Dafny

Dafny provides methods for implementation which may have side-effects and functions for use in specification which are pure. Methods consist of sequences of statements following a familiar imperative style whilst, in contrast, the body of a function is simply an expression. Any side-effecting statements in a method (e.g. assigning an element of an array parameter) must be accounted for by noting which parameters can be mutated in the modifies clause. Dafny also provides a range of immutable collection types including: sequences (e.g. seq<int>), sets (e.g. set<int>) and maps (map<int,int>). In addition, mutable arrays (e.g. array<int>) are provided.

Dafny supports proofs of imperative programs based on the ideas of Hoare logic. The following illustrates many such features in Dafny, including the use of preconditions, postconditions, loop invariants and loop variants.

method max(arr:array<int>) returns(max:int)
 // Array must have at least one element
 requires arr!=null && arr.Length > 0;
 // Return cannot be smaller than any element in array
 ensures (forall j :int :: (j >= 0 && j < arr.Length ==> max >= arr[j]));
 // Return must match some element in array
 ensures (exists j : int :: j>=0 && j < arr.Length && max==arr[j]);
{
  max:=arr[0];
  var i:int :=1;
  //
  while(i < arr.Length)
  // Indent at most arr.Length (needed to show i==arr.Length after loop)
  invariant (i<=arr.Length);
  // No element seen so far larger than max
  invariant (forall j:int :: j>=0 && j<i ==> max >= arr[j]);
  // Some element seen so far matches max
  invariant (exists j:int :: j>=0 && j<i && max == arr[j]);
  // Terminate when i == arr.Length
  decreases (arr.Length-i); 
  {
    // Update max if larger element encountered
    if(arr[i] > max){max := arr[i];}
    // Continue through array
    i := i + 1;
  }
}

This examples computes the maximum element of an array. The method's precondition and postcondition are given with the requires and ensures clauses (respectively). Likewise, the loop invariant and loop variant are given through the invariant and decreases clauses (respectively).

The treatment of loop invariants in Dafny differs from traditional Hoare logic. Variables mutated in a loop are treated such that (most) information known about them prior to the loop is discarded. Information required to prove properties of such variables must be expressed explicitly in the loop invariant. In contrast, variables not mutated in the loop retain all information known about them beforehand. The following provides illustrates:

method sumAndZero(ns:array<int>) returns (sum:nat)
  requires ns != null
  requires forall i :: 0 <= i < ns.Length ==> ns[i] >= 0
  modifies ns 
{
  var i:int := 0;
  var arr:array<int> := ns; // because parameters cannot be assigned
  sum := 0;
  //
  while(i < arr.Length) {
    sum := sum + arr[i];
    arr[i] := arr[i];
    i := i + 1;
  }
}

This fails verification because Dafny cannot establish that (sum + arr[i]) >= 0 holds at the assignment. From the precondition, intuitively, forall i :: 0 <= i < arr.Length ==> arr[i] >= 0 holds in the loop since arr[i] := arr[i]; is a NOP. However, this assignment causes Dafny to treat arr as a mutable variable and drop information known about it from before the loop. To verify this program in Dafny we can either: remove the redundant assignment arr[i] := arr[i];; or, add the loop invariant invariant forall i :: 0 <= i < arr.Length ==> arr[i] >= 0

Dafny additionally employs limited static analysis to infer simple loop invariants where possible. In the example above, it would seem the loop invariant invariant i >= 0 is also required as variable i is mutated within the loop. Whilst the underlying logic does require such an invariant, Dafny automatically infers this and, hence, it can be omitted at the source level.

Dafny includes features which further support its use as a proof assistant. Whilst proofs of simple properties within a function or method (as shown above) are not unusual for tools of this nature, Dafny also allows the proof of properties between one function and another. As is common for a proof assistant, such proofs are often inductive in nature. Dafny is perhaps unusual in employing method invocation as a mechanism for applying the inductive hypothesis. The following illustrates:

datatype List = Nil | Link(data:int,next:List)

function sum(l:List): int {
  match l
    case Nil => 0
    case Link(d,n) => d + sum(n)
}

predicate isNatList(l:List) {
  match l
    case Nil => true
    case Link(d,n) => d >= 0 && isNatList(n)
}

ghost method NatSumLemma(l:List, n:int)
requires isNatList(l) && n == sum(l)
ensures n >= 0 
{
  match l
    case Nil =>
      // Discharged Automatically
    case Link(data,next) => {
      // Apply Inductive Hypothesis
      NatSumLemma(next,sum(next));
      // Check what known by Dafny
      assert data >= 0;
    }
}

Here, NatSumLemma proves a useful property between sum() and isNatList() (i.e. that isNatList(l) ==> (sum(l) >= 0)). The use of a ghost method for encoding lemmas and theorems is standard in Dafny with recursion employed for induction (typically, structural induction). Case analysis is performed using match statements and non-inductive cases are often discharged automatically. The verifier must also have complete access to the contents of a function or predicate in order to unroll them as necessary. This has implications when used in conjunction with access modifiers. Specifically, hiding the contents of a function using the protected modifier can limit what properties can be established about it.

• Dafny: A Language and Program Verifier for Functional Correctness - Microsoft Research
• GitHub - dafny-lang/dafny: Dafny is a verification-aware programming language