Pure Scala

The input to Leon is a purely functional subset of Scala, which we call Pure Scala. Constructs specific for Leon are defined inside Leon’s libraries in package leon and its subpackages. Leon invokes standard scalac compiler on the input file, then performs additional checks to ensure that the given program belongs to Pure Scala.

Pure Scala supports two kinds of top-level declarations:

1. Algebraic Data Type (ADT) definitions in the form of an abstract class and case classes/objects

abstract class MyList
case object MyEmpty extends MyList
case class MyCons(elem: BigInt, rest: MyList) extends MyList

2. Objects/modules, for grouping classes and functions

object Specs {
   def increment(a: BigInt): BigInt = {
      a + 1
   }

   case class Identifier(id: BigInt)
}

Boolean

Booleans are used to express truth conditions in Leon. Unlike some proof assistants, there is no separation at the type level between Boolean values and the truth conditions of conjectures and theorems.

Typical propositional operations are available using Scala notation, along with a new shorthand for implication. The if expression is also present.

a && b
a || b
a == b
!a
a ==> b // Leon syntax for boolean implication

Leon uses short-circuit interpretation of &&, ||, and ==>, which evaluates the second argument only when needed:

a && b   === if (a) b else false

a || b   === if (a) true else b

a ==> b  === if (a) b else true

This aspect is important because of:

1. evaluation of expressions, which is kept compatible with Scala

2. verification condition generation for safety: arguments of Boolean operations may be operations with preconditions; these preconditions apply only in case that the corresponding argument is evaluated.

3. termination checking, which takes into account that only one of the paths in an if expression is evaluated for a given truth value of the condition.

Algebraic Data Types

Abstract Classes

ADT roots need to be defined as abstract, unless the ADT is defined with only one case class/object. Unlike in Scala, abstract classes cannot define fields or constructor arguments.

abstract class MyType

An abstract class can be extended by other abstract classes.

Case Classes

The abstract root can also be extended by a case-class, defining several fields:

case class MyCase1(f: Type, f2: MyType) extends MyType
case class MyCase2(f: Int) extends MyType

Note

You can also define single case-class, for Tuple-like structures.

Case Objects

It is also possible to defined case objects, without fields:

case object BaseCase extends MyType

Generics

Leon supports type parameters for classes and functions.

object Test {
  abstract class List[T]
  case class Cons[T](hd: T, tl: List[T]) extends List[T]
  case class Nil[T]() extends List[T]

  def contains[T](l: List[T], el: T) = { ... }
}

Note

Type parameters are always invariant. It is not possible to define ADTs like:

abstract class List[T]
case class Cons[T](hd: T, tl: List[T]) extends List[T]
case object Nil extends List[Nothing]

Leon in fact restricts type parameters to “simple hierarchies”, where subclasses define the same type parameters in the same order.

Methods

You can define methods in classes.

abstract class List[T] {
  def contains(e: T) = { .. }
}
case class Cons[T](hd: T, tl: List[T]) extends List[T]
case object Nil extends List[Nothing]

def test(a: List[Int]) = a.contains(42)

It is possible to define abstract methods in abstract classes and implement them in case classes. It is also possible to override methods.

abstract class A {
  def x(a: Int): Int
}

abstract class B extends A {
  def x(a: Int) = {
    require(a > 0)
    42
  } ensuring { _ >= 0 }
}

case class C(c: Int) extends B {
  override def x(i: Int) = {
    require(i >= 0)
    if (i == 0) 0
    else c + x(i-1)
  } ensuring ( _ == c * i )
}

case class D() extends B

It is not possible, however, to call methods of a superclass with the super keyword.

Specifications

Leon supports three kinds of specifications to functions and methods:

Preconditions

Preconditions constraint the argument and is expressed using require. It should hold for all calls to the function.

def foo(a: Int, b: Int) = {
  require(a > b)
  ...
}

Postconditions

Postconditions constraint the resulting value, and is expressed using ensuring:

def foo(a: Int): Int = {
  a + 1
} ensuring { res => res > a }

Body Assertsions

Assertions constrain intermediate expressions within the body of a function.

def foo(a: Int): Int = {
  val b = -a
  assert(a >= 0 || b >= 0, "This will fail for -2^31")
  a + 1
}

The error description (last argument of assert) is optional.

Expressions

Leon supports most purely-functional Scala expressions:

Pattern matching

expr match {
  // Simple (nested) patterns:
  case CaseClass( .. , .. , ..) => ...
  case v @ CaseClass( .. , .. , ..) => ...
  case v : CaseClass => ...
  case (t1, t2) => ...
  case 42 => ...
  case _ => ...

  // can also be guarded, e.g.
  case CaseClass(a, b, c) if a > b => ...
}

Custom pattern matching with unapply methods are also supported:

object :: {
  def unapply[A](l: List[A]): Option[(A, List[A])] = l match {
    case Nil() => None()
    case Cons(x, xs) => Some((x, xs))
  }
}

def empty[A](l: List[A]) = l match {
  case x :: xs => false
  case Nil() => true
}

Values

val x = ...

val (x, y) = ...

val Cons(h, _) = ...

Note

The latter two cases are actually syntactic sugar for pattern matching with one case.

Inner Functions

def foo(x: Int) = {
  val y = x + 1
  def bar(z: Int) = {
     z + y
  }
  bar(42)
}

Predefined Types

TupleX

val x = (1,2,3)
val y = 1 -> 2 // alternative Scala syntax for Tuple2
x._1 // == 1

Int

a + b
a - b
-a
a * b
a / b
a % b // a modulo b
a < b
a <= b
a > b
a >= b
a == b

Note

Integers are treated as 32bits integers and are subject to overflows.

BigInt

val a = BigInt(2)
val b = BigInt(3)

-a
a + b
a - b
a * b
a / b
a % b // a modulo b
a < b
a > b
a <= b
a >= b
a == b

Note

BigInt are mathematical integers (arbitrary size, no overflows).

Real

Real represents the mathematical real numbers (different from floating points). It is an extension to Scala which is meant to write programs closer to their true semantics.

val a: Real = Real(2)
val b: Real = Real(3, 5) // 3/5

-a
a + b
a - b
a * b
a / b
a < b
a > b
a <= b
a >= b
a == b

Note

Real have infinite precision, which means their properties differ from Double. For example, the following holds:

def associativity(x: Real, y: Real, z: Real): Boolean = {
  (x + y) + z == x + (y + z)
} holds

While it does not hold with floating point arithmetic.

Set

import leon.lang.Set // Required to have support for Sets

val s1 = Set(1,2,3,1)
val s2 = Set[Int]()

s1 ++ s2 // Set union
s1 & s2  // Set intersection
s1 -- s2 // Set difference
s1 subsetOf s2
s1 contains 42

Functional Array

val a = Array(1,2,3)

a(index)
a.updated(index, value)
a.length

Map

import leon.lang.Map // Required to have support for Maps

val  m = Map[Int, Boolean](42 -> false)

m(index)
m isDefinedAt index
m contains index
m.updated(index, value)
m + (index -> value)
m + (value, index)
m.get(index)
m.getOrElse(index, value2)

Function

val f1 = (x: Int) => x + 1                 // simple anonymous function

val y  = 2
val f2 = (x: Int) => f1(x) + y             // closes over `f1` and `y`
val f3 = (x: Int) => if (x < 0) f1 else f2 // anonymous function returning another function

list.map(f1)      // functions can be passed around ...
list.map(f3(1) _) // ... and partially applied

Note

No operators are defined on function-typed expressions, so specification is currently quite limited.

Symbolic Input-Output examples

Sometimes, a complete formal specification is hard to write, especially when it comes to simple, elementary functions. In such cases, it may be easier to provide a set of IO-examples. On the other hand, IO-examples can never cover all the possible executions of a function, and are thus weaker than a formal specification.

Leon provides a powerful compromise between these two extremes. It introduces symbolic IO-examples, expressed through a specialized passes construct, which resembles pattern-matching:

sealed abstract class List {

  def size: Int = (this match {
    case Nil() => 0
    case Cons(h, t) => 1 + t.size
  }) ensuring { res => (this, res) passes {
    case Nil() => 0
    case Cons(_, Nil()) => 1
    case Cons(_, Cons(_, Nil())) => 2
  }}
}
case class Cons[T](h: T, t: List[T]) extends List[T]
case class Nil[T]() extends List[T]

In the above example, the programmer has chosen to partially specify size through a list of IO-examples, describing what the function should do for lists of size 0, 1 or 2. Notice that the examples are symbolic, in that the elements of the lists are left unconstrained.

The semantics of passes is the following. Let a: A be a tuple of method parameters and/or this, b: B, and for each i pi: A and ei: B. Then

(a, b) passes {
  case p1 => e1
  case p2 => e2
  ...
  case pN => eN
}

is equivalent to

a match {
  case p1 => b == e1
  case p2 => b == e2
  ...
  case pN => b == eN
  case _  => true
}