gapt.proofs.lk.rules.macros.ForallLeftBlock.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.BetaReduction
import gapt.expr.Expr
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.ForallLeftRule

object ForallLeftBlock {

  /**
   * Applies the ForallLeft-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\term1,...,xN\termN], Γ :- Δ
   * ---------------------------------- (∀_l x n)
   *       ∀ x1,..,xN.A, Γ :- Δ
   * 
   *
   * @param subProof The top proof with (sL, A[x1\term1,...,xN\termN] :- sR) as the bottommost sequent.
   * @param main A formula of the form (Forall x1,...,xN.A).
   * @param terms The list of terms with which to instantiate main. The caller of this
   * method has to ensure the correctness of these terms, and, specifically, that
   * A[x1\term1,...,xN\termN] indeed occurs at the bottom of the proof π.
   */
  def apply(subProof: LKProof, main: Formula, terms: Seq[Expr]): LKProof =
    withSequentConnector(subProof, main, terms)._1

  /**
   * Applies the ForallLeft-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\term1,...,xN\termN], Γ :- Δ
   * ---------------------------------- (∀_l x n)
   *       ∀ x1,..,xN.A, Γ :- Δ
   * 
   *
   *
   *
   * @param subProof The top proof with (sL, A[x1\term1,...,xN\termN] :- sR) as the bottommost sequent.
   * @param main A formula of the form (Forall x1,...,xN.A).
   * @param terms The list of terms with which to instantiate main. The caller of this
   * method has to ensure the correctness of these terms, and, specifically, that
   * A[x1\term1,...,xN\termN] 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, terms: Seq[Expr]): (LKProof, SequentConnector) = {
    val partiallyInstantiatedMains = (0 to terms.length).toList.reverse.map(n => BetaReduction.betaNormalize(instantiate(main, terms.take(n))))

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

    (series._1, series._3)
  }
}