| import OAI.Probability.SATComputability.RelaxedCandidates |
| import OAI.Probability.SATComputability.VariableDeletion |
|
|
| namespace OAI |
|
|
| namespace FixedClauseThreshold.Computability |
|
|
| open Finset |
| open scoped Classical |
|
|
| def encodeDeletion {n : ℕ} (D : Finset (Fin n)) (s : Fin n → Bool) : DeletionCandidate n := |
| fun v => if v ∈ D then none else some (s v) |
|
|
| def deletedVariables {n : ℕ} (x : DeletionCandidate n) : Finset (Fin n) := |
| univ.filter (fun v => x v = none) |
|
|
| theorem literalFalse_encode {n : ℕ} (D : Finset (Fin n)) (s : Fin n → Bool) |
| (v : Fin n) (b : Bool) : |
| ¬literalFalse (encodeDeletion D s) (v, b) ↔ v ∈ D ∨ s v = b := by |
| by_cases hv : v ∈ D |
| · simp [literalFalse, encodeDeletion, hv] |
| · cases hb : s v <;> cases b <;> simp [literalFalse, encodeDeletion, hv, hb] |
|
|
| theorem deletedVariables_encode {n : ℕ} (D : Finset (Fin n)) (s : Fin n → Bool) : |
| deletedVariables (encodeDeletion D s) = D := by |
| ext v |
| by_cases hv : v ∈ D <;> simp [deletedVariables, encodeDeletion, hv] |
|
|
| theorem deletionSatisfiable_iff_candidate {n m r : ℕ} |
| (indices : Fin m → Fin 3 → Fin n) (signs : Fin m → Fin 3 → Bool) : |
| deletionSatisfiable r indices signs ↔ |
| ∃ x : DeletionCandidate n, (deletedVariables x).card ≤ r ∧ |
| ∀ j, ∃ l, ¬literalFalse x (indices j l, signs j l) := by |
| constructor |
| · rintro ⟨D, hD, s, hs⟩ |
| refine ⟨encodeDeletion D s, ?_, ?_⟩ |
| · simpa only [deletedVariables_encode] using hD |
| · intro j |
| rcases hs j with ⟨l, hl⟩ | ⟨l, hl⟩ |
| · exact ⟨l, (literalFalse_encode D s _ _).mpr (Or.inl hl)⟩ |
| · exact ⟨l, (literalFalse_encode D s _ _).mpr (Or.inr hl)⟩ |
| · rintro ⟨x, hx, hs⟩ |
| let D := deletedVariables x |
| let s : Fin n → Bool := fun v => (x v).getD false |
| refine ⟨D, hx, s, ?_⟩ |
| intro j |
| obtain ⟨l, hl⟩ := hs j |
| cases hv : x (indices j l) with |
| | none => |
| left |
| exact ⟨l, by simp [D, deletedVariables, hv]⟩ |
| | some b => |
| right |
| refine ⟨l, ?_⟩ |
| have hb : b = signs j l := by |
| cases b <;> cases hsign : signs j l <;> |
| simp_all [literalFalse] |
| simpa only [s, hv, Option.getD_some] using hb |
|
|
| end FixedClauseThreshold.Computability |
|
|
| end OAI |
|
|
