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

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

$\}$

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

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

$\}$

lemma ValidateZerosEquivalence$($s: seq$)$

ensures zeros$1($s zeros$2($s$)$

$\{$

$//$ Prove zeros$1$ implies zeros$2$

if zeros$1($s$)\{$

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

$//$ Assume i $<mtext></mtext><mi>|</mi>$s$|$ and s$[$i$]!=0$

assume i $<mtext></mtext><mi>|</mi>$s$|$ && s$[$i$]!=0$;

$//$ Prove forall j :: j $$ i $==>$ s$[$j$]!=0$

forall j :: j $$ i $==>$ s$[$j$]!=0\{$

assume j $$ i; $//$ Assume the range for j

$//$ From zeros$1$ definition: If s$[$j$]==0,$ then s$[$i$]==0,$ which contradicts s$[$i$]!=0$

assert s$[$j$]!=0$;

$\}$

$\}$

$\}$

$//$ Prove zeros$2$ implies zeros$1$

if zeros$2($s$)\{$

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

$//$ Assume i $$ j $<mtext></mtext><mi>|</mi>$s$|$ and prove s$[$i$]!=0||$ s$[$j$]==0$

assume i $$ j $<mtext></mtext><mi>|</mi>$s$|$;

if s$[$i$]==0\{$

$//$ If s$[$i$]==0,$ then zeros$2$ guarantees that s$[$j$]==0$

assert s$[$j$]==0$;

$\}$ else $\{$

$//$ Otherwise, s$[$i$]!=0$ implies the condition is already satisfied

assert s$[$i$]!=0$;

$\}$

$\}$

$\}$

$\}$

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

ensures zeros$1($t$)$

ensures t $==($s$.$filter$($x $=>$ x $==0)+$ s$.$filter$($x $=>$ x $!=0))$

$\{$

var zeros :$=$ s$.$filter$($x $=>$ x $==0)$;

var nonZeros :$=$ s$.$filter$($x $=>$ x $!=0)$;

t :$=$ zeros $+$ nonZeros;

$\}$

lemma ValidateZeroSortSeq$($s: seq, t: seq$)$

requires t $==$ ZeroSortSeq$($s$)$

ensures zeros$2($t$)$

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

$\{$

assert zeros$2($t$)$;

assert multiset$($t$)==$ multiset$($s$)$;

$\}$

this is my code ; this is the error ; question is attached; it is not a syntax error there is no missiring $\}$ there is something wrong with the code pls help me figure it out make sure your code is right before you senf it to me

n the Linear Sort chapter in the lecture slides, a predicate zerosfirst and method ZeroSort are shown. $1.$ 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 t$.$ This is necessary because the input sequence s 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.