$1.$ Rewrite the predicate zerosfirst to handle a sequence of integers instead of an array. Call this new predicate zeros$1.$

$2.$ Write another predicate, called zeros$2,$ which has the same signature as zeros$1$ and is true if every element in the sequence that is zero is preceded by only elements that are zero and is false otherwise. The predicate should be a precise translation of this statement.

$3.$ Validate that zeros$1$ and zeros$2$ are equivalent. Call this validator anything you like that Dafny accepts.

$4.$ Re$-$write the method ZeroSort from this chapter to sort a sequence of integers instead of an array. The signature of the re$-$written method should be:

method ZeroSortSeq $($s:seq$)$ returns $($t:seq$)$

Notice this method returns the sorted sequence in $\backslash ($ t $\backslash ).$ This is necessary because the input sequence $\backslash ($ s $\backslash )$ cannot be modified $($unlike an array$).$

There are two conditions:

$-$ use the same algorithm to sort the sequence as ZeroSort from lectures

$-$ use predicate zeros$1$ in the specification of this method to verify the method's correctness

$5.$ Write a validator that shows ZeroSortSeq has correctly sorted and not corrupted the input sequence. Use the second predicate zeros$2$ in the validator. You may call the validator anything you like that Dafny accepts.

You are not allowed to use arrays anywhere in this exercise, just sequences.

Submit the file ex$2.$dfy$,$ which should contain $1)$ Dafny predicate zeros$1,2)$ predicate zeros$2,3)$ the alidator proving equivalence, $4)$ method ZeroSortSeq and $5)$ the method's validator.