Problem: gcd

Your task is to implement the Euclidean Algorithm to compute the Greatest Common Divisor (GCD) of two integers and verify its correctness in Dafny. Preconditions: The inputs a and b are u64. Requirements: Implement the function compute_gcd which returns a u64. (1) Algorithm: Use the classic Euclidean algorithm: repeatedly replace (a, b) with (b, a % b) until b becomes 0. The result is a. (2) Functional Correctness: You must verify that your implementation returns exactly the value defined by the specification spec_gcd. Verification Challenges: (1) Termination: This is the primary challenge. You must prove that the loop terminates. This requires proving that the modulo operation a % b strictly decreases the second argument b in every step (i.e., a % b < b). (2) Correctness Invariant: You must verify that the GCD is preserved throughout the transformation. The invariant spec_gcd(current_a, current_b) == spec_gcd(initial_a, initial_b) must hold at every iteration. This relies on the mathematical property that $GCD(a, b) = GCD(b, a \pmod b)$.

// Following is the block for necessary definitions
// <preamble>
// Mathematical definition of divisibility:
// d divides n if there exists some integer k such that d * k = n.
ghost predicate divides(d: nat, n: nat)
{
  exists k: nat :: d * k == n
}

// Predicate defining the properties of the Greatest Common Divisor (g):
// 1. g must be a common divisor of a and b.
// 2. g must be greater than or equal to any other common divisor d.
ghost predicate is_gcd(g: nat, a: nat, b: nat)
{
  divides(g, a) &&
  divides(g, b) &&
  (forall d: nat :: divides(d, a) && divides(d, b) ==> d <= g)
}

// The declarative specification for GCD.
ghost function spec_gcd(a: nat, b: nat): nat
{
  if a == 0 && b == 0 then
    0
  else
    // Dafny requires proof that such a g exists for the Hilbert epsilon operator (:|)
    // We guard it to satisfy the verifier without proving the number theory lemma here.
    if exists g: nat :: is_gcd(g, a, b) then
      var g: nat :| is_gcd(g, a, b); g
    else
      1 // Unreachable mathematically
}
// </preamble>

// Following is the block for potential helper specifications
// <helpers>

// </helpers>

// Following is the block for proofs of lemmas
// <proofs>

// </proofs>

// Following is the block for the main specification
// <spec>
method compute_gcd(a: int, b: int) returns (res: int)
  // Simulate u64: inputs are non-negative
  requires a >= 0 && b >= 0
  ensures res == spec_gcd(a, b)
  // Result is unsigned
  ensures res >= 0
// </spec>
// <code>
{
  // Implement and verify the Euclidean algorithm for GCD here.
}
// </code>

method Main() {
}