| 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 |
|
|
