3.94/1.85	YES
3.94/1.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.94/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.94/1.87	
3.94/1.87	
3.94/1.87	Termination w.r.t. Q of the given QTRS could be proven:
3.94/1.87	
3.94/1.87	(0) QTRS
3.94/1.87	(1) QTRSToCSRProof [SOUND, 0 ms]
3.94/1.87	(2) CSR
3.94/1.87	(3) CSRInnermostProof [EQUIVALENT, 6 ms]
3.94/1.87	(4) CSR
3.94/1.87	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
3.94/1.87	(6) QCSDP
3.94/1.87	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
3.94/1.87	(8) QCSDP
3.94/1.87	(9) QCSDPSubtermProof [EQUIVALENT, 0 ms]
3.94/1.87	(10) QCSDP
3.94/1.87	(11) PIsEmptyProof [EQUIVALENT, 0 ms]
3.94/1.87	(12) YES
3.94/1.87	
3.94/1.87	
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(0)
3.94/1.87	Obligation:
3.94/1.87	Q restricted rewrite system:
3.94/1.87	The TRS R consists of the following rules:
3.94/1.87	
3.94/1.87	   active(primes) -> mark(sieve(from(s(s(0)))))
3.94/1.87	   active(from(X)) -> mark(cons(X, from(s(X))))
3.94/1.87	   active(head(cons(X, Y))) -> mark(X)
3.94/1.87	   active(tail(cons(X, Y))) -> mark(Y)
3.94/1.87	   active(if(true, X, Y)) -> mark(X)
3.94/1.87	   active(if(false, X, Y)) -> mark(Y)
3.94/1.87	   active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
3.94/1.87	   active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y))))
3.94/1.87	   active(sieve(X)) -> sieve(active(X))
3.94/1.87	   active(from(X)) -> from(active(X))
3.94/1.87	   active(s(X)) -> s(active(X))
3.94/1.87	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.94/1.87	   active(head(X)) -> head(active(X))
3.94/1.87	   active(tail(X)) -> tail(active(X))
3.94/1.87	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
3.94/1.87	   active(filter(X1, X2)) -> filter(active(X1), X2)
3.94/1.87	   active(filter(X1, X2)) -> filter(X1, active(X2))
3.94/1.87	   active(divides(X1, X2)) -> divides(active(X1), X2)
3.94/1.87	   active(divides(X1, X2)) -> divides(X1, active(X2))
3.94/1.87	   sieve(mark(X)) -> mark(sieve(X))
3.94/1.87	   from(mark(X)) -> mark(from(X))
3.94/1.87	   s(mark(X)) -> mark(s(X))
3.94/1.87	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.94/1.87	   head(mark(X)) -> mark(head(X))
3.94/1.87	   tail(mark(X)) -> mark(tail(X))
3.94/1.87	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
3.94/1.87	   filter(mark(X1), X2) -> mark(filter(X1, X2))
3.94/1.87	   filter(X1, mark(X2)) -> mark(filter(X1, X2))
3.94/1.87	   divides(mark(X1), X2) -> mark(divides(X1, X2))
3.94/1.87	   divides(X1, mark(X2)) -> mark(divides(X1, X2))
3.94/1.87	   proper(primes) -> ok(primes)
3.94/1.87	   proper(sieve(X)) -> sieve(proper(X))
3.94/1.87	   proper(from(X)) -> from(proper(X))
3.94/1.87	   proper(s(X)) -> s(proper(X))
3.94/1.87	   proper(0) -> ok(0)
3.94/1.87	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.94/1.87	   proper(head(X)) -> head(proper(X))
3.94/1.87	   proper(tail(X)) -> tail(proper(X))
3.94/1.87	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
3.94/1.87	   proper(true) -> ok(true)
3.94/1.87	   proper(false) -> ok(false)
3.94/1.87	   proper(filter(X1, X2)) -> filter(proper(X1), proper(X2))
3.94/1.87	   proper(divides(X1, X2)) -> divides(proper(X1), proper(X2))
3.94/1.87	   sieve(ok(X)) -> ok(sieve(X))
3.94/1.87	   from(ok(X)) -> ok(from(X))
3.94/1.87	   s(ok(X)) -> ok(s(X))
3.94/1.87	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.94/1.87	   head(ok(X)) -> ok(head(X))
3.94/1.87	   tail(ok(X)) -> ok(tail(X))
3.94/1.87	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
3.94/1.87	   filter(ok(X1), ok(X2)) -> ok(filter(X1, X2))
3.94/1.87	   divides(ok(X1), ok(X2)) -> ok(divides(X1, X2))
3.94/1.87	   top(mark(X)) -> top(proper(X))
3.94/1.87	   top(ok(X)) -> top(active(X))
3.94/1.87	
3.94/1.87	The set Q consists of the following terms:
3.94/1.87	
3.94/1.87	   active(primes)
3.94/1.87	   active(from(x0))
3.94/1.87	   active(sieve(x0))
3.94/1.87	   active(s(x0))
3.94/1.87	   active(cons(x0, x1))
3.94/1.87	   active(head(x0))
3.94/1.87	   active(tail(x0))
3.94/1.87	   active(if(x0, x1, x2))
3.94/1.87	   active(filter(x0, x1))
3.94/1.87	   active(divides(x0, x1))
3.94/1.87	   sieve(mark(x0))
3.94/1.87	   from(mark(x0))
3.94/1.87	   s(mark(x0))
3.94/1.87	   cons(mark(x0), x1)
3.94/1.87	   head(mark(x0))
3.94/1.87	   tail(mark(x0))
3.94/1.87	   if(mark(x0), x1, x2)
3.94/1.87	   filter(mark(x0), x1)
3.94/1.87	   filter(x0, mark(x1))
3.94/1.87	   divides(mark(x0), x1)
3.94/1.87	   divides(x0, mark(x1))
3.94/1.87	   proper(primes)
3.94/1.87	   proper(sieve(x0))
3.94/1.87	   proper(from(x0))
3.94/1.87	   proper(s(x0))
3.94/1.87	   proper(0)
3.94/1.87	   proper(cons(x0, x1))
3.94/1.87	   proper(head(x0))
3.94/1.87	   proper(tail(x0))
3.94/1.87	   proper(if(x0, x1, x2))
3.94/1.87	   proper(true)
3.94/1.87	   proper(false)
3.94/1.87	   proper(filter(x0, x1))
3.94/1.87	   proper(divides(x0, x1))
3.94/1.87	   sieve(ok(x0))
3.94/1.87	   from(ok(x0))
3.94/1.87	   s(ok(x0))
3.94/1.87	   cons(ok(x0), ok(x1))
3.94/1.87	   head(ok(x0))
3.94/1.87	   tail(ok(x0))
3.94/1.87	   if(ok(x0), ok(x1), ok(x2))
3.94/1.87	   filter(ok(x0), ok(x1))
3.94/1.87	   divides(ok(x0), ok(x1))
3.94/1.87	   top(mark(x0))
3.94/1.87	   top(ok(x0))
3.94/1.87	
3.94/1.87	
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(1) QTRSToCSRProof (SOUND)
3.94/1.87	The following Q TRS is given: Q restricted rewrite system:
3.94/1.87	The TRS R consists of the following rules:
3.94/1.87	
3.94/1.87	   active(primes) -> mark(sieve(from(s(s(0)))))
3.94/1.87	   active(from(X)) -> mark(cons(X, from(s(X))))
3.94/1.87	   active(head(cons(X, Y))) -> mark(X)
3.94/1.87	   active(tail(cons(X, Y))) -> mark(Y)
3.94/1.87	   active(if(true, X, Y)) -> mark(X)
3.94/1.87	   active(if(false, X, Y)) -> mark(Y)
3.94/1.87	   active(filter(s(s(X)), cons(Y, Z))) -> mark(if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))))
3.94/1.87	   active(sieve(cons(X, Y))) -> mark(cons(X, filter(X, sieve(Y))))
3.94/1.87	   active(sieve(X)) -> sieve(active(X))
3.94/1.87	   active(from(X)) -> from(active(X))
3.94/1.87	   active(s(X)) -> s(active(X))
3.94/1.87	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.94/1.87	   active(head(X)) -> head(active(X))
3.94/1.87	   active(tail(X)) -> tail(active(X))
3.94/1.87	   active(if(X1, X2, X3)) -> if(active(X1), X2, X3)
3.94/1.87	   active(filter(X1, X2)) -> filter(active(X1), X2)
3.94/1.87	   active(filter(X1, X2)) -> filter(X1, active(X2))
3.94/1.87	   active(divides(X1, X2)) -> divides(active(X1), X2)
3.94/1.87	   active(divides(X1, X2)) -> divides(X1, active(X2))
3.94/1.87	   sieve(mark(X)) -> mark(sieve(X))
3.94/1.87	   from(mark(X)) -> mark(from(X))
3.94/1.87	   s(mark(X)) -> mark(s(X))
3.94/1.87	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.94/1.87	   head(mark(X)) -> mark(head(X))
3.94/1.87	   tail(mark(X)) -> mark(tail(X))
3.94/1.87	   if(mark(X1), X2, X3) -> mark(if(X1, X2, X3))
3.94/1.87	   filter(mark(X1), X2) -> mark(filter(X1, X2))
3.94/1.87	   filter(X1, mark(X2)) -> mark(filter(X1, X2))
3.94/1.87	   divides(mark(X1), X2) -> mark(divides(X1, X2))
3.94/1.87	   divides(X1, mark(X2)) -> mark(divides(X1, X2))
3.94/1.87	   proper(primes) -> ok(primes)
3.94/1.87	   proper(sieve(X)) -> sieve(proper(X))
3.94/1.87	   proper(from(X)) -> from(proper(X))
3.94/1.87	   proper(s(X)) -> s(proper(X))
3.94/1.87	   proper(0) -> ok(0)
3.94/1.87	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.94/1.87	   proper(head(X)) -> head(proper(X))
3.94/1.87	   proper(tail(X)) -> tail(proper(X))
3.94/1.87	   proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3))
3.94/1.87	   proper(true) -> ok(true)
3.94/1.87	   proper(false) -> ok(false)
3.94/1.87	   proper(filter(X1, X2)) -> filter(proper(X1), proper(X2))
3.94/1.87	   proper(divides(X1, X2)) -> divides(proper(X1), proper(X2))
3.94/1.87	   sieve(ok(X)) -> ok(sieve(X))
3.94/1.87	   from(ok(X)) -> ok(from(X))
3.94/1.87	   s(ok(X)) -> ok(s(X))
3.94/1.87	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.94/1.87	   head(ok(X)) -> ok(head(X))
3.94/1.87	   tail(ok(X)) -> ok(tail(X))
3.94/1.87	   if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3))
3.94/1.87	   filter(ok(X1), ok(X2)) -> ok(filter(X1, X2))
3.94/1.87	   divides(ok(X1), ok(X2)) -> ok(divides(X1, X2))
3.94/1.87	   top(mark(X)) -> top(proper(X))
3.94/1.87	   top(ok(X)) -> top(active(X))
3.94/1.87	
3.94/1.87	The set Q consists of the following terms:
3.94/1.87	
3.94/1.87	   active(primes)
3.94/1.87	   active(from(x0))
3.94/1.87	   active(sieve(x0))
3.94/1.87	   active(s(x0))
3.94/1.87	   active(cons(x0, x1))
3.94/1.87	   active(head(x0))
3.94/1.87	   active(tail(x0))
3.94/1.87	   active(if(x0, x1, x2))
3.94/1.87	   active(filter(x0, x1))
3.94/1.87	   active(divides(x0, x1))
3.94/1.87	   sieve(mark(x0))
3.94/1.87	   from(mark(x0))
3.94/1.87	   s(mark(x0))
3.94/1.87	   cons(mark(x0), x1)
3.94/1.87	   head(mark(x0))
3.94/1.87	   tail(mark(x0))
3.94/1.87	   if(mark(x0), x1, x2)
3.94/1.87	   filter(mark(x0), x1)
3.94/1.87	   filter(x0, mark(x1))
3.94/1.87	   divides(mark(x0), x1)
3.94/1.87	   divides(x0, mark(x1))
3.94/1.87	   proper(primes)
3.94/1.87	   proper(sieve(x0))
3.94/1.87	   proper(from(x0))
3.94/1.87	   proper(s(x0))
3.94/1.87	   proper(0)
3.94/1.87	   proper(cons(x0, x1))
3.94/1.87	   proper(head(x0))
3.94/1.87	   proper(tail(x0))
3.94/1.87	   proper(if(x0, x1, x2))
3.94/1.87	   proper(true)
3.94/1.87	   proper(false)
3.94/1.87	   proper(filter(x0, x1))
3.94/1.87	   proper(divides(x0, x1))
3.94/1.87	   sieve(ok(x0))
3.94/1.87	   from(ok(x0))
3.94/1.87	   s(ok(x0))
3.94/1.87	   cons(ok(x0), ok(x1))
3.94/1.87	   head(ok(x0))
3.94/1.87	   tail(ok(x0))
3.94/1.87	   if(ok(x0), ok(x1), ok(x2))
3.94/1.87	   filter(ok(x0), ok(x1))
3.94/1.87	   divides(ok(x0), ok(x1))
3.94/1.87	   top(mark(x0))
3.94/1.87	   top(ok(x0))
3.94/1.87	
3.94/1.87	Special symbols used for the transformation (see [GM04]):
3.94/1.87	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.94/1.87	The replacement map contains the following entries:
3.94/1.87	
3.94/1.87	primes: empty set
3.94/1.87	sieve: {1}
3.94/1.87	from: {1}
3.94/1.87	s: {1}
3.94/1.87	0: empty set
3.94/1.87	cons: {1}
3.94/1.87	head: {1}
3.94/1.87	tail: {1}
3.94/1.87	if: {1}
3.94/1.87	true: empty set
3.94/1.87	false: empty set
3.94/1.87	filter: {1, 2}
3.94/1.87	divides: {1, 2}
3.94/1.87	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(2)
3.94/1.87	Obligation:
3.94/1.87	Context-sensitive rewrite system:
3.94/1.87	The TRS R consists of the following rules:
3.94/1.87	
3.94/1.87	   primes -> sieve(from(s(s(0))))
3.94/1.87	   from(X) -> cons(X, from(s(X)))
3.94/1.87	   head(cons(X, Y)) -> X
3.94/1.87	   tail(cons(X, Y)) -> Y
3.94/1.87	   if(true, X, Y) -> X
3.94/1.87	   if(false, X, Y) -> Y
3.94/1.87	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.87	   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
3.94/1.87	
3.94/1.87	The replacement map contains the following entries:
3.94/1.87	
3.94/1.87	primes: empty set
3.94/1.87	sieve: {1}
3.94/1.87	from: {1}
3.94/1.87	s: {1}
3.94/1.87	0: empty set
3.94/1.87	cons: {1}
3.94/1.87	head: {1}
3.94/1.87	tail: {1}
3.94/1.87	if: {1}
3.94/1.87	true: empty set
3.94/1.87	false: empty set
3.94/1.87	filter: {1, 2}
3.94/1.87	divides: {1, 2}
3.94/1.87	
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(3) CSRInnermostProof (EQUIVALENT)
3.94/1.87	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(4)
3.94/1.87	Obligation:
3.94/1.87	Context-sensitive rewrite system:
3.94/1.87	The TRS R consists of the following rules:
3.94/1.87	
3.94/1.87	   primes -> sieve(from(s(s(0))))
3.94/1.87	   from(X) -> cons(X, from(s(X)))
3.94/1.87	   head(cons(X, Y)) -> X
3.94/1.87	   tail(cons(X, Y)) -> Y
3.94/1.87	   if(true, X, Y) -> X
3.94/1.87	   if(false, X, Y) -> Y
3.94/1.87	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.87	   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
3.94/1.87	
3.94/1.87	The replacement map contains the following entries:
3.94/1.87	
3.94/1.87	primes: empty set
3.94/1.87	sieve: {1}
3.94/1.87	from: {1}
3.94/1.87	s: {1}
3.94/1.87	0: empty set
3.94/1.87	cons: {1}
3.94/1.87	head: {1}
3.94/1.87	tail: {1}
3.94/1.87	if: {1}
3.94/1.87	true: empty set
3.94/1.87	false: empty set
3.94/1.87	filter: {1, 2}
3.94/1.87	divides: {1, 2}
3.94/1.87	
3.94/1.87	
3.94/1.87	Innermost Strategy.
3.94/1.87	
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(5) CSDependencyPairsProof (EQUIVALENT)
3.94/1.87	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(6)
3.94/1.87	Obligation:
3.94/1.87	Q-restricted context-sensitive dependency pair problem:
3.94/1.87	The symbols in {sieve_1, from_1, s_1, head_1, tail_1, filter_2, divides_2, SIEVE_1, FROM_1, FILTER_2, TAIL_1} are replacing on all positions.
3.94/1.87	For all symbols f in {cons_2, if_3, IF_3} we have mu(f) = {1}.
3.94/1.87	The symbols in {U_1} are not replacing on any position.
3.94/1.87	
3.94/1.87	The ordinary context-sensitive dependency pairs DP_o are:
3.94/1.87	   PRIMES -> SIEVE(from(s(s(0))))
3.94/1.87	   PRIMES -> FROM(s(s(0)))
3.94/1.87	   FILTER(s(s(X)), cons(Y, Z)) -> IF(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.87	
3.94/1.87	The collapsing dependency pairs are DP_c:
3.94/1.87	   TAIL(cons(X, Y)) -> Y
3.94/1.87	   IF(true, X, Y) -> X
3.94/1.87	   IF(false, X, Y) -> Y
3.94/1.87	
3.94/1.87	
3.94/1.87	The hidden terms of R are:
3.94/1.87	
3.94/1.87	   from(s(x0))
3.94/1.87	   filter(s(s(x0)), x1)
3.94/1.87	   filter(x0, sieve(x1))
3.94/1.87	   sieve(x0)
3.94/1.87	
3.94/1.87	Every hiding context is built from:
3.94/1.87	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@11a21453
3.94/1.87	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@4d3bd416
3.94/1.87	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@705ddca4
3.94/1.87	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@cabbda4
3.94/1.87	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@d2a20ee
3.94/1.87	
3.94/1.87	Hence, the new unhiding pairs DP_u are :
3.94/1.87	   TAIL(cons(X, Y)) -> U(Y)
3.94/1.87	   IF(true, X, Y) -> U(X)
3.94/1.87	   IF(false, X, Y) -> U(Y)
3.94/1.87	   U(s(x_0)) -> U(x_0)
3.94/1.87	   U(from(x_0)) -> U(x_0)
3.94/1.87	   U(filter(x_0, x_1)) -> U(x_0)
3.94/1.87	   U(filter(x_0, x_1)) -> U(x_1)
3.94/1.87	   U(sieve(x_0)) -> U(x_0)
3.94/1.87	   U(cons(x_0, x_1)) -> U(x_0)
3.94/1.87	   U(from(s(x0))) -> FROM(s(x0))
3.94/1.87	   U(filter(s(s(x0)), x1)) -> FILTER(s(s(x0)), x1)
3.94/1.87	   U(filter(x0, sieve(x1))) -> FILTER(x0, sieve(x1))
3.94/1.87	   U(sieve(x0)) -> SIEVE(x0)
3.94/1.87	
3.94/1.87	The TRS R consists of the following rules:
3.94/1.87	
3.94/1.87	   primes -> sieve(from(s(s(0))))
3.94/1.87	   from(X) -> cons(X, from(s(X)))
3.94/1.87	   head(cons(X, Y)) -> X
3.94/1.87	   tail(cons(X, Y)) -> Y
3.94/1.87	   if(true, X, Y) -> X
3.94/1.87	   if(false, X, Y) -> Y
3.94/1.87	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.87	   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
3.94/1.87	
3.94/1.87	The set Q consists of the following terms:
3.94/1.87	
3.94/1.87	   primes
3.94/1.87	   from(x0)
3.94/1.87	   head(cons(x0, x1))
3.94/1.87	   tail(cons(x0, x1))
3.94/1.87	   if(true, x0, x1)
3.94/1.87	   if(false, x0, x1)
3.94/1.87	   filter(s(s(x0)), cons(x1, x2))
3.94/1.87	   sieve(cons(x0, x1))
3.94/1.87	
3.94/1.87	
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(7) QCSDependencyGraphProof (EQUIVALENT)
3.94/1.87	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 8 less nodes.
3.94/1.87	The rules FILTER(s(s(z0)), cons(z1, z2)) -> IF(divides(s(s(z0)), z1), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) and IF(true, x0, x1) -> U(x0) form no chain, because ECap^mu(IF(divides(s(s(z0)), z1), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1))))) = IF(divides(s(s(z0)), z1), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) does not unify with IF(true, x0, x1). The rules FILTER(s(s(z0)), cons(z1, z2)) -> IF(divides(s(s(z0)), z1), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) and IF(false, x0, x1) -> U(x1) form no chain, because ECap^mu(IF(divides(s(s(z0)), z1), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1))))) = IF(divides(s(s(z0)), z1), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) does not unify with IF(false, x0, x1). 
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(8)
3.94/1.87	Obligation:
3.94/1.87	Q-restricted context-sensitive dependency pair problem:
3.94/1.87	The symbols in {sieve_1, from_1, s_1, head_1, tail_1, filter_2, divides_2} are replacing on all positions.
3.94/1.87	For all symbols f in {cons_2, if_3} we have mu(f) = {1}.
3.94/1.87	The symbols in {U_1} are not replacing on any position.
3.94/1.87	
3.94/1.87	The TRS P consists of the following rules:
3.94/1.87	
3.94/1.87	   U(s(x_0)) -> U(x_0)
3.94/1.87	   U(from(x_0)) -> U(x_0)
3.94/1.87	   U(filter(x_0, x_1)) -> U(x_0)
3.94/1.87	   U(filter(x_0, x_1)) -> U(x_1)
3.94/1.87	   U(sieve(x_0)) -> U(x_0)
3.94/1.87	   U(cons(x_0, x_1)) -> U(x_0)
3.94/1.87	
3.94/1.87	The TRS R consists of the following rules:
3.94/1.87	
3.94/1.87	   primes -> sieve(from(s(s(0))))
3.94/1.87	   from(X) -> cons(X, from(s(X)))
3.94/1.87	   head(cons(X, Y)) -> X
3.94/1.87	   tail(cons(X, Y)) -> Y
3.94/1.87	   if(true, X, Y) -> X
3.94/1.87	   if(false, X, Y) -> Y
3.94/1.87	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.87	   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
3.94/1.87	
3.94/1.87	The set Q consists of the following terms:
3.94/1.87	
3.94/1.87	   primes
3.94/1.87	   from(x0)
3.94/1.87	   head(cons(x0, x1))
3.94/1.87	   tail(cons(x0, x1))
3.94/1.87	   if(true, x0, x1)
3.94/1.87	   if(false, x0, x1)
3.94/1.87	   filter(s(s(x0)), cons(x1, x2))
3.94/1.87	   sieve(cons(x0, x1))
3.94/1.87	
3.94/1.87	
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(9) QCSDPSubtermProof (EQUIVALENT)
3.94/1.87	We use the subterm processor [DA_EMMES].
3.94/1.87	
3.94/1.87	
3.94/1.87	The following pairs can be oriented strictly and are deleted.
3.94/1.87	
3.94/1.87	   U(s(x_0)) -> U(x_0)
3.94/1.87	   U(from(x_0)) -> U(x_0)
3.94/1.87	   U(filter(x_0, x_1)) -> U(x_0)
3.94/1.87	   U(filter(x_0, x_1)) -> U(x_1)
3.94/1.87	   U(sieve(x_0)) -> U(x_0)
3.94/1.87	   U(cons(x_0, x_1)) -> U(x_0)
3.94/1.87	The remaining pairs can at least be oriented weakly.
3.94/1.87	none
3.94/1.87	Used ordering:  Combined order from the following AFS and order.
3.94/1.87	U(x1)  =  x1
3.94/1.87	
3.94/1.87	
3.94/1.87	Subterm Order
3.94/1.87	
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(10)
3.94/1.87	Obligation:
3.94/1.87	Q-restricted context-sensitive dependency pair problem:
3.94/1.87	The symbols in {sieve_1, from_1, s_1, head_1, tail_1, filter_2, divides_2} are replacing on all positions.
3.94/1.87	For all symbols f in {cons_2, if_3} we have mu(f) = {1}.
3.94/1.87	
3.94/1.87	The TRS P consists of the following rules:
3.94/1.87	none
3.94/1.87	
3.94/1.87	The TRS R consists of the following rules:
3.94/1.87	
3.94/1.87	   primes -> sieve(from(s(s(0))))
3.94/1.87	   from(X) -> cons(X, from(s(X)))
3.94/1.87	   head(cons(X, Y)) -> X
3.94/1.87	   tail(cons(X, Y)) -> Y
3.94/1.87	   if(true, X, Y) -> X
3.94/1.87	   if(false, X, Y) -> Y
3.94/1.87	   filter(s(s(X)), cons(Y, Z)) -> if(divides(s(s(X)), Y), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y))))
3.94/1.87	   sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
3.94/1.87	
3.94/1.87	The set Q consists of the following terms:
3.94/1.87	
3.94/1.87	   primes
3.94/1.87	   from(x0)
3.94/1.87	   head(cons(x0, x1))
3.94/1.87	   tail(cons(x0, x1))
3.94/1.87	   if(true, x0, x1)
3.94/1.87	   if(false, x0, x1)
3.94/1.87	   filter(s(s(x0)), cons(x1, x2))
3.94/1.87	   sieve(cons(x0, x1))
3.94/1.87	
3.94/1.87	
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(11) PIsEmptyProof (EQUIVALENT)
3.94/1.87	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
3.94/1.87	----------------------------------------
3.94/1.87	
3.94/1.87	(12)
3.94/1.87	YES
4.23/1.89	EOF