gapt.proofs.lk.rules.ExistsRightRule.scala Maven / Gradle / Ivy

Go to download
Show more of this group Show more artifacts with this name
Show all versions of gapt_3 Show documentation
General Architecture for Proof Theory
The newest version!
package gapt.proofs.lk.rules

import gapt.expr.BetaReduction
import gapt.expr.Expr
import gapt.expr.Var
import gapt.expr.formula.Ex
import gapt.expr.formula.Formula
import gapt.expr.subst.Substitution
import gapt.proofs.HOLSequent
import gapt.proofs.Sequent
import gapt.proofs.SequentIndex
import gapt.proofs.Suc
import gapt.proofs.lk.LKProof

/**
 * An LKProof ending with an existential quantifier on the right:
 *  *            (π)
 *      Γ :- Δ, A[x\t]
 *     ----------------∃:r
 *       Γ :- Δ, ∃x.A
 * 
 *
 * @param subProof The proof π.
 * @param aux The index of A[x\t].
 * @param A The formula A.
 * @param term The term t.
 * @param v The variable x.
 */
case class ExistsRightRule(subProof: LKProof, aux: SequentIndex, A: Formula, term: Expr, v: Var)
    extends WeakQuantifierRule {

  validateIndices(premise, Seq(), Seq(aux))

  val mainFormula: Formula = BetaReduction.betaNormalize(Ex(v, A))

  override def name: String = "∃:r"

  def auxIndices: Seq[Seq[SequentIndex]] = Seq(Seq(aux))

  override def mainFormulaSequent: HOLSequent = Sequent() :+ mainFormula
}

object ExistsRightRule extends ConvenienceConstructor("ExistsRightRule") {

  /**
   * Convenience constructor for ∃:r that, given a main formula and a term,
   * will try to construct an inference with that instantiation.
   *
   * @param subProof The subproof.
   * @param mainFormula The formula to be inferred. Must be of the form ∃x.A.
   * @param term A term t such that A[t] occurs in the premise.
   * @return
   */
  def apply(subProof: LKProof, mainFormula: Formula, term: Expr): ExistsRightRule = {
    val premise = subProof.endSequent

    mainFormula match {
      case Ex(v, subFormula) =>
        val auxFormula = BetaReduction.betaNormalize(Substitution(v, term)(subFormula))
        val i = premise.succedent indexOf auxFormula

        if (i == -1)
          throw LKRuleCreationException(s"Formula $auxFormula not found in succedent of $premise.")

        val p = ExistsRightRule(subProof, Suc(i), subFormula, term, v)
        assert(p.mainFormula == mainFormula)
        p

      case _ => throw LKRuleCreationException(s"Proposed main formula $mainFormula is not existentially quantified.")
    }
  }

  /**
   * Convenience constructor for ∃:r that, given a main formula, will try to construct an inference with that formula.
   *
   * @param subProof The subproof.
   * @param mainFormula The formula to be inferred. Must be of the form ∃x.A. The premise must contain A.
   * @return
   */
  def apply(subProof: LKProof, mainFormula: Formula): ExistsRightRule = mainFormula match {
    case Ex(v, _) =>
      val p = apply(subProof, mainFormula, v)
      assert(p.mainFormula == mainFormula)
      p

    case _ => throw LKRuleCreationException(s"Proposed main formula $mainFormula is not existentially quantified.")
  }

  def apply(subProof: LKProof, aux: SequentIndex, mainFormula: Formula, term: Expr): ExistsRightRule =
    mainFormula match {
      case Ex(v, subFormula) =>
        val p = ExistsRightRule(subProof, aux, subFormula, term, v)
        assert(p.mainFormula == mainFormula)
        p
    }
}