gapt.proofs.lk.rules.ForallLeftRule.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.Expr
import gapt.expr.Var
import gapt.expr.formula.All
import gapt.expr.formula.Formula
import gapt.expr.subst.Substitution
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 a universal quantifier on the left:
 *  *            (π)
 *      A[x\t], Γ :- Δ
 *     ----------------∀:l
 *       ∀x.A, Γ :- Δ
 * 
 *
 * @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 ForallLeftRule(subProof: LKProof, aux: SequentIndex, A: Formula, term: Expr, v: Var)
    extends WeakQuantifierRule {

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

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

  override def name: String = "∀:l"

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

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

object ForallLeftRule extends ConvenienceConstructor("ForallLeftRule") {

  /**
   * Convenience constructor for ∀:l 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): ForallLeftRule = {
    val premise = subProof.endSequent

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

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

        val p = ForallLeftRule(subProof, Ant(i), subFormula, term, v)
        assert(p.mainFormula == mainFormula)
        p

      case _ => throw LKRuleCreationException(s"Proposed main formula $mainFormula is not universally 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): ForallLeftRule = mainFormula match {
    case All(v, _) =>
      val p = apply(subProof, mainFormula, v)
      assert(p.mainFormula == mainFormula)
      p

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

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