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

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

import gapt.expr.BetaReduction
import gapt.expr.Var
import gapt.expr.formula.Ex
import gapt.expr.formula.Formula
import gapt.expr.subst.Substitution
import gapt.expr.util.freeVariables
import gapt.proofs.Ant
import gapt.proofs.HOLSequent
import gapt.proofs.Sequent
import gapt.proofs.SequentIndex
import gapt.proofs.lk.LKProof

/**
 * An LKProof ending with an existential quantifier on the left:
 *  *           (π)
 *      A[x\α], Γ :- Δ
 *     ---------------∀:r
 *       ∃x.A Γ :- Δ
 * 
 * This rule is only applicable if the eigenvariable condition is satisfied: α must not occur freely in Γ :- Δ.
 *
 * @param subProof The proof π.
 * @param aux The index of A[x\α].
 * @param eigenVariable The variable α.
 * @param quantifiedVariable The variable x.
 */
case class ExistsLeftRule(subProof: LKProof, aux: SequentIndex, eigenVariable: Var, quantifiedVariable: Var)
    extends StrongQuantifierRule {

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

  val mainFormula: Formula = BetaReduction.betaNormalize(Ex(quantifiedVariable, subFormula))

  override def name: String = "∃:l"

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

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

object ExistsLeftRule extends ConvenienceConstructor("ExistsLeftRule") {

  /**
   * Convenience constructor for ∃:l that, given a main formula and an eigenvariable,
   * 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 eigenVariable A variable α such that A[α] occurs in the premise.
   * @return
   */
  def apply(subProof: LKProof, mainFormula: Formula, eigenVariable: Var): ExistsLeftRule = {
    if (freeVariables(mainFormula) contains eigenVariable) {
      throw LKRuleCreationException(s"Illegal main formula: Eigenvariable $eigenVariable is free in $mainFormula.")
    } else mainFormula match {
      case Ex(v, subFormula) =>
        val auxFormula = Substitution(v, eigenVariable)(subFormula)

        val premise = subProof.endSequent

        val (indices, _) = findAndValidate(premise)(Seq(auxFormula), Seq())
        val p = ExistsLeftRule(subProof, Ant(indices(0)), eigenVariable, v)
        assert(p.mainFormula == mainFormula)
        p

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

  /**
   * Convenience constructor for ∃:l 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): ExistsLeftRule = 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, eigenVariable: Var): ExistsLeftRule =
    mainFormula match {
      case Ex(v, _) => ExistsLeftRule(subProof, aux, eigenVariable, v)
    }
}