4.03/1.99	YES
4.10/2.00	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.10/2.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.10/2.00	
4.10/2.00	
4.10/2.00	Termination w.r.t. Q of the given QTRS could be proven:
4.10/2.00	
4.10/2.00	(0) QTRS
4.10/2.00	(1) QTRSToCSRProof [SOUND, 0 ms]
4.10/2.00	(2) CSR
4.10/2.00	(3) CSRInnermostProof [EQUIVALENT, 0 ms]
4.10/2.00	(4) CSR
4.10/2.00	(5) CSDependencyPairsProof [EQUIVALENT, 1 ms]
4.10/2.00	(6) QCSDP
4.10/2.00	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.10/2.00	(8) AND
4.10/2.00	    (9) QCSDP
4.10/2.00	        (10) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.10/2.00	        (11) QCSDP
4.10/2.00	        (12) PIsEmptyProof [EQUIVALENT, 0 ms]
4.10/2.00	        (13) YES
4.10/2.00	    (14) QCSDP
4.10/2.00	        (15) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.10/2.00	        (16) QCSDP
4.10/2.00	        (17) PIsEmptyProof [EQUIVALENT, 0 ms]
4.10/2.00	        (18) YES
4.10/2.00	    (19) QCSDP
4.10/2.00	        (20) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.10/2.00	        (21) QCSDP
4.10/2.00	        (22) PIsEmptyProof [EQUIVALENT, 0 ms]
4.10/2.00	        (23) YES
4.10/2.00	    (24) QCSDP
4.10/2.00	        (25) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.10/2.00	        (26) QCSDP
4.10/2.00	        (27) PIsEmptyProof [EQUIVALENT, 0 ms]
4.10/2.00	        (28) YES
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(0)
4.10/2.00	Obligation:
4.10/2.00	Q restricted rewrite system:
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   active(from(X)) -> mark(cons(X, from(s(X))))
4.10/2.00	   active(2ndspos(0, Z)) -> mark(rnil)
4.10/2.00	   active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
4.10/2.00	   active(2ndsneg(0, Z)) -> mark(rnil)
4.10/2.00	   active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
4.10/2.00	   active(pi(X)) -> mark(2ndspos(X, from(0)))
4.10/2.00	   active(plus(0, Y)) -> mark(Y)
4.10/2.00	   active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
4.10/2.00	   active(times(0, Y)) -> mark(0)
4.10/2.00	   active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
4.10/2.00	   active(square(X)) -> mark(times(X, X))
4.10/2.00	   active(s(X)) -> s(active(X))
4.10/2.00	   active(posrecip(X)) -> posrecip(active(X))
4.10/2.00	   active(negrecip(X)) -> negrecip(active(X))
4.10/2.00	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.10/2.00	   active(rcons(X1, X2)) -> rcons(active(X1), X2)
4.10/2.00	   active(rcons(X1, X2)) -> rcons(X1, active(X2))
4.10/2.00	   active(from(X)) -> from(active(X))
4.10/2.00	   active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
4.10/2.00	   active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
4.10/2.00	   active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
4.10/2.00	   active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
4.10/2.00	   active(pi(X)) -> pi(active(X))
4.10/2.00	   active(plus(X1, X2)) -> plus(active(X1), X2)
4.10/2.00	   active(plus(X1, X2)) -> plus(X1, active(X2))
4.10/2.00	   active(times(X1, X2)) -> times(active(X1), X2)
4.10/2.00	   active(times(X1, X2)) -> times(X1, active(X2))
4.10/2.00	   active(square(X)) -> square(active(X))
4.10/2.00	   s(mark(X)) -> mark(s(X))
4.10/2.00	   posrecip(mark(X)) -> mark(posrecip(X))
4.10/2.00	   negrecip(mark(X)) -> mark(negrecip(X))
4.10/2.00	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.10/2.00	   rcons(mark(X1), X2) -> mark(rcons(X1, X2))
4.10/2.00	   rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
4.10/2.00	   from(mark(X)) -> mark(from(X))
4.10/2.00	   2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
4.10/2.00	   2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
4.10/2.00	   2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
4.10/2.00	   2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
4.10/2.00	   pi(mark(X)) -> mark(pi(X))
4.10/2.00	   plus(mark(X1), X2) -> mark(plus(X1, X2))
4.10/2.00	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
4.10/2.00	   times(mark(X1), X2) -> mark(times(X1, X2))
4.10/2.00	   times(X1, mark(X2)) -> mark(times(X1, X2))
4.10/2.00	   square(mark(X)) -> mark(square(X))
4.10/2.00	   proper(0) -> ok(0)
4.10/2.00	   proper(s(X)) -> s(proper(X))
4.10/2.00	   proper(posrecip(X)) -> posrecip(proper(X))
4.10/2.00	   proper(negrecip(X)) -> negrecip(proper(X))
4.10/2.00	   proper(nil) -> ok(nil)
4.10/2.00	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.10/2.00	   proper(rnil) -> ok(rnil)
4.10/2.00	   proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
4.10/2.00	   proper(from(X)) -> from(proper(X))
4.10/2.00	   proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
4.10/2.00	   proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
4.10/2.00	   proper(pi(X)) -> pi(proper(X))
4.10/2.00	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
4.10/2.00	   proper(times(X1, X2)) -> times(proper(X1), proper(X2))
4.10/2.00	   proper(square(X)) -> square(proper(X))
4.10/2.00	   s(ok(X)) -> ok(s(X))
4.10/2.00	   posrecip(ok(X)) -> ok(posrecip(X))
4.10/2.00	   negrecip(ok(X)) -> ok(negrecip(X))
4.10/2.00	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.10/2.00	   rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
4.10/2.00	   from(ok(X)) -> ok(from(X))
4.10/2.00	   2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
4.10/2.00	   2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
4.10/2.00	   pi(ok(X)) -> ok(pi(X))
4.10/2.00	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
4.10/2.00	   times(ok(X1), ok(X2)) -> ok(times(X1, X2))
4.10/2.00	   square(ok(X)) -> ok(square(X))
4.10/2.00	   top(mark(X)) -> top(proper(X))
4.10/2.00	   top(ok(X)) -> top(active(X))
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   active(from(x0))
4.10/2.00	   active(pi(x0))
4.10/2.00	   active(square(x0))
4.10/2.00	   active(s(x0))
4.10/2.00	   active(posrecip(x0))
4.10/2.00	   active(negrecip(x0))
4.10/2.00	   active(cons(x0, x1))
4.10/2.00	   active(rcons(x0, x1))
4.10/2.00	   active(2ndspos(x0, x1))
4.10/2.00	   active(2ndsneg(x0, x1))
4.10/2.00	   active(plus(x0, x1))
4.10/2.00	   active(times(x0, x1))
4.10/2.00	   s(mark(x0))
4.10/2.00	   posrecip(mark(x0))
4.10/2.00	   negrecip(mark(x0))
4.10/2.00	   cons(mark(x0), x1)
4.10/2.00	   rcons(mark(x0), x1)
4.10/2.00	   rcons(x0, mark(x1))
4.10/2.00	   from(mark(x0))
4.10/2.00	   2ndspos(mark(x0), x1)
4.10/2.00	   2ndspos(x0, mark(x1))
4.10/2.00	   2ndsneg(mark(x0), x1)
4.10/2.00	   2ndsneg(x0, mark(x1))
4.10/2.00	   pi(mark(x0))
4.10/2.00	   plus(mark(x0), x1)
4.10/2.00	   plus(x0, mark(x1))
4.10/2.00	   times(mark(x0), x1)
4.10/2.00	   times(x0, mark(x1))
4.10/2.00	   square(mark(x0))
4.10/2.00	   proper(0)
4.10/2.00	   proper(s(x0))
4.10/2.00	   proper(posrecip(x0))
4.10/2.00	   proper(negrecip(x0))
4.10/2.00	   proper(nil)
4.10/2.00	   proper(cons(x0, x1))
4.10/2.00	   proper(rnil)
4.10/2.00	   proper(rcons(x0, x1))
4.10/2.00	   proper(from(x0))
4.10/2.00	   proper(2ndspos(x0, x1))
4.10/2.00	   proper(2ndsneg(x0, x1))
4.10/2.00	   proper(pi(x0))
4.10/2.00	   proper(plus(x0, x1))
4.10/2.00	   proper(times(x0, x1))
4.10/2.00	   proper(square(x0))
4.10/2.00	   s(ok(x0))
4.10/2.00	   posrecip(ok(x0))
4.10/2.00	   negrecip(ok(x0))
4.10/2.00	   cons(ok(x0), ok(x1))
4.10/2.00	   rcons(ok(x0), ok(x1))
4.10/2.00	   from(ok(x0))
4.10/2.00	   2ndspos(ok(x0), ok(x1))
4.10/2.00	   2ndsneg(ok(x0), ok(x1))
4.10/2.00	   pi(ok(x0))
4.10/2.00	   plus(ok(x0), ok(x1))
4.10/2.00	   times(ok(x0), ok(x1))
4.10/2.00	   square(ok(x0))
4.10/2.00	   top(mark(x0))
4.10/2.00	   top(ok(x0))
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(1) QTRSToCSRProof (SOUND)
4.10/2.00	The following Q TRS is given: Q restricted rewrite system:
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   active(from(X)) -> mark(cons(X, from(s(X))))
4.10/2.00	   active(2ndspos(0, Z)) -> mark(rnil)
4.10/2.00	   active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
4.10/2.00	   active(2ndsneg(0, Z)) -> mark(rnil)
4.10/2.00	   active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z)))
4.10/2.00	   active(pi(X)) -> mark(2ndspos(X, from(0)))
4.10/2.00	   active(plus(0, Y)) -> mark(Y)
4.10/2.00	   active(plus(s(X), Y)) -> mark(s(plus(X, Y)))
4.10/2.00	   active(times(0, Y)) -> mark(0)
4.10/2.00	   active(times(s(X), Y)) -> mark(plus(Y, times(X, Y)))
4.10/2.00	   active(square(X)) -> mark(times(X, X))
4.10/2.00	   active(s(X)) -> s(active(X))
4.10/2.00	   active(posrecip(X)) -> posrecip(active(X))
4.10/2.00	   active(negrecip(X)) -> negrecip(active(X))
4.10/2.00	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.10/2.00	   active(rcons(X1, X2)) -> rcons(active(X1), X2)
4.10/2.00	   active(rcons(X1, X2)) -> rcons(X1, active(X2))
4.10/2.00	   active(from(X)) -> from(active(X))
4.10/2.00	   active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2)
4.10/2.00	   active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2))
4.10/2.00	   active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2)
4.10/2.00	   active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2))
4.10/2.00	   active(pi(X)) -> pi(active(X))
4.10/2.00	   active(plus(X1, X2)) -> plus(active(X1), X2)
4.10/2.00	   active(plus(X1, X2)) -> plus(X1, active(X2))
4.10/2.00	   active(times(X1, X2)) -> times(active(X1), X2)
4.10/2.00	   active(times(X1, X2)) -> times(X1, active(X2))
4.10/2.00	   active(square(X)) -> square(active(X))
4.10/2.00	   s(mark(X)) -> mark(s(X))
4.10/2.00	   posrecip(mark(X)) -> mark(posrecip(X))
4.10/2.00	   negrecip(mark(X)) -> mark(negrecip(X))
4.10/2.00	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.10/2.00	   rcons(mark(X1), X2) -> mark(rcons(X1, X2))
4.10/2.00	   rcons(X1, mark(X2)) -> mark(rcons(X1, X2))
4.10/2.00	   from(mark(X)) -> mark(from(X))
4.10/2.00	   2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2))
4.10/2.00	   2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2))
4.10/2.00	   2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2))
4.10/2.00	   2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2))
4.10/2.00	   pi(mark(X)) -> mark(pi(X))
4.10/2.00	   plus(mark(X1), X2) -> mark(plus(X1, X2))
4.10/2.00	   plus(X1, mark(X2)) -> mark(plus(X1, X2))
4.10/2.00	   times(mark(X1), X2) -> mark(times(X1, X2))
4.10/2.00	   times(X1, mark(X2)) -> mark(times(X1, X2))
4.10/2.00	   square(mark(X)) -> mark(square(X))
4.10/2.00	   proper(0) -> ok(0)
4.10/2.00	   proper(s(X)) -> s(proper(X))
4.10/2.00	   proper(posrecip(X)) -> posrecip(proper(X))
4.10/2.00	   proper(negrecip(X)) -> negrecip(proper(X))
4.10/2.00	   proper(nil) -> ok(nil)
4.10/2.00	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.10/2.00	   proper(rnil) -> ok(rnil)
4.10/2.00	   proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2))
4.10/2.00	   proper(from(X)) -> from(proper(X))
4.10/2.00	   proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2))
4.10/2.00	   proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2))
4.10/2.00	   proper(pi(X)) -> pi(proper(X))
4.10/2.00	   proper(plus(X1, X2)) -> plus(proper(X1), proper(X2))
4.10/2.00	   proper(times(X1, X2)) -> times(proper(X1), proper(X2))
4.10/2.00	   proper(square(X)) -> square(proper(X))
4.10/2.00	   s(ok(X)) -> ok(s(X))
4.10/2.00	   posrecip(ok(X)) -> ok(posrecip(X))
4.10/2.00	   negrecip(ok(X)) -> ok(negrecip(X))
4.10/2.00	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.10/2.00	   rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2))
4.10/2.00	   from(ok(X)) -> ok(from(X))
4.10/2.00	   2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2))
4.10/2.00	   2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2))
4.10/2.00	   pi(ok(X)) -> ok(pi(X))
4.10/2.00	   plus(ok(X1), ok(X2)) -> ok(plus(X1, X2))
4.10/2.00	   times(ok(X1), ok(X2)) -> ok(times(X1, X2))
4.10/2.00	   square(ok(X)) -> ok(square(X))
4.10/2.00	   top(mark(X)) -> top(proper(X))
4.10/2.00	   top(ok(X)) -> top(active(X))
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   active(from(x0))
4.10/2.00	   active(pi(x0))
4.10/2.00	   active(square(x0))
4.10/2.00	   active(s(x0))
4.10/2.00	   active(posrecip(x0))
4.10/2.00	   active(negrecip(x0))
4.10/2.00	   active(cons(x0, x1))
4.10/2.00	   active(rcons(x0, x1))
4.10/2.00	   active(2ndspos(x0, x1))
4.10/2.00	   active(2ndsneg(x0, x1))
4.10/2.00	   active(plus(x0, x1))
4.10/2.00	   active(times(x0, x1))
4.10/2.00	   s(mark(x0))
4.10/2.00	   posrecip(mark(x0))
4.10/2.00	   negrecip(mark(x0))
4.10/2.00	   cons(mark(x0), x1)
4.10/2.00	   rcons(mark(x0), x1)
4.10/2.00	   rcons(x0, mark(x1))
4.10/2.00	   from(mark(x0))
4.10/2.00	   2ndspos(mark(x0), x1)
4.10/2.00	   2ndspos(x0, mark(x1))
4.10/2.00	   2ndsneg(mark(x0), x1)
4.10/2.00	   2ndsneg(x0, mark(x1))
4.10/2.00	   pi(mark(x0))
4.10/2.00	   plus(mark(x0), x1)
4.10/2.00	   plus(x0, mark(x1))
4.10/2.00	   times(mark(x0), x1)
4.10/2.00	   times(x0, mark(x1))
4.10/2.00	   square(mark(x0))
4.10/2.00	   proper(0)
4.10/2.00	   proper(s(x0))
4.10/2.00	   proper(posrecip(x0))
4.10/2.00	   proper(negrecip(x0))
4.10/2.00	   proper(nil)
4.10/2.00	   proper(cons(x0, x1))
4.10/2.00	   proper(rnil)
4.10/2.00	   proper(rcons(x0, x1))
4.10/2.00	   proper(from(x0))
4.10/2.00	   proper(2ndspos(x0, x1))
4.10/2.00	   proper(2ndsneg(x0, x1))
4.10/2.00	   proper(pi(x0))
4.10/2.00	   proper(plus(x0, x1))
4.10/2.00	   proper(times(x0, x1))
4.10/2.00	   proper(square(x0))
4.10/2.00	   s(ok(x0))
4.10/2.00	   posrecip(ok(x0))
4.10/2.00	   negrecip(ok(x0))
4.10/2.00	   cons(ok(x0), ok(x1))
4.10/2.00	   rcons(ok(x0), ok(x1))
4.10/2.00	   from(ok(x0))
4.10/2.00	   2ndspos(ok(x0), ok(x1))
4.10/2.00	   2ndsneg(ok(x0), ok(x1))
4.10/2.00	   pi(ok(x0))
4.10/2.00	   plus(ok(x0), ok(x1))
4.10/2.00	   times(ok(x0), ok(x1))
4.10/2.00	   square(ok(x0))
4.10/2.00	   top(mark(x0))
4.10/2.00	   top(ok(x0))
4.10/2.00	
4.10/2.00	Special symbols used for the transformation (see [GM04]):
4.10/2.00	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
4.10/2.00	The replacement map contains the following entries:
4.10/2.00	
4.10/2.00	from: {1}
4.10/2.00	cons: {1}
4.10/2.00	s: {1}
4.10/2.00	2ndspos: {1, 2}
4.10/2.00	0: empty set
4.10/2.00	rnil: empty set
4.10/2.00	rcons: {1, 2}
4.10/2.00	posrecip: {1}
4.10/2.00	2ndsneg: {1, 2}
4.10/2.00	negrecip: {1}
4.10/2.00	pi: {1}
4.10/2.00	plus: {1, 2}
4.10/2.00	times: {1, 2}
4.10/2.00	square: {1}
4.10/2.00	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.
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(2)
4.10/2.00	Obligation:
4.10/2.00	Context-sensitive rewrite system:
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The replacement map contains the following entries:
4.10/2.00	
4.10/2.00	from: {1}
4.10/2.00	cons: {1}
4.10/2.00	s: {1}
4.10/2.00	2ndspos: {1, 2}
4.10/2.00	0: empty set
4.10/2.00	rnil: empty set
4.10/2.00	rcons: {1, 2}
4.10/2.00	posrecip: {1}
4.10/2.00	2ndsneg: {1, 2}
4.10/2.00	negrecip: {1}
4.10/2.00	pi: {1}
4.10/2.00	plus: {1, 2}
4.10/2.00	times: {1, 2}
4.10/2.00	square: {1}
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(3) CSRInnermostProof (EQUIVALENT)
4.10/2.00	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(4)
4.10/2.00	Obligation:
4.10/2.00	Context-sensitive rewrite system:
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The replacement map contains the following entries:
4.10/2.00	
4.10/2.00	from: {1}
4.10/2.00	cons: {1}
4.10/2.00	s: {1}
4.10/2.00	2ndspos: {1, 2}
4.10/2.00	0: empty set
4.10/2.00	rnil: empty set
4.10/2.00	rcons: {1, 2}
4.10/2.00	posrecip: {1}
4.10/2.00	2ndsneg: {1, 2}
4.10/2.00	negrecip: {1}
4.10/2.00	pi: {1}
4.10/2.00	plus: {1, 2}
4.10/2.00	times: {1, 2}
4.10/2.00	square: {1}
4.10/2.00	
4.10/2.00	
4.10/2.00	Innermost Strategy.
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(5) CSDependencyPairsProof (EQUIVALENT)
4.10/2.00	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(6)
4.10/2.00	Obligation:
4.10/2.00	Q-restricted context-sensitive dependency pair problem:
4.10/2.00	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1, 2NDSNEG_2, 2NDSPOS_2, PI_1, FROM_1, PLUS_2, TIMES_2, SQUARE_1} are replacing on all positions.
4.10/2.00	For all symbols f in {cons_2} we have mu(f) = {1}.
4.10/2.00	The symbols in {U_1} are not replacing on any position.
4.10/2.00	
4.10/2.00	The ordinary context-sensitive dependency pairs DP_o are:
4.10/2.00	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
4.10/2.00	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
4.10/2.00	   PI(X) -> 2NDSPOS(X, from(0))
4.10/2.00	   PI(X) -> FROM(0)
4.10/2.00	   PLUS(s(X), Y) -> PLUS(X, Y)
4.10/2.00	   TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
4.10/2.00	   TIMES(s(X), Y) -> TIMES(X, Y)
4.10/2.00	   SQUARE(X) -> TIMES(X, X)
4.10/2.00	
4.10/2.00	The collapsing dependency pairs are DP_c:
4.10/2.00	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> Y
4.10/2.00	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> Z
4.10/2.00	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> Y
4.10/2.00	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> Z
4.10/2.00	
4.10/2.00	
4.10/2.00	The hidden terms of R are:
4.10/2.00	
4.10/2.00	   from(s(x0))
4.10/2.00	
4.10/2.00	Every hiding context is built from:
4.10/2.00	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@128530a2
4.10/2.00	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@2a7d8837
4.10/2.00	
4.10/2.00	Hence, the new unhiding pairs DP_u are :
4.10/2.00	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> U(Y)
4.10/2.00	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> U(Z)
4.10/2.00	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> U(Y)
4.10/2.00	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> U(Z)
4.10/2.00	   U(s(x_0)) -> U(x_0)
4.10/2.00	   U(from(x_0)) -> U(x_0)
4.10/2.00	   U(from(s(x0))) -> FROM(s(x0))
4.10/2.00	
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   from(x0)
4.10/2.00	   2ndspos(0, x0)
4.10/2.00	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   2ndsneg(0, x0)
4.10/2.00	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   pi(x0)
4.10/2.00	   plus(0, x0)
4.10/2.00	   plus(s(x0), x1)
4.10/2.00	   times(0, x0)
4.10/2.00	   times(s(x0), x1)
4.10/2.00	   square(x0)
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(7) QCSDependencyGraphProof (EQUIVALENT)
4.10/2.00	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 7 less nodes.
4.10/2.00	The rules PI(x0) -> 2NDSPOS(x0, from(0)) and 2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> 2NDSNEG(z0, z3) form no chain, because ECap^mu_R'(2NDSPOS(s(z0), cons(z1, cons(z2, z3)))) = 2NDSPOS(s(x_1), cons(x_2, cons(z2, z3))) does not unify with 2NDSPOS(x0, from(0)). 
4.10/2.00	R' = 
4.10/2.00	   ( cons(X, from(s(X))), from(X))
4.10/2.00	
4.10/2.00	
4.10/2.00	The rules PI(x0) -> 2NDSPOS(x0, from(0)) and 2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> U(z2) form no chain, because ECap^mu_R'(2NDSPOS(s(z0), cons(z1, cons(z2, z3)))) = 2NDSPOS(s(x_1), cons(x_2, cons(z2, z3))) does not unify with 2NDSPOS(x0, from(0)). 
4.10/2.00	R' = 
4.10/2.00	   ( cons(X, from(s(X))), from(X))
4.10/2.00	
4.10/2.00	
4.10/2.00	The rules PI(x0) -> 2NDSPOS(x0, from(0)) and 2NDSPOS(s(z0), cons(z1, cons(z2, z3))) -> U(z3) form no chain, because ECap^mu_R'(2NDSPOS(s(z0), cons(z1, cons(z2, z3)))) = 2NDSPOS(s(x_1), cons(x_2, cons(z2, z3))) does not unify with 2NDSPOS(x0, from(0)). 
4.10/2.00	R' = 
4.10/2.00	   ( cons(X, from(s(X))), from(X))
4.10/2.00	
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(8)
4.10/2.00	Complex Obligation (AND)
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(9)
4.10/2.00	Obligation:
4.10/2.00	Q-restricted context-sensitive dependency pair problem:
4.10/2.00	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1} are replacing on all positions.
4.10/2.00	For all symbols f in {cons_2} we have mu(f) = {1}.
4.10/2.00	The symbols in {U_1} are not replacing on any position.
4.10/2.00	
4.10/2.00	The TRS P consists of the following rules:
4.10/2.00	
4.10/2.00	   U(s(x_0)) -> U(x_0)
4.10/2.00	   U(from(x_0)) -> U(x_0)
4.10/2.00	
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   from(x0)
4.10/2.00	   2ndspos(0, x0)
4.10/2.00	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   2ndsneg(0, x0)
4.10/2.00	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   pi(x0)
4.10/2.00	   plus(0, x0)
4.10/2.00	   plus(s(x0), x1)
4.10/2.00	   times(0, x0)
4.10/2.00	   times(s(x0), x1)
4.10/2.00	   square(x0)
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(10) QCSDPSubtermProof (EQUIVALENT)
4.10/2.00	We use the subterm processor [DA_EMMES].
4.10/2.00	
4.10/2.00	
4.10/2.00	The following pairs can be oriented strictly and are deleted.
4.10/2.00	
4.10/2.00	   U(s(x_0)) -> U(x_0)
4.10/2.00	   U(from(x_0)) -> U(x_0)
4.10/2.00	The remaining pairs can at least be oriented weakly.
4.10/2.00	none
4.10/2.00	Used ordering:  Combined order from the following AFS and order.
4.10/2.00	U(x1)  =  x1
4.10/2.00	
4.10/2.00	
4.10/2.00	Subterm Order
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(11)
4.10/2.00	Obligation:
4.10/2.00	Q-restricted context-sensitive dependency pair problem:
4.10/2.00	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1} are replacing on all positions.
4.10/2.00	For all symbols f in {cons_2} we have mu(f) = {1}.
4.10/2.00	
4.10/2.00	The TRS P consists of the following rules:
4.10/2.00	none
4.10/2.00	
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   from(x0)
4.10/2.00	   2ndspos(0, x0)
4.10/2.00	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   2ndsneg(0, x0)
4.10/2.00	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   pi(x0)
4.10/2.00	   plus(0, x0)
4.10/2.00	   plus(s(x0), x1)
4.10/2.00	   times(0, x0)
4.10/2.00	   times(s(x0), x1)
4.10/2.00	   square(x0)
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(12) PIsEmptyProof (EQUIVALENT)
4.10/2.00	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(13)
4.10/2.00	YES
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(14)
4.10/2.00	Obligation:
4.10/2.00	Q-restricted context-sensitive dependency pair problem:
4.10/2.00	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1, PLUS_2} are replacing on all positions.
4.10/2.00	For all symbols f in {cons_2} we have mu(f) = {1}.
4.10/2.00	
4.10/2.00	The TRS P consists of the following rules:
4.10/2.00	
4.10/2.00	   PLUS(s(X), Y) -> PLUS(X, Y)
4.10/2.00	
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   from(x0)
4.10/2.00	   2ndspos(0, x0)
4.10/2.00	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   2ndsneg(0, x0)
4.10/2.00	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   pi(x0)
4.10/2.00	   plus(0, x0)
4.10/2.00	   plus(s(x0), x1)
4.10/2.00	   times(0, x0)
4.10/2.00	   times(s(x0), x1)
4.10/2.00	   square(x0)
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(15) QCSDPSubtermProof (EQUIVALENT)
4.10/2.00	We use the subterm processor [DA_EMMES].
4.10/2.00	
4.10/2.00	
4.10/2.00	The following pairs can be oriented strictly and are deleted.
4.10/2.00	
4.10/2.00	   PLUS(s(X), Y) -> PLUS(X, Y)
4.10/2.00	The remaining pairs can at least be oriented weakly.
4.10/2.00	none
4.10/2.00	Used ordering:  Combined order from the following AFS and order.
4.10/2.00	PLUS(x1, x2)  =  x1
4.10/2.00	
4.10/2.00	
4.10/2.00	Subterm Order
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(16)
4.10/2.00	Obligation:
4.10/2.00	Q-restricted context-sensitive dependency pair problem:
4.10/2.00	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1} are replacing on all positions.
4.10/2.00	For all symbols f in {cons_2} we have mu(f) = {1}.
4.10/2.00	
4.10/2.00	The TRS P consists of the following rules:
4.10/2.00	none
4.10/2.00	
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   from(x0)
4.10/2.00	   2ndspos(0, x0)
4.10/2.00	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   2ndsneg(0, x0)
4.10/2.00	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   pi(x0)
4.10/2.00	   plus(0, x0)
4.10/2.00	   plus(s(x0), x1)
4.10/2.00	   times(0, x0)
4.10/2.00	   times(s(x0), x1)
4.10/2.00	   square(x0)
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(17) PIsEmptyProof (EQUIVALENT)
4.10/2.00	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(18)
4.10/2.00	YES
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(19)
4.10/2.00	Obligation:
4.10/2.00	Q-restricted context-sensitive dependency pair problem:
4.10/2.00	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1, TIMES_2} are replacing on all positions.
4.10/2.00	For all symbols f in {cons_2} we have mu(f) = {1}.
4.10/2.00	
4.10/2.00	The TRS P consists of the following rules:
4.10/2.00	
4.10/2.00	   TIMES(s(X), Y) -> TIMES(X, Y)
4.10/2.00	
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   from(x0)
4.10/2.00	   2ndspos(0, x0)
4.10/2.00	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   2ndsneg(0, x0)
4.10/2.00	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   pi(x0)
4.10/2.00	   plus(0, x0)
4.10/2.00	   plus(s(x0), x1)
4.10/2.00	   times(0, x0)
4.10/2.00	   times(s(x0), x1)
4.10/2.00	   square(x0)
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(20) QCSDPSubtermProof (EQUIVALENT)
4.10/2.00	We use the subterm processor [DA_EMMES].
4.10/2.00	
4.10/2.00	
4.10/2.00	The following pairs can be oriented strictly and are deleted.
4.10/2.00	
4.10/2.00	   TIMES(s(X), Y) -> TIMES(X, Y)
4.10/2.00	The remaining pairs can at least be oriented weakly.
4.10/2.00	none
4.10/2.00	Used ordering:  Combined order from the following AFS and order.
4.10/2.00	TIMES(x1, x2)  =  x1
4.10/2.00	
4.10/2.00	
4.10/2.00	Subterm Order
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(21)
4.10/2.00	Obligation:
4.10/2.00	Q-restricted context-sensitive dependency pair problem:
4.10/2.00	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1} are replacing on all positions.
4.10/2.00	For all symbols f in {cons_2} we have mu(f) = {1}.
4.10/2.00	
4.10/2.00	The TRS P consists of the following rules:
4.10/2.00	none
4.10/2.00	
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   from(x0)
4.10/2.00	   2ndspos(0, x0)
4.10/2.00	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   2ndsneg(0, x0)
4.10/2.00	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   pi(x0)
4.10/2.00	   plus(0, x0)
4.10/2.00	   plus(s(x0), x1)
4.10/2.00	   times(0, x0)
4.10/2.00	   times(s(x0), x1)
4.10/2.00	   square(x0)
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(22) PIsEmptyProof (EQUIVALENT)
4.10/2.00	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(23)
4.10/2.00	YES
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(24)
4.10/2.00	Obligation:
4.10/2.00	Q-restricted context-sensitive dependency pair problem:
4.10/2.00	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1, 2NDSPOS_2, 2NDSNEG_2} are replacing on all positions.
4.10/2.00	For all symbols f in {cons_2} we have mu(f) = {1}.
4.10/2.00	
4.10/2.00	The TRS P consists of the following rules:
4.10/2.00	
4.10/2.00	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
4.10/2.00	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
4.10/2.00	
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   from(x0)
4.10/2.00	   2ndspos(0, x0)
4.10/2.00	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   2ndsneg(0, x0)
4.10/2.00	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   pi(x0)
4.10/2.00	   plus(0, x0)
4.10/2.00	   plus(s(x0), x1)
4.10/2.00	   times(0, x0)
4.10/2.00	   times(s(x0), x1)
4.10/2.00	   square(x0)
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(25) QCSDPSubtermProof (EQUIVALENT)
4.10/2.00	We use the subterm processor [DA_EMMES].
4.10/2.00	
4.10/2.00	
4.10/2.00	The following pairs can be oriented strictly and are deleted.
4.10/2.00	
4.10/2.00	   2NDSNEG(s(N), cons(X, cons(Y, Z))) -> 2NDSPOS(N, Z)
4.10/2.00	   2NDSPOS(s(N), cons(X, cons(Y, Z))) -> 2NDSNEG(N, Z)
4.10/2.00	The remaining pairs can at least be oriented weakly.
4.10/2.00	none
4.10/2.00	Used ordering:  Combined order from the following AFS and order.
4.10/2.00	2NDSPOS(x1, x2)  =  x1
4.10/2.00	
4.10/2.00	2NDSNEG(x1, x2)  =  x1
4.10/2.00	
4.10/2.00	
4.10/2.00	Subterm Order
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(26)
4.10/2.00	Obligation:
4.10/2.00	Q-restricted context-sensitive dependency pair problem:
4.10/2.00	The symbols in {from_1, s_1, 2ndspos_2, rcons_2, posrecip_1, 2ndsneg_2, negrecip_1, pi_1, plus_2, times_2, square_1} are replacing on all positions.
4.10/2.00	For all symbols f in {cons_2} we have mu(f) = {1}.
4.10/2.00	
4.10/2.00	The TRS P consists of the following rules:
4.10/2.00	none
4.10/2.00	
4.10/2.00	The TRS R consists of the following rules:
4.10/2.00	
4.10/2.00	   from(X) -> cons(X, from(s(X)))
4.10/2.00	   2ndspos(0, Z) -> rnil
4.10/2.00	   2ndspos(s(N), cons(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, Z))
4.10/2.00	   2ndsneg(0, Z) -> rnil
4.10/2.00	   2ndsneg(s(N), cons(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, Z))
4.10/2.00	   pi(X) -> 2ndspos(X, from(0))
4.10/2.00	   plus(0, Y) -> Y
4.10/2.00	   plus(s(X), Y) -> s(plus(X, Y))
4.10/2.00	   times(0, Y) -> 0
4.10/2.00	   times(s(X), Y) -> plus(Y, times(X, Y))
4.10/2.00	   square(X) -> times(X, X)
4.10/2.00	
4.10/2.00	The set Q consists of the following terms:
4.10/2.00	
4.10/2.00	   from(x0)
4.10/2.00	   2ndspos(0, x0)
4.10/2.00	   2ndspos(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   2ndsneg(0, x0)
4.10/2.00	   2ndsneg(s(x0), cons(x1, cons(x2, x3)))
4.10/2.00	   pi(x0)
4.10/2.00	   plus(0, x0)
4.10/2.00	   plus(s(x0), x1)
4.10/2.00	   times(0, x0)
4.10/2.00	   times(s(x0), x1)
4.10/2.00	   square(x0)
4.10/2.00	
4.10/2.00	
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(27) PIsEmptyProof (EQUIVALENT)
4.10/2.00	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
4.10/2.00	----------------------------------------
4.10/2.00	
4.10/2.00	(28)
4.10/2.00	YES
4.15/2.03	EOF