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Example for the paper "System Description: Translating Scala Programs to Isabelle/HOL"
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| import leon.annotation._ | |
| import leon.collection._ | |
| import leon.lang._ | |
| @isabelle.typ(name = "Nat.nat") | |
| sealed abstract class Nat { | |
| @isabelle.function(term = "op <=") | |
| def <=(that: Nat): Boolean = this match { | |
| case Zero() => true | |
| case Succ(p) => | |
| that match { | |
| case Zero() => false | |
| case Succ(q) => p <= q | |
| } | |
| } | |
| @isabelle.function(term = "op +") | |
| def +(that: Nat): Nat = (this match { | |
| case Zero() => that | |
| case Succ(pred) => Succ(pred + that) | |
| }) ensuring { res => | |
| this <= res && that <= res | |
| } | |
| @isabelle.function(term = "op *") | |
| def *(that: Nat): Nat = this match { | |
| case Zero() => Zero() | |
| case Succ(pred) => that + pred * that | |
| } | |
| } | |
| @isabelle.constructor(name = "Groups.zero_class.zero") | |
| case class Zero() extends Nat | |
| @isabelle.constructor(name = "Nat.Suc") | |
| case class Succ(pred: Nat) extends Nat | |
| object Nats { | |
| @isabelle.function(term = "Groups_List.monoid_add_class.listsum") | |
| def listSum(xs: List[Nat]): Nat = xs match { | |
| case Nil() => Zero() | |
| case Cons(x, xs) => x + listSum(xs) | |
| } | |
| @isabelle.function(term = "length") | |
| def length[A](xs: List[A]): Nat = xs match { | |
| case Nil() => Zero() | |
| case Cons(x, xs) => Succ(length(xs)) | |
| } | |
| @induct | |
| @isabelle.script( | |
| name = "Map_Fst_Zip", | |
| source = "declare map_fst_zip[simp del]" | |
| ) | |
| @isabelle.proof(method = """(clarsimp, induct rule: list_induct2, auto)""") | |
| def mapFstZip[A, B](xs: List[A], ys: List[B]) = { | |
| require(length(xs) == length(ys)) | |
| xs.zip(ys).map(_._1) | |
| } ensuring { _ == xs } | |
| @induct | |
| def addCommute(x: Nat, y: Nat) = | |
| (x + y == y + x).holds | |
| @induct | |
| def sumReverse[A](xs: List[Nat]) = | |
| (listSum(xs) == listSum(xs.reverse)).holds | |
| @induct | |
| def sumZero[A](xs: List[A]) = | |
| (listSum(xs.map(_ => Zero())) == Zero()).holds | |
| @isabelle.proof(method = """(induct "<var xs>", auto)""") | |
| @induct | |
| def sumConstant[A](xs: List[A], k: Nat) = | |
| (listSum(xs.map(_ => k)) == length(xs) * k).holds | |
| } |
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