The dafny program below keeps outputting the error attached. This is the code:

predicate zeros$1($s: seq$)$

$\{$

forall i$,$ j :: i $<mtext></mtext><mi>|</mi>$s$|$ && j $<mtext></mtext><mi>|</mi>$s$|$ && s$[$i$]==0$ && s$[$j$]!=0)==>$ i $$ j

$\}$

predicate zeros$2($s: seq$)$

$\{$

forall i :: i $<mtext></mtext><mi>|</mi>$s$|$ && s$[$i$]==0==>($forall j :: j $$ i $==>$ s$[$j$]==0)$

$\}$

lemma ValidateEquivalence$()$

ensures forall s: seq :: zeros$1($s$)==>$ zeros$2($s$)$

ensures forall s: seq :: zeros$2($s$)==>$ zeros$1($s$)$

$\{$

$//$ Proof: Show zeros$1==>$ zeros$2$

forall s $|$ zeros$1($s$)$

ensures zeros$2($s$)$

$\{$

$//$ zeros$1$ implies that if s$[$i$]==0$ and s$[$j$]!=0,$ then i $$ j

$//$ Therefore, all zeros are at the start as specified in zeros$2.$

forall i i $<mtext></mtext><mi>|</mi>$s$|$ && s$[$i$]==0$

ensures forall j :: j $$ i $==>$ s$[$j$]==0$

$\{$

forall j j $$ i

ensures s$[$j$]==0$

$\{$

assert s$[$j$]==0$; $//$ Since j $$ i and s$[$i$]==0,$ s$[$j$]$ cannot be non$-$zero

$\}$

$\}$

$\}$

$//$ Proof: Show zeros$2==>$ zeros$1$

forall s $|$ zeros$2($s$)$

ensures zeros$1($s$)$

$\{$

$//$ zeros$2$ implies that if s$[$i$]==0,$ then all elements before i are also $0$

$//$ Thus, zeros$1$ condition $($i $$ j for s$[$i$]==0$ and s$[$j$]!=0)$ holds.

forall i$,$ j i $<mtext></mtext><mi>|</mi>$s$|$ && j $<mtext></mtext><mi>|</mi>$s$|$ && s$[$i$]==0$ && s$[$j$]!=0$

ensures i $$ j

$\{$

assert i $$ j; $//$ As all zeros are at the start, j must be after all zeros.

$\}$

$\}$

$\}$

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

ensures zeros$1($t$)$

ensures multiset$($t$)==$ multiset$($s$)$

$\{$

t :$=$ s;

var nonz :$=0$;

var next :$=0$;

while next $<mtext></mtext><mi>|</mi>$t$|$

invariant nonz next t$|$

invariant forall i :: i $$ nonz $==>$ t$[$i$]==0$

invariant forall i :: nonz i $$ next $==>$ t$[$i$]!=0$

$//$ COMMENT

invariant multiset$($t$)==$ multiset$($s$)$

$\{$

if t$[$next$]==0\{$

$//$ Move the zero to the beginning of the nonz section

t :$=$ t$[0..$nonz$]+[$t$[$next$]]+$ t$[$nonz$..$next$]+$ t$[$next $+1..]$;

nonz :$=$ nonz $+1$;

$\}$

next :$=$ next $+1$;

$\}$

$\}$

predicate ValidateZeroSort$($s: seq, t: seq$)$

$\{$

zeros$2($t$)$ && multiset$($s$)==$ multiset$($t$)$

$\}$

method Main$()$

$\{$

$//$ test cases

var s$1$ :$=[2,0,1,5,3,0,4,0]$;

var t$1$ :$=$ ZeroSortSeq$($s$1)$;

assert ValidateZeroSort$($s$1,$ t$1)$;

print "Test $1$ passed: $",$ s$1,"->",$ t$1,"$

$"$;

var s$2$ :$=[0,0,0]$;

var t$2$ :$=$ ZeroSortSeq$($s$2)$;

assert ValidateZeroSort$($s$2,$ t$2)$;

print "Test $2$ passed: $",$ s$2,"->",$ t$2,"$

$"$;

var s$3$ :$=[1,2,3]$;

var t$3$ :$=$ ZeroSortSeq$($s$3)$;

assert ValidateZeroSort$($s$3,$ t$3)$;

print "Test $3$ passed: $",$ s$3,"->",$ t$3,"$

$"$;

var s$4$ :$=[]$;

var t$4$ :$=$ ZeroSortSeq$($s$4)$;

assert ValidateZeroSort$($s$4,$ t$4)$;

print "Test $4$ passed: $",$ s$4,"->",$ t$4,"$

$"$;

$\}$