gapt.proofs.lk.transformations.acnf.scala Maven / Gradle / Ivy

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General Architecture for Proof Theory
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package gapt.proofs.lk.transformations

import gapt.expr.formula.Formula
import gapt.expr.formula.hol.isAtom
import gapt.proofs.lk.LKProof
import gapt.proofs.lk.reductions.CutReduction
import gapt.proofs.lk.reductions.gradeReduction
import gapt.proofs.lk.reductions.leftRankReduction
import gapt.proofs.lk.reductions.rightRankReduction
import gapt.proofs.lk.rules.CutRule
import gapt.proofs.lk.rules.InitialSequent
import gapt.proofs.lk.rules.LogicalAxiom
import gapt.proofs.lk.rules.WeakeningLeftRule
import gapt.proofs.lk.rules.WeakeningRightRule

object acnf {

  /**
   * Transforms a given proof to a proof in ACNF.
   *
   * @param proof The proof to be transformed to ACNF.
   * @return A proof of the same end-sequent in ACNF.
   */
  def apply(proof: LKProof): LKProof =
    (new UppermostFirstStrategy(acnfReduction)).run(proof)

  private val acnfReduction = new CutReduction {
    def reduce(proof: CutRule): Option[LKProof] = proof match {
      case cut @ CutRule(lsb, l, rsb, _) if !isAtom(lsb.endSequent(l)) && isAcnf(lsb) && isAcnf(rsb) =>
        if (isAtom(lsb.endSequent(l)))
          (leftRankReduction orElse rightRankReduction).reduce(cut)
        else
          (gradeReduction orElse leftRankReduction orElse rightRankReduction).reduce(cut)
      case _ => None
    }
  }
}

object isAcnf {

  /**
   * This method checks whether a proof is in ACNF
   *
   * @param proof The proof to check for in ACNF.
   * @return true if proof is in ACNF, false otherwise.
   */
  def apply(proof: LKProof): Boolean = proof match {
    case InitialSequent(_) => true
    case CutRule(lsb, l, rsb, r) =>
      if (isAtom(lsb.endSequent(l))) isAcnf(lsb) && isAcnf(rsb)
      else false
    case _ => proof.immediateSubProofs.forall(isAcnf(_))
  }
}

object acnfTop {

  /**
   * Transforms a proof to a proof in ACNF Top.
   *
   * @param proof The proof to be transformed to ACNF Top
   * @return A proof of the same end-sequent in ACNF Top.
   */
  def apply(proof: LKProof): LKProof =
    (new UppermostFirstStrategy(acnfTopReduction)).run(proof)

  private val acnfTopReduction = new CutReduction {
    def reduce(proof: CutRule): Option[LKProof] =
      proof match {
        case cut @ CutRule(lsb, l, rsb, r) if isAtomicCut(cut) =>
          if (!(introOrCut(lsb, lsb.endSequent(l)) && introOrCut(rsb, rsb.endSequent(r)))) {
            if (introOrCut(lsb, lsb.endSequent(l)))
              rightRankReduction.reduce(cut)
            else
              (leftRankReduction orElse rightRankReduction).reduce(cut)
          } else {
            None
          }
        case cut @ CutRule(lsb, _, rsb, _) if isAcnfTop(lsb) && isAcnfTop(rsb) =>
          (gradeReduction orElse leftRankReduction orElse rightRankReduction).reduce(cut)
        case _ => None
      }
  }
}

object isAcnfTop {

  /**
   * This method checks whether a proof is in ACNF top
   *
   * @param proof The proof to check for in ACNF top.
   * @return true if proof is in ACNF, false otherwise.
   */
  def apply(proof: LKProof): Boolean = proof match {
    case InitialSequent(_) => true
    case CutRule(lsb, l, rsb, r) =>
      if (isAtom(lsb.endSequent(l)))
        if (introOrCut(lsb, lsb.endSequent(l)) && introOrCut(rsb, rsb.endSequent(r)))
          isAcnfTop(lsb) && isAcnfTop(rsb)
        else false
      else false
    case _ => proof.immediateSubProofs.forall(isAcnfTop(_))
  }
}

object introOrCut {

  /**
   * Checks if the last rule in proof is a leaf, a cut rule, or a weakening rule on
   * the given formula.
   *
   * @param proof   The proof we are checking.
   * @param formula The formula we are checking.
   * @return True is structure is correct or false if not.
   */
  def apply(proof: LKProof, formula: Formula): Boolean = proof match {
    case LogicalAxiom(_)             => true
    case CutRule(lsb, l, rsb, r)     => true
    case WeakeningRightRule(_, main) => if (main == formula) true else false
    case WeakeningLeftRule(_, main)  => if (main == formula) true else false
    case _                           => false
  }
}

object isAtomicCut {
  def apply(cut: CutRule): Boolean = isAtom(cut.cutFormula)
}