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



fn f1_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 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(arr: &Vec<i32>, key: i32) -> (result: bool)
{
    let mut index = 0;
    while index < arr.len()
    {
        if (arr[index] == key) {
            return true;
        }
        index += 1;
    }
    false
}

fn f2_find_dissimilar(arr1: &Vec<i32>, arr2: &Vec<i32>) -> (result: Vec<i32>)
{
    let mut result = Vec::new();

    let mut index = 0;
    while index < arr1.len()
    {
        if (!f2_contains(arr2, arr1[index]) && !f2_contains(&result, arr1[index])) {
            
            result.push(arr1[index]);

        }
        index += 1;
    }
    let mut index = 0;
    while index < arr2.len()
    {
        if (!f2_contains(arr1, arr2[index]) && !f2_contains(&result, arr2[index])) {
            
            result.push(arr2[index]);
        }
        index += 1;
    }

    result
}



spec fn step1(__hp0: Seq<i32>, __hp1: i32, outp: Seq<i32>) -> bool {
    (outp.len() == (2 * __hp0.len()))
    && (forall|k: int| 0 <= k < __hp0.len() ==> #[trigger] outp[2 * k] == __hp1)
    && (forall|k: int| 0 <= k < __hp0.len() ==> #[trigger] outp[2 * k + 1] == __hp0[k])
}

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(arr: &Vec<i32>, elem: i32, p2_key: i32) -> (r_final: bool)
    ensures
        exists|v1: Seq<i32>| #[trigger] step1(arr@, elem, v1) && step2(v1, p2_key, r_final),
{
    let x1 = f1_insert_before_each(arr, elem);
    let x2 = f2_contains(&x1, p2_key);
    proof {
        assert(step1(arr@, elem, x1@));
        assert(step2(x1@, p2_key, x2));
    }
    x2
}

} // verus!