use vstd::prelude::*;
fn main() {}
verus!{



fn f1_find_odd_numbers(arr: &Vec<u32>) -> (odd_numbers: Vec<u32>)
{
    let mut odd_numbers: Vec<u32> = Vec::new();
    let input_len = arr.len();

    let mut index = 0;
    while index < arr.len()
    {
        if (arr[index] % 2 != 0) {
            odd_numbers.push(arr[index]);
        }
        index += 1;
    }
    odd_numbers
}





spec fn f2_is_even(n: u32) -> bool {
    (n % 2) == 0
}

fn f2_is_product_even(arr: &Vec<u32>) -> (result: bool)
{
    let mut index = 0;
    while index < arr.len()
    {
        if (arr[index] % 2 == 0) {
            return true;
        }
        index += 1;
    }
    false
}



spec fn step1(__hp0: Seq<u32>, outp: Seq<u32>) -> bool {
    (outp == __hp0.filter(|x: u32| x % 2 != 0))
}

spec fn step2(inp: Seq<u32>, outp: bool) -> bool {
    (outp <==> (exists|k: int| 0 <= k < inp.len() && f2_is_even(#[trigger] inp[k])))
}

fn chain_driver(arr: &Vec<u32>) -> (r_final: bool)
    ensures
        exists|v1: Seq<u32>| #[trigger] step1(arr@, v1) && step2(v1, r_final),
{
    let x1 = f1_find_odd_numbers(arr);
    let x2 = f2_is_product_even(&x1);
    proof {
        assert(step1(arr@, x1@));
        assert(step2(x1@, x2));
    }
    x2
}

} // verus!