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
}





fn f2_insert_before_each(arr: &Vec<i32>, elem: i32) -> (result: Vec<i32>)
{
    let mut result: Vec<i32> = Vec::new();
    let mut index = 0;

    while index < arr.len()
    {
        result.push(elem);
        result.push(arr[index]);
        index += 1;
    }
    result
}





fn f3_has_only_one_distinct_element(arr: &Vec<i32>) -> (result: bool)
{
    if arr.len() <= 1 {
        return true;
    }
    let mut index = 1;
    while index < arr.len()
    {
        if arr[0] != arr[index] {
            return false;
        }
        index += 1;
    }
    true
}



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_elem: i32, outp: Seq<i32>) -> bool {
    (outp.len() == (2 * inp.len()))
    && (forall|k: int| 0 <= k < inp.len() ==> #[trigger] outp[2 * k] == p2_elem)
    && (forall|k: int| 0 <= k < inp.len() ==> #[trigger] outp[2 * k + 1] == inp[k])
}

spec fn step3(inp: Seq<i32>, outp: bool) -> bool {
    (outp == (forall|i: int| 1 <= i < inp.len() ==> inp[0] == #[trigger] inp[i]))
}

fn chain_driver(nums: &Vec<i32>, p2_elem: 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>, v2: Seq<i32>| #[trigger] step1(nums@, v1) && #[trigger] step2(v1, p2_elem, v2) && step3(v2, r_final),
{
    let x1 = f1_cube_element(nums);
    let x2 = f2_insert_before_each(&x1, p2_elem);
    let x3 = f3_has_only_one_distinct_element(&x2);
    proof {
        assert(step1(nums@, x1@));
        assert(step2(x1@, p2_elem, x2@));
        assert(step3(x2@, x3));
    }
    x3
}

} // verus!