Problem: maximum_subarray_sum

Your task is to implement the algorithm for the Maximum Subarray Sum problem (commonly known as Kadane's Algorithm) and verify its correctness in Dafny. For simplicity, the sequence contains signed integers, and we consider contiguous subarrays (including the empty subarray if applicable). The algorithm needs to return the maximum sum possible from any contiguous subarray. In the incomplete code, it contains the specification for the sum of a sequence slice (spec_sum). Your task is to implement the maximum subarray sum algorithm and verify that the code indeed returns the global maximum sum, i.e., any contiguous subarray has a sum at most the return of the algorithm, and there exists a contiguous subarray that has a sum exactly equal to the returned value.


// Following is the block for necessary definitions
// <preamble>
// Recursive definition of sum for a sequence slice [start, end)
function spec_sum(s: seq<int>, start: int, end: int): int
    requires 0 <= start <= end <= |s|
    decreases end - start
{
    if start >= end then
        0
    else
        s[end - 1] + spec_sum(s, start, end - 1)
}
// </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 max_subarray_sum(s: seq<int>) returns (result: int)
    requires |s| > 0
    requires |s| <= 100000
    ensures
        forall i: int, j: int :: 0 <= i <= j <= |s| ==> spec_sum(s, i, j) <= result
    ensures
        exists i: int, j: int :: 0 <= i <= j <= |s| && spec_sum(s, i, j) == result
// </spec>
// <code>
{
    // Implement and verify the maximum subarray sum algorithm here.
}
// </code>