Problem: maximum_subarray_sum

Your task is to implement the Maximum Subarray Sum algorithm (Kadane's Algorithm) and verify its correctness in Lean. Given a sequence of integers, find the maximum sum of any contiguous subarray. Postconditions: Result is greater than or equal to the sum of any contiguous subarray, and there exists a contiguous subarray with that sum. Basic case handles empty sequence as sum 0 or handling non-empty requirement. Spec assumes non-empty list.


import Mathlib

-- Precondition definitions
@[reducible, simp]
def maxSubarraySum_precond (seq : List Int) : Prop :=
  -- !benchmark @start precond
  0 < seq.length ∧ seq.length ≤ 100000
  -- !benchmark @end precond

def spec_sum (seq : List Int) (i j : Nat) : Int :=
  ((seq.drop i).take (j - i)).foldl (fun acc x => acc + x) 0

-- !benchmark @start auxcode
-- !benchmark @end auxcode

-- Main function definition
def maxSubarraySum (seq : List Int) (h_precond : maxSubarraySum_precond (seq)) : Int :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def maxSubarraySum_postcond (seq : List Int) (result: Int) (h_precond : maxSubarraySum_precond (seq)) : Prop :=
  -- !benchmark @start postcond
  (∀ i j : Nat, i ≤ j → j ≤ seq.length → spec_sum seq i j ≤ result) ∧
  (∃ i j : Nat, i ≤ j ∧ j ≤ seq.length ∧ spec_sum seq i j = result)
  -- !benchmark @end postcond


-- !benchmark @start lemma
-- !benchmark @end lemma

-- Proof content
theorem maxSubarraySum_postcond_satisfied (seq: List Int) (h_precond : maxSubarraySum_precond (seq)) :
    maxSubarraySum_postcond (seq) (maxSubarraySum (seq) h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof