gapt.proofs.lk.rules.EqualityRightRule.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.Abs
import gapt.expr.App
import gapt.expr.BetaReduction
import gapt.expr.formula.Eq
import gapt.expr.formula.Formula
import gapt.expr.util.replacementContext
import gapt.proofs.Ant
import gapt.proofs.HOLSequent
import gapt.proofs.IndexOrFormula
import gapt.proofs.Sequent
import gapt.proofs.SequentIndex
import gapt.proofs.Suc
import gapt.proofs.lk.LKProof

/**
 * An LKProof ending with a right equality rule.
 * There are two possible cases according to which direction the rule is applied in:
 *
 *  *            (π)                            (π)
 *    s = t, Γ :- Δ, A[s]            s = t, Γ :- Δ, A[t]
 *   ---------------------eq:r      ---------------------eq:r
 *    s = t, Γ :- Δ, A[t]            s = t, Γ :- Δ, A[s]
 * 
 *
 * @param subProof The subproof π.
 * @param eq The index of s = t.
 * @param aux The index of the formula in which the replacement is to be performed.
 * @param replacementContext A term λx.A[x] that designates the positions to be replaced.
 */
case class EqualityRightRule(subProof: LKProof, eq: SequentIndex, aux: SequentIndex, replacementContext: Abs)
    extends EqualityRule {

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

  override def name: String = "eq:r"

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

object EqualityRightRule extends ConvenienceConstructor("EqualityRightRule") {

  /**
   * Convenience constructor for eq:r.
   * Each of the aux formulas can be given as an index or a formula. If it is given as a formula, the constructor
   * will attempt to find an appropriate index on its own.
   *
   * @param subProof The subproof.
   * @param eqFormula The index of the equation or the equation itself.
   * @param auxFormula The index of the auxiliary formula or the formula itself.
   * @param replacementContext A term λx.A[x] that designates the positions to be replaced.
   * @return
   */
  def apply(subProof: LKProof, eqFormula: IndexOrFormula, auxFormula: IndexOrFormula, replacementContext: Abs): EqualityRightRule = {
    val premise = subProof.endSequent

    val (indicesAnt, indicesSuc) = findAndValidate(premise)(Seq(eqFormula), Seq(auxFormula))

    EqualityRightRule(subProof, Ant(indicesAnt(0)), Suc(indicesSuc(0)), replacementContext)

  }

  /**
   * Convenience constructor for eq:r.
   * Each of the aux formulas can be given as an index or a formula. If it is given as a formula, the constructor
   * will attempt to find an appropriate index on its own.
   *
   * @param subProof The subproof.
   * @param eq The index of the equation or the equation itself.
   * @param aux The index of the auxiliary formula or the formula itself.
   * @param mainFormula The proposed main formula.
   * @return
   */
  def apply(subProof: LKProof, eq: IndexOrFormula, aux: IndexOrFormula, mainFormula: Formula): EqualityRightRule = {
    val premise = subProof.endSequent
    val (indicesAnt, indicesSuc) = findAndValidate(premise)(Seq(eq), Seq(aux))
    val (eqFormula, auxFormula) = (premise(Ant(indicesAnt(0))), premise(Suc(indicesSuc(0))))

    eqFormula match {
      case Eq(s, t) =>
        if (s == t && auxFormula == mainFormula) {
          val repContext = replacementContext.abstractTerm(auxFormula)(s)

          val Abs(_, _) = repContext
          if (auxFormula.find(s).isEmpty)
            throw LKRuleCreationException(s"Eq is trivial, but term $s does not occur in $auxFormula.")

          EqualityRightRule(subProof, eq, aux, repContext)

        } else if (s == t && auxFormula != mainFormula) {
          throw LKRuleCreationException(s"Eq is trivial, but aux formula $auxFormula and main formula $mainFormula.")

        } else if (s != t && auxFormula == mainFormula) {
          throw LKRuleCreationException("Nontrivial equation, but aux and main formula are equal.")

        } else {
          val contextS = replacementContext(s.ty, auxFormula, auxFormula.find(s) intersect mainFormula.find(t))
          val contextT = replacementContext(t.ty, auxFormula, auxFormula.find(t) intersect mainFormula.find(s))

          val Abs(vS, restS) = contextS
          val Abs(vT, restT) = contextT

          if (restS.find(vS).isEmpty && restT.find(vT).isEmpty)
            throw LKRuleCreationException(s"Neither $s nor $t found in formula $auxFormula.")

          val p =
            if (
              BetaReduction.betaNormalize(App(contextS, t)) ==
                BetaReduction.betaNormalize(mainFormula)
            ) {
              EqualityRightRule(subProof, eq, aux, contextS)
            } else if (
              BetaReduction.betaNormalize(App(contextT, s)) ==
                BetaReduction.betaNormalize(mainFormula)
            ) {
              EqualityRightRule(subProof, eq, aux, contextT)
            } else throw LKRuleCreationException("Replacement in neither direction leads to proposed main formula.")

          assert(p.mainFormula == mainFormula)
          p
        }

      case _ => throw LKRuleCreationException(s"Formula $eqFormula is not an equation.")
    }
  }
}