4.00/1.86	YES
4.00/1.87	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.00/1.87	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.00/1.87	
4.00/1.87	
4.00/1.87	Termination w.r.t. Q of the given QTRS could be proven:
4.00/1.87	
4.00/1.87	(0) QTRS
4.00/1.87	(1) QTRSToCSRProof [SOUND, 0 ms]
4.00/1.87	(2) CSR
4.00/1.87	(3) CSRInnermostProof [EQUIVALENT, 0 ms]
4.00/1.87	(4) CSR
4.00/1.87	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
4.00/1.87	(6) QCSDP
4.00/1.87	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.00/1.87	(8) AND
4.00/1.87	    (9) QCSDP
4.00/1.87	        (10) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.00/1.87	        (11) QCSDP
4.00/1.87	        (12) PIsEmptyProof [EQUIVALENT, 0 ms]
4.00/1.87	        (13) YES
4.00/1.87	    (14) QCSDP
4.00/1.87	        (15) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.00/1.87	        (16) QCSDP
4.00/1.87	        (17) PIsEmptyProof [EQUIVALENT, 0 ms]
4.00/1.87	        (18) YES
4.00/1.87	
4.00/1.87	
4.00/1.87	----------------------------------------
4.00/1.87	
4.00/1.87	(0)
4.00/1.87	Obligation:
4.00/1.87	Q restricted rewrite system:
4.00/1.87	The TRS R consists of the following rules:
4.00/1.87	
4.00/1.87	   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
4.00/1.87	   active(fst(pair(XS, YS))) -> mark(XS)
4.00/1.87	   active(snd(pair(XS, YS))) -> mark(YS)
4.00/1.87	   active(splitAt(0, XS)) -> mark(pair(nil, XS))
4.00/1.87	   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
4.00/1.87	   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
4.00/1.87	   active(head(cons(N, XS))) -> mark(N)
4.00/1.87	   active(tail(cons(N, XS))) -> mark(XS)
4.00/1.87	   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
4.00/1.87	   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
4.00/1.87	   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
4.00/1.87	   active(natsFrom(X)) -> natsFrom(active(X))
4.00/1.87	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.00/1.87	   active(s(X)) -> s(active(X))
4.00/1.87	   active(fst(X)) -> fst(active(X))
4.00/1.87	   active(pair(X1, X2)) -> pair(active(X1), X2)
4.00/1.87	   active(pair(X1, X2)) -> pair(X1, active(X2))
4.00/1.87	   active(snd(X)) -> snd(active(X))
4.00/1.87	   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
4.00/1.87	   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
4.00/1.87	   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
4.00/1.87	   active(head(X)) -> head(active(X))
4.00/1.87	   active(tail(X)) -> tail(active(X))
4.00/1.87	   active(sel(X1, X2)) -> sel(active(X1), X2)
4.00/1.87	   active(sel(X1, X2)) -> sel(X1, active(X2))
4.00/1.87	   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
4.00/1.87	   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
4.00/1.87	   active(take(X1, X2)) -> take(active(X1), X2)
4.00/1.87	   active(take(X1, X2)) -> take(X1, active(X2))
4.00/1.87	   natsFrom(mark(X)) -> mark(natsFrom(X))
4.00/1.87	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.00/1.87	   s(mark(X)) -> mark(s(X))
4.00/1.87	   fst(mark(X)) -> mark(fst(X))
4.00/1.87	   pair(mark(X1), X2) -> mark(pair(X1, X2))
4.00/1.87	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
4.00/1.87	   snd(mark(X)) -> mark(snd(X))
4.00/1.87	   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
4.00/1.87	   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
4.00/1.87	   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
4.00/1.87	   head(mark(X)) -> mark(head(X))
4.00/1.87	   tail(mark(X)) -> mark(tail(X))
4.00/1.87	   sel(mark(X1), X2) -> mark(sel(X1, X2))
4.00/1.87	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
4.00/1.87	   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
4.00/1.87	   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
4.00/1.87	   take(mark(X1), X2) -> mark(take(X1, X2))
4.00/1.87	   take(X1, mark(X2)) -> mark(take(X1, X2))
4.00/1.87	   proper(natsFrom(X)) -> natsFrom(proper(X))
4.00/1.87	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.00/1.87	   proper(s(X)) -> s(proper(X))
4.00/1.87	   proper(fst(X)) -> fst(proper(X))
4.00/1.87	   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
4.00/1.87	   proper(snd(X)) -> snd(proper(X))
4.00/1.87	   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
4.00/1.87	   proper(0) -> ok(0)
4.00/1.87	   proper(nil) -> ok(nil)
4.00/1.87	   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
4.00/1.87	   proper(head(X)) -> head(proper(X))
4.00/1.87	   proper(tail(X)) -> tail(proper(X))
4.00/1.87	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
4.00/1.87	   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
4.00/1.87	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
4.00/1.87	   natsFrom(ok(X)) -> ok(natsFrom(X))
4.00/1.87	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.00/1.87	   s(ok(X)) -> ok(s(X))
4.00/1.87	   fst(ok(X)) -> ok(fst(X))
4.00/1.87	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
4.00/1.87	   snd(ok(X)) -> ok(snd(X))
4.00/1.87	   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
4.00/1.87	   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
4.00/1.87	   head(ok(X)) -> ok(head(X))
4.00/1.87	   tail(ok(X)) -> ok(tail(X))
4.00/1.87	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
4.00/1.87	   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
4.00/1.87	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
4.00/1.87	   top(mark(X)) -> top(proper(X))
4.00/1.87	   top(ok(X)) -> top(active(X))
4.00/1.87	
4.00/1.87	The set Q consists of the following terms:
4.00/1.87	
4.00/1.87	   active(natsFrom(x0))
4.00/1.87	   active(sel(x0, x1))
4.00/1.87	   active(take(x0, x1))
4.00/1.87	   active(afterNth(x0, x1))
4.00/1.87	   active(cons(x0, x1))
4.00/1.87	   active(s(x0))
4.00/1.87	   active(fst(x0))
4.00/1.87	   active(pair(x0, x1))
4.00/1.87	   active(snd(x0))
4.00/1.87	   active(splitAt(x0, x1))
4.00/1.87	   active(u(x0, x1, x2, x3))
4.00/1.87	   active(head(x0))
4.00/1.87	   active(tail(x0))
4.00/1.87	   natsFrom(mark(x0))
4.00/1.87	   cons(mark(x0), x1)
4.00/1.87	   s(mark(x0))
4.00/1.87	   fst(mark(x0))
4.00/1.87	   pair(mark(x0), x1)
4.00/1.87	   pair(x0, mark(x1))
4.00/1.87	   snd(mark(x0))
4.00/1.87	   splitAt(mark(x0), x1)
4.00/1.87	   splitAt(x0, mark(x1))
4.00/1.87	   u(mark(x0), x1, x2, x3)
4.00/1.87	   head(mark(x0))
4.00/1.87	   tail(mark(x0))
4.00/1.87	   sel(mark(x0), x1)
4.00/1.87	   sel(x0, mark(x1))
4.00/1.87	   afterNth(mark(x0), x1)
4.00/1.87	   afterNth(x0, mark(x1))
4.00/1.87	   take(mark(x0), x1)
4.00/1.87	   take(x0, mark(x1))
4.00/1.87	   proper(natsFrom(x0))
4.00/1.87	   proper(cons(x0, x1))
4.00/1.87	   proper(s(x0))
4.00/1.87	   proper(fst(x0))
4.00/1.87	   proper(pair(x0, x1))
4.00/1.87	   proper(snd(x0))
4.00/1.87	   proper(splitAt(x0, x1))
4.00/1.87	   proper(0)
4.00/1.87	   proper(nil)
4.00/1.87	   proper(u(x0, x1, x2, x3))
4.00/1.87	   proper(head(x0))
4.00/1.87	   proper(tail(x0))
4.00/1.87	   proper(sel(x0, x1))
4.00/1.87	   proper(afterNth(x0, x1))
4.00/1.87	   proper(take(x0, x1))
4.00/1.87	   natsFrom(ok(x0))
4.00/1.87	   cons(ok(x0), ok(x1))
4.00/1.87	   s(ok(x0))
4.00/1.87	   fst(ok(x0))
4.00/1.87	   pair(ok(x0), ok(x1))
4.00/1.87	   snd(ok(x0))
4.00/1.87	   splitAt(ok(x0), ok(x1))
4.00/1.87	   u(ok(x0), ok(x1), ok(x2), ok(x3))
4.00/1.87	   head(ok(x0))
4.00/1.87	   tail(ok(x0))
4.00/1.87	   sel(ok(x0), ok(x1))
4.00/1.87	   afterNth(ok(x0), ok(x1))
4.00/1.87	   take(ok(x0), ok(x1))
4.00/1.87	   top(mark(x0))
4.00/1.87	   top(ok(x0))
4.00/1.87	
4.00/1.87	
4.00/1.87	----------------------------------------
4.00/1.87	
4.00/1.87	(1) QTRSToCSRProof (SOUND)
4.00/1.87	The following Q TRS is given: Q restricted rewrite system:
4.00/1.87	The TRS R consists of the following rules:
4.00/1.87	
4.00/1.87	   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
4.00/1.87	   active(fst(pair(XS, YS))) -> mark(XS)
4.00/1.87	   active(snd(pair(XS, YS))) -> mark(YS)
4.00/1.87	   active(splitAt(0, XS)) -> mark(pair(nil, XS))
4.00/1.87	   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
4.00/1.87	   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
4.00/1.87	   active(head(cons(N, XS))) -> mark(N)
4.00/1.87	   active(tail(cons(N, XS))) -> mark(XS)
4.00/1.87	   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
4.00/1.87	   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
4.00/1.87	   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
4.00/1.87	   active(natsFrom(X)) -> natsFrom(active(X))
4.00/1.87	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.00/1.87	   active(s(X)) -> s(active(X))
4.00/1.87	   active(fst(X)) -> fst(active(X))
4.00/1.87	   active(pair(X1, X2)) -> pair(active(X1), X2)
4.00/1.87	   active(pair(X1, X2)) -> pair(X1, active(X2))
4.00/1.87	   active(snd(X)) -> snd(active(X))
4.00/1.87	   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
4.00/1.87	   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
4.00/1.87	   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
4.00/1.87	   active(head(X)) -> head(active(X))
4.00/1.87	   active(tail(X)) -> tail(active(X))
4.00/1.87	   active(sel(X1, X2)) -> sel(active(X1), X2)
4.00/1.87	   active(sel(X1, X2)) -> sel(X1, active(X2))
4.00/1.87	   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
4.00/1.87	   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
4.00/1.87	   active(take(X1, X2)) -> take(active(X1), X2)
4.00/1.87	   active(take(X1, X2)) -> take(X1, active(X2))
4.00/1.87	   natsFrom(mark(X)) -> mark(natsFrom(X))
4.00/1.87	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.00/1.87	   s(mark(X)) -> mark(s(X))
4.00/1.87	   fst(mark(X)) -> mark(fst(X))
4.00/1.87	   pair(mark(X1), X2) -> mark(pair(X1, X2))
4.00/1.87	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
4.00/1.87	   snd(mark(X)) -> mark(snd(X))
4.00/1.87	   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
4.00/1.87	   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
4.00/1.87	   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
4.00/1.87	   head(mark(X)) -> mark(head(X))
4.00/1.87	   tail(mark(X)) -> mark(tail(X))
4.00/1.87	   sel(mark(X1), X2) -> mark(sel(X1, X2))
4.00/1.87	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
4.00/1.87	   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
4.00/1.87	   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
4.00/1.87	   take(mark(X1), X2) -> mark(take(X1, X2))
4.00/1.87	   take(X1, mark(X2)) -> mark(take(X1, X2))
4.00/1.87	   proper(natsFrom(X)) -> natsFrom(proper(X))
4.00/1.87	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.00/1.87	   proper(s(X)) -> s(proper(X))
4.00/1.87	   proper(fst(X)) -> fst(proper(X))
4.00/1.87	   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
4.00/1.87	   proper(snd(X)) -> snd(proper(X))
4.00/1.87	   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
4.00/1.87	   proper(0) -> ok(0)
4.00/1.87	   proper(nil) -> ok(nil)
4.00/1.87	   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
4.00/1.87	   proper(head(X)) -> head(proper(X))
4.00/1.87	   proper(tail(X)) -> tail(proper(X))
4.00/1.87	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
4.00/1.87	   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
4.00/1.87	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
4.00/1.87	   natsFrom(ok(X)) -> ok(natsFrom(X))
4.00/1.87	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.00/1.87	   s(ok(X)) -> ok(s(X))
4.00/1.87	   fst(ok(X)) -> ok(fst(X))
4.00/1.87	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
4.00/1.87	   snd(ok(X)) -> ok(snd(X))
4.00/1.87	   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
4.00/1.87	   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
4.00/1.87	   head(ok(X)) -> ok(head(X))
4.00/1.87	   tail(ok(X)) -> ok(tail(X))
4.00/1.87	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
4.00/1.87	   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
4.00/1.87	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
4.00/1.88	   top(mark(X)) -> top(proper(X))
4.00/1.88	   top(ok(X)) -> top(active(X))
4.00/1.88	
4.00/1.88	The set Q consists of the following terms:
4.00/1.88	
4.00/1.88	   active(natsFrom(x0))
4.00/1.88	   active(sel(x0, x1))
4.00/1.88	   active(take(x0, x1))
4.00/1.88	   active(afterNth(x0, x1))
4.00/1.88	   active(cons(x0, x1))
4.00/1.88	   active(s(x0))
4.00/1.88	   active(fst(x0))
4.00/1.88	   active(pair(x0, x1))
4.00/1.88	   active(snd(x0))
4.00/1.88	   active(splitAt(x0, x1))
4.00/1.88	   active(u(x0, x1, x2, x3))
4.00/1.88	   active(head(x0))
4.00/1.88	   active(tail(x0))
4.00/1.88	   natsFrom(mark(x0))
4.00/1.88	   cons(mark(x0), x1)
4.00/1.88	   s(mark(x0))
4.00/1.88	   fst(mark(x0))
4.00/1.88	   pair(mark(x0), x1)
4.00/1.88	   pair(x0, mark(x1))
4.00/1.88	   snd(mark(x0))
4.00/1.88	   splitAt(mark(x0), x1)
4.00/1.88	   splitAt(x0, mark(x1))
4.00/1.88	   u(mark(x0), x1, x2, x3)
4.00/1.88	   head(mark(x0))
4.00/1.88	   tail(mark(x0))
4.00/1.88	   sel(mark(x0), x1)
4.00/1.88	   sel(x0, mark(x1))
4.00/1.88	   afterNth(mark(x0), x1)
4.00/1.88	   afterNth(x0, mark(x1))
4.00/1.88	   take(mark(x0), x1)
4.00/1.88	   take(x0, mark(x1))
4.00/1.88	   proper(natsFrom(x0))
4.00/1.88	   proper(cons(x0, x1))
4.00/1.88	   proper(s(x0))
4.00/1.88	   proper(fst(x0))
4.00/1.88	   proper(pair(x0, x1))
4.00/1.88	   proper(snd(x0))
4.00/1.88	   proper(splitAt(x0, x1))
4.00/1.88	   proper(0)
4.00/1.88	   proper(nil)
4.00/1.88	   proper(u(x0, x1, x2, x3))
4.00/1.88	   proper(head(x0))
4.00/1.88	   proper(tail(x0))
4.00/1.88	   proper(sel(x0, x1))
4.00/1.88	   proper(afterNth(x0, x1))
4.00/1.88	   proper(take(x0, x1))
4.00/1.88	   natsFrom(ok(x0))
4.00/1.88	   cons(ok(x0), ok(x1))
4.00/1.88	   s(ok(x0))
4.00/1.88	   fst(ok(x0))
4.00/1.88	   pair(ok(x0), ok(x1))
4.00/1.88	   snd(ok(x0))
4.00/1.88	   splitAt(ok(x0), ok(x1))
4.00/1.88	   u(ok(x0), ok(x1), ok(x2), ok(x3))
4.00/1.88	   head(ok(x0))
4.00/1.88	   tail(ok(x0))
4.00/1.88	   sel(ok(x0), ok(x1))
4.00/1.88	   afterNth(ok(x0), ok(x1))
4.00/1.88	   take(ok(x0), ok(x1))
4.00/1.88	   top(mark(x0))
4.00/1.88	   top(ok(x0))
4.00/1.88	
4.00/1.88	Special symbols used for the transformation (see [GM04]):
4.00/1.88	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
4.00/1.88	The replacement map contains the following entries:
4.00/1.88	
4.00/1.88	natsFrom: {1}
4.00/1.88	cons: {1}
4.00/1.88	s: {1}
4.00/1.88	fst: {1}
4.00/1.88	pair: {1, 2}
4.00/1.88	snd: {1}
4.00/1.88	splitAt: {1, 2}
4.00/1.88	0: empty set
4.00/1.88	nil: empty set
4.00/1.88	u: {1}
4.00/1.88	head: {1}
4.00/1.88	tail: {1}
4.00/1.88	sel: {1, 2}
4.00/1.88	afterNth: {1, 2}
4.00/1.88	take: {1, 2}
4.00/1.88	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.
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(2)
4.00/1.88	Obligation:
4.00/1.88	Context-sensitive rewrite system:
4.00/1.88	The TRS R consists of the following rules:
4.00/1.88	
4.00/1.88	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.00/1.88	   fst(pair(XS, YS)) -> XS
4.00/1.88	   snd(pair(XS, YS)) -> YS
4.00/1.88	   splitAt(0, XS) -> pair(nil, XS)
4.00/1.88	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
4.00/1.88	   u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)
4.00/1.88	   head(cons(N, XS)) -> N
4.00/1.88	   tail(cons(N, XS)) -> XS
4.00/1.88	   sel(N, XS) -> head(afterNth(N, XS))
4.00/1.88	   take(N, XS) -> fst(splitAt(N, XS))
4.00/1.88	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.00/1.88	
4.00/1.88	The replacement map contains the following entries:
4.00/1.88	
4.00/1.88	natsFrom: {1}
4.00/1.88	cons: {1}
4.00/1.88	s: {1}
4.00/1.88	fst: {1}
4.00/1.88	pair: {1, 2}
4.00/1.88	snd: {1}
4.00/1.88	splitAt: {1, 2}
4.00/1.88	0: empty set
4.00/1.88	nil: empty set
4.00/1.88	u: {1}
4.00/1.88	head: {1}
4.00/1.88	tail: {1}
4.00/1.88	sel: {1, 2}
4.00/1.88	afterNth: {1, 2}
4.00/1.88	take: {1, 2}
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(3) CSRInnermostProof (EQUIVALENT)
4.00/1.88	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(4)
4.00/1.88	Obligation:
4.00/1.88	Context-sensitive rewrite system:
4.00/1.88	The TRS R consists of the following rules:
4.00/1.88	
4.00/1.88	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.00/1.88	   fst(pair(XS, YS)) -> XS
4.00/1.88	   snd(pair(XS, YS)) -> YS
4.00/1.88	   splitAt(0, XS) -> pair(nil, XS)
4.00/1.88	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
4.00/1.88	   u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)
4.00/1.88	   head(cons(N, XS)) -> N
4.00/1.88	   tail(cons(N, XS)) -> XS
4.00/1.88	   sel(N, XS) -> head(afterNth(N, XS))
4.00/1.88	   take(N, XS) -> fst(splitAt(N, XS))
4.00/1.88	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.00/1.88	
4.00/1.88	The replacement map contains the following entries:
4.00/1.88	
4.00/1.88	natsFrom: {1}
4.00/1.88	cons: {1}
4.00/1.88	s: {1}
4.00/1.88	fst: {1}
4.00/1.88	pair: {1, 2}
4.00/1.88	snd: {1}
4.00/1.88	splitAt: {1, 2}
4.00/1.88	0: empty set
4.00/1.88	nil: empty set
4.00/1.88	u: {1}
4.00/1.88	head: {1}
4.00/1.88	tail: {1}
4.00/1.88	sel: {1, 2}
4.00/1.88	afterNth: {1, 2}
4.00/1.88	take: {1, 2}
4.00/1.88	
4.00/1.88	
4.00/1.88	Innermost Strategy.
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(5) CSDependencyPairsProof (EQUIVALENT)
4.00/1.88	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(6)
4.00/1.88	Obligation:
4.00/1.88	Q-restricted context-sensitive dependency pair problem:
4.00/1.88	The symbols in {natsFrom_1, s_1, fst_1, pair_2, snd_1, splitAt_2, head_1, tail_1, sel_2, afterNth_2, take_2, SPLITAT_2, HEAD_1, SEL_2, AFTERNTH_2, FST_1, TAKE_2, SND_1, TAIL_1, NATSFROM_1} are replacing on all positions.
4.00/1.88	For all symbols f in {cons_2, u_4, U_4} we have mu(f) = {1}.
4.00/1.88	The symbols in {U'_1} are not replacing on any position.
4.00/1.88	
4.00/1.88	The ordinary context-sensitive dependency pairs DP_o are:
4.00/1.88	   SPLITAT(s(N), cons(X, XS)) -> U(splitAt(N, XS), N, X, XS)
4.00/1.88	   SPLITAT(s(N), cons(X, XS)) -> SPLITAT(N, XS)
4.00/1.88	   SEL(N, XS) -> HEAD(afterNth(N, XS))
4.00/1.88	   SEL(N, XS) -> AFTERNTH(N, XS)
4.00/1.88	   TAKE(N, XS) -> FST(splitAt(N, XS))
4.00/1.88	   TAKE(N, XS) -> SPLITAT(N, XS)
4.00/1.88	   AFTERNTH(N, XS) -> SND(splitAt(N, XS))
4.00/1.88	   AFTERNTH(N, XS) -> SPLITAT(N, XS)
4.00/1.88	
4.00/1.88	The collapsing dependency pairs are DP_c:
4.00/1.88	   SPLITAT(s(N), cons(X, XS)) -> XS
4.00/1.88	   U(pair(YS, ZS), N, X, XS) -> X
4.00/1.88	   TAIL(cons(N, XS)) -> XS
4.00/1.88	
4.00/1.88	
4.00/1.88	The hidden terms of R are:
4.00/1.88	
4.00/1.88	   natsFrom(s(x0))
4.00/1.88	
4.00/1.88	Every hiding context is built from:
4.00/1.88	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@27ca33b8
4.00/1.88	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@261467fe
4.00/1.88	
4.00/1.88	Hence, the new unhiding pairs DP_u are :
4.00/1.88	   SPLITAT(s(N), cons(X, XS)) -> U'(XS)
4.00/1.88	   U(pair(YS, ZS), N, X, XS) -> U'(X)
4.00/1.88	   TAIL(cons(N, XS)) -> U'(XS)
4.00/1.88	   U'(s(x_0)) -> U'(x_0)
4.00/1.88	   U'(natsFrom(x_0)) -> U'(x_0)
4.00/1.88	   U'(natsFrom(s(x0))) -> NATSFROM(s(x0))
4.00/1.88	
4.00/1.88	The TRS R consists of the following rules:
4.00/1.88	
4.00/1.88	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.00/1.88	   fst(pair(XS, YS)) -> XS
4.00/1.88	   snd(pair(XS, YS)) -> YS
4.00/1.88	   splitAt(0, XS) -> pair(nil, XS)
4.00/1.88	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
4.00/1.88	   u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)
4.00/1.88	   head(cons(N, XS)) -> N
4.00/1.88	   tail(cons(N, XS)) -> XS
4.00/1.88	   sel(N, XS) -> head(afterNth(N, XS))
4.00/1.88	   take(N, XS) -> fst(splitAt(N, XS))
4.00/1.88	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.00/1.88	
4.00/1.88	The set Q consists of the following terms:
4.00/1.88	
4.00/1.88	   natsFrom(x0)
4.00/1.88	   fst(pair(x0, x1))
4.00/1.88	   snd(pair(x0, x1))
4.00/1.88	   splitAt(0, x0)
4.00/1.88	   splitAt(s(x0), cons(x1, x2))
4.00/1.88	   u(pair(x0, x1), x2, x3, x4)
4.00/1.88	   head(cons(x0, x1))
4.00/1.88	   tail(cons(x0, x1))
4.00/1.88	   sel(x0, x1)
4.00/1.88	   take(x0, x1)
4.00/1.88	   afterNth(x0, x1)
4.00/1.88	
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(7) QCSDependencyGraphProof (EQUIVALENT)
4.00/1.88	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 9 less nodes.
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(8)
4.00/1.88	Complex Obligation (AND)
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(9)
4.00/1.88	Obligation:
4.00/1.88	Q-restricted context-sensitive dependency pair problem:
4.00/1.88	The symbols in {natsFrom_1, s_1, fst_1, pair_2, snd_1, splitAt_2, head_1, tail_1, sel_2, afterNth_2, take_2} are replacing on all positions.
4.00/1.88	For all symbols f in {cons_2, u_4} we have mu(f) = {1}.
4.00/1.88	The symbols in {U'_1} are not replacing on any position.
4.00/1.88	
4.00/1.88	The TRS P consists of the following rules:
4.00/1.88	
4.00/1.88	   U'(s(x_0)) -> U'(x_0)
4.00/1.88	   U'(natsFrom(x_0)) -> U'(x_0)
4.00/1.88	
4.00/1.88	The TRS R consists of the following rules:
4.00/1.88	
4.00/1.88	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.00/1.88	   fst(pair(XS, YS)) -> XS
4.00/1.88	   snd(pair(XS, YS)) -> YS
4.00/1.88	   splitAt(0, XS) -> pair(nil, XS)
4.00/1.88	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
4.00/1.88	   u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)
4.00/1.88	   head(cons(N, XS)) -> N
4.00/1.88	   tail(cons(N, XS)) -> XS
4.00/1.88	   sel(N, XS) -> head(afterNth(N, XS))
4.00/1.88	   take(N, XS) -> fst(splitAt(N, XS))
4.00/1.88	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.00/1.88	
4.00/1.88	The set Q consists of the following terms:
4.00/1.88	
4.00/1.88	   natsFrom(x0)
4.00/1.88	   fst(pair(x0, x1))
4.00/1.88	   snd(pair(x0, x1))
4.00/1.88	   splitAt(0, x0)
4.00/1.88	   splitAt(s(x0), cons(x1, x2))
4.00/1.88	   u(pair(x0, x1), x2, x3, x4)
4.00/1.88	   head(cons(x0, x1))
4.00/1.88	   tail(cons(x0, x1))
4.00/1.88	   sel(x0, x1)
4.00/1.88	   take(x0, x1)
4.00/1.88	   afterNth(x0, x1)
4.00/1.88	
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(10) QCSDPSubtermProof (EQUIVALENT)
4.00/1.88	We use the subterm processor [DA_EMMES].
4.00/1.88	
4.00/1.88	
4.00/1.88	The following pairs can be oriented strictly and are deleted.
4.00/1.88	
4.00/1.88	   U'(s(x_0)) -> U'(x_0)
4.00/1.88	   U'(natsFrom(x_0)) -> U'(x_0)
4.00/1.88	The remaining pairs can at least be oriented weakly.
4.00/1.88	none
4.00/1.88	Used ordering:  Combined order from the following AFS and order.
4.00/1.88	U'(x1)  =  x1
4.00/1.88	
4.00/1.88	
4.00/1.88	Subterm Order
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(11)
4.00/1.88	Obligation:
4.00/1.88	Q-restricted context-sensitive dependency pair problem:
4.00/1.88	The symbols in {natsFrom_1, s_1, fst_1, pair_2, snd_1, splitAt_2, head_1, tail_1, sel_2, afterNth_2, take_2} are replacing on all positions.
4.00/1.88	For all symbols f in {cons_2, u_4} we have mu(f) = {1}.
4.00/1.88	
4.00/1.88	The TRS P consists of the following rules:
4.00/1.88	none
4.00/1.88	
4.00/1.88	The TRS R consists of the following rules:
4.00/1.88	
4.00/1.88	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.00/1.88	   fst(pair(XS, YS)) -> XS
4.00/1.88	   snd(pair(XS, YS)) -> YS
4.00/1.88	   splitAt(0, XS) -> pair(nil, XS)
4.00/1.88	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
4.00/1.88	   u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)
4.00/1.88	   head(cons(N, XS)) -> N
4.00/1.88	   tail(cons(N, XS)) -> XS
4.00/1.88	   sel(N, XS) -> head(afterNth(N, XS))
4.00/1.88	   take(N, XS) -> fst(splitAt(N, XS))
4.00/1.88	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.00/1.88	
4.00/1.88	The set Q consists of the following terms:
4.00/1.88	
4.00/1.88	   natsFrom(x0)
4.00/1.88	   fst(pair(x0, x1))
4.00/1.88	   snd(pair(x0, x1))
4.00/1.88	   splitAt(0, x0)
4.00/1.88	   splitAt(s(x0), cons(x1, x2))
4.00/1.88	   u(pair(x0, x1), x2, x3, x4)
4.00/1.88	   head(cons(x0, x1))
4.00/1.88	   tail(cons(x0, x1))
4.00/1.88	   sel(x0, x1)
4.00/1.88	   take(x0, x1)
4.00/1.88	   afterNth(x0, x1)
4.00/1.88	
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(12) PIsEmptyProof (EQUIVALENT)
4.00/1.88	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(13)
4.00/1.88	YES
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(14)
4.00/1.88	Obligation:
4.00/1.88	Q-restricted context-sensitive dependency pair problem:
4.00/1.88	The symbols in {natsFrom_1, s_1, fst_1, pair_2, snd_1, splitAt_2, head_1, tail_1, sel_2, afterNth_2, take_2, SPLITAT_2} are replacing on all positions.
4.00/1.88	For all symbols f in {cons_2, u_4} we have mu(f) = {1}.
4.00/1.88	
4.00/1.88	The TRS P consists of the following rules:
4.00/1.88	
4.00/1.88	   SPLITAT(s(N), cons(X, XS)) -> SPLITAT(N, XS)
4.00/1.88	
4.00/1.88	The TRS R consists of the following rules:
4.00/1.88	
4.00/1.88	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.00/1.88	   fst(pair(XS, YS)) -> XS
4.00/1.88	   snd(pair(XS, YS)) -> YS
4.00/1.88	   splitAt(0, XS) -> pair(nil, XS)
4.00/1.88	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
4.00/1.88	   u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)
4.00/1.88	   head(cons(N, XS)) -> N
4.00/1.88	   tail(cons(N, XS)) -> XS
4.00/1.88	   sel(N, XS) -> head(afterNth(N, XS))
4.00/1.88	   take(N, XS) -> fst(splitAt(N, XS))
4.00/1.88	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.00/1.88	
4.00/1.88	The set Q consists of the following terms:
4.00/1.88	
4.00/1.88	   natsFrom(x0)
4.00/1.88	   fst(pair(x0, x1))
4.00/1.88	   snd(pair(x0, x1))
4.00/1.88	   splitAt(0, x0)
4.00/1.88	   splitAt(s(x0), cons(x1, x2))
4.00/1.88	   u(pair(x0, x1), x2, x3, x4)
4.00/1.88	   head(cons(x0, x1))
4.00/1.88	   tail(cons(x0, x1))
4.00/1.88	   sel(x0, x1)
4.00/1.88	   take(x0, x1)
4.00/1.88	   afterNth(x0, x1)
4.00/1.88	
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(15) QCSDPSubtermProof (EQUIVALENT)
4.00/1.88	We use the subterm processor [DA_EMMES].
4.00/1.88	
4.00/1.88	
4.00/1.88	The following pairs can be oriented strictly and are deleted.
4.00/1.88	
4.00/1.88	   SPLITAT(s(N), cons(X, XS)) -> SPLITAT(N, XS)
4.00/1.88	The remaining pairs can at least be oriented weakly.
4.00/1.88	none
4.00/1.88	Used ordering:  Combined order from the following AFS and order.
4.00/1.88	SPLITAT(x1, x2)  =  x1
4.00/1.88	
4.00/1.88	
4.00/1.88	Subterm Order
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(16)
4.00/1.88	Obligation:
4.00/1.88	Q-restricted context-sensitive dependency pair problem:
4.00/1.88	The symbols in {natsFrom_1, s_1, fst_1, pair_2, snd_1, splitAt_2, head_1, tail_1, sel_2, afterNth_2, take_2} are replacing on all positions.
4.00/1.88	For all symbols f in {cons_2, u_4} we have mu(f) = {1}.
4.00/1.88	
4.00/1.88	The TRS P consists of the following rules:
4.00/1.88	none
4.00/1.88	
4.00/1.88	The TRS R consists of the following rules:
4.00/1.88	
4.00/1.88	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.00/1.88	   fst(pair(XS, YS)) -> XS
4.00/1.88	   snd(pair(XS, YS)) -> YS
4.00/1.88	   splitAt(0, XS) -> pair(nil, XS)
4.00/1.88	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, XS), N, X, XS)
4.00/1.88	   u(pair(YS, ZS), N, X, XS) -> pair(cons(X, YS), ZS)
4.00/1.88	   head(cons(N, XS)) -> N
4.00/1.88	   tail(cons(N, XS)) -> XS
4.00/1.88	   sel(N, XS) -> head(afterNth(N, XS))
4.00/1.88	   take(N, XS) -> fst(splitAt(N, XS))
4.00/1.88	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.00/1.88	
4.00/1.88	The set Q consists of the following terms:
4.00/1.88	
4.00/1.88	   natsFrom(x0)
4.00/1.88	   fst(pair(x0, x1))
4.00/1.88	   snd(pair(x0, x1))
4.00/1.88	   splitAt(0, x0)
4.00/1.88	   splitAt(s(x0), cons(x1, x2))
4.00/1.88	   u(pair(x0, x1), x2, x3, x4)
4.00/1.88	   head(cons(x0, x1))
4.00/1.88	   tail(cons(x0, x1))
4.00/1.88	   sel(x0, x1)
4.00/1.88	   take(x0, x1)
4.00/1.88	   afterNth(x0, x1)
4.00/1.88	
4.00/1.88	
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(17) PIsEmptyProof (EQUIVALENT)
4.00/1.88	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
4.00/1.88	----------------------------------------
4.00/1.88	
4.00/1.88	(18)
4.00/1.88	YES
4.00/1.91	EOF