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



fn f1_replace_last_element(first: &Vec<i32>, second: &Vec<i32>) -> (replaced_list: Vec<i32>)
    requires
        first.len() > 0,
{
    let mut replaced_list = Vec::new();
    let first_end = first.len() - 1;
    let mut index = 0;

    while index < first_end
    {
        replaced_list.push(first[index]);
        index += 1;
    }
    index = 0;
    while index < second.len()
    {
        replaced_list.push(second[index]);
        index += 1;
    }
    replaced_list
}





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
}





proof fn f3_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 f3_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 f3_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 (!f3_contains(arr2, arr1[index])) {
            
            output_str.push(arr1[index]);
        }
        index += 1;
    }
    output_str
}



spec fn step1(__hp0: Seq<i32>, __hp1: Seq<i32>, outp: Seq<i32>) -> bool {
    (outp == __hp0.subrange(0, __hp0.len() - 1).add(__hp1))
}

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>, p3_key: i32, outp: bool) -> bool {
    (outp <==> (exists|i: int| 0 <= i < inp.len() && (inp[i] == p3_key)))
}

fn chain_driver(first: &Vec<i32>, second: &Vec<i32>, p2_elem: i32, p3_key: i32) -> (r_final: bool)
    requires
        first.len() > 0,
    ensures
        exists|v1: Seq<i32>, v2: Seq<i32>| #[trigger] step1(first@, second@, v1) && #[trigger] step2(v1, p2_elem, v2) && step3(v2, p3_key, r_final),
{
    let x1 = f1_replace_last_element(first, second);
    let x2 = f2_insert_before_each(&x1, p2_elem);
    let x3 = f3_contains(&x2, p3_key);
    proof {
        assert(step1(first@, second@, x1@));
        assert(step2(x1@, p2_elem, x2@));
        assert(step3(x2@, p3_key, x3));
    }
    x3
}

} // verus!