gapt.expr.util.longNormalForm.scala Maven / Gradle / Ivy

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package gapt.expr.util

import gapt.expr.Abs
import gapt.expr.Apps
import gapt.expr.Expr
import gapt.expr.ExprNameGenerator
import gapt.expr.Var
import gapt.expr.ty.FunctionType
import gapt.utils.NameGenerator

/**
 * If t = λx₁...λxₘ(y u₁ ... uₚ) is a normal term of type T₁ → ... → Tₙ → U, with
 * U atomic and m ≤ n, then its long normal form is the term
 *
 * λx₁...λxₘλxₘ₊₁...λxₙ(y u₁'... uₚ' xₘ₊₁' ... xₙ'),
 *
 * where uᵢ' is the long normal form of uᵢ and xⱼ' is the long normal form of xⱼ.
 *
 * Implemented according to Definition 2.25, Higher-Order Unification and
 * Matching by Gilles Dowek.
 */
object longNormalForm {

  /**
   * Computes the long normal form.
   *
   * @param term A term.
   * @return The long normal form of `term`. Note that η-expansion is applied
   * only to expressions in β-normal form.
   */
  def apply(e: Expr): Expr =
    e match {
      case Abs(v, e_) => Abs(v, longNormalForm(e_))
      case Apps(e, es) =>
        (e.ty: @unchecked) match {
          case FunctionType(_, ts) =>
            val names = new NameGenerator(freeVariables(e).map { _.name })
            val etaVars = ts.drop(es.length).map { t => names.fresh(Var("η", t)) }
            Abs.Block(
              etaVars,
              Apps(
                e,
                es.map { longNormalForm(_) } ++ etaVars.map { longNormalForm(_) }
              )
            )
        }
    }
}