4.03/1.96	YES
4.03/1.97	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.03/1.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.03/1.97	
4.03/1.97	
4.03/1.97	Termination w.r.t. Q of the given QTRS could be proven:
4.03/1.97	
4.03/1.97	(0) QTRS
4.03/1.97	(1) QTRSToCSRProof [SOUND, 0 ms]
4.03/1.97	(2) CSR
4.03/1.97	(3) CSRInnermostProof [EQUIVALENT, 0 ms]
4.03/1.97	(4) CSR
4.03/1.97	(5) CSDependencyPairsProof [EQUIVALENT, 1 ms]
4.03/1.97	(6) QCSDP
4.03/1.97	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.03/1.97	(8) AND
4.03/1.97	    (9) QCSDP
4.03/1.97	        (10) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.03/1.97	        (11) QCSDP
4.03/1.97	        (12) PIsEmptyProof [EQUIVALENT, 0 ms]
4.03/1.97	        (13) YES
4.03/1.97	    (14) QCSDP
4.03/1.97	        (15) QCSDPSubtermProof [EQUIVALENT, 0 ms]
4.03/1.97	        (16) QCSDP
4.03/1.97	        (17) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
4.03/1.97	        (18) TRUE
4.03/1.97	
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(0)
4.03/1.97	Obligation:
4.03/1.97	Q restricted rewrite system:
4.03/1.97	The TRS R consists of the following rules:
4.03/1.97	
4.03/1.97	   active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X))
4.03/1.97	   active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
4.03/1.97	   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
4.03/1.97	   active(and(tt, X)) -> mark(X)
4.03/1.97	   active(fst(pair(X, Y))) -> mark(X)
4.03/1.97	   active(head(cons(N, XS))) -> mark(N)
4.03/1.97	   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
4.03/1.97	   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
4.03/1.97	   active(snd(pair(X, Y))) -> mark(Y)
4.03/1.97	   active(splitAt(0, XS)) -> mark(pair(nil, XS))
4.03/1.97	   active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS))
4.03/1.97	   active(tail(cons(N, XS))) -> mark(XS)
4.03/1.97	   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
4.03/1.97	   active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4)
4.03/1.97	   active(U12(X1, X2)) -> U12(active(X1), X2)
4.03/1.97	   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
4.03/1.97	   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
4.03/1.97	   active(pair(X1, X2)) -> pair(active(X1), X2)
4.03/1.97	   active(pair(X1, X2)) -> pair(X1, active(X2))
4.03/1.97	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.03/1.97	   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
4.03/1.97	   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
4.03/1.97	   active(snd(X)) -> snd(active(X))
4.03/1.97	   active(and(X1, X2)) -> and(active(X1), X2)
4.03/1.97	   active(fst(X)) -> fst(active(X))
4.03/1.97	   active(head(X)) -> head(active(X))
4.03/1.97	   active(natsFrom(X)) -> natsFrom(active(X))
4.03/1.97	   active(s(X)) -> s(active(X))
4.03/1.97	   active(sel(X1, X2)) -> sel(active(X1), X2)
4.03/1.97	   active(sel(X1, X2)) -> sel(X1, active(X2))
4.03/1.97	   active(tail(X)) -> tail(active(X))
4.03/1.97	   active(take(X1, X2)) -> take(active(X1), X2)
4.03/1.97	   active(take(X1, X2)) -> take(X1, active(X2))
4.03/1.97	   U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4))
4.03/1.97	   U12(mark(X1), X2) -> mark(U12(X1, X2))
4.03/1.97	   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
4.03/1.97	   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
4.03/1.97	   pair(mark(X1), X2) -> mark(pair(X1, X2))
4.03/1.97	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
4.03/1.97	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.03/1.97	   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
4.03/1.97	   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
4.03/1.97	   snd(mark(X)) -> mark(snd(X))
4.03/1.97	   and(mark(X1), X2) -> mark(and(X1, X2))
4.03/1.97	   fst(mark(X)) -> mark(fst(X))
4.03/1.97	   head(mark(X)) -> mark(head(X))
4.03/1.97	   natsFrom(mark(X)) -> mark(natsFrom(X))
4.03/1.97	   s(mark(X)) -> mark(s(X))
4.03/1.97	   sel(mark(X1), X2) -> mark(sel(X1, X2))
4.03/1.97	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
4.03/1.97	   tail(mark(X)) -> mark(tail(X))
4.03/1.97	   take(mark(X1), X2) -> mark(take(X1, X2))
4.03/1.97	   take(X1, mark(X2)) -> mark(take(X1, X2))
4.03/1.97	   proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4))
4.03/1.97	   proper(tt) -> ok(tt)
4.03/1.97	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
4.03/1.97	   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
4.03/1.97	   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
4.03/1.97	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.03/1.97	   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
4.03/1.97	   proper(snd(X)) -> snd(proper(X))
4.03/1.97	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
4.03/1.97	   proper(fst(X)) -> fst(proper(X))
4.03/1.97	   proper(head(X)) -> head(proper(X))
4.03/1.97	   proper(natsFrom(X)) -> natsFrom(proper(X))
4.03/1.97	   proper(s(X)) -> s(proper(X))
4.03/1.97	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
4.03/1.97	   proper(0) -> ok(0)
4.03/1.97	   proper(nil) -> ok(nil)
4.03/1.97	   proper(tail(X)) -> tail(proper(X))
4.03/1.97	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
4.03/1.97	   U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4))
4.03/1.97	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
4.03/1.97	   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
4.03/1.97	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
4.03/1.97	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.03/1.97	   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
4.03/1.97	   snd(ok(X)) -> ok(snd(X))
4.03/1.97	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
4.03/1.97	   fst(ok(X)) -> ok(fst(X))
4.03/1.97	   head(ok(X)) -> ok(head(X))
4.03/1.97	   natsFrom(ok(X)) -> ok(natsFrom(X))
4.03/1.97	   s(ok(X)) -> ok(s(X))
4.03/1.97	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
4.03/1.97	   tail(ok(X)) -> ok(tail(X))
4.03/1.97	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
4.03/1.97	   top(mark(X)) -> top(proper(X))
4.03/1.97	   top(ok(X)) -> top(active(X))
4.03/1.97	
4.03/1.97	The set Q consists of the following terms:
4.03/1.97	
4.03/1.97	   active(afterNth(x0, x1))
4.03/1.97	   active(natsFrom(x0))
4.03/1.97	   active(sel(x0, x1))
4.03/1.97	   active(take(x0, x1))
4.03/1.97	   active(U11(x0, x1, x2, x3))
4.03/1.97	   active(U12(x0, x1))
4.03/1.97	   active(splitAt(x0, x1))
4.03/1.97	   active(pair(x0, x1))
4.03/1.97	   active(cons(x0, x1))
4.03/1.97	   active(snd(x0))
4.03/1.97	   active(and(x0, x1))
4.03/1.97	   active(fst(x0))
4.03/1.97	   active(head(x0))
4.03/1.97	   active(s(x0))
4.03/1.97	   active(tail(x0))
4.03/1.97	   U11(mark(x0), x1, x2, x3)
4.03/1.97	   U12(mark(x0), x1)
4.03/1.97	   splitAt(mark(x0), x1)
4.03/1.97	   splitAt(x0, mark(x1))
4.03/1.97	   pair(mark(x0), x1)
4.03/1.97	   pair(x0, mark(x1))
4.03/1.97	   cons(mark(x0), x1)
4.03/1.97	   afterNth(mark(x0), x1)
4.03/1.97	   afterNth(x0, mark(x1))
4.03/1.97	   snd(mark(x0))
4.03/1.97	   and(mark(x0), x1)
4.03/1.97	   fst(mark(x0))
4.03/1.97	   head(mark(x0))
4.03/1.97	   natsFrom(mark(x0))
4.03/1.97	   s(mark(x0))
4.03/1.97	   sel(mark(x0), x1)
4.03/1.97	   sel(x0, mark(x1))
4.03/1.97	   tail(mark(x0))
4.03/1.97	   take(mark(x0), x1)
4.03/1.97	   take(x0, mark(x1))
4.03/1.97	   proper(U11(x0, x1, x2, x3))
4.03/1.97	   proper(tt)
4.03/1.97	   proper(U12(x0, x1))
4.03/1.97	   proper(splitAt(x0, x1))
4.03/1.97	   proper(pair(x0, x1))
4.03/1.97	   proper(cons(x0, x1))
4.03/1.97	   proper(afterNth(x0, x1))
4.03/1.97	   proper(snd(x0))
4.03/1.97	   proper(and(x0, x1))
4.03/1.97	   proper(fst(x0))
4.03/1.97	   proper(head(x0))
4.03/1.97	   proper(natsFrom(x0))
4.03/1.97	   proper(s(x0))
4.03/1.97	   proper(sel(x0, x1))
4.03/1.97	   proper(0)
4.03/1.97	   proper(nil)
4.03/1.97	   proper(tail(x0))
4.03/1.97	   proper(take(x0, x1))
4.03/1.97	   U11(ok(x0), ok(x1), ok(x2), ok(x3))
4.03/1.97	   U12(ok(x0), ok(x1))
4.03/1.97	   splitAt(ok(x0), ok(x1))
4.03/1.97	   pair(ok(x0), ok(x1))
4.03/1.97	   cons(ok(x0), ok(x1))
4.03/1.97	   afterNth(ok(x0), ok(x1))
4.03/1.97	   snd(ok(x0))
4.03/1.97	   and(ok(x0), ok(x1))
4.03/1.97	   fst(ok(x0))
4.03/1.97	   head(ok(x0))
4.03/1.97	   natsFrom(ok(x0))
4.03/1.97	   s(ok(x0))
4.03/1.97	   sel(ok(x0), ok(x1))
4.03/1.97	   tail(ok(x0))
4.03/1.97	   take(ok(x0), ok(x1))
4.03/1.97	   top(mark(x0))
4.03/1.97	   top(ok(x0))
4.03/1.97	
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(1) QTRSToCSRProof (SOUND)
4.03/1.97	The following Q TRS is given: Q restricted rewrite system:
4.03/1.97	The TRS R consists of the following rules:
4.03/1.97	
4.03/1.97	   active(U11(tt, N, X, XS)) -> mark(U12(splitAt(N, XS), X))
4.03/1.97	   active(U12(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS))
4.03/1.97	   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
4.03/1.97	   active(and(tt, X)) -> mark(X)
4.03/1.97	   active(fst(pair(X, Y))) -> mark(X)
4.03/1.97	   active(head(cons(N, XS))) -> mark(N)
4.03/1.97	   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
4.03/1.97	   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
4.03/1.97	   active(snd(pair(X, Y))) -> mark(Y)
4.03/1.97	   active(splitAt(0, XS)) -> mark(pair(nil, XS))
4.03/1.97	   active(splitAt(s(N), cons(X, XS))) -> mark(U11(tt, N, X, XS))
4.03/1.97	   active(tail(cons(N, XS))) -> mark(XS)
4.03/1.97	   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
4.03/1.97	   active(U11(X1, X2, X3, X4)) -> U11(active(X1), X2, X3, X4)
4.03/1.97	   active(U12(X1, X2)) -> U12(active(X1), X2)
4.03/1.97	   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
4.03/1.97	   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
4.03/1.97	   active(pair(X1, X2)) -> pair(active(X1), X2)
4.03/1.97	   active(pair(X1, X2)) -> pair(X1, active(X2))
4.03/1.97	   active(cons(X1, X2)) -> cons(active(X1), X2)
4.03/1.97	   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
4.03/1.97	   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
4.03/1.97	   active(snd(X)) -> snd(active(X))
4.03/1.97	   active(and(X1, X2)) -> and(active(X1), X2)
4.03/1.97	   active(fst(X)) -> fst(active(X))
4.03/1.97	   active(head(X)) -> head(active(X))
4.03/1.97	   active(natsFrom(X)) -> natsFrom(active(X))
4.03/1.97	   active(s(X)) -> s(active(X))
4.03/1.97	   active(sel(X1, X2)) -> sel(active(X1), X2)
4.03/1.97	   active(sel(X1, X2)) -> sel(X1, active(X2))
4.03/1.97	   active(tail(X)) -> tail(active(X))
4.03/1.97	   active(take(X1, X2)) -> take(active(X1), X2)
4.03/1.97	   active(take(X1, X2)) -> take(X1, active(X2))
4.03/1.97	   U11(mark(X1), X2, X3, X4) -> mark(U11(X1, X2, X3, X4))
4.03/1.97	   U12(mark(X1), X2) -> mark(U12(X1, X2))
4.03/1.97	   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
4.03/1.97	   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
4.03/1.97	   pair(mark(X1), X2) -> mark(pair(X1, X2))
4.03/1.97	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
4.03/1.97	   cons(mark(X1), X2) -> mark(cons(X1, X2))
4.03/1.97	   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
4.03/1.97	   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
4.03/1.97	   snd(mark(X)) -> mark(snd(X))
4.03/1.97	   and(mark(X1), X2) -> mark(and(X1, X2))
4.03/1.97	   fst(mark(X)) -> mark(fst(X))
4.03/1.97	   head(mark(X)) -> mark(head(X))
4.03/1.97	   natsFrom(mark(X)) -> mark(natsFrom(X))
4.03/1.97	   s(mark(X)) -> mark(s(X))
4.03/1.97	   sel(mark(X1), X2) -> mark(sel(X1, X2))
4.03/1.97	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
4.03/1.97	   tail(mark(X)) -> mark(tail(X))
4.03/1.97	   take(mark(X1), X2) -> mark(take(X1, X2))
4.03/1.97	   take(X1, mark(X2)) -> mark(take(X1, X2))
4.03/1.97	   proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4))
4.03/1.97	   proper(tt) -> ok(tt)
4.03/1.97	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
4.03/1.97	   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
4.03/1.97	   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
4.03/1.97	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
4.03/1.97	   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
4.03/1.97	   proper(snd(X)) -> snd(proper(X))
4.03/1.97	   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
4.03/1.97	   proper(fst(X)) -> fst(proper(X))
4.03/1.97	   proper(head(X)) -> head(proper(X))
4.03/1.97	   proper(natsFrom(X)) -> natsFrom(proper(X))
4.03/1.97	   proper(s(X)) -> s(proper(X))
4.03/1.97	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
4.03/1.97	   proper(0) -> ok(0)
4.03/1.97	   proper(nil) -> ok(nil)
4.03/1.97	   proper(tail(X)) -> tail(proper(X))
4.03/1.97	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
4.03/1.97	   U11(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U11(X1, X2, X3, X4))
4.03/1.97	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
4.03/1.97	   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
4.03/1.97	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
4.03/1.97	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
4.03/1.97	   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
4.03/1.97	   snd(ok(X)) -> ok(snd(X))
4.03/1.97	   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
4.03/1.97	   fst(ok(X)) -> ok(fst(X))
4.03/1.97	   head(ok(X)) -> ok(head(X))
4.03/1.97	   natsFrom(ok(X)) -> ok(natsFrom(X))
4.03/1.97	   s(ok(X)) -> ok(s(X))
4.03/1.97	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
4.03/1.97	   tail(ok(X)) -> ok(tail(X))
4.03/1.97	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
4.03/1.97	   top(mark(X)) -> top(proper(X))
4.03/1.97	   top(ok(X)) -> top(active(X))
4.03/1.97	
4.03/1.97	The set Q consists of the following terms:
4.03/1.97	
4.03/1.97	   active(afterNth(x0, x1))
4.03/1.97	   active(natsFrom(x0))
4.03/1.97	   active(sel(x0, x1))
4.03/1.97	   active(take(x0, x1))
4.03/1.97	   active(U11(x0, x1, x2, x3))
4.03/1.97	   active(U12(x0, x1))
4.03/1.97	   active(splitAt(x0, x1))
4.03/1.97	   active(pair(x0, x1))
4.03/1.97	   active(cons(x0, x1))
4.03/1.97	   active(snd(x0))
4.03/1.97	   active(and(x0, x1))
4.03/1.97	   active(fst(x0))
4.03/1.97	   active(head(x0))
4.03/1.97	   active(s(x0))
4.03/1.97	   active(tail(x0))
4.03/1.97	   U11(mark(x0), x1, x2, x3)
4.03/1.97	   U12(mark(x0), x1)
4.03/1.97	   splitAt(mark(x0), x1)
4.03/1.97	   splitAt(x0, mark(x1))
4.03/1.97	   pair(mark(x0), x1)
4.03/1.97	   pair(x0, mark(x1))
4.03/1.97	   cons(mark(x0), x1)
4.03/1.97	   afterNth(mark(x0), x1)
4.03/1.97	   afterNth(x0, mark(x1))
4.03/1.97	   snd(mark(x0))
4.03/1.97	   and(mark(x0), x1)
4.03/1.97	   fst(mark(x0))
4.03/1.97	   head(mark(x0))
4.03/1.97	   natsFrom(mark(x0))
4.03/1.97	   s(mark(x0))
4.03/1.97	   sel(mark(x0), x1)
4.03/1.97	   sel(x0, mark(x1))
4.03/1.97	   tail(mark(x0))
4.03/1.97	   take(mark(x0), x1)
4.03/1.97	   take(x0, mark(x1))
4.03/1.97	   proper(U11(x0, x1, x2, x3))
4.03/1.97	   proper(tt)
4.03/1.97	   proper(U12(x0, x1))
4.03/1.97	   proper(splitAt(x0, x1))
4.03/1.97	   proper(pair(x0, x1))
4.03/1.97	   proper(cons(x0, x1))
4.03/1.97	   proper(afterNth(x0, x1))
4.03/1.97	   proper(snd(x0))
4.03/1.97	   proper(and(x0, x1))
4.03/1.97	   proper(fst(x0))
4.03/1.97	   proper(head(x0))
4.03/1.97	   proper(natsFrom(x0))
4.03/1.97	   proper(s(x0))
4.03/1.97	   proper(sel(x0, x1))
4.03/1.97	   proper(0)
4.03/1.97	   proper(nil)
4.03/1.97	   proper(tail(x0))
4.03/1.97	   proper(take(x0, x1))
4.03/1.97	   U11(ok(x0), ok(x1), ok(x2), ok(x3))
4.03/1.97	   U12(ok(x0), ok(x1))
4.03/1.97	   splitAt(ok(x0), ok(x1))
4.03/1.97	   pair(ok(x0), ok(x1))
4.03/1.97	   cons(ok(x0), ok(x1))
4.03/1.97	   afterNth(ok(x0), ok(x1))
4.03/1.97	   snd(ok(x0))
4.03/1.97	   and(ok(x0), ok(x1))
4.03/1.97	   fst(ok(x0))
4.03/1.97	   head(ok(x0))
4.03/1.97	   natsFrom(ok(x0))
4.03/1.97	   s(ok(x0))
4.03/1.97	   sel(ok(x0), ok(x1))
4.03/1.97	   tail(ok(x0))
4.03/1.97	   take(ok(x0), ok(x1))
4.03/1.97	   top(mark(x0))
4.03/1.97	   top(ok(x0))
4.03/1.97	
4.03/1.97	Special symbols used for the transformation (see [GM04]):
4.03/1.97	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
4.03/1.97	The replacement map contains the following entries:
4.03/1.97	
4.03/1.97	U11: {1}
4.03/1.97	tt: empty set
4.03/1.97	U12: {1}
4.03/1.97	splitAt: {1, 2}
4.03/1.97	pair: {1, 2}
4.03/1.97	cons: {1}
4.03/1.97	afterNth: {1, 2}
4.03/1.97	snd: {1}
4.03/1.97	and: {1}
4.03/1.97	fst: {1}
4.03/1.97	head: {1}
4.03/1.97	natsFrom: {1}
4.03/1.97	s: {1}
4.03/1.97	sel: {1, 2}
4.03/1.97	0: empty set
4.03/1.97	nil: empty set
4.03/1.97	tail: {1}
4.03/1.97	take: {1, 2}
4.03/1.97	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.
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(2)
4.03/1.97	Obligation:
4.03/1.97	Context-sensitive rewrite system:
4.03/1.97	The TRS R consists of the following rules:
4.03/1.97	
4.03/1.97	   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
4.03/1.97	   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
4.03/1.97	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.03/1.97	   and(tt, X) -> X
4.03/1.97	   fst(pair(X, Y)) -> X
4.03/1.97	   head(cons(N, XS)) -> N
4.03/1.97	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.03/1.97	   sel(N, XS) -> head(afterNth(N, XS))
4.03/1.97	   snd(pair(X, Y)) -> Y
4.03/1.97	   splitAt(0, XS) -> pair(nil, XS)
4.03/1.97	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
4.03/1.97	   tail(cons(N, XS)) -> XS
4.03/1.97	   take(N, XS) -> fst(splitAt(N, XS))
4.03/1.97	
4.03/1.97	The replacement map contains the following entries:
4.03/1.97	
4.03/1.97	U11: {1}
4.03/1.97	tt: empty set
4.03/1.97	U12: {1}
4.03/1.97	splitAt: {1, 2}
4.03/1.97	pair: {1, 2}
4.03/1.97	cons: {1}
4.03/1.97	afterNth: {1, 2}
4.03/1.97	snd: {1}
4.03/1.97	and: {1}
4.03/1.97	fst: {1}
4.03/1.97	head: {1}
4.03/1.97	natsFrom: {1}
4.03/1.97	s: {1}
4.03/1.97	sel: {1, 2}
4.03/1.97	0: empty set
4.03/1.97	nil: empty set
4.03/1.97	tail: {1}
4.03/1.97	take: {1, 2}
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(3) CSRInnermostProof (EQUIVALENT)
4.03/1.97	The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(4)
4.03/1.97	Obligation:
4.03/1.97	Context-sensitive rewrite system:
4.03/1.97	The TRS R consists of the following rules:
4.03/1.97	
4.03/1.97	   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
4.03/1.97	   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
4.03/1.97	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.03/1.97	   and(tt, X) -> X
4.03/1.97	   fst(pair(X, Y)) -> X
4.03/1.97	   head(cons(N, XS)) -> N
4.03/1.97	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.03/1.97	   sel(N, XS) -> head(afterNth(N, XS))
4.03/1.97	   snd(pair(X, Y)) -> Y
4.03/1.97	   splitAt(0, XS) -> pair(nil, XS)
4.03/1.97	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
4.03/1.97	   tail(cons(N, XS)) -> XS
4.03/1.97	   take(N, XS) -> fst(splitAt(N, XS))
4.03/1.97	
4.03/1.97	The replacement map contains the following entries:
4.03/1.97	
4.03/1.97	U11: {1}
4.03/1.97	tt: empty set
4.03/1.97	U12: {1}
4.03/1.97	splitAt: {1, 2}
4.03/1.97	pair: {1, 2}
4.03/1.97	cons: {1}
4.03/1.97	afterNth: {1, 2}
4.03/1.97	snd: {1}
4.03/1.97	and: {1}
4.03/1.97	fst: {1}
4.03/1.97	head: {1}
4.03/1.97	natsFrom: {1}
4.03/1.97	s: {1}
4.03/1.97	sel: {1, 2}
4.03/1.97	0: empty set
4.03/1.97	nil: empty set
4.03/1.97	tail: {1}
4.03/1.97	take: {1, 2}
4.03/1.97	
4.03/1.97	
4.03/1.97	Innermost Strategy.
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(5) CSDependencyPairsProof (EQUIVALENT)
4.03/1.97	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(6)
4.03/1.97	Obligation:
4.03/1.97	Q-restricted context-sensitive dependency pair problem:
4.03/1.97	The symbols in {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2, SPLITAT_2, SND_1, AFTERNTH_2, HEAD_1, SEL_2, FST_1, TAKE_2, TAIL_1, NATSFROM_1} are replacing on all positions.
4.03/1.97	For all symbols f in {U11_4, U12_2, cons_2, and_2, U12'_2, U11'_4, AND_2} we have mu(f) = {1}.
4.03/1.97	The symbols in {U_1} are not replacing on any position.
4.03/1.97	
4.03/1.97	The ordinary context-sensitive dependency pairs DP_o are:
4.03/1.97	   U11'(tt, N, X, XS) -> U12'(splitAt(N, XS), X)
4.03/1.97	   U11'(tt, N, X, XS) -> SPLITAT(N, XS)
4.03/1.97	   AFTERNTH(N, XS) -> SND(splitAt(N, XS))
4.03/1.97	   AFTERNTH(N, XS) -> SPLITAT(N, XS)
4.03/1.97	   SEL(N, XS) -> HEAD(afterNth(N, XS))
4.03/1.97	   SEL(N, XS) -> AFTERNTH(N, XS)
4.03/1.97	   SPLITAT(s(N), cons(X, XS)) -> U11'(tt, N, X, XS)
4.03/1.97	   TAKE(N, XS) -> FST(splitAt(N, XS))
4.03/1.97	   TAKE(N, XS) -> SPLITAT(N, XS)
4.03/1.97	
4.03/1.97	The collapsing dependency pairs are DP_c:
4.03/1.97	   U11'(tt, N, X, XS) -> N
4.03/1.97	   U11'(tt, N, X, XS) -> XS
4.03/1.97	   U12'(pair(YS, ZS), X) -> X
4.03/1.97	   AND(tt, X) -> X
4.03/1.97	   TAIL(cons(N, XS)) -> XS
4.03/1.97	
4.03/1.97	
4.03/1.97	The hidden terms of R are:
4.03/1.97	
4.03/1.97	   natsFrom(s(x0))
4.03/1.97	
4.03/1.97	Every hiding context is built from:
4.03/1.97	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@133d4b6c
4.03/1.97	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@46981015
4.03/1.97	
4.03/1.97	Hence, the new unhiding pairs DP_u are :
4.03/1.97	   U11'(tt, N, X, XS) -> U(N)
4.03/1.97	   U11'(tt, N, X, XS) -> U(XS)
4.03/1.97	   U12'(pair(YS, ZS), X) -> U(X)
4.03/1.97	   AND(tt, X) -> U(X)
4.03/1.97	   TAIL(cons(N, XS)) -> U(XS)
4.03/1.97	   U(s(x_0)) -> U(x_0)
4.03/1.97	   U(natsFrom(x_0)) -> U(x_0)
4.03/1.97	   U(natsFrom(s(x0))) -> NATSFROM(s(x0))
4.03/1.97	
4.03/1.97	The TRS R consists of the following rules:
4.03/1.97	
4.03/1.97	   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
4.03/1.97	   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
4.03/1.97	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.03/1.97	   and(tt, X) -> X
4.03/1.97	   fst(pair(X, Y)) -> X
4.03/1.97	   head(cons(N, XS)) -> N
4.03/1.97	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.03/1.97	   sel(N, XS) -> head(afterNth(N, XS))
4.03/1.97	   snd(pair(X, Y)) -> Y
4.03/1.97	   splitAt(0, XS) -> pair(nil, XS)
4.03/1.97	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
4.03/1.97	   tail(cons(N, XS)) -> XS
4.03/1.97	   take(N, XS) -> fst(splitAt(N, XS))
4.03/1.97	
4.03/1.97	The set Q consists of the following terms:
4.03/1.97	
4.03/1.97	   U11(tt, x0, x1, x2)
4.03/1.97	   U12(pair(x0, x1), x2)
4.03/1.97	   afterNth(x0, x1)
4.03/1.97	   and(tt, x0)
4.03/1.97	   fst(pair(x0, x1))
4.03/1.97	   head(cons(x0, x1))
4.03/1.97	   natsFrom(x0)
4.03/1.97	   sel(x0, x1)
4.03/1.97	   snd(pair(x0, x1))
4.03/1.97	   splitAt(0, x0)
4.03/1.97	   splitAt(s(x0), cons(x1, x2))
4.03/1.97	   tail(cons(x0, x1))
4.03/1.97	   take(x0, x1)
4.03/1.97	
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(7) QCSDependencyGraphProof (EQUIVALENT)
4.03/1.97	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 11 less nodes.
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(8)
4.03/1.97	Complex Obligation (AND)
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(9)
4.03/1.97	Obligation:
4.03/1.97	Q-restricted context-sensitive dependency pair problem:
4.03/1.97	The symbols in {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2} are replacing on all positions.
4.03/1.97	For all symbols f in {U11_4, U12_2, cons_2, and_2} we have mu(f) = {1}.
4.03/1.97	The symbols in {U_1} are not replacing on any position.
4.03/1.97	
4.03/1.97	The TRS P consists of the following rules:
4.03/1.97	
4.03/1.97	   U(s(x_0)) -> U(x_0)
4.03/1.97	   U(natsFrom(x_0)) -> U(x_0)
4.03/1.97	
4.03/1.97	The TRS R consists of the following rules:
4.03/1.97	
4.03/1.97	   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
4.03/1.97	   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
4.03/1.97	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.03/1.97	   and(tt, X) -> X
4.03/1.97	   fst(pair(X, Y)) -> X
4.03/1.97	   head(cons(N, XS)) -> N
4.03/1.97	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.03/1.97	   sel(N, XS) -> head(afterNth(N, XS))
4.03/1.97	   snd(pair(X, Y)) -> Y
4.03/1.97	   splitAt(0, XS) -> pair(nil, XS)
4.03/1.97	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
4.03/1.97	   tail(cons(N, XS)) -> XS
4.03/1.97	   take(N, XS) -> fst(splitAt(N, XS))
4.03/1.97	
4.03/1.97	The set Q consists of the following terms:
4.03/1.97	
4.03/1.97	   U11(tt, x0, x1, x2)
4.03/1.97	   U12(pair(x0, x1), x2)
4.03/1.97	   afterNth(x0, x1)
4.03/1.97	   and(tt, x0)
4.03/1.97	   fst(pair(x0, x1))
4.03/1.97	   head(cons(x0, x1))
4.03/1.97	   natsFrom(x0)
4.03/1.97	   sel(x0, x1)
4.03/1.97	   snd(pair(x0, x1))
4.03/1.97	   splitAt(0, x0)
4.03/1.97	   splitAt(s(x0), cons(x1, x2))
4.03/1.97	   tail(cons(x0, x1))
4.03/1.97	   take(x0, x1)
4.03/1.97	
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(10) QCSDPSubtermProof (EQUIVALENT)
4.03/1.97	We use the subterm processor [DA_EMMES].
4.03/1.97	
4.03/1.97	
4.03/1.97	The following pairs can be oriented strictly and are deleted.
4.03/1.97	
4.03/1.97	   U(s(x_0)) -> U(x_0)
4.03/1.97	   U(natsFrom(x_0)) -> U(x_0)
4.03/1.97	The remaining pairs can at least be oriented weakly.
4.03/1.97	none
4.03/1.97	Used ordering:  Combined order from the following AFS and order.
4.03/1.97	U(x1)  =  x1
4.03/1.97	
4.03/1.97	
4.03/1.97	Subterm Order
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(11)
4.03/1.97	Obligation:
4.03/1.97	Q-restricted context-sensitive dependency pair problem:
4.03/1.97	The symbols in {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2} are replacing on all positions.
4.03/1.97	For all symbols f in {U11_4, U12_2, cons_2, and_2} we have mu(f) = {1}.
4.03/1.97	
4.03/1.97	The TRS P consists of the following rules:
4.03/1.97	none
4.03/1.97	
4.03/1.97	The TRS R consists of the following rules:
4.03/1.97	
4.03/1.97	   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
4.03/1.97	   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
4.03/1.97	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.03/1.97	   and(tt, X) -> X
4.03/1.97	   fst(pair(X, Y)) -> X
4.03/1.97	   head(cons(N, XS)) -> N
4.03/1.97	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.03/1.97	   sel(N, XS) -> head(afterNth(N, XS))
4.03/1.97	   snd(pair(X, Y)) -> Y
4.03/1.97	   splitAt(0, XS) -> pair(nil, XS)
4.03/1.97	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
4.03/1.97	   tail(cons(N, XS)) -> XS
4.03/1.97	   take(N, XS) -> fst(splitAt(N, XS))
4.03/1.97	
4.03/1.97	The set Q consists of the following terms:
4.03/1.97	
4.03/1.97	   U11(tt, x0, x1, x2)
4.03/1.97	   U12(pair(x0, x1), x2)
4.03/1.97	   afterNth(x0, x1)
4.03/1.97	   and(tt, x0)
4.03/1.97	   fst(pair(x0, x1))
4.03/1.97	   head(cons(x0, x1))
4.03/1.97	   natsFrom(x0)
4.03/1.97	   sel(x0, x1)
4.03/1.97	   snd(pair(x0, x1))
4.03/1.97	   splitAt(0, x0)
4.03/1.97	   splitAt(s(x0), cons(x1, x2))
4.03/1.97	   tail(cons(x0, x1))
4.03/1.97	   take(x0, x1)
4.03/1.97	
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(12) PIsEmptyProof (EQUIVALENT)
4.03/1.97	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(13)
4.03/1.97	YES
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(14)
4.03/1.97	Obligation:
4.03/1.97	Q-restricted context-sensitive dependency pair problem:
4.03/1.97	The symbols in {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2, SPLITAT_2} are replacing on all positions.
4.03/1.97	For all symbols f in {U11_4, U12_2, cons_2, and_2, U11'_4} we have mu(f) = {1}.
4.03/1.97	
4.03/1.97	The TRS P consists of the following rules:
4.03/1.97	
4.03/1.97	   U11'(tt, N, X, XS) -> SPLITAT(N, XS)
4.03/1.97	   SPLITAT(s(N), cons(X, XS)) -> U11'(tt, N, X, XS)
4.03/1.97	
4.03/1.97	The TRS R consists of the following rules:
4.03/1.97	
4.03/1.97	   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
4.03/1.97	   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
4.03/1.97	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.03/1.97	   and(tt, X) -> X
4.03/1.97	   fst(pair(X, Y)) -> X
4.03/1.97	   head(cons(N, XS)) -> N
4.03/1.97	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.03/1.97	   sel(N, XS) -> head(afterNth(N, XS))
4.03/1.97	   snd(pair(X, Y)) -> Y
4.03/1.97	   splitAt(0, XS) -> pair(nil, XS)
4.03/1.97	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
4.03/1.97	   tail(cons(N, XS)) -> XS
4.03/1.97	   take(N, XS) -> fst(splitAt(N, XS))
4.03/1.97	
4.03/1.97	The set Q consists of the following terms:
4.03/1.97	
4.03/1.97	   U11(tt, x0, x1, x2)
4.03/1.97	   U12(pair(x0, x1), x2)
4.03/1.97	   afterNth(x0, x1)
4.03/1.97	   and(tt, x0)
4.03/1.97	   fst(pair(x0, x1))
4.03/1.97	   head(cons(x0, x1))
4.03/1.97	   natsFrom(x0)
4.03/1.97	   sel(x0, x1)
4.03/1.97	   snd(pair(x0, x1))
4.03/1.97	   splitAt(0, x0)
4.03/1.97	   splitAt(s(x0), cons(x1, x2))
4.03/1.97	   tail(cons(x0, x1))
4.03/1.97	   take(x0, x1)
4.03/1.97	
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(15) QCSDPSubtermProof (EQUIVALENT)
4.03/1.97	We use the subterm processor [DA_EMMES].
4.03/1.97	
4.03/1.97	
4.03/1.97	The following pairs can be oriented strictly and are deleted.
4.03/1.97	
4.03/1.97	   SPLITAT(s(N), cons(X, XS)) -> U11'(tt, N, X, XS)
4.03/1.97	The remaining pairs can at least be oriented weakly.
4.03/1.97	
4.03/1.97	   U11'(tt, N, X, XS) -> SPLITAT(N, XS)
4.03/1.97	Used ordering:  Combined order from the following AFS and order.
4.03/1.97	SPLITAT(x1, x2)  =  x1
4.03/1.97	
4.03/1.97	U11'(x1, x2, x3, x4)  =  x2
4.03/1.97	
4.03/1.97	
4.03/1.97	Subterm Order
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(16)
4.03/1.97	Obligation:
4.03/1.97	Q-restricted context-sensitive dependency pair problem:
4.03/1.97	The symbols in {splitAt_2, pair_2, afterNth_2, snd_1, fst_1, head_1, natsFrom_1, s_1, sel_2, tail_1, take_2, SPLITAT_2} are replacing on all positions.
4.03/1.97	For all symbols f in {U11_4, U12_2, cons_2, and_2, U11'_4} we have mu(f) = {1}.
4.03/1.97	
4.03/1.97	The TRS P consists of the following rules:
4.03/1.97	
4.03/1.97	   U11'(tt, N, X, XS) -> SPLITAT(N, XS)
4.03/1.97	
4.03/1.97	The TRS R consists of the following rules:
4.03/1.97	
4.03/1.97	   U11(tt, N, X, XS) -> U12(splitAt(N, XS), X)
4.03/1.97	   U12(pair(YS, ZS), X) -> pair(cons(X, YS), ZS)
4.03/1.97	   afterNth(N, XS) -> snd(splitAt(N, XS))
4.03/1.97	   and(tt, X) -> X
4.03/1.97	   fst(pair(X, Y)) -> X
4.03/1.97	   head(cons(N, XS)) -> N
4.03/1.97	   natsFrom(N) -> cons(N, natsFrom(s(N)))
4.03/1.97	   sel(N, XS) -> head(afterNth(N, XS))
4.03/1.97	   snd(pair(X, Y)) -> Y
4.03/1.97	   splitAt(0, XS) -> pair(nil, XS)
4.03/1.97	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, XS)
4.03/1.97	   tail(cons(N, XS)) -> XS
4.03/1.97	   take(N, XS) -> fst(splitAt(N, XS))
4.03/1.97	
4.03/1.97	The set Q consists of the following terms:
4.03/1.97	
4.03/1.97	   U11(tt, x0, x1, x2)
4.03/1.97	   U12(pair(x0, x1), x2)
4.03/1.97	   afterNth(x0, x1)
4.03/1.97	   and(tt, x0)
4.03/1.97	   fst(pair(x0, x1))
4.03/1.97	   head(cons(x0, x1))
4.03/1.97	   natsFrom(x0)
4.03/1.97	   sel(x0, x1)
4.03/1.97	   snd(pair(x0, x1))
4.03/1.97	   splitAt(0, x0)
4.03/1.97	   splitAt(s(x0), cons(x1, x2))
4.03/1.97	   tail(cons(x0, x1))
4.03/1.97	   take(x0, x1)
4.03/1.97	
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(17) QCSDependencyGraphProof (EQUIVALENT)
4.03/1.97	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.
4.03/1.97	
4.03/1.97	----------------------------------------
4.03/1.97	
4.03/1.97	(18)
4.03/1.97	TRUE
4.27/2.01	EOF