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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 | import OAI.Probability.SATComputability.ActiveClauses
import OAI.Probability.SATComputability.PoissonMaskCounts
namespace OAI
namespace FixedClauseThreshold.Computability
open scoped Classical
theorem countsMask_mono {A X : Type*} [Fintype A] [Fintype X]
{f g : A → Finset X} (h : ∀ a, f a ⊆ g a) (c : A → ℕ) :
countsMask f c ⊆ countsMask g c := by
intro x hx
apply Finset.mem_filter.mpr
refine ⟨Finset.mem_univ _,?_⟩
intro a
exact ((Finset.mem_filter.mp hx).2 a).imp_right (fun hx => h a hx)
theorem countsMask_equiv {A B X : Type*} [Fintype A] [Fintype B] [Fintype X]
(e : A ≃ B) (f : B → Finset X) (c : A → ℕ) :
countsMask (f ∘ e) c = countsMask f (c ∘ e.symm) := by
ext x
simp only [countsMask, Finset.mem_filter, Finset.mem_univ, true_and, Function.comp_def]
constructor
· intro h b
simpa only [Equiv.apply_symm_apply] using h (e.symm b)
· intro h a
simpa only [Equiv.symm_apply_apply] using h (e a)
theorem countsMask_active {n : ℕ} (c : TripleClasses (SignedLiteral n) → ℕ) :
countsMask (fun a => clauseMask (classClause (activeClass a))) (c ∘ activeClass) =
countsMask (fun a => clauseMask (classClause a)) c := by
ext x
simp only [countsMask, Finset.mem_filter, Finset.mem_univ, true_and, Function.comp_def]
constructor
· intro h a
by_cases ha : a ∈ Set.range (activeClass (n := n))
· obtain ⟨b,rfl⟩ := ha
exact h b
· exact Or.inr (by rw [inactiveClass_trivial a ha]; exact Finset.mem_univ _)
· intro h a
exact h (activeClass a)
theorem countsMask_stronger {n : ℕ} (c : ActiveClause n → ℕ) :
countsMask strongerClassMask c =
restoredMask
(countsMask (fun a => clauseMask ((tripleVectorEquiv _).symm a)) (fun a => c (.inl a)))
(countsMask incidentMask (fun a => c (.inr (true,a))))
(countsMask incidentMask (fun a => c (.inr (false,a)))) := by
ext x
rcases h0 : x 0 with _ | (_ | _)
all_goals
simp [countsMask, strongerClassMask, restoredMask, Sum.forall, Prod.forall,
Bool.forall_bool, h0]
end FixedClauseThreshold.Computability
end OAI
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