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



fn f1_cube_element(nums: &Vec<i32>) -> (cubed: Vec<i32>)
    requires
        forall|k: int|
            0 <= k < nums.len() ==> (i32::MIN <= #[trigger] nums[k] * #[trigger] nums[k]
                <= i32::MAX),
        forall|k: int|
            0 <= k < nums.len() ==> (i32::MIN <= #[trigger] nums[k] * #[trigger] nums[k]
                * #[trigger] nums[k] <= i32::MAX),
{
    let mut cubed_array: Vec<i32> = Vec::new();
    let mut i = 0;
    while i < nums.len()
    {
        cubed_array.push(nums[i] * nums[i] * nums[i]);
        i += 1;
    }
    cubed_array
}





proof fn f2_lemma_vec_push<T>(vec: Seq<T>, i: T, l: usize)
    requires
        l == vec.len(),
    ensures
        forall|k: int| 0 <= k < vec.len() ==> #[trigger] vec[k] == vec.push(i)[k],
        vec.push(i).index(l as int) == i,
{
}

fn f2_contains(str: &Vec<i32>, key: i32) -> (result: bool)
{
    let mut i = 0;
    while i < str.len()
    {
        if (str[i] == key) {
            return true;
        }
        i += 1;
    }
    false
}

fn f2_remove_elements(arr1: &Vec<i32>, arr2: &Vec<i32>) -> (result: Vec<i32>)
{
    let mut output_str = Vec::new();
    let mut index: usize = 0;

    while index < arr1.len()
    {
        if (!f2_contains(arr2, arr1[index])) {
            
            output_str.push(arr1[index]);
        }
        index += 1;
    }
    output_str
}



spec fn step1(__hp0: Seq<i32>, outp: Seq<i32>) -> bool {
    (forall|i: int|
            0 <= i < __hp0.len() ==> outp[i] == #[trigger] __hp0[i] * #[trigger] __hp0[i]
                * #[trigger] __hp0[i])
}

spec fn step2(inp: Seq<i32>, p2_key: i32, outp: bool) -> bool {
    (outp <==> (exists|i: int| 0 <= i < inp.len() && (inp[i] == p2_key)))
}

fn chain_driver(nums: &Vec<i32>, p2_key: i32) -> (r_final: bool)
    requires
        forall|k: int|
            0 <= k < nums.len() ==> (i32::MIN <= #[trigger] nums[k] * #[trigger] nums[k]
                <= i32::MAX),
        forall|k: int|
            0 <= k < nums.len() ==> (i32::MIN <= #[trigger] nums[k] * #[trigger] nums[k]
                * #[trigger] nums[k] <= i32::MAX),
    ensures
        exists|v1: Seq<i32>| #[trigger] step1(nums@, v1) && step2(v1, p2_key, r_final),
{
    let x1 = f1_cube_element(nums);
    let x2 = f2_contains(&x1, p2_key);
    proof {
        assert(step1(nums@, x1@));
        assert(step2(x1@, p2_key, x2));
    }
    x2
}

} // verus!