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



proof fn f1_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 f1_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 f1_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 (!f1_contains(arr2, arr1[index])) {
            
            output_str.push(arr1[index]);
        }
        index += 1;
    }
    output_str
}





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

fn f2_intersection(arr1: &Vec<i32>, arr2: &Vec<i32>) -> (result: Vec<i32>)
{
    let mut output_arr = Vec::new();

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



spec fn step1(__hp0: Seq<i32>, __hp1: Seq<i32>, outp: Seq<i32>) -> bool {
    (forall|i: int|
            0 <= i < outp.len() ==> (__hp0.contains(#[trigger] outp[i]) && !__hp1.contains(
                #[trigger] outp[i],
            )))
    && (forall|i: int|
            0 <= i < __hp0.len() ==> (__hp1.contains(#[trigger] __hp0[i]) || outp.contains(
                #[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(arr1: &Vec<i32>, arr2: &Vec<i32>, p2_key: i32) -> (r_final: bool)
    ensures
        exists|v1: Seq<i32>| #[trigger] step1(arr1@, arr2@, v1) && step2(v1, p2_key, r_final),
{
    let x1 = f1_remove_elements(arr1, arr2);
    let x2 = f2_contains(&x1, p2_key);
    proof {
        assert(step1(arr1@, arr2@, x1@));
        assert(step2(x1@, p2_key, x2));
    }
    x2
}

} // verus!