gapt.proofs.lk.rules.macros.ExistsLeftBlock.scala Maven / Gradle / Ivy

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

import gapt.expr.Var
import gapt.expr.formula.Formula
import gapt.expr.formula.hol.instantiate
import gapt.proofs.SequentConnector
import gapt.proofs.lk.LKProof
import gapt.proofs.lk.rules.ExistsLeftRule

object ExistsLeftBlock {

  /**
   * Applies the ExistsLeft-rule n times.
   * This method expects a formula main with
   * a quantifier block, and a proof s1 which has a fully
   * instantiated version of main on the left side of its
   * bottommost sequent.
   *
   * The rule:
   *    *              (π)
   *    A[x1\y1,...,xN\yN], Γ :- Δ
   * ---------------------------------- (∀_r x n)
   *     ∃x1,..,xN.A, Γ :- Δ
   *
   * where y1,...,yN are eigenvariables.
   * 
   *
   * @param subProof The proof π with (A[x1\y1,...,xN\yN], Γ :- Δ) as the bottommost sequent.
   * @param main A formula of the form (∃ x1,...,xN.A).
   * @param eigenvariables The list of eigenvariables with which to instantiate main. The caller of this
   * method has to ensure the correctness of these terms, and, specifically, that
   * A[x1\y1,...,xN\yN] indeed occurs at the bottom of the proof π.
   */
  def apply(subProof: LKProof, main: Formula, eigenvariables: Seq[Var]): LKProof =
    withSequentConnector(subProof, main, eigenvariables)._1

  /**
   * Applies the ExistsLeft-rule n times.
   * This method expects a formula main with
   * a quantifier block, and a proof s1 which has a fully
   * instantiated version of main on the left side of its
   * bottommost sequent.
   *
   * The rule:
   *    *              (π)
   *    A[x1\y1,...,xN\yN], Γ :- Δ
   * ---------------------------------- (∀_r x n)
   *     ∃x1,..,xN.A, Γ :- Δ
   *
   * where y1,...,yN are eigenvariables.
   * 
   *
   * @param subProof The proof π with (A[x1\y1,...,xN\yN], Γ :- Δ) as the bottommost sequent.
   * @param main A formula of the form (∃ x1,...,xN.A).
   * @param eigenvariables The list of eigenvariables with which to instantiate main. The caller of this
   * method has to ensure the correctness of these terms, and, specifically, that
   * A[x1\y1,...,xN\yN] indeed occurs at the bottom of the proof π.
   * @return A pair consisting of an LKProof and an SequentConnector.
   */
  def withSequentConnector(subProof: LKProof, main: Formula, eigenvariables: Seq[Var]): (LKProof, SequentConnector) = {
    val partiallyInstantiatedMains = (0 to eigenvariables.length).toList.reverse.map(n => instantiate(main, eigenvariables.take(n)))

    val series = eigenvariables.reverse.foldLeft(
      (subProof, partiallyInstantiatedMains, SequentConnector(subProof.endSequent))
    ) { (acc, ai) =>
      val newSubProof = ExistsLeftRule(acc._1, acc._2.tail.head, ai)
      val newSequentConnector = newSubProof.getSequentConnector * acc._3
      (newSubProof, acc._2.tail, newSequentConnector)
    }

    (series._1, series._3)
  }
}