Problem: quick_sort

Your task is to implement the Quick Sort algorithm and verify its correctness in Lean. Given a list of integers, return a new list containing the same elements sorted in non-decreasing order. Postconditions: (1) Output is sorted. (2) Output is a permutation of input. Preconditions: None. Implementation: Recursive divide-and-conquer. Pick a pivot (e.g., head), partition the remaining elements into those less-than-or-equal and those greater-than, then recursively sort and append. Use 'partial' or prove termination (decreasing size).


import Mathlib

-- Precondition definitions
@[reducible, simp]
def quickSort_precond (v : List Int) : Prop :=
  -- !benchmark @start precond
  True
  -- !benchmark @end precond

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

-- Main function definitions
def quickSort (v : List Int) (h_precond : quickSort_precond v) : List Int :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def quickSort_postcond (v : List Int) (result : List Int) (h_precond : quickSort_precond v) : Prop :=
  -- !benchmark @start postcond
   List.Sorted (· ≤ ·) result ∧ result.Perm v
  -- !benchmark @end postcond

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

-- Proof content
theorem quickSort_postcond_satisfied (v : List Int) (h_precond : quickSort_precond v) :
    quickSort_postcond v (quickSort v h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof