6.73/2.52	YES
6.73/2.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.73/2.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.73/2.55	
6.73/2.55	
6.73/2.55	Termination w.r.t. Q of the given QTRS could be proven:
6.73/2.55	
6.73/2.55	(0) QTRS
6.73/2.55	(1) QTRSToCSRProof [SOUND, 0 ms]
6.73/2.55	(2) CSR
6.73/2.55	(3) CSRRRRProof [EQUIVALENT, 92 ms]
6.73/2.55	(4) CSR
6.73/2.55	(5) CSRRRRProof [EQUIVALENT, 0 ms]
6.73/2.55	(6) CSR
6.73/2.55	(7) CSDependencyPairsProof [EQUIVALENT, 0 ms]
6.73/2.55	(8) QCSDP
6.73/2.55	(9) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
6.73/2.55	(10) AND
6.73/2.55	    (11) QCSDP
6.73/2.55	        (12) QCSUsableRulesProof [EQUIVALENT, 0 ms]
6.73/2.55	        (13) QCSDP
6.73/2.55	        (14) QCSDPMuMonotonicPoloProof [EQUIVALENT, 30 ms]
6.73/2.55	        (15) QCSDP
6.73/2.55	        (16) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
6.73/2.55	        (17) QCSDP
6.73/2.55	        (18) QCSDPSubtermProof [EQUIVALENT, 0 ms]
6.73/2.55	        (19) QCSDP
6.73/2.55	        (20) PIsEmptyProof [EQUIVALENT, 0 ms]
6.73/2.55	        (21) YES
6.73/2.55	    (22) QCSDP
6.73/2.55	        (23) QCSDPSubtermProof [EQUIVALENT, 2 ms]
6.73/2.55	        (24) QCSDP
6.73/2.55	        (25) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
6.73/2.55	        (26) TRUE
6.73/2.55	    (27) QCSDP
6.73/2.55	        (28) QCSDPReductionPairProof [EQUIVALENT, 26 ms]
6.73/2.55	        (29) QCSDP
6.73/2.55	        (30) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
6.73/2.55	        (31) TRUE
6.73/2.55	
6.73/2.55	
6.73/2.55	----------------------------------------
6.73/2.55	
6.73/2.55	(0)
6.73/2.55	Obligation:
6.73/2.55	Q restricted rewrite system:
6.73/2.55	The TRS R consists of the following rules:
6.73/2.55	
6.73/2.55	   active(zeros) -> mark(cons(0, zeros))
6.73/2.55	   active(U11(tt)) -> mark(tt)
6.73/2.55	   active(U21(tt)) -> mark(tt)
6.73/2.55	   active(U31(tt)) -> mark(tt)
6.73/2.55	   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
6.73/2.55	   active(U42(tt)) -> mark(tt)
6.73/2.55	   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
6.73/2.55	   active(U52(tt)) -> mark(tt)
6.73/2.55	   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
6.73/2.55	   active(U62(tt)) -> mark(tt)
6.73/2.55	   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
6.73/2.55	   active(U72(tt, L)) -> mark(s(length(L)))
6.73/2.55	   active(U81(tt)) -> mark(nil)
6.73/2.55	   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
6.73/2.55	   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
6.73/2.55	   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
6.73/2.55	   active(isNat(0)) -> mark(tt)
6.73/2.55	   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
6.73/2.55	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
6.73/2.55	   active(isNatIList(V)) -> mark(U31(isNatList(V)))
6.73/2.55	   active(isNatIList(zeros)) -> mark(tt)
6.73/2.55	   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
6.73/2.55	   active(isNatList(nil)) -> mark(tt)
6.73/2.55	   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
6.73/2.55	   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
6.73/2.55	   active(length(nil)) -> mark(0)
6.73/2.55	   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
6.73/2.55	   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
6.73/2.55	   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
6.73/2.55	   active(cons(X1, X2)) -> cons(active(X1), X2)
6.73/2.55	   active(U11(X)) -> U11(active(X))
6.73/2.55	   active(U21(X)) -> U21(active(X))
6.73/2.55	   active(U31(X)) -> U31(active(X))
6.73/2.55	   active(U41(X1, X2)) -> U41(active(X1), X2)
6.73/2.55	   active(U42(X)) -> U42(active(X))
6.73/2.55	   active(U51(X1, X2)) -> U51(active(X1), X2)
6.73/2.55	   active(U52(X)) -> U52(active(X))
6.73/2.55	   active(U61(X1, X2)) -> U61(active(X1), X2)
6.73/2.55	   active(U62(X)) -> U62(active(X))
6.73/2.55	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
6.73/2.55	   active(U72(X1, X2)) -> U72(active(X1), X2)
6.73/2.55	   active(s(X)) -> s(active(X))
6.73/2.55	   active(length(X)) -> length(active(X))
6.73/2.55	   active(U81(X)) -> U81(active(X))
6.73/2.55	   active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
6.73/2.55	   active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
6.73/2.55	   active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
6.73/2.55	   active(take(X1, X2)) -> take(active(X1), X2)
6.73/2.55	   active(take(X1, X2)) -> take(X1, active(X2))
6.73/2.55	   cons(mark(X1), X2) -> mark(cons(X1, X2))
6.73/2.55	   U11(mark(X)) -> mark(U11(X))
6.73/2.55	   U21(mark(X)) -> mark(U21(X))
6.73/2.55	   U31(mark(X)) -> mark(U31(X))
6.73/2.55	   U41(mark(X1), X2) -> mark(U41(X1, X2))
6.73/2.55	   U42(mark(X)) -> mark(U42(X))
6.73/2.55	   U51(mark(X1), X2) -> mark(U51(X1, X2))
6.73/2.55	   U52(mark(X)) -> mark(U52(X))
6.73/2.55	   U61(mark(X1), X2) -> mark(U61(X1, X2))
6.73/2.55	   U62(mark(X)) -> mark(U62(X))
6.73/2.55	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
6.73/2.55	   U72(mark(X1), X2) -> mark(U72(X1, X2))
6.73/2.55	   s(mark(X)) -> mark(s(X))
6.73/2.55	   length(mark(X)) -> mark(length(X))
6.73/2.55	   U81(mark(X)) -> mark(U81(X))
6.73/2.55	   U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
6.73/2.55	   U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
6.73/2.55	   U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
6.73/2.55	   take(mark(X1), X2) -> mark(take(X1, X2))
6.73/2.55	   take(X1, mark(X2)) -> mark(take(X1, X2))
6.73/2.55	   proper(zeros) -> ok(zeros)
6.73/2.55	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
6.73/2.55	   proper(0) -> ok(0)
6.73/2.55	   proper(U11(X)) -> U11(proper(X))
6.73/2.55	   proper(tt) -> ok(tt)
6.73/2.55	   proper(U21(X)) -> U21(proper(X))
6.73/2.55	   proper(U31(X)) -> U31(proper(X))
6.73/2.55	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
6.73/2.55	   proper(U42(X)) -> U42(proper(X))
6.73/2.55	   proper(isNatIList(X)) -> isNatIList(proper(X))
6.73/2.55	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
6.73/2.55	   proper(U52(X)) -> U52(proper(X))
6.73/2.55	   proper(isNatList(X)) -> isNatList(proper(X))
6.73/2.55	   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
6.73/2.55	   proper(U62(X)) -> U62(proper(X))
6.73/2.55	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
6.73/2.55	   proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
6.73/2.55	   proper(isNat(X)) -> isNat(proper(X))
6.73/2.55	   proper(s(X)) -> s(proper(X))
6.73/2.55	   proper(length(X)) -> length(proper(X))
6.73/2.55	   proper(U81(X)) -> U81(proper(X))
6.73/2.55	   proper(nil) -> ok(nil)
6.73/2.55	   proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
6.73/2.55	   proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
6.73/2.55	   proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
6.73/2.55	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
6.73/2.55	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
6.73/2.55	   U11(ok(X)) -> ok(U11(X))
6.73/2.55	   U21(ok(X)) -> ok(U21(X))
6.73/2.55	   U31(ok(X)) -> ok(U31(X))
6.73/2.55	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
6.73/2.55	   U42(ok(X)) -> ok(U42(X))
6.73/2.55	   isNatIList(ok(X)) -> ok(isNatIList(X))
6.73/2.55	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
6.73/2.55	   U52(ok(X)) -> ok(U52(X))
6.73/2.55	   isNatList(ok(X)) -> ok(isNatList(X))
6.73/2.55	   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
6.73/2.55	   U62(ok(X)) -> ok(U62(X))
6.73/2.55	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
6.73/2.55	   U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
6.73/2.55	   isNat(ok(X)) -> ok(isNat(X))
6.73/2.55	   s(ok(X)) -> ok(s(X))
6.73/2.55	   length(ok(X)) -> ok(length(X))
6.73/2.55	   U81(ok(X)) -> ok(U81(X))
6.73/2.55	   U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
6.73/2.55	   U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
6.73/2.55	   U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
6.73/2.55	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
6.73/2.55	   top(mark(X)) -> top(proper(X))
6.73/2.55	   top(ok(X)) -> top(active(X))
6.73/2.55	
6.73/2.55	The set Q consists of the following terms:
6.73/2.55	
6.73/2.55	   active(zeros)
6.73/2.55	   active(isNat(0))
6.73/2.55	   active(isNat(length(x0)))
6.73/2.55	   active(isNat(s(x0)))
6.73/2.55	   active(isNatIList(x0))
6.73/2.55	   active(isNatList(nil))
6.73/2.55	   active(isNatList(cons(x0, x1)))
6.73/2.55	   active(isNatList(take(x0, x1)))
6.73/2.55	   active(cons(x0, x1))
6.73/2.55	   active(U11(x0))
6.73/2.55	   active(U21(x0))
6.73/2.55	   active(U31(x0))
6.73/2.55	   active(U41(x0, x1))
6.73/2.55	   active(U42(x0))
6.73/2.55	   active(U51(x0, x1))
6.73/2.55	   active(U52(x0))
6.73/2.55	   active(U61(x0, x1))
6.73/2.55	   active(U62(x0))
6.73/2.55	   active(U71(x0, x1, x2))
6.73/2.55	   active(U72(x0, x1))
6.73/2.55	   active(s(x0))
6.73/2.55	   active(length(x0))
6.73/2.55	   active(U81(x0))
6.73/2.55	   active(U91(x0, x1, x2, x3))
6.73/2.55	   active(U92(x0, x1, x2, x3))
6.73/2.55	   active(U93(x0, x1, x2, x3))
6.73/2.55	   active(take(x0, x1))
6.73/2.55	   cons(mark(x0), x1)
6.73/2.55	   U11(mark(x0))
6.73/2.55	   U21(mark(x0))
6.73/2.55	   U31(mark(x0))
6.73/2.55	   U41(mark(x0), x1)
6.73/2.55	   U42(mark(x0))
6.73/2.55	   U51(mark(x0), x1)
6.73/2.55	   U52(mark(x0))
6.73/2.55	   U61(mark(x0), x1)
6.73/2.55	   U62(mark(x0))
6.73/2.55	   U71(mark(x0), x1, x2)
6.73/2.55	   U72(mark(x0), x1)
6.73/2.55	   s(mark(x0))
6.73/2.55	   length(mark(x0))
6.73/2.55	   U81(mark(x0))
6.73/2.55	   U91(mark(x0), x1, x2, x3)
6.73/2.55	   U92(mark(x0), x1, x2, x3)
6.73/2.55	   U93(mark(x0), x1, x2, x3)
6.73/2.55	   take(mark(x0), x1)
6.73/2.55	   take(x0, mark(x1))
6.73/2.55	   proper(zeros)
6.73/2.55	   proper(cons(x0, x1))
6.73/2.55	   proper(0)
6.73/2.55	   proper(U11(x0))
6.73/2.55	   proper(tt)
6.73/2.55	   proper(U21(x0))
6.73/2.55	   proper(U31(x0))
6.73/2.55	   proper(U41(x0, x1))
6.73/2.55	   proper(U42(x0))
6.73/2.55	   proper(isNatIList(x0))
6.73/2.55	   proper(U51(x0, x1))
6.73/2.55	   proper(U52(x0))
6.73/2.55	   proper(isNatList(x0))
6.73/2.55	   proper(U61(x0, x1))
6.73/2.55	   proper(U62(x0))
6.73/2.55	   proper(U71(x0, x1, x2))
6.73/2.55	   proper(U72(x0, x1))
6.73/2.55	   proper(isNat(x0))
6.73/2.55	   proper(s(x0))
6.73/2.55	   proper(length(x0))
6.73/2.55	   proper(U81(x0))
6.73/2.55	   proper(nil)
6.73/2.55	   proper(U91(x0, x1, x2, x3))
6.73/2.55	   proper(U92(x0, x1, x2, x3))
6.73/2.55	   proper(U93(x0, x1, x2, x3))
6.73/2.55	   proper(take(x0, x1))
6.73/2.55	   cons(ok(x0), ok(x1))
6.73/2.55	   U11(ok(x0))
6.73/2.55	   U21(ok(x0))
6.73/2.55	   U31(ok(x0))
6.73/2.55	   U41(ok(x0), ok(x1))
6.73/2.55	   U42(ok(x0))
6.73/2.55	   isNatIList(ok(x0))
6.73/2.55	   U51(ok(x0), ok(x1))
6.73/2.55	   U52(ok(x0))
6.73/2.55	   isNatList(ok(x0))
6.73/2.55	   U61(ok(x0), ok(x1))
6.73/2.55	   U62(ok(x0))
6.73/2.55	   U71(ok(x0), ok(x1), ok(x2))
6.73/2.55	   U72(ok(x0), ok(x1))
6.73/2.55	   isNat(ok(x0))
6.73/2.55	   s(ok(x0))
6.73/2.55	   length(ok(x0))
6.73/2.55	   U81(ok(x0))
6.73/2.55	   U91(ok(x0), ok(x1), ok(x2), ok(x3))
6.73/2.55	   U92(ok(x0), ok(x1), ok(x2), ok(x3))
6.73/2.55	   U93(ok(x0), ok(x1), ok(x2), ok(x3))
6.73/2.55	   take(ok(x0), ok(x1))
6.73/2.55	   top(mark(x0))
6.73/2.55	   top(ok(x0))
6.73/2.55	
6.73/2.55	
6.73/2.55	----------------------------------------
6.73/2.55	
6.73/2.55	(1) QTRSToCSRProof (SOUND)
6.73/2.55	The following Q TRS is given: Q restricted rewrite system:
6.73/2.55	The TRS R consists of the following rules:
6.73/2.55	
6.73/2.55	   active(zeros) -> mark(cons(0, zeros))
6.73/2.55	   active(U11(tt)) -> mark(tt)
6.73/2.55	   active(U21(tt)) -> mark(tt)
6.73/2.55	   active(U31(tt)) -> mark(tt)
6.73/2.55	   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
6.73/2.55	   active(U42(tt)) -> mark(tt)
6.73/2.55	   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
6.73/2.55	   active(U52(tt)) -> mark(tt)
6.73/2.55	   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
6.73/2.55	   active(U62(tt)) -> mark(tt)
6.73/2.55	   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
6.73/2.55	   active(U72(tt, L)) -> mark(s(length(L)))
6.73/2.55	   active(U81(tt)) -> mark(nil)
6.73/2.55	   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
6.73/2.55	   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
6.73/2.55	   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
6.73/2.55	   active(isNat(0)) -> mark(tt)
6.73/2.55	   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
6.73/2.55	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
6.73/2.55	   active(isNatIList(V)) -> mark(U31(isNatList(V)))
6.73/2.55	   active(isNatIList(zeros)) -> mark(tt)
6.73/2.55	   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
6.73/2.55	   active(isNatList(nil)) -> mark(tt)
6.73/2.55	   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
6.73/2.55	   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
6.73/2.55	   active(length(nil)) -> mark(0)
6.73/2.55	   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
6.73/2.55	   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
6.73/2.55	   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
6.73/2.55	   active(cons(X1, X2)) -> cons(active(X1), X2)
6.73/2.55	   active(U11(X)) -> U11(active(X))
6.73/2.55	   active(U21(X)) -> U21(active(X))
6.73/2.55	   active(U31(X)) -> U31(active(X))
6.73/2.55	   active(U41(X1, X2)) -> U41(active(X1), X2)
6.73/2.55	   active(U42(X)) -> U42(active(X))
6.73/2.55	   active(U51(X1, X2)) -> U51(active(X1), X2)
6.73/2.55	   active(U52(X)) -> U52(active(X))
6.73/2.55	   active(U61(X1, X2)) -> U61(active(X1), X2)
6.73/2.55	   active(U62(X)) -> U62(active(X))
6.73/2.55	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
6.73/2.55	   active(U72(X1, X2)) -> U72(active(X1), X2)
6.73/2.55	   active(s(X)) -> s(active(X))
6.73/2.55	   active(length(X)) -> length(active(X))
6.73/2.55	   active(U81(X)) -> U81(active(X))
6.73/2.55	   active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
6.73/2.55	   active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
6.73/2.55	   active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
6.73/2.55	   active(take(X1, X2)) -> take(active(X1), X2)
6.73/2.55	   active(take(X1, X2)) -> take(X1, active(X2))
6.73/2.55	   cons(mark(X1), X2) -> mark(cons(X1, X2))
6.73/2.55	   U11(mark(X)) -> mark(U11(X))
6.73/2.55	   U21(mark(X)) -> mark(U21(X))
6.73/2.55	   U31(mark(X)) -> mark(U31(X))
6.73/2.55	   U41(mark(X1), X2) -> mark(U41(X1, X2))
6.73/2.55	   U42(mark(X)) -> mark(U42(X))
6.73/2.55	   U51(mark(X1), X2) -> mark(U51(X1, X2))
6.73/2.55	   U52(mark(X)) -> mark(U52(X))
6.73/2.55	   U61(mark(X1), X2) -> mark(U61(X1, X2))
6.73/2.55	   U62(mark(X)) -> mark(U62(X))
6.73/2.55	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
6.73/2.55	   U72(mark(X1), X2) -> mark(U72(X1, X2))
6.73/2.55	   s(mark(X)) -> mark(s(X))
6.73/2.55	   length(mark(X)) -> mark(length(X))
6.73/2.55	   U81(mark(X)) -> mark(U81(X))
6.73/2.55	   U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
6.73/2.55	   U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
6.73/2.55	   U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
6.73/2.55	   take(mark(X1), X2) -> mark(take(X1, X2))
6.73/2.55	   take(X1, mark(X2)) -> mark(take(X1, X2))
6.73/2.55	   proper(zeros) -> ok(zeros)
6.73/2.55	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
6.73/2.55	   proper(0) -> ok(0)
6.73/2.55	   proper(U11(X)) -> U11(proper(X))
6.73/2.55	   proper(tt) -> ok(tt)
6.73/2.55	   proper(U21(X)) -> U21(proper(X))
6.73/2.55	   proper(U31(X)) -> U31(proper(X))
6.73/2.55	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
6.73/2.55	   proper(U42(X)) -> U42(proper(X))
6.73/2.55	   proper(isNatIList(X)) -> isNatIList(proper(X))
6.73/2.55	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
6.73/2.55	   proper(U52(X)) -> U52(proper(X))
6.73/2.55	   proper(isNatList(X)) -> isNatList(proper(X))
6.73/2.55	   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
6.73/2.55	   proper(U62(X)) -> U62(proper(X))
6.73/2.55	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
6.73/2.55	   proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
6.73/2.55	   proper(isNat(X)) -> isNat(proper(X))
6.73/2.55	   proper(s(X)) -> s(proper(X))
6.73/2.55	   proper(length(X)) -> length(proper(X))
6.73/2.55	   proper(U81(X)) -> U81(proper(X))
6.73/2.55	   proper(nil) -> ok(nil)
6.73/2.55	   proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
6.73/2.55	   proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
6.73/2.55	   proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
6.73/2.55	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
6.73/2.55	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
6.73/2.55	   U11(ok(X)) -> ok(U11(X))
6.73/2.55	   U21(ok(X)) -> ok(U21(X))
6.73/2.55	   U31(ok(X)) -> ok(U31(X))
6.73/2.55	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
6.73/2.55	   U42(ok(X)) -> ok(U42(X))
6.73/2.55	   isNatIList(ok(X)) -> ok(isNatIList(X))
6.73/2.55	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
6.73/2.55	   U52(ok(X)) -> ok(U52(X))
6.73/2.55	   isNatList(ok(X)) -> ok(isNatList(X))
6.73/2.55	   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
6.73/2.55	   U62(ok(X)) -> ok(U62(X))
6.73/2.55	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
6.73/2.55	   U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
6.73/2.55	   isNat(ok(X)) -> ok(isNat(X))
6.73/2.55	   s(ok(X)) -> ok(s(X))
6.73/2.55	   length(ok(X)) -> ok(length(X))
6.73/2.55	   U81(ok(X)) -> ok(U81(X))
6.73/2.55	   U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
6.73/2.55	   U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
6.73/2.55	   U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
6.73/2.55	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
6.73/2.55	   top(mark(X)) -> top(proper(X))
6.73/2.55	   top(ok(X)) -> top(active(X))
6.73/2.55	
6.73/2.55	The set Q consists of the following terms:
6.73/2.55	
6.73/2.55	   active(zeros)
6.73/2.55	   active(isNat(0))
6.73/2.55	   active(isNat(length(x0)))
6.73/2.55	   active(isNat(s(x0)))
6.73/2.55	   active(isNatIList(x0))
6.73/2.55	   active(isNatList(nil))
6.73/2.55	   active(isNatList(cons(x0, x1)))
6.73/2.55	   active(isNatList(take(x0, x1)))
6.73/2.55	   active(cons(x0, x1))
6.73/2.55	   active(U11(x0))
6.73/2.55	   active(U21(x0))
6.73/2.55	   active(U31(x0))
6.73/2.55	   active(U41(x0, x1))
6.73/2.55	   active(U42(x0))
6.73/2.55	   active(U51(x0, x1))
6.73/2.55	   active(U52(x0))
6.73/2.55	   active(U61(x0, x1))
6.73/2.55	   active(U62(x0))
6.73/2.55	   active(U71(x0, x1, x2))
6.73/2.55	   active(U72(x0, x1))
6.73/2.55	   active(s(x0))
6.73/2.55	   active(length(x0))
6.73/2.55	   active(U81(x0))
6.73/2.55	   active(U91(x0, x1, x2, x3))
6.73/2.55	   active(U92(x0, x1, x2, x3))
6.73/2.55	   active(U93(x0, x1, x2, x3))
6.73/2.55	   active(take(x0, x1))
6.73/2.55	   cons(mark(x0), x1)
6.73/2.55	   U11(mark(x0))
6.73/2.55	   U21(mark(x0))
6.73/2.55	   U31(mark(x0))
6.73/2.55	   U41(mark(x0), x1)
6.73/2.55	   U42(mark(x0))
6.73/2.55	   U51(mark(x0), x1)
6.73/2.55	   U52(mark(x0))
6.73/2.55	   U61(mark(x0), x1)
6.73/2.55	   U62(mark(x0))
6.73/2.55	   U71(mark(x0), x1, x2)
6.73/2.55	   U72(mark(x0), x1)
6.73/2.55	   s(mark(x0))
6.73/2.55	   length(mark(x0))
6.73/2.55	   U81(mark(x0))
6.73/2.55	   U91(mark(x0), x1, x2, x3)
6.73/2.55	   U92(mark(x0), x1, x2, x3)
6.73/2.55	   U93(mark(x0), x1, x2, x3)
6.73/2.55	   take(mark(x0), x1)
6.73/2.55	   take(x0, mark(x1))
6.73/2.55	   proper(zeros)
6.73/2.55	   proper(cons(x0, x1))
6.73/2.55	   proper(0)
6.73/2.55	   proper(U11(x0))
6.73/2.55	   proper(tt)
6.73/2.55	   proper(U21(x0))
6.73/2.55	   proper(U31(x0))
6.73/2.55	   proper(U41(x0, x1))
6.73/2.55	   proper(U42(x0))
6.73/2.55	   proper(isNatIList(x0))
6.73/2.55	   proper(U51(x0, x1))
6.73/2.55	   proper(U52(x0))
6.73/2.55	   proper(isNatList(x0))
6.73/2.55	   proper(U61(x0, x1))
6.73/2.55	   proper(U62(x0))
6.73/2.55	   proper(U71(x0, x1, x2))
6.73/2.55	   proper(U72(x0, x1))
6.73/2.55	   proper(isNat(x0))
6.73/2.55	   proper(s(x0))
6.73/2.55	   proper(length(x0))
6.73/2.55	   proper(U81(x0))
6.73/2.55	   proper(nil)
6.73/2.55	   proper(U91(x0, x1, x2, x3))
6.73/2.55	   proper(U92(x0, x1, x2, x3))
6.73/2.55	   proper(U93(x0, x1, x2, x3))
6.73/2.55	   proper(take(x0, x1))
6.73/2.55	   cons(ok(x0), ok(x1))
6.73/2.55	   U11(ok(x0))
6.73/2.55	   U21(ok(x0))
6.73/2.55	   U31(ok(x0))
6.73/2.55	   U41(ok(x0), ok(x1))
6.73/2.55	   U42(ok(x0))
6.73/2.55	   isNatIList(ok(x0))
6.73/2.55	   U51(ok(x0), ok(x1))
6.73/2.55	   U52(ok(x0))
6.73/2.55	   isNatList(ok(x0))
6.73/2.55	   U61(ok(x0), ok(x1))
6.73/2.55	   U62(ok(x0))
6.73/2.55	   U71(ok(x0), ok(x1), ok(x2))
6.73/2.55	   U72(ok(x0), ok(x1))
6.73/2.55	   isNat(ok(x0))
6.73/2.55	   s(ok(x0))
6.73/2.55	   length(ok(x0))
6.73/2.55	   U81(ok(x0))
6.73/2.55	   U91(ok(x0), ok(x1), ok(x2), ok(x3))
6.73/2.55	   U92(ok(x0), ok(x1), ok(x2), ok(x3))
6.73/2.55	   U93(ok(x0), ok(x1), ok(x2), ok(x3))
6.73/2.55	   take(ok(x0), ok(x1))
6.73/2.55	   top(mark(x0))
6.73/2.55	   top(ok(x0))
6.73/2.55	
6.73/2.55	Special symbols used for the transformation (see [GM04]):
6.73/2.55	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
7.05/2.56	The replacement map contains the following entries:
7.05/2.56	
7.05/2.56	zeros: empty set
7.05/2.56	cons: {1}
7.05/2.56	0: empty set
7.05/2.56	U11: {1}
7.05/2.56	tt: empty set
7.05/2.56	U21: {1}
7.05/2.56	U31: {1}
7.05/2.56	U41: {1}
7.05/2.56	U42: {1}
7.05/2.56	isNatIList: empty set
7.05/2.56	U51: {1}
7.05/2.56	U52: {1}
7.05/2.56	isNatList: empty set
7.05/2.56	U61: {1}
7.05/2.56	U62: {1}
7.05/2.56	U71: {1}
7.05/2.56	U72: {1}
7.05/2.56	isNat: empty set
7.05/2.56	s: {1}
7.05/2.56	length: {1}
7.05/2.56	U81: {1}
7.05/2.56	nil: empty set
7.05/2.56	U91: {1}
7.05/2.56	U92: {1}
7.05/2.56	U93: {1}
7.05/2.56	take: {1, 2}
7.05/2.56	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.
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(2)
7.05/2.56	Obligation:
7.05/2.56	Context-sensitive rewrite system:
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U81(tt) -> nil
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(nil) -> 0
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(0, IL) -> U81(isNatIList(IL))
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	The replacement map contains the following entries:
7.05/2.56	
7.05/2.56	zeros: empty set
7.05/2.56	cons: {1}
7.05/2.56	0: empty set
7.05/2.56	U11: {1}
7.05/2.56	tt: empty set
7.05/2.56	U21: {1}
7.05/2.56	U31: {1}
7.05/2.56	U41: {1}
7.05/2.56	U42: {1}
7.05/2.56	isNatIList: empty set
7.05/2.56	U51: {1}
7.05/2.56	U52: {1}
7.05/2.56	isNatList: empty set
7.05/2.56	U61: {1}
7.05/2.56	U62: {1}
7.05/2.56	U71: {1}
7.05/2.56	U72: {1}
7.05/2.56	isNat: empty set
7.05/2.56	s: {1}
7.05/2.56	length: {1}
7.05/2.56	U81: {1}
7.05/2.56	nil: empty set
7.05/2.56	U91: {1}
7.05/2.56	U92: {1}
7.05/2.56	U93: {1}
7.05/2.56	take: {1, 2}
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(3) CSRRRRProof (EQUIVALENT)
7.05/2.56	The following CSR is given: Context-sensitive rewrite system:
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U81(tt) -> nil
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(nil) -> 0
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(0, IL) -> U81(isNatIList(IL))
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	The replacement map contains the following entries:
7.05/2.56	
7.05/2.56	zeros: empty set
7.05/2.56	cons: {1}
7.05/2.56	0: empty set
7.05/2.56	U11: {1}
7.05/2.56	tt: empty set
7.05/2.56	U21: {1}
7.05/2.56	U31: {1}
7.05/2.56	U41: {1}
7.05/2.56	U42: {1}
7.05/2.56	isNatIList: empty set
7.05/2.56	U51: {1}
7.05/2.56	U52: {1}
7.05/2.56	isNatList: empty set
7.05/2.56	U61: {1}
7.05/2.56	U62: {1}
7.05/2.56	U71: {1}
7.05/2.56	U72: {1}
7.05/2.56	isNat: empty set
7.05/2.56	s: {1}
7.05/2.56	length: {1}
7.05/2.56	U81: {1}
7.05/2.56	nil: empty set
7.05/2.56	U91: {1}
7.05/2.56	U92: {1}
7.05/2.56	U93: {1}
7.05/2.56	take: {1, 2}
7.05/2.56	Used ordering:
7.05/2.56	Polynomial interpretation [POLO]:
7.05/2.56	
7.05/2.56	   POL(0) = 0
7.05/2.56	   POL(U11(x_1)) = x_1
7.05/2.56	   POL(U21(x_1)) = x_1
7.05/2.56	   POL(U31(x_1)) = x_1
7.05/2.56	   POL(U41(x_1, x_2)) = x_1
7.05/2.56	   POL(U42(x_1)) = x_1
7.05/2.56	   POL(U51(x_1, x_2)) = x_1
7.05/2.56	   POL(U52(x_1)) = x_1
7.05/2.56	   POL(U61(x_1, x_2)) = x_1
7.05/2.56	   POL(U62(x_1)) = x_1
7.05/2.56	   POL(U71(x_1, x_2, x_3)) = x_1 + x_2
7.05/2.56	   POL(U72(x_1, x_2)) = x_1 + x_2
7.05/2.56	   POL(U81(x_1)) = x_1
7.05/2.56	   POL(U91(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
7.05/2.56	   POL(U92(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
7.05/2.56	   POL(U93(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
7.05/2.56	   POL(cons(x_1, x_2)) = x_1 + x_2
7.05/2.56	   POL(isNat(x_1)) = 1
7.05/2.56	   POL(isNatIList(x_1)) = 1
7.05/2.56	   POL(isNatList(x_1)) = 1
7.05/2.56	   POL(length(x_1)) = 1 + x_1
7.05/2.56	   POL(nil) = 1
7.05/2.56	   POL(s(x_1)) = x_1
7.05/2.56	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
7.05/2.56	   POL(tt) = 1
7.05/2.56	   POL(zeros) = 1
7.05/2.56	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.05/2.56	
7.05/2.56	   length(nil) -> 0
7.05/2.56	
7.05/2.56	
7.05/2.56	
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(4)
7.05/2.56	Obligation:
7.05/2.56	Context-sensitive rewrite system:
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U81(tt) -> nil
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(0, IL) -> U81(isNatIList(IL))
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	The replacement map contains the following entries:
7.05/2.56	
7.05/2.56	zeros: empty set
7.05/2.56	cons: {1}
7.05/2.56	0: empty set
7.05/2.56	U11: {1}
7.05/2.56	tt: empty set
7.05/2.56	U21: {1}
7.05/2.56	U31: {1}
7.05/2.56	U41: {1}
7.05/2.56	U42: {1}
7.05/2.56	isNatIList: empty set
7.05/2.56	U51: {1}
7.05/2.56	U52: {1}
7.05/2.56	isNatList: empty set
7.05/2.56	U61: {1}
7.05/2.56	U62: {1}
7.05/2.56	U71: {1}
7.05/2.56	U72: {1}
7.05/2.56	isNat: empty set
7.05/2.56	s: {1}
7.05/2.56	length: {1}
7.05/2.56	U81: {1}
7.05/2.56	nil: empty set
7.05/2.56	U91: {1}
7.05/2.56	U92: {1}
7.05/2.56	U93: {1}
7.05/2.56	take: {1, 2}
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(5) CSRRRRProof (EQUIVALENT)
7.05/2.56	The following CSR is given: Context-sensitive rewrite system:
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U81(tt) -> nil
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(0, IL) -> U81(isNatIList(IL))
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	The replacement map contains the following entries:
7.05/2.56	
7.05/2.56	zeros: empty set
7.05/2.56	cons: {1}
7.05/2.56	0: empty set
7.05/2.56	U11: {1}
7.05/2.56	tt: empty set
7.05/2.56	U21: {1}
7.05/2.56	U31: {1}
7.05/2.56	U41: {1}
7.05/2.56	U42: {1}
7.05/2.56	isNatIList: empty set
7.05/2.56	U51: {1}
7.05/2.56	U52: {1}
7.05/2.56	isNatList: empty set
7.05/2.56	U61: {1}
7.05/2.56	U62: {1}
7.05/2.56	U71: {1}
7.05/2.56	U72: {1}
7.05/2.56	isNat: empty set
7.05/2.56	s: {1}
7.05/2.56	length: {1}
7.05/2.56	U81: {1}
7.05/2.56	nil: empty set
7.05/2.56	U91: {1}
7.05/2.56	U92: {1}
7.05/2.56	U93: {1}
7.05/2.56	take: {1, 2}
7.05/2.56	Used ordering:
7.05/2.56	Polynomial interpretation [POLO]:
7.05/2.56	
7.05/2.56	   POL(0) = 0
7.05/2.56	   POL(U11(x_1)) = x_1
7.05/2.56	   POL(U21(x_1)) = x_1
7.05/2.56	   POL(U31(x_1)) = 2*x_1
7.05/2.56	   POL(U41(x_1, x_2)) = 2*x_1
7.05/2.56	   POL(U42(x_1)) = 2*x_1
7.05/2.56	   POL(U51(x_1, x_2)) = x_1
7.05/2.56	   POL(U52(x_1)) = x_1
7.05/2.56	   POL(U61(x_1, x_2)) = 2*x_1
7.05/2.56	   POL(U62(x_1)) = 2*x_1
7.05/2.56	   POL(U71(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
7.05/2.56	   POL(U72(x_1, x_2)) = 2*x_1 + x_2
7.05/2.56	   POL(U81(x_1)) = 1 + 2*x_1
7.05/2.56	   POL(U91(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 + x_3 + 2*x_4
7.05/2.56	   POL(U92(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 + x_3 + 2*x_4
7.05/2.56	   POL(U93(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 + x_3 + 2*x_4
7.05/2.56	   POL(cons(x_1, x_2)) = 2*x_1 + x_2
7.05/2.56	   POL(isNat(x_1)) = 0
7.05/2.56	   POL(isNatIList(x_1)) = 0
7.05/2.56	   POL(isNatList(x_1)) = 0
7.05/2.56	   POL(length(x_1)) = x_1
7.05/2.56	   POL(nil) = 0
7.05/2.56	   POL(s(x_1)) = x_1
7.05/2.56	   POL(take(x_1, x_2)) = 2 + x_1 + x_2
7.05/2.56	   POL(tt) = 0
7.05/2.56	   POL(zeros) = 0
7.05/2.56	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
7.05/2.56	
7.05/2.56	   U81(tt) -> nil
7.05/2.56	   take(0, IL) -> U81(isNatIList(IL))
7.05/2.56	
7.05/2.56	
7.05/2.56	
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(6)
7.05/2.56	Obligation:
7.05/2.56	Context-sensitive rewrite system:
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	The replacement map contains the following entries:
7.05/2.56	
7.05/2.56	zeros: empty set
7.05/2.56	cons: {1}
7.05/2.56	0: empty set
7.05/2.56	U11: {1}
7.05/2.56	tt: empty set
7.05/2.56	U21: {1}
7.05/2.56	U31: {1}
7.05/2.56	U41: {1}
7.05/2.56	U42: {1}
7.05/2.56	isNatIList: empty set
7.05/2.56	U51: {1}
7.05/2.56	U52: {1}
7.05/2.56	isNatList: empty set
7.05/2.56	U61: {1}
7.05/2.56	U62: {1}
7.05/2.56	U71: {1}
7.05/2.56	U72: {1}
7.05/2.56	isNat: empty set
7.05/2.56	s: {1}
7.05/2.56	length: {1}
7.05/2.56	nil: empty set
7.05/2.56	U91: {1}
7.05/2.56	U92: {1}
7.05/2.56	U93: {1}
7.05/2.56	take: {1, 2}
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(7) CSDependencyPairsProof (EQUIVALENT)
7.05/2.56	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(8)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {U11_1, U21_1, U31_1, U42_1, U52_1, U62_1, s_1, length_1, take_2, U42'_1, U52'_1, U62'_1, LENGTH_1, U11'_1, U21'_1, U31'_1, TAKE_2} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U41_2, U51_2, U61_2, U71_3, U72_2, U91_4, U92_4, U93_4, U41'_2, U51'_2, U61'_2, U72'_2, U71'_3, U92'_4, U91'_4, U93'_4} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNatIList_1, isNatList_1, isNat_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1, U_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The ordinary context-sensitive dependency pairs DP_o are:
7.05/2.56	   U41'(tt, V2) -> U42'(isNatIList(V2))
7.05/2.56	   U41'(tt, V2) -> ISNATILIST(V2)
7.05/2.56	   U51'(tt, V2) -> U52'(isNatList(V2))
7.05/2.56	   U51'(tt, V2) -> ISNATLIST(V2)
7.05/2.56	   U61'(tt, V2) -> U62'(isNatIList(V2))
7.05/2.56	   U61'(tt, V2) -> ISNATILIST(V2)
7.05/2.56	   U71'(tt, L, N) -> U72'(isNat(N), L)
7.05/2.56	   U71'(tt, L, N) -> ISNAT(N)
7.05/2.56	   U72'(tt, L) -> LENGTH(L)
7.05/2.56	   U91'(tt, IL, M, N) -> U92'(isNat(M), IL, M, N)
7.05/2.56	   U91'(tt, IL, M, N) -> ISNAT(M)
7.05/2.56	   U92'(tt, IL, M, N) -> U93'(isNat(N), IL, M, N)
7.05/2.56	   U92'(tt, IL, M, N) -> ISNAT(N)
7.05/2.56	   ISNAT(length(V1)) -> U11'(isNatList(V1))
7.05/2.56	   ISNAT(length(V1)) -> ISNATLIST(V1)
7.05/2.56	   ISNAT(s(V1)) -> U21'(isNat(V1))
7.05/2.56	   ISNAT(s(V1)) -> ISNAT(V1)
7.05/2.56	   ISNATILIST(V) -> U31'(isNatList(V))
7.05/2.56	   ISNATILIST(V) -> ISNATLIST(V)
7.05/2.56	   ISNATILIST(cons(V1, V2)) -> U41'(isNat(V1), V2)
7.05/2.56	   ISNATILIST(cons(V1, V2)) -> ISNAT(V1)
7.05/2.56	   ISNATLIST(cons(V1, V2)) -> U51'(isNat(V1), V2)
7.05/2.56	   ISNATLIST(cons(V1, V2)) -> ISNAT(V1)
7.05/2.56	   ISNATLIST(take(V1, V2)) -> U61'(isNat(V1), V2)
7.05/2.56	   ISNATLIST(take(V1, V2)) -> ISNAT(V1)
7.05/2.56	   LENGTH(cons(N, L)) -> U71'(isNatList(L), L, N)
7.05/2.56	   LENGTH(cons(N, L)) -> ISNATLIST(L)
7.05/2.56	   TAKE(s(M), cons(N, IL)) -> U91'(isNatIList(IL), IL, M, N)
7.05/2.56	   TAKE(s(M), cons(N, IL)) -> ISNATILIST(IL)
7.05/2.56	
7.05/2.56	The collapsing dependency pairs are DP_c:
7.05/2.56	   U72'(tt, L) -> L
7.05/2.56	   U93'(tt, IL, M, N) -> N
7.05/2.56	
7.05/2.56	
7.05/2.56	The hidden terms of R are:
7.05/2.56	
7.05/2.56	   zeros
7.05/2.56	   take(x0, x1)
7.05/2.56	
7.05/2.56	Every hiding context is built from:
7.05/2.56	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@4aa9ea16
7.05/2.56	
7.05/2.56	Hence, the new unhiding pairs DP_u are :
7.05/2.56	   U72'(tt, L) -> U(L)
7.05/2.56	   U93'(tt, IL, M, N) -> U(N)
7.05/2.56	   U(take(x_0, x_1)) -> U(x_0)
7.05/2.56	   U(take(x_0, x_1)) -> U(x_1)
7.05/2.56	   U(zeros) -> ZEROS
7.05/2.56	   U(take(x0, x1)) -> TAKE(x0, x1)
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(9) QCSDependencyGraphProof (EQUIVALENT)
7.05/2.56	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 13 less nodes.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(10)
7.05/2.56	Complex Obligation (AND)
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(11)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {U11_1, U21_1, U31_1, U42_1, U52_1, U62_1, s_1, length_1, take_2} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U41_2, U51_2, U61_2, U71_3, U72_2, U91_4, U92_4, U93_4, U51'_2, U61'_2, U41'_2} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNatIList_1, isNatList_1, isNat_1, ISNATLIST_1, ISNATILIST_1, ISNAT_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The TRS P consists of the following rules:
7.05/2.56	
7.05/2.56	   ISNATILIST(V) -> ISNATLIST(V)
7.05/2.56	   ISNATLIST(cons(V1, V2)) -> U51'(isNat(V1), V2)
7.05/2.56	   U51'(tt, V2) -> ISNATLIST(V2)
7.05/2.56	   ISNATLIST(cons(V1, V2)) -> ISNAT(V1)
7.05/2.56	   ISNAT(length(V1)) -> ISNATLIST(V1)
7.05/2.56	   ISNATLIST(take(V1, V2)) -> U61'(isNat(V1), V2)
7.05/2.56	   U61'(tt, V2) -> ISNATILIST(V2)
7.05/2.56	   ISNATILIST(cons(V1, V2)) -> U41'(isNat(V1), V2)
7.05/2.56	   U41'(tt, V2) -> ISNATILIST(V2)
7.05/2.56	   ISNATILIST(cons(V1, V2)) -> ISNAT(V1)
7.05/2.56	   ISNAT(s(V1)) -> ISNAT(V1)
7.05/2.56	   ISNATLIST(take(V1, V2)) -> ISNAT(V1)
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(12) QCSUsableRulesProof (EQUIVALENT)
7.05/2.56	The following rules are not useable [DA_EMMES] and can be deleted:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U71(tt, x0, x1) -> U72(isNat(x1), x0)
7.05/2.56	   U72(tt, x0) -> s(length(x0))
7.05/2.56	   U91(tt, x0, x1, x2) -> U92(isNat(x1), x0, x1, x2)
7.05/2.56	   U92(tt, x0, x1, x2) -> U93(isNat(x2), x0, x1, x2)
7.05/2.56	   U93(tt, x0, x1, x2) -> cons(x2, take(x1, x0))
7.05/2.56	   length(cons(x0, x1)) -> U71(isNatList(x1), x1, x0)
7.05/2.56	   take(s(x0), cons(x1, x2)) -> U91(isNatIList(x2), x2, x0, x1)
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(13)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {length_1, U11_1, s_1, U21_1, U52_1, take_2, U62_1, U31_1, U42_1} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U51_2, U61_2, U41_2, U51'_2, U61'_2, U41'_2} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNat_1, isNatList_1, isNatIList_1, ISNATLIST_1, ISNATILIST_1, ISNAT_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The TRS P consists of the following rules:
7.05/2.56	
7.05/2.56	   ISNATILIST(V) -> ISNATLIST(V)
7.05/2.56	   ISNATLIST(cons(V1, V2)) -> U51'(isNat(V1), V2)
7.05/2.56	   U51'(tt, V2) -> ISNATLIST(V2)
7.05/2.56	   ISNATLIST(cons(V1, V2)) -> ISNAT(V1)
7.05/2.56	   ISNAT(length(V1)) -> ISNATLIST(V1)
7.05/2.56	   ISNATLIST(take(V1, V2)) -> U61'(isNat(V1), V2)
7.05/2.56	   U61'(tt, V2) -> ISNATILIST(V2)
7.05/2.56	   ISNATILIST(cons(V1, V2)) -> U41'(isNat(V1), V2)
7.05/2.56	   U41'(tt, V2) -> ISNATILIST(V2)
7.05/2.56	   ISNATILIST(cons(V1, V2)) -> ISNAT(V1)
7.05/2.56	   ISNAT(s(V1)) -> ISNAT(V1)
7.05/2.56	   ISNATLIST(take(V1, V2)) -> ISNAT(V1)
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U11(tt) -> tt
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(14) QCSDPMuMonotonicPoloProof (EQUIVALENT)
7.05/2.56	By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted.
7.05/2.56	
7.05/2.56	Strictly oriented dependency pairs:
7.05/2.56	
7.05/2.56	   ISNATLIST(cons(V1, V2)) -> U51'(isNat(V1), V2)
7.05/2.56	   U51'(tt, V2) -> ISNATLIST(V2)
7.05/2.56	   ISNATLIST(cons(V1, V2)) -> ISNAT(V1)
7.05/2.56	   ISNAT(length(V1)) -> ISNATLIST(V1)
7.05/2.56	   ISNATLIST(take(V1, V2)) -> U61'(isNat(V1), V2)
7.05/2.56	   U61'(tt, V2) -> ISNATILIST(V2)
7.05/2.56	   ISNATILIST(cons(V1, V2)) -> U41'(isNat(V1), V2)
7.05/2.56	   U41'(tt, V2) -> ISNATILIST(V2)
7.05/2.56	   ISNATILIST(cons(V1, V2)) -> ISNAT(V1)
7.05/2.56	   ISNATLIST(take(V1, V2)) -> ISNAT(V1)
7.05/2.56	
7.05/2.56	
7.05/2.56	Used ordering: POLO with Polynomial interpretation [POLO]:
7.05/2.56	
7.05/2.56	   POL(0) = 2
7.05/2.56	   POL(ISNAT(x_1)) = 2 + x_1
7.05/2.56	   POL(ISNATILIST(x_1)) = 1 + 2*x_1
7.05/2.56	   POL(ISNATLIST(x_1)) = 1 + 2*x_1
7.05/2.56	   POL(U11(x_1)) = 2*x_1
7.05/2.56	   POL(U21(x_1)) = 2*x_1
7.05/2.56	   POL(U31(x_1)) = 2*x_1
7.05/2.56	   POL(U41(x_1, x_2)) = 2*x_1
7.05/2.56	   POL(U41'(x_1, x_2)) = 2 + x_1 + 2*x_2
7.05/2.56	   POL(U42(x_1)) = 2*x_1
7.05/2.56	   POL(U51(x_1, x_2)) = 2*x_1
7.05/2.56	   POL(U51'(x_1, x_2)) = 2 + x_1 + 2*x_2
7.05/2.56	   POL(U52(x_1)) = x_1
7.05/2.56	   POL(U61(x_1, x_2)) = 2*x_1
7.05/2.56	   POL(U61'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
7.05/2.56	   POL(U62(x_1)) = 2*x_1
7.05/2.56	   POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
7.05/2.56	   POL(isNat(x_1)) = 0
7.05/2.56	   POL(isNatIList(x_1)) = 0
7.05/2.56	   POL(isNatList(x_1)) = 0
7.05/2.56	   POL(length(x_1)) = 2*x_1
7.05/2.56	   POL(nil) = 2
7.05/2.56	   POL(s(x_1)) = x_1
7.05/2.56	   POL(take(x_1, x_2)) = 2 + x_1 + 2*x_2
7.05/2.56	   POL(tt) = 0
7.05/2.56	   POL(zeros) = 2
7.05/2.56	
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(15)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {length_1, U11_1, s_1, U21_1, U52_1, take_2, U62_1, U31_1, U42_1} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U51_2, U61_2, U41_2} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNat_1, isNatList_1, isNatIList_1, ISNATLIST_1, ISNATILIST_1, ISNAT_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The TRS P consists of the following rules:
7.05/2.56	
7.05/2.56	   ISNATILIST(V) -> ISNATLIST(V)
7.05/2.56	   ISNAT(s(V1)) -> ISNAT(V1)
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U11(tt) -> tt
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(16) QCSDependencyGraphProof (EQUIVALENT)
7.05/2.56	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 1 less node.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(17)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {length_1, U11_1, s_1, U21_1, U52_1, take_2, U62_1, U31_1, U42_1} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U51_2, U61_2, U41_2} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNat_1, isNatList_1, isNatIList_1, ISNAT_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The TRS P consists of the following rules:
7.05/2.56	
7.05/2.56	   ISNAT(s(V1)) -> ISNAT(V1)
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U11(tt) -> tt
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(18) QCSDPSubtermProof (EQUIVALENT)
7.05/2.56	We use the subterm processor [DA_EMMES].
7.05/2.56	
7.05/2.56	
7.05/2.56	The following pairs can be oriented strictly and are deleted.
7.05/2.56	
7.05/2.56	   ISNAT(s(V1)) -> ISNAT(V1)
7.05/2.56	The remaining pairs can at least be oriented weakly.
7.05/2.56	none
7.05/2.56	Used ordering:  Combined order from the following AFS and order.
7.05/2.56	ISNAT(x1)  =  x1
7.05/2.56	
7.05/2.56	
7.05/2.56	Subterm Order
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(19)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {length_1, U11_1, s_1, U21_1, U52_1, take_2, U62_1, U31_1, U42_1} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U51_2, U61_2, U41_2} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNat_1, isNatList_1, isNatIList_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The TRS P consists of the following rules:
7.05/2.56	none
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U11(tt) -> tt
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(20) PIsEmptyProof (EQUIVALENT)
7.05/2.56	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(21)
7.05/2.56	YES
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(22)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {U11_1, U21_1, U31_1, U42_1, U52_1, U62_1, s_1, length_1, take_2, TAKE_2} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U41_2, U51_2, U61_2, U71_3, U72_2, U91_4, U92_4, U93_4, U92'_4, U91'_4, U93'_4} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNatIList_1, isNatList_1, isNat_1, U_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The TRS P consists of the following rules:
7.05/2.56	
7.05/2.56	   U91'(tt, IL, M, N) -> U92'(isNat(M), IL, M, N)
7.05/2.56	   U92'(tt, IL, M, N) -> U93'(isNat(N), IL, M, N)
7.05/2.56	   U93'(tt, IL, M, N) -> U(N)
7.05/2.56	   U(take(x_0, x_1)) -> U(x_0)
7.05/2.56	   U(take(x_0, x_1)) -> U(x_1)
7.05/2.56	   U(take(x0, x1)) -> TAKE(x0, x1)
7.05/2.56	   TAKE(s(M), cons(N, IL)) -> U91'(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(23) QCSDPSubtermProof (EQUIVALENT)
7.05/2.56	We use the subterm processor [DA_EMMES].
7.05/2.56	
7.05/2.56	
7.05/2.56	The following pairs can be oriented strictly and are deleted.
7.05/2.56	
7.05/2.56	   U(take(x_0, x_1)) -> U(x_0)
7.05/2.56	   U(take(x_0, x_1)) -> U(x_1)
7.05/2.56	   U(take(x0, x1)) -> TAKE(x0, x1)
7.05/2.56	   TAKE(s(M), cons(N, IL)) -> U91'(isNatIList(IL), IL, M, N)
7.05/2.56	The remaining pairs can at least be oriented weakly.
7.05/2.56	
7.05/2.56	   U91'(tt, IL, M, N) -> U92'(isNat(M), IL, M, N)
7.05/2.56	   U92'(tt, IL, M, N) -> U93'(isNat(N), IL, M, N)
7.05/2.56	   U93'(tt, IL, M, N) -> U(N)
7.05/2.56	Used ordering:  Combined order from the following AFS and order.
7.05/2.56	U92'(x1, x2, x3, x4)  =  x4
7.05/2.56	
7.05/2.56	U91'(x1, x2, x3, x4)  =  x4
7.05/2.56	
7.05/2.56	U93'(x1, x2, x3, x4)  =  x4
7.05/2.56	
7.05/2.56	U(x1)  =  x1
7.05/2.56	
7.05/2.56	TAKE(x1, x2)  =  x2
7.05/2.56	
7.05/2.56	
7.05/2.56	Subterm Order
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(24)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {U11_1, U21_1, U31_1, U42_1, U52_1, U62_1, s_1, length_1, take_2} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U41_2, U51_2, U61_2, U71_3, U72_2, U91_4, U92_4, U93_4, U92'_4, U91'_4, U93'_4} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNatIList_1, isNatList_1, isNat_1, U_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The TRS P consists of the following rules:
7.05/2.56	
7.05/2.56	   U91'(tt, IL, M, N) -> U92'(isNat(M), IL, M, N)
7.05/2.56	   U92'(tt, IL, M, N) -> U93'(isNat(N), IL, M, N)
7.05/2.56	   U93'(tt, IL, M, N) -> U(N)
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(25) QCSDependencyGraphProof (EQUIVALENT)
7.05/2.56	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(26)
7.05/2.56	TRUE
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(27)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {U11_1, U21_1, U31_1, U42_1, U52_1, U62_1, s_1, length_1, take_2, LENGTH_1} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U41_2, U51_2, U61_2, U71_3, U72_2, U91_4, U92_4, U93_4, U72'_2, U71'_3} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNatIList_1, isNatList_1, isNat_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The TRS P consists of the following rules:
7.05/2.56	
7.05/2.56	   U71'(tt, L, N) -> U72'(isNat(N), L)
7.05/2.56	   U72'(tt, L) -> LENGTH(L)
7.05/2.56	   LENGTH(cons(N, L)) -> U71'(isNatList(L), L, N)
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(28) QCSDPReductionPairProof (EQUIVALENT)
7.05/2.56	Using the order
7.05/2.56	
7.05/2.56	Polynomial interpretation [POLO]:
7.05/2.56	
7.05/2.56	   POL(0) = 2
7.05/2.56	   POL(LENGTH(x_1)) = 2*x_1
7.05/2.56	   POL(U11(x_1)) = x_1
7.05/2.56	   POL(U21(x_1)) = x_1
7.05/2.56	   POL(U31(x_1)) = 2
7.05/2.56	   POL(U41(x_1, x_2)) = 2
7.05/2.56	   POL(U42(x_1)) = x_1
7.05/2.56	   POL(U51(x_1, x_2)) = 2*x_2
7.05/2.56	   POL(U52(x_1)) = x_1
7.05/2.56	   POL(U61(x_1, x_2)) = x_1
7.05/2.56	   POL(U62(x_1)) = x_1
7.05/2.56	   POL(U71(x_1, x_2, x_3)) = 2*x_2
7.05/2.56	   POL(U71'(x_1, x_2, x_3)) = x_1 + 2*x_2
7.05/2.56	   POL(U72(x_1, x_2)) = 2*x_2
7.05/2.56	   POL(U72'(x_1, x_2)) = 1 + 2*x_2
7.05/2.56	   POL(U91(x_1, x_2, x_3, x_4)) = 2*x_3
7.05/2.56	   POL(U92(x_1, x_2, x_3, x_4)) = 2*x_3
7.05/2.56	   POL(U93(x_1, x_2, x_3, x_4)) = 2*x_3
7.05/2.56	   POL(cons(x_1, x_2)) = 2*x_2
7.05/2.56	   POL(isNat(x_1)) = 2*x_1
7.05/2.56	   POL(isNatIList(x_1)) = 2
7.05/2.56	   POL(isNatList(x_1)) = 2*x_1
7.05/2.56	   POL(length(x_1)) = x_1
7.05/2.56	   POL(nil) = 2
7.05/2.56	   POL(s(x_1)) = 2*x_1
7.05/2.56	   POL(take(x_1, x_2)) = x_1
7.05/2.56	   POL(tt) = 2
7.05/2.56	   POL(zeros) = 0
7.05/2.56	
7.05/2.56	
7.05/2.56	the following usable rules
7.05/2.56	
7.05/2.56	
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	
7.05/2.56	
7.05/2.56	could all be oriented weakly.
7.05/2.56	
7.05/2.56	Furthermore, the pairs
7.05/2.56	
7.05/2.56	
7.05/2.56	   U71'(tt, L, N) -> U72'(isNat(N), L)
7.05/2.56	   U72'(tt, L) -> LENGTH(L)
7.05/2.56	
7.05/2.56	
7.05/2.56	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
7.05/2.56	
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(29)
7.05/2.56	Obligation:
7.05/2.56	Q-restricted context-sensitive dependency pair problem:
7.05/2.56	The symbols in {U11_1, U21_1, U31_1, U42_1, U52_1, U62_1, s_1, length_1, take_2, LENGTH_1} are replacing on all positions.
7.05/2.56	For all symbols f in {cons_2, U41_2, U51_2, U61_2, U71_3, U72_2, U91_4, U92_4, U93_4, U71'_3} we have mu(f) = {1}.
7.05/2.56	The symbols in {isNatIList_1, isNatList_1, isNat_1} are not replacing on any position.
7.05/2.56	
7.05/2.56	The TRS P consists of the following rules:
7.05/2.56	
7.05/2.56	   LENGTH(cons(N, L)) -> U71'(isNatList(L), L, N)
7.05/2.56	
7.05/2.56	The TRS R consists of the following rules:
7.05/2.56	
7.05/2.56	   zeros -> cons(0, zeros)
7.05/2.56	   U11(tt) -> tt
7.05/2.56	   U21(tt) -> tt
7.05/2.56	   U31(tt) -> tt
7.05/2.56	   U41(tt, V2) -> U42(isNatIList(V2))
7.05/2.56	   U42(tt) -> tt
7.05/2.56	   U51(tt, V2) -> U52(isNatList(V2))
7.05/2.56	   U52(tt) -> tt
7.05/2.56	   U61(tt, V2) -> U62(isNatIList(V2))
7.05/2.56	   U62(tt) -> tt
7.05/2.56	   U71(tt, L, N) -> U72(isNat(N), L)
7.05/2.56	   U72(tt, L) -> s(length(L))
7.05/2.56	   U91(tt, IL, M, N) -> U92(isNat(M), IL, M, N)
7.05/2.56	   U92(tt, IL, M, N) -> U93(isNat(N), IL, M, N)
7.05/2.56	   U93(tt, IL, M, N) -> cons(N, take(M, IL))
7.05/2.56	   isNat(0) -> tt
7.05/2.56	   isNat(length(V1)) -> U11(isNatList(V1))
7.05/2.56	   isNat(s(V1)) -> U21(isNat(V1))
7.05/2.56	   isNatIList(V) -> U31(isNatList(V))
7.05/2.56	   isNatIList(zeros) -> tt
7.05/2.56	   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
7.05/2.56	   isNatList(nil) -> tt
7.05/2.56	   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
7.05/2.56	   isNatList(take(V1, V2)) -> U61(isNat(V1), V2)
7.05/2.56	   length(cons(N, L)) -> U71(isNatList(L), L, N)
7.05/2.56	   take(s(M), cons(N, IL)) -> U91(isNatIList(IL), IL, M, N)
7.05/2.56	
7.05/2.56	Q is empty.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(30) QCSDependencyGraphProof (EQUIVALENT)
7.05/2.56	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.
7.05/2.56	
7.05/2.56	----------------------------------------
7.05/2.56	
7.05/2.56	(31)
7.05/2.56	TRUE
7.18/2.63	EOF