How to check this sequence element is equal to that sequence element in Dafny - dafny

Question

I have 2 sequences a:seq and b:seq, I wonder if we use the function, how we can determine that the element at this index in seq a is equal to element at this index in seq b
function test(s:seq<nat>, u:seq<nat>): nat
ensures |s|>0
ensures |u|>0
ensures |s| == |u|
{
// Code
}
method Testing()
{
var sys:seq<nat> := [4,2,9,3,1];
var usr:seq<nat> := [1,2,3,4,5];
assert test(sys, usr) == 1
// The element at the index 2 of sys and usr are equal, so it have 1 element that match in both 2 sequence
}
Because of the function I could not create a while loop, so I can not do the basic logic on that, so I wonder if there's something that fit the requirement.

Answer 1

After researching and working by Python to find the recursion in Python, finally I found the answer for this:
function bullspec(s:seq<nat>, u:seq<nat>): nat
requires |s| > 0
requires |u| > 0
requires |s| == |u|
{
var index:=0;
if |s| == 1 then (
if s[0]==u[0]
then 1 else 0
) else (
if s[index] != u[index]
then bullspec(s[index+1..],u[index+1..])
else 1+bullspec(s[index+1..],u[index+1..])
)
}

Answer 2

This is a wonderful problem to solve with Dafny.
Let me state the problem in clear:
Given two sequences of the same length, find the first index at which these sequences are equal. Otherwise return the length of the sequences.
That formulation makes it possible to not require that sequences are non-empty.
Thus, we can start with the following definition
function bullspec(s:seq<nat>, u:seq<nat>): (r: nat)
requires |s| == |u|
// Ensures r is either a sequence index or the sequence length
ensures r <= |s|
// All the elements before r are different
ensures forall i: nat | i < r :: s[i] != u[i]
// Either r is the sequence length or the elements at index r are equal
ensures r == |s| || s[r] == u[r]
{
Now, if you manage to prove this function, you will have prove that this function does what you want it to do.
To obtain the body of the function, you usually have to check whether the sequence if empty. In our case, we can return 0, which is the length of the sequence.
if |s| == 0 then 0 else
If the sequence is not empty, then we can compare the first elements. If they are equal, then we return the index 0
if s[0] == u[0] then 0 else
Otherwise, what happens if we recurse into bullspec(s[1..],u[1..])? We will obtain an index that is offset by 1 ! So we only need to add 1 to it.
1 + bullspec(s[1..],u[1..])
}
With this, you can verify that your function does exactly what you intended it to do.