : ℕ) :
MeasurableSingletonClass (DeletionCandidate n) := ⟨fun _ => trivial⟩
theorem gridMaskLaw_pi_of_expect {A : Type*} [Fintype A] {n M : ℕ}
(r : ℝ≥0) (mask : A → Finset (DeletionCandidate n))
(Q : FiniteLaw (Finset (DeletionCandidate n)))
(hQ : ∀ f, Q.expect f = ∫ c, f (countsMask mask c) ∂poissonCountLaw A r) :
gridMaskLaw M r mask = FiniteLaw.pi (fun _ : Fin M => Q) := by
classical
let μ := poissonCountLaw A r
have hsingle : finitePushforward μ (countsMask mask) (measurable_of_countable _) = Q := by
apply finiteLaw_eq_of_expect
intro f
rw [finitePushforward_expect]
exact (hQ f).symm
apply finiteLaw_eq_of_expect
intro f
rw [gridMaskLaw_expect]
have htranspose := poissonCountLaw_equiv (Equiv.prodComm (Fin M) A) r
(fun c => f (maskGrid mask c))
rw [← htranspose, ← poissonCountLaw_uncurry_integral]
change (∫ c : Fin M → A → ℕ, f (fun j => countsMask mask (c j))
∂Measure.pi (fun _ : Fin M => μ)) = _
rw [← finitePushforward_expect (Measure.pi (fun _ : Fin M => μ))
(fun c j => countsMask mask (c j))
(Measurable.of_eval (fun j => (measurable_of_countable (countsMask mask)).comp
(measurable_pi_apply j))) f,
finitePushforward_pi _ _ (fun _ => measurable_of_countable _)]
simp only [hsingle]
theorem gridMaskLaw_equiv {A B : Type} [Fintype A] [Fintype B] {n M : ℕ}
(e : A ≃ B) (r : ℝ≥0) (mask : B → Finset (DeletionCandidate n)) :
gridMaskLaw M r (mask ∘ e) = gridMaskLaw M r mask := by
apply finiteLaw_eq_of_expect
intro f
simp only [gridMaskLaw_expect]
have he (c : A × Fin M → ℕ) :
maskGrid (mask ∘ e) c = maskGrid mask (c ∘ (e.prodCongr (Equiv.refl (Fin M))).symm) := by
funext j
exact countsMask_equiv e mask (fun a => c (a,j))
simp only [he]
simpa only [Function.comp_def] using
poissonCountLaw_equiv (e.prodCongr (Equiv.refl (Fin M))) r (fun c => f (maskGrid mask c))
theorem gridMaskLaw_clauses {n : ℕ} [NeZero n] (M k : ℕ) (r : ℝ≥0) :
gridMaskLaw M r (clauseMask (n := n) (k := k)) =
FiniteLaw.pi (fun _ : Fin M => candidateBlock
(r*Fintype.card (Fin k → SignedLiteral n)) k Finset.univ) := by
apply gridMaskLaw_pi_of_expect
intro f
rw [candidateBlock_counts]
simp only [mul_div_cancel_right₀ _ (show
(Fintype.card (Fin k → SignedLiteral n) : ℝ≥0) ≠ 0 by
exact_mod_cast Fintype.card_ne_zero)]
rfl
end FixedClauseThreshold.Computability
end OAI
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