11.49/3.81	YES
11.68/3.83	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
11.68/3.83	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.68/3.83	
11.68/3.83	
11.68/3.83	Termination w.r.t. Q of the given QTRS could be proven:
11.68/3.83	
11.68/3.83	(0) QTRS
11.68/3.83	(1) QTRSToCSRProof [SOUND, 0 ms]
11.68/3.83	(2) CSR
11.68/3.83	(3) CSRRRRProof [EQUIVALENT, 161 ms]
11.68/3.83	(4) CSR
11.68/3.83	(5) CSRRRRProof [EQUIVALENT, 37 ms]
11.68/3.83	(6) CSR
11.68/3.83	(7) CSRRRRProof [EQUIVALENT, 48 ms]
11.68/3.83	(8) CSR
11.68/3.83	(9) CSDependencyPairsProof [EQUIVALENT, 95 ms]
11.68/3.83	(10) QCSDP
11.68/3.83	(11) QCSDependencyGraphProof [EQUIVALENT, 1 ms]
11.68/3.83	(12) AND
11.68/3.83	    (13) QCSDP
11.68/3.83	        (14) QCSUsableRulesProof [EQUIVALENT, 6 ms]
11.68/3.83	        (15) QCSDP
11.68/3.83	        (16) QCSDPMuMonotonicPoloProof [EQUIVALENT, 11 ms]
11.68/3.83	        (17) QCSDP
11.68/3.83	        (18) PIsEmptyProof [EQUIVALENT, 0 ms]
11.68/3.83	        (19) YES
11.68/3.83	    (20) QCSDP
11.68/3.83	        (21) QCSUsableRulesProof [EQUIVALENT, 21 ms]
11.68/3.83	        (22) QCSDP
11.68/3.83	        (23) QCSDPMuMonotonicPoloProof [EQUIVALENT, 78 ms]
11.68/3.83	        (24) QCSDP
11.68/3.83	        (25) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.68/3.83	        (26) QCSDP
11.68/3.83	        (27) QCSDPSubtermProof [EQUIVALENT, 0 ms]
11.68/3.83	        (28) QCSDP
11.68/3.83	        (29) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.68/3.83	        (30) TRUE
11.68/3.83	    (31) QCSDP
11.68/3.83	        (32) QCSDPSubtermProof [EQUIVALENT, 9 ms]
11.68/3.83	        (33) QCSDP
11.68/3.83	        (34) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.68/3.83	        (35) TRUE
11.68/3.83	    (36) QCSDP
11.68/3.83	        (37) QCSDPReductionPairProof [EQUIVALENT, 87 ms]
11.68/3.83	        (38) QCSDP
11.68/3.83	        (39) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
11.68/3.83	        (40) TRUE
11.68/3.83	
11.68/3.83	
11.68/3.83	----------------------------------------
11.68/3.83	
11.68/3.83	(0)
11.68/3.83	Obligation:
11.68/3.83	Q restricted rewrite system:
11.68/3.83	The TRS R consists of the following rules:
11.68/3.83	
11.68/3.83	   active(zeros) -> mark(cons(0, zeros))
11.68/3.83	   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
11.68/3.83	   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
11.68/3.83	   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
11.68/3.83	   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
11.68/3.83	   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
11.68/3.83	   active(U106(tt)) -> mark(tt)
11.68/3.83	   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
11.68/3.83	   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
11.68/3.83	   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
11.68/3.83	   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
11.68/3.83	   active(U114(tt, L)) -> mark(s(length(L)))
11.68/3.83	   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
11.68/3.83	   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
11.68/3.83	   active(U122(tt)) -> mark(nil)
11.68/3.83	   active(U13(tt)) -> mark(tt)
11.68/3.83	   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
11.68/3.83	   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
11.68/3.83	   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
11.68/3.83	   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
11.68/3.83	   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
11.68/3.83	   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
11.68/3.83	   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
11.68/3.83	   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
11.68/3.83	   active(U23(tt)) -> mark(tt)
11.68/3.83	   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
11.68/3.83	   active(U32(tt, V)) -> mark(U33(isNatList(V)))
11.68/3.83	   active(U33(tt)) -> mark(tt)
11.68/3.83	   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
11.68/3.83	   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
11.68/3.83	   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
11.68/3.83	   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
11.68/3.83	   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
11.68/3.83	   active(U46(tt)) -> mark(tt)
11.68/3.83	   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
11.68/3.83	   active(U52(tt)) -> mark(tt)
11.68/3.83	   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
11.68/3.83	   active(U62(tt)) -> mark(tt)
11.68/3.83	   active(U71(tt)) -> mark(tt)
11.68/3.83	   active(U81(tt)) -> mark(tt)
11.68/3.83	   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
11.68/3.83	   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
11.68/3.83	   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
11.68/3.83	   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
11.68/3.83	   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
11.68/3.83	   active(U96(tt)) -> mark(tt)
11.68/3.83	   active(isNat(0)) -> mark(tt)
11.68/3.83	   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
11.68/3.83	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
11.68/3.83	   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
11.68/3.83	   active(isNatIList(zeros)) -> mark(tt)
11.68/3.83	   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
11.68/3.83	   active(isNatIListKind(nil)) -> mark(tt)
11.68/3.83	   active(isNatIListKind(zeros)) -> mark(tt)
11.68/3.83	   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
11.68/3.83	   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
11.68/3.83	   active(isNatKind(0)) -> mark(tt)
11.68/3.83	   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
11.68/3.83	   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
11.68/3.83	   active(isNatList(nil)) -> mark(tt)
11.68/3.83	   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
11.68/3.83	   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
11.68/3.83	   active(length(nil)) -> mark(0)
11.68/3.83	   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
11.68/3.83	   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
11.68/3.83	   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
11.68/3.83	   active(cons(X1, X2)) -> cons(active(X1), X2)
11.68/3.83	   active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
11.68/3.83	   active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
11.68/3.83	   active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
11.68/3.83	   active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
11.68/3.83	   active(U105(X1, X2)) -> U105(active(X1), X2)
11.68/3.83	   active(U106(X)) -> U106(active(X))
11.68/3.83	   active(U11(X1, X2)) -> U11(active(X1), X2)
11.68/3.83	   active(U12(X1, X2)) -> U12(active(X1), X2)
11.68/3.83	   active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
11.68/3.83	   active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
11.68/3.83	   active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
11.68/3.83	   active(U114(X1, X2)) -> U114(active(X1), X2)
11.68/3.83	   active(s(X)) -> s(active(X))
11.68/3.83	   active(length(X)) -> length(active(X))
11.68/3.83	   active(U13(X)) -> U13(active(X))
11.68/3.83	   active(U121(X1, X2)) -> U121(active(X1), X2)
11.68/3.83	   active(U122(X)) -> U122(active(X))
11.68/3.83	   active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
11.68/3.83	   active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
11.68/3.83	   active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
11.68/3.83	   active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
11.68/3.83	   active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
11.68/3.83	   active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
11.68/3.83	   active(take(X1, X2)) -> take(active(X1), X2)
11.68/3.83	   active(take(X1, X2)) -> take(X1, active(X2))
11.68/3.83	   active(U21(X1, X2)) -> U21(active(X1), X2)
11.68/3.83	   active(U22(X1, X2)) -> U22(active(X1), X2)
11.68/3.83	   active(U23(X)) -> U23(active(X))
11.68/3.83	   active(U31(X1, X2)) -> U31(active(X1), X2)
11.68/3.83	   active(U32(X1, X2)) -> U32(active(X1), X2)
11.68/3.83	   active(U33(X)) -> U33(active(X))
11.68/3.83	   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
11.68/3.83	   active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
11.68/3.83	   active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
11.68/3.83	   active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
11.68/3.83	   active(U45(X1, X2)) -> U45(active(X1), X2)
11.68/3.83	   active(U46(X)) -> U46(active(X))
11.68/3.83	   active(U51(X1, X2)) -> U51(active(X1), X2)
11.68/3.83	   active(U52(X)) -> U52(active(X))
11.68/3.83	   active(U61(X1, X2)) -> U61(active(X1), X2)
11.68/3.83	   active(U62(X)) -> U62(active(X))
11.68/3.83	   active(U71(X)) -> U71(active(X))
11.68/3.83	   active(U81(X)) -> U81(active(X))
11.68/3.83	   active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
11.68/3.83	   active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
11.68/3.83	   active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
11.68/3.83	   active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
11.68/3.83	   active(U95(X1, X2)) -> U95(active(X1), X2)
11.68/3.83	   active(U96(X)) -> U96(active(X))
11.68/3.83	   cons(mark(X1), X2) -> mark(cons(X1, X2))
11.68/3.83	   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
11.68/3.83	   U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
11.68/3.83	   U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
11.68/3.83	   U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
11.68/3.83	   U105(mark(X1), X2) -> mark(U105(X1, X2))
11.68/3.83	   U106(mark(X)) -> mark(U106(X))
11.68/3.83	   U11(mark(X1), X2) -> mark(U11(X1, X2))
11.68/3.83	   U12(mark(X1), X2) -> mark(U12(X1, X2))
11.68/3.83	   U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
11.68/3.83	   U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
11.68/3.83	   U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
11.68/3.83	   U114(mark(X1), X2) -> mark(U114(X1, X2))
11.68/3.83	   s(mark(X)) -> mark(s(X))
11.73/3.84	   length(mark(X)) -> mark(length(X))
11.73/3.84	   U13(mark(X)) -> mark(U13(X))
11.73/3.84	   U121(mark(X1), X2) -> mark(U121(X1, X2))
11.73/3.84	   U122(mark(X)) -> mark(U122(X))
11.73/3.84	   U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
11.73/3.84	   U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
11.73/3.84	   U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
11.73/3.84	   U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
11.73/3.84	   U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
11.73/3.84	   U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
11.73/3.84	   take(mark(X1), X2) -> mark(take(X1, X2))
11.73/3.84	   take(X1, mark(X2)) -> mark(take(X1, X2))
11.73/3.84	   U21(mark(X1), X2) -> mark(U21(X1, X2))
11.73/3.84	   U22(mark(X1), X2) -> mark(U22(X1, X2))
11.73/3.84	   U23(mark(X)) -> mark(U23(X))
11.73/3.84	   U31(mark(X1), X2) -> mark(U31(X1, X2))
11.73/3.84	   U32(mark(X1), X2) -> mark(U32(X1, X2))
11.73/3.84	   U33(mark(X)) -> mark(U33(X))
11.73/3.84	   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
11.73/3.84	   U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
11.73/3.84	   U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
11.73/3.84	   U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
11.73/3.84	   U45(mark(X1), X2) -> mark(U45(X1, X2))
11.73/3.84	   U46(mark(X)) -> mark(U46(X))
11.73/3.84	   U51(mark(X1), X2) -> mark(U51(X1, X2))
11.73/3.84	   U52(mark(X)) -> mark(U52(X))
11.73/3.84	   U61(mark(X1), X2) -> mark(U61(X1, X2))
11.73/3.84	   U62(mark(X)) -> mark(U62(X))
11.73/3.84	   U71(mark(X)) -> mark(U71(X))
11.73/3.84	   U81(mark(X)) -> mark(U81(X))
11.73/3.84	   U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
11.73/3.84	   U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
11.73/3.84	   U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
11.73/3.84	   U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
11.73/3.84	   U95(mark(X1), X2) -> mark(U95(X1, X2))
11.73/3.84	   U96(mark(X)) -> mark(U96(X))
11.73/3.84	   proper(zeros) -> ok(zeros)
11.73/3.84	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
11.73/3.84	   proper(0) -> ok(0)
11.73/3.84	   proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(tt) -> ok(tt)
11.73/3.84	   proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(isNatKind(X)) -> isNatKind(proper(X))
11.73/3.84	   proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
11.73/3.84	   proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
11.73/3.84	   proper(isNat(X)) -> isNat(proper(X))
11.73/3.84	   proper(U106(X)) -> U106(proper(X))
11.73/3.84	   proper(isNatIList(X)) -> isNatIList(proper(X))
11.73/3.84	   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
11.73/3.84	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
11.73/3.84	   proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
11.73/3.84	   proper(s(X)) -> s(proper(X))
11.73/3.84	   proper(length(X)) -> length(proper(X))
11.73/3.84	   proper(U13(X)) -> U13(proper(X))
11.73/3.84	   proper(isNatList(X)) -> isNatList(proper(X))
11.73/3.84	   proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
11.73/3.84	   proper(U122(X)) -> U122(proper(X))
11.73/3.84	   proper(nil) -> ok(nil)
11.73/3.84	   proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.84	   proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.84	   proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.84	   proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.84	   proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.84	   proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.84	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
11.73/3.84	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
11.73/3.84	   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
11.73/3.84	   proper(U23(X)) -> U23(proper(X))
11.73/3.84	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
11.73/3.84	   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
11.73/3.84	   proper(U33(X)) -> U33(proper(X))
11.73/3.84	   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
11.73/3.84	   proper(U46(X)) -> U46(proper(X))
11.73/3.84	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
11.73/3.84	   proper(U52(X)) -> U52(proper(X))
11.73/3.84	   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
11.73/3.84	   proper(U62(X)) -> U62(proper(X))
11.73/3.84	   proper(U71(X)) -> U71(proper(X))
11.73/3.84	   proper(U81(X)) -> U81(proper(X))
11.73/3.84	   proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
11.73/3.84	   proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
11.73/3.84	   proper(U96(X)) -> U96(proper(X))
11.73/3.84	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
11.73/3.84	   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
11.73/3.84	   U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
11.73/3.84	   isNatKind(ok(X)) -> ok(isNatKind(X))
11.73/3.84	   U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
11.73/3.84	   isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
11.73/3.84	   U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
11.73/3.84	   U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
11.73/3.84	   isNat(ok(X)) -> ok(isNat(X))
11.73/3.84	   U106(ok(X)) -> ok(U106(X))
11.73/3.84	   isNatIList(ok(X)) -> ok(isNatIList(X))
11.73/3.84	   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
11.73/3.84	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
11.73/3.84	   U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
11.73/3.84	   U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
11.73/3.84	   U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
11.73/3.84	   U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
11.73/3.84	   s(ok(X)) -> ok(s(X))
11.73/3.84	   length(ok(X)) -> ok(length(X))
11.73/3.84	   U13(ok(X)) -> ok(U13(X))
11.73/3.84	   isNatList(ok(X)) -> ok(isNatList(X))
11.73/3.84	   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
11.73/3.84	   U122(ok(X)) -> ok(U122(X))
11.73/3.84	   U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
11.73/3.84	   U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
11.73/3.84	   U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
11.73/3.84	   U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
11.73/3.84	   U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
11.73/3.84	   U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
11.73/3.84	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
11.73/3.84	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
11.73/3.84	   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
11.73/3.84	   U23(ok(X)) -> ok(U23(X))
11.73/3.84	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
11.73/3.84	   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
11.73/3.84	   U33(ok(X)) -> ok(U33(X))
11.73/3.84	   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
11.73/3.84	   U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
11.73/3.84	   U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
11.73/3.84	   U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
11.73/3.84	   U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
11.73/3.84	   U46(ok(X)) -> ok(U46(X))
11.73/3.84	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
11.73/3.84	   U52(ok(X)) -> ok(U52(X))
11.73/3.84	   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
11.73/3.84	   U62(ok(X)) -> ok(U62(X))
11.73/3.84	   U71(ok(X)) -> ok(U71(X))
11.73/3.84	   U81(ok(X)) -> ok(U81(X))
11.73/3.84	   U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
11.73/3.84	   U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
11.73/3.84	   U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
11.73/3.84	   U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
11.73/3.84	   U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
11.73/3.84	   U96(ok(X)) -> ok(U96(X))
11.73/3.84	   top(mark(X)) -> top(proper(X))
11.73/3.84	   top(ok(X)) -> top(active(X))
11.73/3.84	
11.73/3.84	The set Q consists of the following terms:
11.73/3.84	
11.73/3.84	   active(zeros)
11.73/3.84	   active(isNat(0))
11.73/3.84	   active(isNat(length(x0)))
11.73/3.84	   active(isNat(s(x0)))
11.73/3.84	   active(isNatIList(x0))
11.73/3.84	   active(isNatIListKind(nil))
11.73/3.84	   active(isNatIListKind(zeros))
11.73/3.84	   active(isNatIListKind(cons(x0, x1)))
11.73/3.84	   active(isNatIListKind(take(x0, x1)))
11.73/3.84	   active(isNatKind(0))
11.73/3.84	   active(isNatKind(length(x0)))
11.73/3.84	   active(isNatKind(s(x0)))
11.73/3.84	   active(isNatList(nil))
11.73/3.84	   active(isNatList(cons(x0, x1)))
11.73/3.84	   active(isNatList(take(x0, x1)))
11.73/3.84	   active(cons(x0, x1))
11.73/3.84	   active(U101(x0, x1, x2))
11.73/3.84	   active(U102(x0, x1, x2))
11.73/3.84	   active(U103(x0, x1, x2))
11.73/3.84	   active(U104(x0, x1, x2))
11.73/3.84	   active(U105(x0, x1))
11.73/3.84	   active(U106(x0))
11.73/3.84	   active(U11(x0, x1))
11.73/3.84	   active(U12(x0, x1))
11.73/3.84	   active(U111(x0, x1, x2))
11.73/3.84	   active(U112(x0, x1, x2))
11.73/3.84	   active(U113(x0, x1, x2))
11.73/3.84	   active(U114(x0, x1))
11.73/3.84	   active(s(x0))
11.73/3.84	   active(length(x0))
11.73/3.84	   active(U13(x0))
11.73/3.84	   active(U121(x0, x1))
11.73/3.84	   active(U122(x0))
11.73/3.84	   active(U131(x0, x1, x2, x3))
11.73/3.84	   active(U132(x0, x1, x2, x3))
11.73/3.84	   active(U133(x0, x1, x2, x3))
11.73/3.84	   active(U134(x0, x1, x2, x3))
11.73/3.84	   active(U135(x0, x1, x2, x3))
11.73/3.84	   active(U136(x0, x1, x2, x3))
11.73/3.84	   active(take(x0, x1))
11.73/3.84	   active(U21(x0, x1))
11.73/3.84	   active(U22(x0, x1))
11.73/3.84	   active(U23(x0))
11.73/3.84	   active(U31(x0, x1))
11.73/3.84	   active(U32(x0, x1))
11.73/3.84	   active(U33(x0))
11.73/3.84	   active(U41(x0, x1, x2))
11.73/3.84	   active(U42(x0, x1, x2))
11.73/3.84	   active(U43(x0, x1, x2))
11.73/3.84	   active(U44(x0, x1, x2))
11.73/3.84	   active(U45(x0, x1))
11.73/3.84	   active(U46(x0))
11.73/3.84	   active(U51(x0, x1))
11.73/3.84	   active(U52(x0))
11.73/3.84	   active(U61(x0, x1))
11.73/3.84	   active(U62(x0))
11.73/3.84	   active(U71(x0))
11.73/3.84	   active(U81(x0))
11.73/3.84	   active(U91(x0, x1, x2))
11.73/3.84	   active(U92(x0, x1, x2))
11.73/3.84	   active(U93(x0, x1, x2))
11.73/3.84	   active(U94(x0, x1, x2))
11.73/3.84	   active(U95(x0, x1))
11.73/3.84	   active(U96(x0))
11.73/3.84	   cons(mark(x0), x1)
11.73/3.84	   U101(mark(x0), x1, x2)
11.73/3.84	   U102(mark(x0), x1, x2)
11.73/3.84	   U103(mark(x0), x1, x2)
11.73/3.84	   U104(mark(x0), x1, x2)
11.73/3.84	   U105(mark(x0), x1)
11.73/3.84	   U106(mark(x0))
11.73/3.84	   U11(mark(x0), x1)
11.73/3.84	   U12(mark(x0), x1)
11.73/3.84	   U111(mark(x0), x1, x2)
11.73/3.84	   U112(mark(x0), x1, x2)
11.73/3.84	   U113(mark(x0), x1, x2)
11.73/3.84	   U114(mark(x0), x1)
11.73/3.84	   s(mark(x0))
11.73/3.84	   length(mark(x0))
11.73/3.84	   U13(mark(x0))
11.73/3.84	   U121(mark(x0), x1)
11.73/3.84	   U122(mark(x0))
11.73/3.84	   U131(mark(x0), x1, x2, x3)
11.73/3.84	   U132(mark(x0), x1, x2, x3)
11.73/3.84	   U133(mark(x0), x1, x2, x3)
11.73/3.84	   U134(mark(x0), x1, x2, x3)
11.73/3.84	   U135(mark(x0), x1, x2, x3)
11.73/3.84	   U136(mark(x0), x1, x2, x3)
11.73/3.84	   take(mark(x0), x1)
11.73/3.84	   take(x0, mark(x1))
11.73/3.84	   U21(mark(x0), x1)
11.73/3.84	   U22(mark(x0), x1)
11.73/3.84	   U23(mark(x0))
11.73/3.84	   U31(mark(x0), x1)
11.73/3.84	   U32(mark(x0), x1)
11.73/3.84	   U33(mark(x0))
11.73/3.84	   U41(mark(x0), x1, x2)
11.73/3.84	   U42(mark(x0), x1, x2)
11.73/3.84	   U43(mark(x0), x1, x2)
11.73/3.84	   U44(mark(x0), x1, x2)
11.73/3.84	   U45(mark(x0), x1)
11.73/3.84	   U46(mark(x0))
11.73/3.84	   U51(mark(x0), x1)
11.73/3.84	   U52(mark(x0))
11.73/3.84	   U61(mark(x0), x1)
11.73/3.84	   U62(mark(x0))
11.73/3.84	   U71(mark(x0))
11.73/3.84	   U81(mark(x0))
11.73/3.84	   U91(mark(x0), x1, x2)
11.73/3.84	   U92(mark(x0), x1, x2)
11.73/3.84	   U93(mark(x0), x1, x2)
11.73/3.84	   U94(mark(x0), x1, x2)
11.73/3.84	   U95(mark(x0), x1)
11.73/3.84	   U96(mark(x0))
11.73/3.84	   proper(zeros)
11.73/3.84	   proper(cons(x0, x1))
11.73/3.84	   proper(0)
11.73/3.84	   proper(U101(x0, x1, x2))
11.73/3.84	   proper(tt)
11.73/3.84	   proper(U102(x0, x1, x2))
11.73/3.84	   proper(isNatKind(x0))
11.73/3.84	   proper(U103(x0, x1, x2))
11.73/3.84	   proper(isNatIListKind(x0))
11.73/3.84	   proper(U104(x0, x1, x2))
11.73/3.84	   proper(U105(x0, x1))
11.73/3.84	   proper(isNat(x0))
11.73/3.84	   proper(U106(x0))
11.73/3.84	   proper(isNatIList(x0))
11.73/3.84	   proper(U11(x0, x1))
11.73/3.84	   proper(U12(x0, x1))
11.73/3.84	   proper(U111(x0, x1, x2))
11.73/3.84	   proper(U112(x0, x1, x2))
11.73/3.84	   proper(U113(x0, x1, x2))
11.73/3.84	   proper(U114(x0, x1))
11.73/3.84	   proper(s(x0))
11.73/3.84	   proper(length(x0))
11.73/3.84	   proper(U13(x0))
11.73/3.84	   proper(isNatList(x0))
11.73/3.84	   proper(U121(x0, x1))
11.73/3.84	   proper(U122(x0))
11.73/3.84	   proper(nil)
11.73/3.84	   proper(U131(x0, x1, x2, x3))
11.73/3.84	   proper(U132(x0, x1, x2, x3))
11.73/3.84	   proper(U133(x0, x1, x2, x3))
11.73/3.84	   proper(U134(x0, x1, x2, x3))
11.73/3.84	   proper(U135(x0, x1, x2, x3))
11.73/3.84	   proper(U136(x0, x1, x2, x3))
11.73/3.84	   proper(take(x0, x1))
11.73/3.84	   proper(U21(x0, x1))
11.73/3.84	   proper(U22(x0, x1))
11.73/3.84	   proper(U23(x0))
11.73/3.84	   proper(U31(x0, x1))
11.73/3.84	   proper(U32(x0, x1))
11.73/3.84	   proper(U33(x0))
11.73/3.84	   proper(U41(x0, x1, x2))
11.73/3.84	   proper(U42(x0, x1, x2))
11.73/3.84	   proper(U43(x0, x1, x2))
11.73/3.84	   proper(U44(x0, x1, x2))
11.73/3.84	   proper(U45(x0, x1))
11.73/3.84	   proper(U46(x0))
11.73/3.84	   proper(U51(x0, x1))
11.73/3.84	   proper(U52(x0))
11.73/3.84	   proper(U61(x0, x1))
11.73/3.84	   proper(U62(x0))
11.73/3.84	   proper(U71(x0))
11.73/3.84	   proper(U81(x0))
11.73/3.84	   proper(U91(x0, x1, x2))
11.73/3.84	   proper(U92(x0, x1, x2))
11.73/3.84	   proper(U93(x0, x1, x2))
11.73/3.84	   proper(U94(x0, x1, x2))
11.73/3.84	   proper(U95(x0, x1))
11.73/3.84	   proper(U96(x0))
11.73/3.84	   cons(ok(x0), ok(x1))
11.73/3.84	   U101(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U102(ok(x0), ok(x1), ok(x2))
11.73/3.84	   isNatKind(ok(x0))
11.73/3.84	   U103(ok(x0), ok(x1), ok(x2))
11.73/3.84	   isNatIListKind(ok(x0))
11.73/3.84	   U104(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U105(ok(x0), ok(x1))
11.73/3.84	   isNat(ok(x0))
11.73/3.84	   U106(ok(x0))
11.73/3.84	   isNatIList(ok(x0))
11.73/3.84	   U11(ok(x0), ok(x1))
11.73/3.84	   U12(ok(x0), ok(x1))
11.73/3.84	   U111(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U112(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U113(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U114(ok(x0), ok(x1))
11.73/3.84	   s(ok(x0))
11.73/3.84	   length(ok(x0))
11.73/3.84	   U13(ok(x0))
11.73/3.84	   isNatList(ok(x0))
11.73/3.84	   U121(ok(x0), ok(x1))
11.73/3.84	   U122(ok(x0))
11.73/3.84	   U131(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.84	   U132(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.84	   U133(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.84	   U134(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.84	   U135(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.84	   U136(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.84	   take(ok(x0), ok(x1))
11.73/3.84	   U21(ok(x0), ok(x1))
11.73/3.84	   U22(ok(x0), ok(x1))
11.73/3.84	   U23(ok(x0))
11.73/3.84	   U31(ok(x0), ok(x1))
11.73/3.84	   U32(ok(x0), ok(x1))
11.73/3.84	   U33(ok(x0))
11.73/3.84	   U41(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U42(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U43(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U44(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U45(ok(x0), ok(x1))
11.73/3.84	   U46(ok(x0))
11.73/3.84	   U51(ok(x0), ok(x1))
11.73/3.84	   U52(ok(x0))
11.73/3.84	   U61(ok(x0), ok(x1))
11.73/3.84	   U62(ok(x0))
11.73/3.84	   U71(ok(x0))
11.73/3.84	   U81(ok(x0))
11.73/3.84	   U91(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U92(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U93(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U94(ok(x0), ok(x1), ok(x2))
11.73/3.84	   U95(ok(x0), ok(x1))
11.73/3.84	   U96(ok(x0))
11.73/3.84	   top(mark(x0))
11.73/3.84	   top(ok(x0))
11.73/3.84	
11.73/3.84	
11.73/3.84	----------------------------------------
11.73/3.84	
11.73/3.84	(1) QTRSToCSRProof (SOUND)
11.73/3.84	The following Q TRS is given: Q restricted rewrite system:
11.73/3.84	The TRS R consists of the following rules:
11.73/3.84	
11.73/3.84	   active(zeros) -> mark(cons(0, zeros))
11.73/3.84	   active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
11.73/3.84	   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
11.73/3.84	   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
11.73/3.84	   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
11.73/3.84	   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
11.73/3.84	   active(U106(tt)) -> mark(tt)
11.73/3.84	   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
11.73/3.84	   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
11.73/3.84	   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
11.73/3.84	   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
11.73/3.84	   active(U114(tt, L)) -> mark(s(length(L)))
11.73/3.84	   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
11.73/3.84	   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
11.73/3.84	   active(U122(tt)) -> mark(nil)
11.73/3.84	   active(U13(tt)) -> mark(tt)
11.73/3.84	   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
11.73/3.84	   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
11.73/3.84	   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
11.73/3.84	   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
11.73/3.84	   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
11.73/3.84	   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
11.73/3.85	   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
11.73/3.85	   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
11.73/3.85	   active(U23(tt)) -> mark(tt)
11.73/3.85	   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
11.73/3.85	   active(U32(tt, V)) -> mark(U33(isNatList(V)))
11.73/3.85	   active(U33(tt)) -> mark(tt)
11.73/3.85	   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
11.73/3.85	   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
11.73/3.85	   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
11.73/3.85	   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
11.73/3.85	   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
11.73/3.85	   active(U46(tt)) -> mark(tt)
11.73/3.85	   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
11.73/3.85	   active(U52(tt)) -> mark(tt)
11.73/3.85	   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
11.73/3.85	   active(U62(tt)) -> mark(tt)
11.73/3.85	   active(U71(tt)) -> mark(tt)
11.73/3.85	   active(U81(tt)) -> mark(tt)
11.73/3.85	   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
11.73/3.85	   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
11.73/3.85	   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
11.73/3.85	   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
11.73/3.85	   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
11.73/3.85	   active(U96(tt)) -> mark(tt)
11.73/3.85	   active(isNat(0)) -> mark(tt)
11.73/3.85	   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
11.73/3.85	   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
11.73/3.85	   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
11.73/3.85	   active(isNatIList(zeros)) -> mark(tt)
11.73/3.85	   active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2))
11.73/3.85	   active(isNatIListKind(nil)) -> mark(tt)
11.73/3.85	   active(isNatIListKind(zeros)) -> mark(tt)
11.73/3.85	   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
11.73/3.85	   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
11.73/3.85	   active(isNatKind(0)) -> mark(tt)
11.73/3.85	   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
11.73/3.85	   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
11.73/3.85	   active(isNatList(nil)) -> mark(tt)
11.73/3.85	   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
11.73/3.85	   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
11.73/3.85	   active(length(nil)) -> mark(0)
11.73/3.85	   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
11.73/3.85	   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
11.73/3.85	   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
11.73/3.85	   active(cons(X1, X2)) -> cons(active(X1), X2)
11.73/3.85	   active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
11.73/3.85	   active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
11.73/3.85	   active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
11.73/3.85	   active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
11.73/3.85	   active(U105(X1, X2)) -> U105(active(X1), X2)
11.73/3.85	   active(U106(X)) -> U106(active(X))
11.73/3.85	   active(U11(X1, X2)) -> U11(active(X1), X2)
11.73/3.85	   active(U12(X1, X2)) -> U12(active(X1), X2)
11.73/3.85	   active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
11.73/3.85	   active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
11.73/3.85	   active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
11.73/3.85	   active(U114(X1, X2)) -> U114(active(X1), X2)
11.73/3.85	   active(s(X)) -> s(active(X))
11.73/3.85	   active(length(X)) -> length(active(X))
11.73/3.85	   active(U13(X)) -> U13(active(X))
11.73/3.85	   active(U121(X1, X2)) -> U121(active(X1), X2)
11.73/3.85	   active(U122(X)) -> U122(active(X))
11.73/3.85	   active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
11.73/3.85	   active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
11.73/3.85	   active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
11.73/3.85	   active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
11.73/3.85	   active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
11.73/3.85	   active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
11.73/3.85	   active(take(X1, X2)) -> take(active(X1), X2)
11.73/3.85	   active(take(X1, X2)) -> take(X1, active(X2))
11.73/3.85	   active(U21(X1, X2)) -> U21(active(X1), X2)
11.73/3.85	   active(U22(X1, X2)) -> U22(active(X1), X2)
11.73/3.85	   active(U23(X)) -> U23(active(X))
11.73/3.85	   active(U31(X1, X2)) -> U31(active(X1), X2)
11.73/3.85	   active(U32(X1, X2)) -> U32(active(X1), X2)
11.73/3.85	   active(U33(X)) -> U33(active(X))
11.73/3.85	   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
11.73/3.85	   active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
11.73/3.85	   active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
11.73/3.85	   active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
11.73/3.85	   active(U45(X1, X2)) -> U45(active(X1), X2)
11.73/3.85	   active(U46(X)) -> U46(active(X))
11.73/3.85	   active(U51(X1, X2)) -> U51(active(X1), X2)
11.73/3.85	   active(U52(X)) -> U52(active(X))
11.73/3.85	   active(U61(X1, X2)) -> U61(active(X1), X2)
11.73/3.85	   active(U62(X)) -> U62(active(X))
11.73/3.85	   active(U71(X)) -> U71(active(X))
11.73/3.85	   active(U81(X)) -> U81(active(X))
11.73/3.85	   active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
11.73/3.85	   active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
11.73/3.85	   active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
11.73/3.85	   active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
11.73/3.85	   active(U95(X1, X2)) -> U95(active(X1), X2)
11.73/3.85	   active(U96(X)) -> U96(active(X))
11.73/3.85	   cons(mark(X1), X2) -> mark(cons(X1, X2))
11.73/3.85	   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
11.73/3.85	   U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
11.73/3.85	   U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
11.73/3.85	   U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
11.73/3.85	   U105(mark(X1), X2) -> mark(U105(X1, X2))
11.73/3.85	   U106(mark(X)) -> mark(U106(X))
11.73/3.85	   U11(mark(X1), X2) -> mark(U11(X1, X2))
11.73/3.85	   U12(mark(X1), X2) -> mark(U12(X1, X2))
11.73/3.85	   U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
11.73/3.85	   U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
11.73/3.85	   U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
11.73/3.85	   U114(mark(X1), X2) -> mark(U114(X1, X2))
11.73/3.85	   s(mark(X)) -> mark(s(X))
11.73/3.85	   length(mark(X)) -> mark(length(X))
11.73/3.85	   U13(mark(X)) -> mark(U13(X))
11.73/3.85	   U121(mark(X1), X2) -> mark(U121(X1, X2))
11.73/3.85	   U122(mark(X)) -> mark(U122(X))
11.73/3.85	   U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
11.73/3.85	   U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
11.73/3.85	   U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
11.73/3.85	   U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
11.73/3.85	   U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
11.73/3.85	   U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
11.73/3.85	   take(mark(X1), X2) -> mark(take(X1, X2))
11.73/3.85	   take(X1, mark(X2)) -> mark(take(X1, X2))
11.73/3.85	   U21(mark(X1), X2) -> mark(U21(X1, X2))
11.73/3.85	   U22(mark(X1), X2) -> mark(U22(X1, X2))
11.73/3.85	   U23(mark(X)) -> mark(U23(X))
11.73/3.85	   U31(mark(X1), X2) -> mark(U31(X1, X2))
11.73/3.85	   U32(mark(X1), X2) -> mark(U32(X1, X2))
11.73/3.85	   U33(mark(X)) -> mark(U33(X))
11.73/3.85	   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
11.73/3.85	   U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
11.73/3.85	   U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
11.73/3.85	   U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
11.73/3.85	   U45(mark(X1), X2) -> mark(U45(X1, X2))
11.73/3.85	   U46(mark(X)) -> mark(U46(X))
11.73/3.85	   U51(mark(X1), X2) -> mark(U51(X1, X2))
11.73/3.85	   U52(mark(X)) -> mark(U52(X))
11.73/3.85	   U61(mark(X1), X2) -> mark(U61(X1, X2))
11.73/3.85	   U62(mark(X)) -> mark(U62(X))
11.73/3.85	   U71(mark(X)) -> mark(U71(X))
11.73/3.85	   U81(mark(X)) -> mark(U81(X))
11.73/3.85	   U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
11.73/3.85	   U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
11.73/3.85	   U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
11.73/3.85	   U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
11.73/3.85	   U95(mark(X1), X2) -> mark(U95(X1, X2))
11.73/3.85	   U96(mark(X)) -> mark(U96(X))
11.73/3.85	   proper(zeros) -> ok(zeros)
11.73/3.85	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
11.73/3.85	   proper(0) -> ok(0)
11.73/3.85	   proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(tt) -> ok(tt)
11.73/3.85	   proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(isNatKind(X)) -> isNatKind(proper(X))
11.73/3.85	   proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
11.73/3.85	   proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
11.73/3.85	   proper(isNat(X)) -> isNat(proper(X))
11.73/3.85	   proper(U106(X)) -> U106(proper(X))
11.73/3.85	   proper(isNatIList(X)) -> isNatIList(proper(X))
11.73/3.85	   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
11.73/3.85	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
11.73/3.85	   proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
11.73/3.85	   proper(s(X)) -> s(proper(X))
11.73/3.85	   proper(length(X)) -> length(proper(X))
11.73/3.85	   proper(U13(X)) -> U13(proper(X))
11.73/3.85	   proper(isNatList(X)) -> isNatList(proper(X))
11.73/3.85	   proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
11.73/3.85	   proper(U122(X)) -> U122(proper(X))
11.73/3.85	   proper(nil) -> ok(nil)
11.73/3.85	   proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.85	   proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.85	   proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.85	   proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.85	   proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.85	   proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
11.73/3.85	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
11.73/3.85	   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
11.73/3.85	   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
11.73/3.85	   proper(U23(X)) -> U23(proper(X))
11.73/3.85	   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
11.73/3.85	   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
11.73/3.85	   proper(U33(X)) -> U33(proper(X))
11.73/3.85	   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
11.73/3.85	   proper(U46(X)) -> U46(proper(X))
11.73/3.85	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
11.73/3.85	   proper(U52(X)) -> U52(proper(X))
11.73/3.85	   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
11.73/3.85	   proper(U62(X)) -> U62(proper(X))
11.73/3.85	   proper(U71(X)) -> U71(proper(X))
11.73/3.85	   proper(U81(X)) -> U81(proper(X))
11.73/3.85	   proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
11.73/3.85	   proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
11.73/3.85	   proper(U96(X)) -> U96(proper(X))
11.73/3.85	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
11.73/3.85	   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
11.73/3.85	   U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
11.73/3.85	   isNatKind(ok(X)) -> ok(isNatKind(X))
11.73/3.85	   U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
11.73/3.85	   isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
11.73/3.85	   U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
11.73/3.85	   U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
11.73/3.85	   isNat(ok(X)) -> ok(isNat(X))
11.73/3.85	   U106(ok(X)) -> ok(U106(X))
11.73/3.85	   isNatIList(ok(X)) -> ok(isNatIList(X))
11.73/3.85	   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
11.73/3.85	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
11.73/3.85	   U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
11.73/3.85	   U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
11.73/3.85	   U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
11.73/3.85	   U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
11.73/3.85	   s(ok(X)) -> ok(s(X))
11.73/3.85	   length(ok(X)) -> ok(length(X))
11.73/3.85	   U13(ok(X)) -> ok(U13(X))
11.73/3.85	   isNatList(ok(X)) -> ok(isNatList(X))
11.73/3.85	   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
11.73/3.85	   U122(ok(X)) -> ok(U122(X))
11.73/3.85	   U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
11.73/3.85	   U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
11.73/3.85	   U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
11.73/3.85	   U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
11.73/3.85	   U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
11.73/3.85	   U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
11.73/3.85	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
11.73/3.85	   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
11.73/3.85	   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
11.73/3.85	   U23(ok(X)) -> ok(U23(X))
11.73/3.85	   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
11.73/3.85	   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
11.73/3.85	   U33(ok(X)) -> ok(U33(X))
11.73/3.85	   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
11.73/3.85	   U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
11.73/3.85	   U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
11.73/3.85	   U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
11.73/3.85	   U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
11.73/3.85	   U46(ok(X)) -> ok(U46(X))
11.73/3.85	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
11.73/3.85	   U52(ok(X)) -> ok(U52(X))
11.73/3.85	   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
11.73/3.85	   U62(ok(X)) -> ok(U62(X))
11.73/3.85	   U71(ok(X)) -> ok(U71(X))
11.73/3.85	   U81(ok(X)) -> ok(U81(X))
11.73/3.85	   U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
11.73/3.85	   U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
11.73/3.85	   U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
11.73/3.85	   U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
11.73/3.85	   U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
11.73/3.85	   U96(ok(X)) -> ok(U96(X))
11.73/3.85	   top(mark(X)) -> top(proper(X))
11.73/3.85	   top(ok(X)) -> top(active(X))
11.73/3.85	
11.73/3.85	The set Q consists of the following terms:
11.73/3.85	
11.73/3.85	   active(zeros)
11.73/3.85	   active(isNat(0))
11.73/3.85	   active(isNat(length(x0)))
11.73/3.85	   active(isNat(s(x0)))
11.73/3.85	   active(isNatIList(x0))
11.73/3.85	   active(isNatIListKind(nil))
11.73/3.85	   active(isNatIListKind(zeros))
11.73/3.85	   active(isNatIListKind(cons(x0, x1)))
11.73/3.85	   active(isNatIListKind(take(x0, x1)))
11.73/3.85	   active(isNatKind(0))
11.73/3.85	   active(isNatKind(length(x0)))
11.73/3.85	   active(isNatKind(s(x0)))
11.73/3.85	   active(isNatList(nil))
11.73/3.85	   active(isNatList(cons(x0, x1)))
11.73/3.85	   active(isNatList(take(x0, x1)))
11.73/3.85	   active(cons(x0, x1))
11.73/3.85	   active(U101(x0, x1, x2))
11.73/3.85	   active(U102(x0, x1, x2))
11.73/3.85	   active(U103(x0, x1, x2))
11.73/3.85	   active(U104(x0, x1, x2))
11.73/3.85	   active(U105(x0, x1))
11.73/3.85	   active(U106(x0))
11.73/3.85	   active(U11(x0, x1))
11.73/3.85	   active(U12(x0, x1))
11.73/3.85	   active(U111(x0, x1, x2))
11.73/3.85	   active(U112(x0, x1, x2))
11.73/3.85	   active(U113(x0, x1, x2))
11.73/3.85	   active(U114(x0, x1))
11.73/3.85	   active(s(x0))
11.73/3.85	   active(length(x0))
11.73/3.85	   active(U13(x0))
11.73/3.85	   active(U121(x0, x1))
11.73/3.85	   active(U122(x0))
11.73/3.85	   active(U131(x0, x1, x2, x3))
11.73/3.85	   active(U132(x0, x1, x2, x3))
11.73/3.85	   active(U133(x0, x1, x2, x3))
11.73/3.85	   active(U134(x0, x1, x2, x3))
11.73/3.85	   active(U135(x0, x1, x2, x3))
11.73/3.85	   active(U136(x0, x1, x2, x3))
11.73/3.85	   active(take(x0, x1))
11.73/3.85	   active(U21(x0, x1))
11.73/3.85	   active(U22(x0, x1))
11.73/3.85	   active(U23(x0))
11.73/3.85	   active(U31(x0, x1))
11.73/3.85	   active(U32(x0, x1))
11.73/3.85	   active(U33(x0))
11.73/3.85	   active(U41(x0, x1, x2))
11.73/3.85	   active(U42(x0, x1, x2))
11.73/3.85	   active(U43(x0, x1, x2))
11.73/3.85	   active(U44(x0, x1, x2))
11.73/3.85	   active(U45(x0, x1))
11.73/3.85	   active(U46(x0))
11.73/3.85	   active(U51(x0, x1))
11.73/3.85	   active(U52(x0))
11.73/3.85	   active(U61(x0, x1))
11.73/3.85	   active(U62(x0))
11.73/3.85	   active(U71(x0))
11.73/3.85	   active(U81(x0))
11.73/3.85	   active(U91(x0, x1, x2))
11.73/3.85	   active(U92(x0, x1, x2))
11.73/3.85	   active(U93(x0, x1, x2))
11.73/3.85	   active(U94(x0, x1, x2))
11.73/3.85	   active(U95(x0, x1))
11.73/3.85	   active(U96(x0))
11.73/3.85	   cons(mark(x0), x1)
11.73/3.85	   U101(mark(x0), x1, x2)
11.73/3.85	   U102(mark(x0), x1, x2)
11.73/3.85	   U103(mark(x0), x1, x2)
11.73/3.85	   U104(mark(x0), x1, x2)
11.73/3.85	   U105(mark(x0), x1)
11.73/3.85	   U106(mark(x0))
11.73/3.85	   U11(mark(x0), x1)
11.73/3.85	   U12(mark(x0), x1)
11.73/3.85	   U111(mark(x0), x1, x2)
11.73/3.85	   U112(mark(x0), x1, x2)
11.73/3.85	   U113(mark(x0), x1, x2)
11.73/3.85	   U114(mark(x0), x1)
11.73/3.85	   s(mark(x0))
11.73/3.85	   length(mark(x0))
11.73/3.85	   U13(mark(x0))
11.73/3.85	   U121(mark(x0), x1)
11.73/3.85	   U122(mark(x0))
11.73/3.85	   U131(mark(x0), x1, x2, x3)
11.73/3.85	   U132(mark(x0), x1, x2, x3)
11.73/3.85	   U133(mark(x0), x1, x2, x3)
11.73/3.85	   U134(mark(x0), x1, x2, x3)
11.73/3.85	   U135(mark(x0), x1, x2, x3)
11.73/3.85	   U136(mark(x0), x1, x2, x3)
11.73/3.85	   take(mark(x0), x1)
11.73/3.85	   take(x0, mark(x1))
11.73/3.85	   U21(mark(x0), x1)
11.73/3.85	   U22(mark(x0), x1)
11.73/3.85	   U23(mark(x0))
11.73/3.85	   U31(mark(x0), x1)
11.73/3.85	   U32(mark(x0), x1)
11.73/3.85	   U33(mark(x0))
11.73/3.85	   U41(mark(x0), x1, x2)
11.73/3.85	   U42(mark(x0), x1, x2)
11.73/3.85	   U43(mark(x0), x1, x2)
11.73/3.85	   U44(mark(x0), x1, x2)
11.73/3.85	   U45(mark(x0), x1)
11.73/3.85	   U46(mark(x0))
11.73/3.85	   U51(mark(x0), x1)
11.73/3.85	   U52(mark(x0))
11.73/3.85	   U61(mark(x0), x1)
11.73/3.85	   U62(mark(x0))
11.73/3.85	   U71(mark(x0))
11.73/3.85	   U81(mark(x0))
11.73/3.85	   U91(mark(x0), x1, x2)
11.73/3.85	   U92(mark(x0), x1, x2)
11.73/3.85	   U93(mark(x0), x1, x2)
11.73/3.85	   U94(mark(x0), x1, x2)
11.73/3.85	   U95(mark(x0), x1)
11.73/3.85	   U96(mark(x0))
11.73/3.85	   proper(zeros)
11.73/3.85	   proper(cons(x0, x1))
11.73/3.85	   proper(0)
11.73/3.85	   proper(U101(x0, x1, x2))
11.73/3.85	   proper(tt)
11.73/3.85	   proper(U102(x0, x1, x2))
11.73/3.85	   proper(isNatKind(x0))
11.73/3.85	   proper(U103(x0, x1, x2))
11.73/3.85	   proper(isNatIListKind(x0))
11.73/3.85	   proper(U104(x0, x1, x2))
11.73/3.85	   proper(U105(x0, x1))
11.73/3.85	   proper(isNat(x0))
11.73/3.85	   proper(U106(x0))
11.73/3.85	   proper(isNatIList(x0))
11.73/3.85	   proper(U11(x0, x1))
11.73/3.85	   proper(U12(x0, x1))
11.73/3.85	   proper(U111(x0, x1, x2))
11.73/3.85	   proper(U112(x0, x1, x2))
11.73/3.85	   proper(U113(x0, x1, x2))
11.73/3.85	   proper(U114(x0, x1))
11.73/3.85	   proper(s(x0))
11.73/3.85	   proper(length(x0))
11.73/3.85	   proper(U13(x0))
11.73/3.85	   proper(isNatList(x0))
11.73/3.85	   proper(U121(x0, x1))
11.73/3.85	   proper(U122(x0))
11.73/3.85	   proper(nil)
11.73/3.85	   proper(U131(x0, x1, x2, x3))
11.73/3.85	   proper(U132(x0, x1, x2, x3))
11.73/3.85	   proper(U133(x0, x1, x2, x3))
11.73/3.85	   proper(U134(x0, x1, x2, x3))
11.73/3.85	   proper(U135(x0, x1, x2, x3))
11.73/3.85	   proper(U136(x0, x1, x2, x3))
11.73/3.85	   proper(take(x0, x1))
11.73/3.85	   proper(U21(x0, x1))
11.73/3.85	   proper(U22(x0, x1))
11.73/3.85	   proper(U23(x0))
11.73/3.85	   proper(U31(x0, x1))
11.73/3.85	   proper(U32(x0, x1))
11.73/3.85	   proper(U33(x0))
11.73/3.85	   proper(U41(x0, x1, x2))
11.73/3.85	   proper(U42(x0, x1, x2))
11.73/3.85	   proper(U43(x0, x1, x2))
11.73/3.85	   proper(U44(x0, x1, x2))
11.73/3.85	   proper(U45(x0, x1))
11.73/3.85	   proper(U46(x0))
11.73/3.85	   proper(U51(x0, x1))
11.73/3.85	   proper(U52(x0))
11.73/3.85	   proper(U61(x0, x1))
11.73/3.85	   proper(U62(x0))
11.73/3.85	   proper(U71(x0))
11.73/3.85	   proper(U81(x0))
11.73/3.85	   proper(U91(x0, x1, x2))
11.73/3.85	   proper(U92(x0, x1, x2))
11.73/3.85	   proper(U93(x0, x1, x2))
11.73/3.85	   proper(U94(x0, x1, x2))
11.73/3.85	   proper(U95(x0, x1))
11.73/3.85	   proper(U96(x0))
11.73/3.85	   cons(ok(x0), ok(x1))
11.73/3.85	   U101(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U102(ok(x0), ok(x1), ok(x2))
11.73/3.85	   isNatKind(ok(x0))
11.73/3.85	   U103(ok(x0), ok(x1), ok(x2))
11.73/3.85	   isNatIListKind(ok(x0))
11.73/3.85	   U104(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U105(ok(x0), ok(x1))
11.73/3.85	   isNat(ok(x0))
11.73/3.85	   U106(ok(x0))
11.73/3.85	   isNatIList(ok(x0))
11.73/3.85	   U11(ok(x0), ok(x1))
11.73/3.85	   U12(ok(x0), ok(x1))
11.73/3.85	   U111(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U112(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U113(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U114(ok(x0), ok(x1))
11.73/3.85	   s(ok(x0))
11.73/3.85	   length(ok(x0))
11.73/3.85	   U13(ok(x0))
11.73/3.85	   isNatList(ok(x0))
11.73/3.85	   U121(ok(x0), ok(x1))
11.73/3.85	   U122(ok(x0))
11.73/3.85	   U131(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.85	   U132(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.85	   U133(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.85	   U134(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.85	   U135(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.85	   U136(ok(x0), ok(x1), ok(x2), ok(x3))
11.73/3.85	   take(ok(x0), ok(x1))
11.73/3.85	   U21(ok(x0), ok(x1))
11.73/3.85	   U22(ok(x0), ok(x1))
11.73/3.85	   U23(ok(x0))
11.73/3.85	   U31(ok(x0), ok(x1))
11.73/3.85	   U32(ok(x0), ok(x1))
11.73/3.85	   U33(ok(x0))
11.73/3.85	   U41(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U42(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U43(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U44(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U45(ok(x0), ok(x1))
11.73/3.85	   U46(ok(x0))
11.73/3.85	   U51(ok(x0), ok(x1))
11.73/3.85	   U52(ok(x0))
11.73/3.85	   U61(ok(x0), ok(x1))
11.73/3.85	   U62(ok(x0))
11.73/3.85	   U71(ok(x0))
11.73/3.85	   U81(ok(x0))
11.73/3.85	   U91(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U92(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U93(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U94(ok(x0), ok(x1), ok(x2))
11.73/3.85	   U95(ok(x0), ok(x1))
11.73/3.85	   U96(ok(x0))
11.73/3.85	   top(mark(x0))
11.73/3.85	   top(ok(x0))
11.73/3.85	
11.73/3.85	Special symbols used for the transformation (see [GM04]):
11.73/3.85	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
11.73/3.85	The replacement map contains the following entries:
11.73/3.85	
11.73/3.85	zeros: empty set
11.73/3.85	cons: {1}
11.73/3.85	0: empty set
11.73/3.85	U101: {1}
11.73/3.85	tt: empty set
11.73/3.85	U102: {1}
11.73/3.85	isNatKind: empty set
11.73/3.85	U103: {1}
11.73/3.85	isNatIListKind: empty set
11.73/3.85	U104: {1}
11.73/3.85	U105: {1}
11.73/3.85	isNat: empty set
11.73/3.85	U106: {1}
11.73/3.85	isNatIList: empty set
11.73/3.85	U11: {1}
11.73/3.85	U12: {1}
11.73/3.85	U111: {1}
11.73/3.85	U112: {1}
11.73/3.85	U113: {1}
11.73/3.85	U114: {1}
11.73/3.85	s: {1}
11.73/3.85	length: {1}
11.73/3.85	U13: {1}
11.73/3.85	isNatList: empty set
11.73/3.85	U121: {1}
11.73/3.85	U122: {1}
11.73/3.85	nil: empty set
11.73/3.85	U131: {1}
11.73/3.85	U132: {1}
11.73/3.85	U133: {1}
11.73/3.85	U134: {1}
11.73/3.85	U135: {1}
11.73/3.85	U136: {1}
11.73/3.85	take: {1, 2}
11.73/3.85	U21: {1}
11.73/3.85	U22: {1}
11.73/3.85	U23: {1}
11.73/3.85	U31: {1}
11.73/3.85	U32: {1}
11.73/3.85	U33: {1}
11.73/3.85	U41: {1}
11.73/3.85	U42: {1}
11.73/3.85	U43: {1}
11.73/3.85	U44: {1}
11.73/3.85	U45: {1}
11.73/3.85	U46: {1}
11.73/3.85	U51: {1}
11.73/3.85	U52: {1}
11.73/3.85	U61: {1}
11.73/3.85	U62: {1}
11.73/3.85	U71: {1}
11.73/3.85	U81: {1}
11.73/3.85	U91: {1}
11.73/3.85	U92: {1}
11.73/3.85	U93: {1}
11.73/3.85	U94: {1}
11.73/3.85	U95: {1}
11.73/3.85	U96: {1}
11.73/3.85	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.
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(2)
11.73/3.85	Obligation:
11.73/3.85	Context-sensitive rewrite system:
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.85	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.85	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.85	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.85	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.85	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.85	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.85	   U114(tt, L) -> s(length(L))
11.73/3.85	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.85	   U121(tt, IL) -> U122(isNatIListKind(IL))
11.73/3.85	   U122(tt) -> nil
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.85	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.85	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.85	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.85	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.85	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.85	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.85	   U32(tt, V) -> U33(isNatList(V))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.85	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.85	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.85	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.85	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.85	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.85	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.85	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.85	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.85	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.85	   length(nil) -> 0
11.73/3.85	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.85	   take(0, IL) -> U121(isNatIList(IL), IL)
11.73/3.85	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.85	
11.73/3.85	The replacement map contains the following entries:
11.73/3.85	
11.73/3.85	zeros: empty set
11.73/3.85	cons: {1}
11.73/3.85	0: empty set
11.73/3.85	U101: {1}
11.73/3.85	tt: empty set
11.73/3.85	U102: {1}
11.73/3.85	isNatKind: empty set
11.73/3.85	U103: {1}
11.73/3.85	isNatIListKind: empty set
11.73/3.85	U104: {1}
11.73/3.85	U105: {1}
11.73/3.85	isNat: empty set
11.73/3.85	U106: {1}
11.73/3.85	isNatIList: empty set
11.73/3.85	U11: {1}
11.73/3.85	U12: {1}
11.73/3.85	U111: {1}
11.73/3.85	U112: {1}
11.73/3.85	U113: {1}
11.73/3.85	U114: {1}
11.73/3.85	s: {1}
11.73/3.85	length: {1}
11.73/3.85	U13: {1}
11.73/3.85	isNatList: empty set
11.73/3.85	U121: {1}
11.73/3.85	U122: {1}
11.73/3.85	nil: empty set
11.73/3.85	U131: {1}
11.73/3.85	U132: {1}
11.73/3.85	U133: {1}
11.73/3.85	U134: {1}
11.73/3.85	U135: {1}
11.73/3.85	U136: {1}
11.73/3.85	take: {1, 2}
11.73/3.85	U21: {1}
11.73/3.85	U22: {1}
11.73/3.85	U23: {1}
11.73/3.85	U31: {1}
11.73/3.85	U32: {1}
11.73/3.85	U33: {1}
11.73/3.85	U41: {1}
11.73/3.85	U42: {1}
11.73/3.85	U43: {1}
11.73/3.85	U44: {1}
11.73/3.85	U45: {1}
11.73/3.85	U46: {1}
11.73/3.85	U51: {1}
11.73/3.85	U52: {1}
11.73/3.85	U61: {1}
11.73/3.85	U62: {1}
11.73/3.85	U71: {1}
11.73/3.85	U81: {1}
11.73/3.85	U91: {1}
11.73/3.85	U92: {1}
11.73/3.85	U93: {1}
11.73/3.85	U94: {1}
11.73/3.85	U95: {1}
11.73/3.85	U96: {1}
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(3) CSRRRRProof (EQUIVALENT)
11.73/3.85	The following CSR is given: Context-sensitive rewrite system:
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.85	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.85	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.85	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.85	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.85	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.85	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.85	   U114(tt, L) -> s(length(L))
11.73/3.85	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.85	   U121(tt, IL) -> U122(isNatIListKind(IL))
11.73/3.85	   U122(tt) -> nil
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.85	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.85	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.85	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.85	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.85	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.85	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.85	   U32(tt, V) -> U33(isNatList(V))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.85	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.85	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.85	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.85	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.85	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.85	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.85	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.85	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.85	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.85	   length(nil) -> 0
11.73/3.85	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.85	   take(0, IL) -> U121(isNatIList(IL), IL)
11.73/3.85	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.85	
11.73/3.85	The replacement map contains the following entries:
11.73/3.85	
11.73/3.85	zeros: empty set
11.73/3.85	cons: {1}
11.73/3.85	0: empty set
11.73/3.85	U101: {1}
11.73/3.85	tt: empty set
11.73/3.85	U102: {1}
11.73/3.85	isNatKind: empty set
11.73/3.85	U103: {1}
11.73/3.85	isNatIListKind: empty set
11.73/3.85	U104: {1}
11.73/3.85	U105: {1}
11.73/3.85	isNat: empty set
11.73/3.85	U106: {1}
11.73/3.85	isNatIList: empty set
11.73/3.85	U11: {1}
11.73/3.85	U12: {1}
11.73/3.85	U111: {1}
11.73/3.85	U112: {1}
11.73/3.85	U113: {1}
11.73/3.85	U114: {1}
11.73/3.85	s: {1}
11.73/3.85	length: {1}
11.73/3.85	U13: {1}
11.73/3.85	isNatList: empty set
11.73/3.85	U121: {1}
11.73/3.85	U122: {1}
11.73/3.85	nil: empty set
11.73/3.85	U131: {1}
11.73/3.85	U132: {1}
11.73/3.85	U133: {1}
11.73/3.85	U134: {1}
11.73/3.85	U135: {1}
11.73/3.85	U136: {1}
11.73/3.85	take: {1, 2}
11.73/3.85	U21: {1}
11.73/3.85	U22: {1}
11.73/3.85	U23: {1}
11.73/3.85	U31: {1}
11.73/3.85	U32: {1}
11.73/3.85	U33: {1}
11.73/3.85	U41: {1}
11.73/3.85	U42: {1}
11.73/3.85	U43: {1}
11.73/3.85	U44: {1}
11.73/3.85	U45: {1}
11.73/3.85	U46: {1}
11.73/3.85	U51: {1}
11.73/3.85	U52: {1}
11.73/3.85	U61: {1}
11.73/3.85	U62: {1}
11.73/3.85	U71: {1}
11.73/3.85	U81: {1}
11.73/3.85	U91: {1}
11.73/3.85	U92: {1}
11.73/3.85	U93: {1}
11.73/3.85	U94: {1}
11.73/3.85	U95: {1}
11.73/3.85	U96: {1}
11.73/3.85	Used ordering:
11.73/3.85	Polynomial interpretation [POLO]:
11.73/3.85	
11.73/3.85	   POL(0) = 0
11.73/3.85	   POL(U101(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U102(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U103(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U104(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U105(x_1, x_2)) = x_1
11.73/3.85	   POL(U106(x_1)) = x_1
11.73/3.85	   POL(U11(x_1, x_2)) = x_1
11.73/3.85	   POL(U111(x_1, x_2, x_3)) = x_1 + x_2 + x_3
11.73/3.85	   POL(U112(x_1, x_2, x_3)) = x_1 + x_2 + x_3
11.73/3.85	   POL(U113(x_1, x_2, x_3)) = x_1 + x_2 + x_3
11.73/3.85	   POL(U114(x_1, x_2)) = x_1 + x_2
11.73/3.85	   POL(U12(x_1, x_2)) = x_1
11.73/3.85	   POL(U121(x_1, x_2)) = 1 + x_1 + x_2
11.73/3.85	   POL(U122(x_1)) = x_1
11.73/3.85	   POL(U13(x_1)) = x_1
11.73/3.85	   POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U21(x_1, x_2)) = x_1
11.73/3.85	   POL(U22(x_1, x_2)) = x_1
11.73/3.85	   POL(U23(x_1)) = x_1
11.73/3.85	   POL(U31(x_1, x_2)) = x_1
11.73/3.85	   POL(U32(x_1, x_2)) = x_1
11.73/3.85	   POL(U33(x_1)) = x_1
11.73/3.85	   POL(U41(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U42(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U43(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U44(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U45(x_1, x_2)) = x_1
11.73/3.85	   POL(U46(x_1)) = x_1
11.73/3.85	   POL(U51(x_1, x_2)) = x_1
11.73/3.85	   POL(U52(x_1)) = x_1
11.73/3.85	   POL(U61(x_1, x_2)) = x_1
11.73/3.85	   POL(U62(x_1)) = x_1
11.73/3.85	   POL(U71(x_1)) = x_1
11.73/3.85	   POL(U81(x_1)) = x_1
11.73/3.85	   POL(U91(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U92(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U93(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U94(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U95(x_1, x_2)) = x_1
11.73/3.85	   POL(U96(x_1)) = x_1
11.73/3.85	   POL(cons(x_1, x_2)) = x_1 + x_2
11.73/3.85	   POL(isNat(x_1)) = 0
11.73/3.85	   POL(isNatIList(x_1)) = 0
11.73/3.85	   POL(isNatIListKind(x_1)) = 0
11.73/3.85	   POL(isNatKind(x_1)) = 0
11.73/3.85	   POL(isNatList(x_1)) = 0
11.73/3.85	   POL(length(x_1)) = x_1
11.73/3.85	   POL(nil) = 0
11.73/3.85	   POL(s(x_1)) = x_1
11.73/3.85	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
11.73/3.85	   POL(tt) = 0
11.73/3.85	   POL(zeros) = 1
11.73/3.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
11.73/3.85	
11.73/3.85	   U121(tt, IL) -> U122(isNatIListKind(IL))
11.73/3.85	
11.73/3.85	
11.73/3.85	
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(4)
11.73/3.85	Obligation:
11.73/3.85	Context-sensitive rewrite system:
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.85	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.85	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.85	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.85	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.85	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.85	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.85	   U114(tt, L) -> s(length(L))
11.73/3.85	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.85	   U122(tt) -> nil
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.85	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.85	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.85	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.85	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.85	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.85	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.85	   U32(tt, V) -> U33(isNatList(V))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.85	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.85	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.85	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.85	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.85	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.85	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.85	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.85	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.85	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.85	   length(nil) -> 0
11.73/3.85	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.85	   take(0, IL) -> U121(isNatIList(IL), IL)
11.73/3.85	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.85	
11.73/3.85	The replacement map contains the following entries:
11.73/3.85	
11.73/3.85	zeros: empty set
11.73/3.85	cons: {1}
11.73/3.85	0: empty set
11.73/3.85	U101: {1}
11.73/3.85	tt: empty set
11.73/3.85	U102: {1}
11.73/3.85	isNatKind: empty set
11.73/3.85	U103: {1}
11.73/3.85	isNatIListKind: empty set
11.73/3.85	U104: {1}
11.73/3.85	U105: {1}
11.73/3.85	isNat: empty set
11.73/3.85	U106: {1}
11.73/3.85	isNatIList: empty set
11.73/3.85	U11: {1}
11.73/3.85	U12: {1}
11.73/3.85	U111: {1}
11.73/3.85	U112: {1}
11.73/3.85	U113: {1}
11.73/3.85	U114: {1}
11.73/3.85	s: {1}
11.73/3.85	length: {1}
11.73/3.85	U13: {1}
11.73/3.85	isNatList: empty set
11.73/3.85	U121: {1}
11.73/3.85	U122: {1}
11.73/3.85	nil: empty set
11.73/3.85	U131: {1}
11.73/3.85	U132: {1}
11.73/3.85	U133: {1}
11.73/3.85	U134: {1}
11.73/3.85	U135: {1}
11.73/3.85	U136: {1}
11.73/3.85	take: {1, 2}
11.73/3.85	U21: {1}
11.73/3.85	U22: {1}
11.73/3.85	U23: {1}
11.73/3.85	U31: {1}
11.73/3.85	U32: {1}
11.73/3.85	U33: {1}
11.73/3.85	U41: {1}
11.73/3.85	U42: {1}
11.73/3.85	U43: {1}
11.73/3.85	U44: {1}
11.73/3.85	U45: {1}
11.73/3.85	U46: {1}
11.73/3.85	U51: {1}
11.73/3.85	U52: {1}
11.73/3.85	U61: {1}
11.73/3.85	U62: {1}
11.73/3.85	U71: {1}
11.73/3.85	U81: {1}
11.73/3.85	U91: {1}
11.73/3.85	U92: {1}
11.73/3.85	U93: {1}
11.73/3.85	U94: {1}
11.73/3.85	U95: {1}
11.73/3.85	U96: {1}
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(5) CSRRRRProof (EQUIVALENT)
11.73/3.85	The following CSR is given: Context-sensitive rewrite system:
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.85	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.85	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.85	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.85	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.85	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.85	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.85	   U114(tt, L) -> s(length(L))
11.73/3.85	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.85	   U122(tt) -> nil
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.85	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.85	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.85	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.85	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.85	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.85	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.85	   U32(tt, V) -> U33(isNatList(V))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.85	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.85	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.85	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.85	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.85	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.85	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.85	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.85	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.85	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.85	   length(nil) -> 0
11.73/3.85	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.85	   take(0, IL) -> U121(isNatIList(IL), IL)
11.73/3.85	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.85	
11.73/3.85	The replacement map contains the following entries:
11.73/3.85	
11.73/3.85	zeros: empty set
11.73/3.85	cons: {1}
11.73/3.85	0: empty set
11.73/3.85	U101: {1}
11.73/3.85	tt: empty set
11.73/3.85	U102: {1}
11.73/3.85	isNatKind: empty set
11.73/3.85	U103: {1}
11.73/3.85	isNatIListKind: empty set
11.73/3.85	U104: {1}
11.73/3.85	U105: {1}
11.73/3.85	isNat: empty set
11.73/3.85	U106: {1}
11.73/3.85	isNatIList: empty set
11.73/3.85	U11: {1}
11.73/3.85	U12: {1}
11.73/3.85	U111: {1}
11.73/3.85	U112: {1}
11.73/3.85	U113: {1}
11.73/3.85	U114: {1}
11.73/3.85	s: {1}
11.73/3.85	length: {1}
11.73/3.85	U13: {1}
11.73/3.85	isNatList: empty set
11.73/3.85	U121: {1}
11.73/3.85	U122: {1}
11.73/3.85	nil: empty set
11.73/3.85	U131: {1}
11.73/3.85	U132: {1}
11.73/3.85	U133: {1}
11.73/3.85	U134: {1}
11.73/3.85	U135: {1}
11.73/3.85	U136: {1}
11.73/3.85	take: {1, 2}
11.73/3.85	U21: {1}
11.73/3.85	U22: {1}
11.73/3.85	U23: {1}
11.73/3.85	U31: {1}
11.73/3.85	U32: {1}
11.73/3.85	U33: {1}
11.73/3.85	U41: {1}
11.73/3.85	U42: {1}
11.73/3.85	U43: {1}
11.73/3.85	U44: {1}
11.73/3.85	U45: {1}
11.73/3.85	U46: {1}
11.73/3.85	U51: {1}
11.73/3.85	U52: {1}
11.73/3.85	U61: {1}
11.73/3.85	U62: {1}
11.73/3.85	U71: {1}
11.73/3.85	U81: {1}
11.73/3.85	U91: {1}
11.73/3.85	U92: {1}
11.73/3.85	U93: {1}
11.73/3.85	U94: {1}
11.73/3.85	U95: {1}
11.73/3.85	U96: {1}
11.73/3.85	Used ordering:
11.73/3.85	Polynomial interpretation [POLO]:
11.73/3.85	
11.73/3.85	   POL(0) = 0
11.73/3.85	   POL(U101(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U102(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U103(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U104(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U105(x_1, x_2)) = x_1
11.73/3.85	   POL(U106(x_1)) = x_1
11.73/3.85	   POL(U11(x_1, x_2)) = x_1
11.73/3.85	   POL(U111(x_1, x_2, x_3)) = x_1 + x_2 + x_3
11.73/3.85	   POL(U112(x_1, x_2, x_3)) = x_1 + x_2 + x_3
11.73/3.85	   POL(U113(x_1, x_2, x_3)) = x_1 + x_2 + x_3
11.73/3.85	   POL(U114(x_1, x_2)) = x_1 + x_2
11.73/3.85	   POL(U12(x_1, x_2)) = x_1
11.73/3.85	   POL(U121(x_1, x_2)) = x_1 + x_2
11.73/3.85	   POL(U122(x_1)) = 1 + x_1
11.73/3.85	   POL(U13(x_1)) = x_1
11.73/3.85	   POL(U131(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U132(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U133(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U134(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U135(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U136(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U21(x_1, x_2)) = x_1
11.73/3.85	   POL(U22(x_1, x_2)) = x_1
11.73/3.85	   POL(U23(x_1)) = x_1
11.73/3.85	   POL(U31(x_1, x_2)) = x_1
11.73/3.85	   POL(U32(x_1, x_2)) = x_1
11.73/3.85	   POL(U33(x_1)) = x_1
11.73/3.85	   POL(U41(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U42(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U43(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U44(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U45(x_1, x_2)) = x_1
11.73/3.85	   POL(U46(x_1)) = x_1
11.73/3.85	   POL(U51(x_1, x_2)) = x_1
11.73/3.85	   POL(U52(x_1)) = x_1
11.73/3.85	   POL(U61(x_1, x_2)) = x_1
11.73/3.85	   POL(U62(x_1)) = x_1
11.73/3.85	   POL(U71(x_1)) = x_1
11.73/3.85	   POL(U81(x_1)) = x_1
11.73/3.85	   POL(U91(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U92(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U93(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U94(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U95(x_1, x_2)) = x_1
11.73/3.85	   POL(U96(x_1)) = x_1
11.73/3.85	   POL(cons(x_1, x_2)) = x_1 + x_2
11.73/3.85	   POL(isNat(x_1)) = 1
11.73/3.85	   POL(isNatIList(x_1)) = 1
11.73/3.85	   POL(isNatIListKind(x_1)) = 1
11.73/3.85	   POL(isNatKind(x_1)) = 1
11.73/3.85	   POL(isNatList(x_1)) = 1
11.73/3.85	   POL(length(x_1)) = 1 + x_1
11.73/3.85	   POL(nil) = 0
11.73/3.85	   POL(s(x_1)) = x_1
11.73/3.85	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
11.73/3.85	   POL(tt) = 1
11.73/3.85	   POL(zeros) = 1
11.73/3.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
11.73/3.85	
11.73/3.85	   U122(tt) -> nil
11.73/3.85	   length(nil) -> 0
11.73/3.85	
11.73/3.85	
11.73/3.85	
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(6)
11.73/3.85	Obligation:
11.73/3.85	Context-sensitive rewrite system:
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.85	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.85	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.85	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.85	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.85	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.85	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.85	   U114(tt, L) -> s(length(L))
11.73/3.85	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.85	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.85	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.85	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.85	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.85	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.85	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.85	   U32(tt, V) -> U33(isNatList(V))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.85	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.85	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.85	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.85	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.85	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.85	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.85	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.85	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.85	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.85	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.85	   take(0, IL) -> U121(isNatIList(IL), IL)
11.73/3.85	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.85	
11.73/3.85	The replacement map contains the following entries:
11.73/3.85	
11.73/3.85	zeros: empty set
11.73/3.85	cons: {1}
11.73/3.85	0: empty set
11.73/3.85	U101: {1}
11.73/3.85	tt: empty set
11.73/3.85	U102: {1}
11.73/3.85	isNatKind: empty set
11.73/3.85	U103: {1}
11.73/3.85	isNatIListKind: empty set
11.73/3.85	U104: {1}
11.73/3.85	U105: {1}
11.73/3.85	isNat: empty set
11.73/3.85	U106: {1}
11.73/3.85	isNatIList: empty set
11.73/3.85	U11: {1}
11.73/3.85	U12: {1}
11.73/3.85	U111: {1}
11.73/3.85	U112: {1}
11.73/3.85	U113: {1}
11.73/3.85	U114: {1}
11.73/3.85	s: {1}
11.73/3.85	length: {1}
11.73/3.85	U13: {1}
11.73/3.85	isNatList: empty set
11.73/3.85	U121: {1}
11.73/3.85	nil: empty set
11.73/3.85	U131: {1}
11.73/3.85	U132: {1}
11.73/3.85	U133: {1}
11.73/3.85	U134: {1}
11.73/3.85	U135: {1}
11.73/3.85	U136: {1}
11.73/3.85	take: {1, 2}
11.73/3.85	U21: {1}
11.73/3.85	U22: {1}
11.73/3.85	U23: {1}
11.73/3.85	U31: {1}
11.73/3.85	U32: {1}
11.73/3.85	U33: {1}
11.73/3.85	U41: {1}
11.73/3.85	U42: {1}
11.73/3.85	U43: {1}
11.73/3.85	U44: {1}
11.73/3.85	U45: {1}
11.73/3.85	U46: {1}
11.73/3.85	U51: {1}
11.73/3.85	U52: {1}
11.73/3.85	U61: {1}
11.73/3.85	U62: {1}
11.73/3.85	U71: {1}
11.73/3.85	U81: {1}
11.73/3.85	U91: {1}
11.73/3.85	U92: {1}
11.73/3.85	U93: {1}
11.73/3.85	U94: {1}
11.73/3.85	U95: {1}
11.73/3.85	U96: {1}
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(7) CSRRRRProof (EQUIVALENT)
11.73/3.85	The following CSR is given: Context-sensitive rewrite system:
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.85	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.85	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.85	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.85	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.85	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.85	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.85	   U114(tt, L) -> s(length(L))
11.73/3.85	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.85	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.85	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.85	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.85	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.85	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.85	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.85	   U32(tt, V) -> U33(isNatList(V))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.85	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.85	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.85	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.85	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.85	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.85	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.85	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.85	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.85	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.85	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.85	   take(0, IL) -> U121(isNatIList(IL), IL)
11.73/3.85	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.85	
11.73/3.85	The replacement map contains the following entries:
11.73/3.85	
11.73/3.85	zeros: empty set
11.73/3.85	cons: {1}
11.73/3.85	0: empty set
11.73/3.85	U101: {1}
11.73/3.85	tt: empty set
11.73/3.85	U102: {1}
11.73/3.85	isNatKind: empty set
11.73/3.85	U103: {1}
11.73/3.85	isNatIListKind: empty set
11.73/3.85	U104: {1}
11.73/3.85	U105: {1}
11.73/3.85	isNat: empty set
11.73/3.85	U106: {1}
11.73/3.85	isNatIList: empty set
11.73/3.85	U11: {1}
11.73/3.85	U12: {1}
11.73/3.85	U111: {1}
11.73/3.85	U112: {1}
11.73/3.85	U113: {1}
11.73/3.85	U114: {1}
11.73/3.85	s: {1}
11.73/3.85	length: {1}
11.73/3.85	U13: {1}
11.73/3.85	isNatList: empty set
11.73/3.85	U121: {1}
11.73/3.85	nil: empty set
11.73/3.85	U131: {1}
11.73/3.85	U132: {1}
11.73/3.85	U133: {1}
11.73/3.85	U134: {1}
11.73/3.85	U135: {1}
11.73/3.85	U136: {1}
11.73/3.85	take: {1, 2}
11.73/3.85	U21: {1}
11.73/3.85	U22: {1}
11.73/3.85	U23: {1}
11.73/3.85	U31: {1}
11.73/3.85	U32: {1}
11.73/3.85	U33: {1}
11.73/3.85	U41: {1}
11.73/3.85	U42: {1}
11.73/3.85	U43: {1}
11.73/3.85	U44: {1}
11.73/3.85	U45: {1}
11.73/3.85	U46: {1}
11.73/3.85	U51: {1}
11.73/3.85	U52: {1}
11.73/3.85	U61: {1}
11.73/3.85	U62: {1}
11.73/3.85	U71: {1}
11.73/3.85	U81: {1}
11.73/3.85	U91: {1}
11.73/3.85	U92: {1}
11.73/3.85	U93: {1}
11.73/3.85	U94: {1}
11.73/3.85	U95: {1}
11.73/3.85	U96: {1}
11.73/3.85	Used ordering:
11.73/3.85	Polynomial interpretation [POLO]:
11.73/3.85	
11.73/3.85	   POL(0) = 0
11.73/3.85	   POL(U101(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U102(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U103(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U104(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U105(x_1, x_2)) = x_1
11.73/3.85	   POL(U106(x_1)) = x_1
11.73/3.85	   POL(U11(x_1, x_2)) = x_1
11.73/3.85	   POL(U111(x_1, x_2, x_3)) = x_1 + x_2 + x_3
11.73/3.85	   POL(U112(x_1, x_2, x_3)) = x_1 + x_2 + x_3
11.73/3.85	   POL(U113(x_1, x_2, x_3)) = x_1 + x_2 + x_3
11.73/3.85	   POL(U114(x_1, x_2)) = x_1 + x_2
11.73/3.85	   POL(U12(x_1, x_2)) = x_1
11.73/3.85	   POL(U121(x_1, x_2)) = x_1 + x_2
11.73/3.85	   POL(U13(x_1)) = x_1
11.73/3.85	   POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
11.73/3.85	   POL(U21(x_1, x_2)) = x_1
11.73/3.85	   POL(U22(x_1, x_2)) = x_1
11.73/3.85	   POL(U23(x_1)) = x_1
11.73/3.85	   POL(U31(x_1, x_2)) = x_1
11.73/3.85	   POL(U32(x_1, x_2)) = x_1
11.73/3.85	   POL(U33(x_1)) = x_1
11.73/3.85	   POL(U41(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U42(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U43(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U44(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U45(x_1, x_2)) = x_1
11.73/3.85	   POL(U46(x_1)) = x_1
11.73/3.85	   POL(U51(x_1, x_2)) = x_1
11.73/3.85	   POL(U52(x_1)) = x_1
11.73/3.85	   POL(U61(x_1, x_2)) = x_1
11.73/3.85	   POL(U62(x_1)) = x_1
11.73/3.85	   POL(U71(x_1)) = x_1
11.73/3.85	   POL(U81(x_1)) = x_1
11.73/3.85	   POL(U91(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U92(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U93(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U94(x_1, x_2, x_3)) = x_1
11.73/3.85	   POL(U95(x_1, x_2)) = x_1
11.73/3.85	   POL(U96(x_1)) = x_1
11.73/3.85	   POL(cons(x_1, x_2)) = x_1 + x_2
11.73/3.85	   POL(isNat(x_1)) = 0
11.73/3.85	   POL(isNatIList(x_1)) = 0
11.73/3.85	   POL(isNatIListKind(x_1)) = 0
11.73/3.85	   POL(isNatKind(x_1)) = 0
11.73/3.85	   POL(isNatList(x_1)) = 0
11.73/3.85	   POL(length(x_1)) = x_1
11.73/3.85	   POL(nil) = 1
11.73/3.85	   POL(s(x_1)) = x_1
11.73/3.85	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
11.73/3.85	   POL(tt) = 0
11.73/3.85	   POL(zeros) = 1
11.73/3.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
11.73/3.85	
11.73/3.85	   take(0, IL) -> U121(isNatIList(IL), IL)
11.73/3.85	
11.73/3.85	
11.73/3.85	
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(8)
11.73/3.85	Obligation:
11.73/3.85	Context-sensitive rewrite system:
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.85	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.85	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.85	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.85	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.85	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.85	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.85	   U114(tt, L) -> s(length(L))
11.73/3.85	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.85	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.85	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.85	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.85	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.85	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.85	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.85	   U32(tt, V) -> U33(isNatList(V))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.85	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.85	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.85	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.85	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.85	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.85	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.85	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.85	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.85	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.85	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.85	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.85	
11.73/3.85	The replacement map contains the following entries:
11.73/3.85	
11.73/3.85	zeros: empty set
11.73/3.85	cons: {1}
11.73/3.85	0: empty set
11.73/3.85	U101: {1}
11.73/3.85	tt: empty set
11.73/3.85	U102: {1}
11.73/3.85	isNatKind: empty set
11.73/3.85	U103: {1}
11.73/3.85	isNatIListKind: empty set
11.73/3.85	U104: {1}
11.73/3.85	U105: {1}
11.73/3.85	isNat: empty set
11.73/3.85	U106: {1}
11.73/3.85	isNatIList: empty set
11.73/3.85	U11: {1}
11.73/3.85	U12: {1}
11.73/3.85	U111: {1}
11.73/3.85	U112: {1}
11.73/3.85	U113: {1}
11.73/3.85	U114: {1}
11.73/3.85	s: {1}
11.73/3.85	length: {1}
11.73/3.85	U13: {1}
11.73/3.85	isNatList: empty set
11.73/3.85	nil: empty set
11.73/3.85	U131: {1}
11.73/3.85	U132: {1}
11.73/3.85	U133: {1}
11.73/3.85	U134: {1}
11.73/3.85	U135: {1}
11.73/3.85	U136: {1}
11.73/3.85	take: {1, 2}
11.73/3.85	U21: {1}
11.73/3.85	U22: {1}
11.73/3.85	U23: {1}
11.73/3.85	U31: {1}
11.73/3.85	U32: {1}
11.73/3.85	U33: {1}
11.73/3.85	U41: {1}
11.73/3.85	U42: {1}
11.73/3.85	U43: {1}
11.73/3.85	U44: {1}
11.73/3.85	U45: {1}
11.73/3.85	U46: {1}
11.73/3.85	U51: {1}
11.73/3.85	U52: {1}
11.73/3.85	U61: {1}
11.73/3.85	U62: {1}
11.73/3.85	U71: {1}
11.73/3.85	U81: {1}
11.73/3.85	U91: {1}
11.73/3.85	U92: {1}
11.73/3.85	U93: {1}
11.73/3.85	U94: {1}
11.73/3.85	U95: {1}
11.73/3.85	U96: {1}
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(9) CSDependencyPairsProof (EQUIVALENT)
11.73/3.85	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(10)
11.73/3.85	Obligation:
11.73/3.85	Q-restricted context-sensitive dependency pair problem:
11.73/3.85	The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, U106'_1, LENGTH_1, U13'_1, U23'_1, U33'_1, U46'_1, U52'_1, U62'_1, U96'_1, U71'_1, U81'_1, TAKE_2} are replacing on all positions.
11.73/3.85	For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U102'_3, U101'_3, U103'_3, U104'_3, U105'_2, U12'_2, U11'_2, U112'_3, U111'_3, U113'_3, U114'_2, U132'_4, U131'_4, U133'_4, U134'_4, U135'_4, U136'_4, U22'_2, U21'_2, U32'_2, U31'_2, U42'_3, U41'_3, U43'_3, U44'_3, U45'_2, U51'_2, U61'_2, U92'_3, U91'_3, U93'_3, U94'_3, U95'_2} we have mu(f) = {1}.
11.73/3.85	The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, ISNATKIND_1, ISNATILISTKIND_1, ISNAT_1, ISNATILIST_1, ISNATLIST_1, U_1} are not replacing on any position.
11.73/3.85	
11.73/3.85	The ordinary context-sensitive dependency pairs DP_o are:
11.73/3.85	   U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2)
11.73/3.85	   U101'(tt, V1, V2) -> ISNATKIND(V1)
11.73/3.85	   U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2)
11.73/3.85	   U102'(tt, V1, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103'(tt, V1, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   U104'(tt, V1, V2) -> U105'(isNat(V1), V2)
11.73/3.85	   U104'(tt, V1, V2) -> ISNAT(V1)
11.73/3.85	   U105'(tt, V2) -> U106'(isNatIList(V2))
11.73/3.85	   U105'(tt, V2) -> ISNATILIST(V2)
11.73/3.85	   U11'(tt, V1) -> U12'(isNatIListKind(V1), V1)
11.73/3.85	   U11'(tt, V1) -> ISNATILISTKIND(V1)
11.73/3.85	   U111'(tt, L, N) -> U112'(isNatIListKind(L), L, N)
11.73/3.85	   U111'(tt, L, N) -> ISNATILISTKIND(L)
11.73/3.85	   U112'(tt, L, N) -> U113'(isNat(N), L, N)
11.73/3.85	   U112'(tt, L, N) -> ISNAT(N)
11.73/3.85	   U113'(tt, L, N) -> U114'(isNatKind(N), L)
11.73/3.85	   U113'(tt, L, N) -> ISNATKIND(N)
11.73/3.85	   U114'(tt, L) -> LENGTH(L)
11.73/3.85	   U12'(tt, V1) -> U13'(isNatList(V1))
11.73/3.85	   U12'(tt, V1) -> ISNATLIST(V1)
11.73/3.85	   U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U131'(tt, IL, M, N) -> ISNATILISTKIND(IL)
11.73/3.85	   U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N)
11.73/3.85	   U132'(tt, IL, M, N) -> ISNAT(M)
11.73/3.85	   U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N)
11.73/3.85	   U133'(tt, IL, M, N) -> ISNATKIND(M)
11.73/3.85	   U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N)
11.73/3.85	   U134'(tt, IL, M, N) -> ISNAT(N)
11.73/3.85	   U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N)
11.73/3.85	   U135'(tt, IL, M, N) -> ISNATKIND(N)
11.73/3.85	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
11.73/3.85	   U21'(tt, V1) -> ISNATKIND(V1)
11.73/3.85	   U22'(tt, V1) -> U23'(isNat(V1))
11.73/3.85	   U22'(tt, V1) -> ISNAT(V1)
11.73/3.85	   U31'(tt, V) -> U32'(isNatIListKind(V), V)
11.73/3.85	   U31'(tt, V) -> ISNATILISTKIND(V)
11.73/3.85	   U32'(tt, V) -> U33'(isNatList(V))
11.73/3.85	   U32'(tt, V) -> ISNATLIST(V)
11.73/3.85	   U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2)
11.73/3.85	   U41'(tt, V1, V2) -> ISNATKIND(V1)
11.73/3.85	   U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2)
11.73/3.85	   U42'(tt, V1, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43'(tt, V1, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   U44'(tt, V1, V2) -> U45'(isNat(V1), V2)
11.73/3.85	   U44'(tt, V1, V2) -> ISNAT(V1)
11.73/3.85	   U45'(tt, V2) -> U46'(isNatIList(V2))
11.73/3.85	   U45'(tt, V2) -> ISNATILIST(V2)
11.73/3.85	   U51'(tt, V2) -> U52'(isNatIListKind(V2))
11.73/3.85	   U51'(tt, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   U61'(tt, V2) -> U62'(isNatIListKind(V2))
11.73/3.85	   U61'(tt, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2)
11.73/3.85	   U91'(tt, V1, V2) -> ISNATKIND(V1)
11.73/3.85	   U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2)
11.73/3.85	   U92'(tt, V1, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93'(tt, V1, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   U94'(tt, V1, V2) -> U95'(isNat(V1), V2)
11.73/3.85	   U94'(tt, V1, V2) -> ISNAT(V1)
11.73/3.85	   U95'(tt, V2) -> U96'(isNatList(V2))
11.73/3.85	   U95'(tt, V2) -> ISNATLIST(V2)
11.73/3.85	   ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1)
11.73/3.85	   ISNAT(length(V1)) -> ISNATILISTKIND(V1)
11.73/3.85	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.73/3.85	   ISNAT(s(V1)) -> ISNATKIND(V1)
11.73/3.85	   ISNATILIST(V) -> U31'(isNatIListKind(V), V)
11.73/3.85	   ISNATILIST(V) -> ISNATILISTKIND(V)
11.73/3.85	   ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2)
11.73/3.85	   ISNATILIST(cons(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	   ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2)
11.73/3.85	   ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	   ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2)
11.73/3.85	   ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	   ISNATKIND(length(V1)) -> U71'(isNatIListKind(V1))
11.73/3.85	   ISNATKIND(length(V1)) -> ISNATILISTKIND(V1)
11.73/3.85	   ISNATKIND(s(V1)) -> U81'(isNatKind(V1))
11.73/3.85	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
11.73/3.85	   ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2)
11.73/3.85	   ISNATLIST(cons(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	   ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2)
11.73/3.85	   ISNATLIST(take(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	   LENGTH(cons(N, L)) -> U111'(isNatList(L), L, N)
11.73/3.85	   LENGTH(cons(N, L)) -> ISNATLIST(L)
11.73/3.85	   TAKE(s(M), cons(N, IL)) -> U131'(isNatIList(IL), IL, M, N)
11.73/3.85	   TAKE(s(M), cons(N, IL)) -> ISNATILIST(IL)
11.73/3.85	
11.73/3.85	The collapsing dependency pairs are DP_c:
11.73/3.85	   U114'(tt, L) -> L
11.73/3.85	   U136'(tt, IL, M, N) -> N
11.73/3.85	
11.73/3.85	
11.73/3.85	The hidden terms of R are:
11.73/3.85	
11.73/3.85	   zeros
11.73/3.85	   take(x0, x1)
11.73/3.85	
11.73/3.85	Every hiding context is built from:
11.73/3.85	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@497f43e6
11.73/3.85	
11.73/3.85	Hence, the new unhiding pairs DP_u are :
11.73/3.85	   U114'(tt, L) -> U(L)
11.73/3.85	   U136'(tt, IL, M, N) -> U(N)
11.73/3.85	   U(take(x_0, x_1)) -> U(x_0)
11.73/3.85	   U(take(x_0, x_1)) -> U(x_1)
11.73/3.85	   U(zeros) -> ZEROS
11.73/3.85	   U(take(x0, x1)) -> TAKE(x0, x1)
11.73/3.85	
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.85	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.85	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.85	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.85	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.85	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.85	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.85	   U114(tt, L) -> s(length(L))
11.73/3.85	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.85	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.85	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.85	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.85	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.85	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.85	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.85	   U32(tt, V) -> U33(isNatList(V))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.85	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.85	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.85	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.85	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.85	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.85	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.85	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.85	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.85	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.85	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.85	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.85	
11.73/3.85	Q is empty.
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(11) QCSDependencyGraphProof (EQUIVALENT)
11.73/3.85	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 40 less nodes.
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(12)
11.73/3.85	Complex Obligation (AND)
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(13)
11.73/3.85	Obligation:
11.73/3.85	Q-restricted context-sensitive dependency pair problem:
11.73/3.85	The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1} are replacing on all positions.
11.73/3.85	For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U51'_2, U61'_2} we have mu(f) = {1}.
11.73/3.85	The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, ISNATILISTKIND_1, ISNATKIND_1} are not replacing on any position.
11.73/3.85	
11.73/3.85	The TRS P consists of the following rules:
11.73/3.85	
11.73/3.85	   ISNATKIND(length(V1)) -> ISNATILISTKIND(V1)
11.73/3.85	   ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2)
11.73/3.85	   U51'(tt, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
11.73/3.85	   ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2)
11.73/3.85	   U61'(tt, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.85	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.85	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.85	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.85	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.85	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.85	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.85	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.85	   U114(tt, L) -> s(length(L))
11.73/3.85	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.85	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.85	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.85	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.85	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.85	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.85	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.85	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.85	   U32(tt, V) -> U33(isNatList(V))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.85	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.85	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.85	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.85	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.85	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.85	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.85	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.85	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.85	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.85	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.85	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.85	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.85	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.85	
11.73/3.85	Q is empty.
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(14) QCSUsableRulesProof (EQUIVALENT)
11.73/3.85	The following rules are not useable [DA_EMMES] and can be deleted:
11.73/3.85	
11.73/3.85	   zeros -> cons(0, zeros)
11.73/3.85	   U101(tt, x0, x1) -> U102(isNatKind(x0), x0, x1)
11.73/3.85	   U102(tt, x0, x1) -> U103(isNatIListKind(x1), x0, x1)
11.73/3.85	   U103(tt, x0, x1) -> U104(isNatIListKind(x1), x0, x1)
11.73/3.85	   U104(tt, x0, x1) -> U105(isNat(x0), x1)
11.73/3.85	   U105(tt, x0) -> U106(isNatIList(x0))
11.73/3.85	   U106(tt) -> tt
11.73/3.85	   U11(tt, x0) -> U12(isNatIListKind(x0), x0)
11.73/3.85	   U111(tt, x0, x1) -> U112(isNatIListKind(x0), x0, x1)
11.73/3.85	   U112(tt, x0, x1) -> U113(isNat(x1), x0, x1)
11.73/3.85	   U113(tt, x0, x1) -> U114(isNatKind(x1), x0)
11.73/3.85	   U114(tt, x0) -> s(length(x0))
11.73/3.85	   U12(tt, x0) -> U13(isNatList(x0))
11.73/3.85	   U13(tt) -> tt
11.73/3.85	   U131(tt, x0, x1, x2) -> U132(isNatIListKind(x0), x0, x1, x2)
11.73/3.85	   U132(tt, x0, x1, x2) -> U133(isNat(x1), x0, x1, x2)
11.73/3.85	   U133(tt, x0, x1, x2) -> U134(isNatKind(x1), x0, x1, x2)
11.73/3.85	   U134(tt, x0, x1, x2) -> U135(isNat(x2), x0, x1, x2)
11.73/3.85	   U135(tt, x0, x1, x2) -> U136(isNatKind(x2), x0, x1, x2)
11.73/3.85	   U136(tt, x0, x1, x2) -> cons(x2, take(x1, x0))
11.73/3.85	   U21(tt, x0) -> U22(isNatKind(x0), x0)
11.73/3.85	   U22(tt, x0) -> U23(isNat(x0))
11.73/3.85	   U23(tt) -> tt
11.73/3.85	   U31(tt, x0) -> U32(isNatIListKind(x0), x0)
11.73/3.85	   U32(tt, x0) -> U33(isNatList(x0))
11.73/3.85	   U33(tt) -> tt
11.73/3.85	   U41(tt, x0, x1) -> U42(isNatKind(x0), x0, x1)
11.73/3.85	   U42(tt, x0, x1) -> U43(isNatIListKind(x1), x0, x1)
11.73/3.85	   U43(tt, x0, x1) -> U44(isNatIListKind(x1), x0, x1)
11.73/3.85	   U44(tt, x0, x1) -> U45(isNat(x0), x1)
11.73/3.85	   U45(tt, x0) -> U46(isNatIList(x0))
11.73/3.85	   U46(tt) -> tt
11.73/3.85	   U91(tt, x0, x1) -> U92(isNatKind(x0), x0, x1)
11.73/3.85	   U92(tt, x0, x1) -> U93(isNatIListKind(x1), x0, x1)
11.73/3.85	   U93(tt, x0, x1) -> U94(isNatIListKind(x1), x0, x1)
11.73/3.85	   U94(tt, x0, x1) -> U95(isNat(x0), x1)
11.73/3.85	   U95(tt, x0) -> U96(isNatList(x0))
11.73/3.85	   U96(tt) -> tt
11.73/3.85	   isNat(0) -> tt
11.73/3.85	   isNat(length(x0)) -> U11(isNatIListKind(x0), x0)
11.73/3.85	   isNat(s(x0)) -> U21(isNatKind(x0), x0)
11.73/3.85	   isNatIList(x0) -> U31(isNatIListKind(x0), x0)
11.73/3.85	   isNatIList(zeros) -> tt
11.73/3.85	   isNatIList(cons(x0, x1)) -> U41(isNatKind(x0), x0, x1)
11.73/3.85	   isNatList(nil) -> tt
11.73/3.85	   isNatList(cons(x0, x1)) -> U91(isNatKind(x0), x0, x1)
11.73/3.85	   isNatList(take(x0, x1)) -> U101(isNatKind(x0), x0, x1)
11.73/3.85	   length(cons(x0, x1)) -> U111(isNatList(x1), x1, x0)
11.73/3.85	   take(s(x0), cons(x1, x2)) -> U131(isNatIList(x2), x2, x0, x1)
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(15)
11.73/3.85	Obligation:
11.73/3.85	Q-restricted context-sensitive dependency pair problem:
11.73/3.85	The symbols in {length_1, U71_1, s_1, U81_1, U52_1, take_2, U62_1} are replacing on all positions.
11.73/3.85	For all symbols f in {cons_2, U51_2, U61_2, U51'_2, U61'_2} we have mu(f) = {1}.
11.73/3.85	The symbols in {isNatKind_1, isNatIListKind_1, ISNATILISTKIND_1, ISNATKIND_1} are not replacing on any position.
11.73/3.85	
11.73/3.85	The TRS P consists of the following rules:
11.73/3.85	
11.73/3.85	   ISNATKIND(length(V1)) -> ISNATILISTKIND(V1)
11.73/3.85	   ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2)
11.73/3.85	   U51'(tt, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
11.73/3.85	   ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2)
11.73/3.85	   U61'(tt, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	
11.73/3.85	Q is empty.
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(16) QCSDPMuMonotonicPoloProof (EQUIVALENT)
11.73/3.85	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.
11.73/3.85	
11.73/3.85	Strictly oriented dependency pairs:
11.73/3.85	
11.73/3.85	   ISNATKIND(length(V1)) -> ISNATILISTKIND(V1)
11.73/3.85	   ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2)
11.73/3.85	   U51'(tt, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	   ISNATKIND(s(V1)) -> ISNATKIND(V1)
11.73/3.85	   ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2)
11.73/3.85	   U61'(tt, V2) -> ISNATILISTKIND(V2)
11.73/3.85	   ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1)
11.73/3.85	
11.73/3.85	Strictly oriented rules of the TRS R:
11.73/3.85	
11.73/3.85	   isNatKind(0) -> tt
11.73/3.85	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.85	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.85	   U81(tt) -> tt
11.73/3.85	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.85	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.85	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.85	   U62(tt) -> tt
11.73/3.85	   U52(tt) -> tt
11.73/3.85	   U71(tt) -> tt
11.73/3.85	
11.73/3.85	Used ordering: POLO with Polynomial interpretation [POLO]:
11.73/3.85	
11.73/3.85	   POL(0) = 2
11.73/3.85	   POL(ISNATILISTKIND(x_1)) = 1 + 2*x_1
11.73/3.85	   POL(ISNATKIND(x_1)) = 2 + 2*x_1
11.73/3.85	   POL(U51(x_1, x_2)) = 2 + x_1 + 2*x_2
11.73/3.85	   POL(U51'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
11.73/3.85	   POL(U52(x_1)) = 1 + 2*x_1
11.73/3.85	   POL(U61(x_1, x_2)) = x_1 + 2*x_2
11.73/3.85	   POL(U61'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
11.73/3.85	   POL(U62(x_1)) = 1 + 2*x_1
11.73/3.85	   POL(U71(x_1)) = 1 + 2*x_1
11.73/3.85	   POL(U81(x_1)) = 1 + 2*x_1
11.73/3.85	   POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
11.73/3.85	   POL(isNatIListKind(x_1)) = x_1
11.73/3.85	   POL(isNatKind(x_1)) = 2*x_1
11.73/3.85	   POL(length(x_1)) = 2 + 2*x_1
11.73/3.85	   POL(nil) = 2
11.73/3.85	   POL(s(x_1)) = 2 + 2*x_1
11.73/3.85	   POL(take(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
11.73/3.85	   POL(tt) = 2
11.73/3.85	   POL(zeros) = 2
11.73/3.85	
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(17)
11.73/3.85	Obligation:
11.73/3.85	Q-restricted context-sensitive dependency pair problem:
11.73/3.85	For all symbols f in {cons_2, U51_2} we have mu(f) = {1}.
11.73/3.85	The symbols in {isNatIListKind_1, isNatKind_1} are not replacing on any position.
11.73/3.85	
11.73/3.85	The TRS P consists of the following rules:
11.73/3.85	none
11.73/3.85	
11.73/3.85	The TRS R consists of the following rules:
11.73/3.85	
11.73/3.85	   isNatIListKind(nil) -> tt
11.73/3.85	   isNatIListKind(zeros) -> tt
11.73/3.85	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.85	
11.73/3.85	Q is empty.
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(18) PIsEmptyProof (EQUIVALENT)
11.73/3.85	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
11.73/3.85	----------------------------------------
11.73/3.85	
11.73/3.85	(19)
11.73/3.85	YES
11.73/3.85	
11.73/3.85	----------------------------------------
11.73/3.86	
11.73/3.86	(20)
11.73/3.86	Obligation:
11.73/3.86	Q-restricted context-sensitive dependency pair problem:
11.73/3.86	The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1} are replacing on all positions.
11.73/3.86	For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U103'_3, U102'_3, U104'_3, U105'_2, U31'_2, U32'_2, U91'_3, U92'_3, U93'_3, U94'_3, U95'_2, U101'_3, U11'_2, U12'_2, U21'_2, U22'_2, U41'_3, U42'_3, U43'_3, U44'_3, U45'_2} we have mu(f) = {1}.
11.73/3.86	The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1} are not replacing on any position.
11.73/3.86	
11.73/3.86	The TRS P consists of the following rules:
11.73/3.86	
11.73/3.86	   U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104'(tt, V1, V2) -> U105'(isNat(V1), V2)
11.73/3.86	   U105'(tt, V2) -> ISNATILIST(V2)
11.73/3.86	   ISNATILIST(V) -> U31'(isNatIListKind(V), V)
11.73/3.86	   U31'(tt, V) -> U32'(isNatIListKind(V), V)
11.73/3.86	   U32'(tt, V) -> ISNATLIST(V)
11.73/3.86	   ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2)
11.73/3.86	   U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2)
11.73/3.86	   U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94'(tt, V1, V2) -> U95'(isNat(V1), V2)
11.73/3.86	   U95'(tt, V2) -> ISNATLIST(V2)
11.73/3.86	   ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2)
11.73/3.86	   U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2)
11.73/3.86	   U94'(tt, V1, V2) -> ISNAT(V1)
11.73/3.86	   ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1)
11.73/3.86	   U11'(tt, V1) -> U12'(isNatIListKind(V1), V1)
11.73/3.86	   U12'(tt, V1) -> ISNATLIST(V1)
11.73/3.86	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.73/3.86	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
11.73/3.86	   U22'(tt, V1) -> ISNAT(V1)
11.73/3.86	   ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2)
11.73/3.86	   U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2)
11.73/3.86	   U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44'(tt, V1, V2) -> U45'(isNat(V1), V2)
11.73/3.86	   U45'(tt, V2) -> ISNATILIST(V2)
11.73/3.86	   U44'(tt, V1, V2) -> ISNAT(V1)
11.73/3.86	   U104'(tt, V1, V2) -> ISNAT(V1)
11.73/3.86	
11.73/3.86	The TRS R consists of the following rules:
11.73/3.86	
11.73/3.86	   zeros -> cons(0, zeros)
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.86	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.86	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.86	   U114(tt, L) -> s(length(L))
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   U13(tt) -> tt
11.73/3.86	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.86	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.86	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.86	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.86	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.86	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.86	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.86	
11.73/3.86	Q is empty.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(21) QCSUsableRulesProof (EQUIVALENT)
11.73/3.86	The following rules are not useable [DA_EMMES] and can be deleted:
11.73/3.86	
11.73/3.86	   zeros -> cons(0, zeros)
11.73/3.86	   U111(tt, x0, x1) -> U112(isNatIListKind(x0), x0, x1)
11.73/3.86	   U112(tt, x0, x1) -> U113(isNat(x1), x0, x1)
11.73/3.86	   U113(tt, x0, x1) -> U114(isNatKind(x1), x0)
11.73/3.86	   U114(tt, x0) -> s(length(x0))
11.73/3.86	   U131(tt, x0, x1, x2) -> U132(isNatIListKind(x0), x0, x1, x2)
11.73/3.86	   U132(tt, x0, x1, x2) -> U133(isNat(x1), x0, x1, x2)
11.73/3.86	   U133(tt, x0, x1, x2) -> U134(isNatKind(x1), x0, x1, x2)
11.73/3.86	   U134(tt, x0, x1, x2) -> U135(isNat(x2), x0, x1, x2)
11.73/3.86	   U135(tt, x0, x1, x2) -> U136(isNatKind(x2), x0, x1, x2)
11.73/3.86	   U136(tt, x0, x1, x2) -> cons(x2, take(x1, x0))
11.73/3.86	   length(cons(x0, x1)) -> U111(isNatList(x1), x1, x0)
11.73/3.86	   take(s(x0), cons(x1, x2)) -> U131(isNatIList(x2), x2, x0, x1)
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(22)
11.73/3.86	Obligation:
11.73/3.86	Q-restricted context-sensitive dependency pair problem:
11.73/3.86	The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions.
11.73/3.86	For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U103'_3, U102'_3, U104'_3, U105'_2, U31'_2, U32'_2, U91'_3, U92'_3, U93'_3, U94'_3, U95'_2, U101'_3, U11'_2, U12'_2, U21'_2, U22'_2, U41'_3, U42'_3, U43'_3, U44'_3, U45'_2} we have mu(f) = {1}.
11.73/3.86	The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1} are not replacing on any position.
11.73/3.86	
11.73/3.86	The TRS P consists of the following rules:
11.73/3.86	
11.73/3.86	   U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104'(tt, V1, V2) -> U105'(isNat(V1), V2)
11.73/3.86	   U105'(tt, V2) -> ISNATILIST(V2)
11.73/3.86	   ISNATILIST(V) -> U31'(isNatIListKind(V), V)
11.73/3.86	   U31'(tt, V) -> U32'(isNatIListKind(V), V)
11.73/3.86	   U32'(tt, V) -> ISNATLIST(V)
11.73/3.86	   ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2)
11.73/3.86	   U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2)
11.73/3.86	   U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94'(tt, V1, V2) -> U95'(isNat(V1), V2)
11.73/3.86	   U95'(tt, V2) -> ISNATLIST(V2)
11.73/3.86	   ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2)
11.73/3.86	   U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2)
11.73/3.86	   U94'(tt, V1, V2) -> ISNAT(V1)
11.73/3.86	   ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1)
11.73/3.86	   U11'(tt, V1) -> U12'(isNatIListKind(V1), V1)
11.73/3.86	   U12'(tt, V1) -> ISNATLIST(V1)
11.73/3.86	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.73/3.86	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
11.73/3.86	   U22'(tt, V1) -> ISNAT(V1)
11.73/3.86	   ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2)
11.73/3.86	   U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2)
11.73/3.86	   U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44'(tt, V1, V2) -> U45'(isNat(V1), V2)
11.73/3.86	   U45'(tt, V2) -> ISNATILIST(V2)
11.73/3.86	   U44'(tt, V1, V2) -> ISNAT(V1)
11.73/3.86	   U104'(tt, V1, V2) -> ISNAT(V1)
11.73/3.86	
11.73/3.86	The TRS R consists of the following rules:
11.73/3.86	
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   U13(tt) -> tt
11.73/3.86	
11.73/3.86	Q is empty.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(23) QCSDPMuMonotonicPoloProof (EQUIVALENT)
11.73/3.86	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.
11.73/3.86	
11.73/3.86	Strictly oriented dependency pairs:
11.73/3.86	
11.73/3.86	   U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2)
11.73/3.86	   ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2)
11.73/3.86	   U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U95'(tt, V2) -> ISNATLIST(V2)
11.73/3.86	   ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2)
11.73/3.86	   U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2)
11.73/3.86	   U94'(tt, V1, V2) -> ISNAT(V1)
11.73/3.86	   ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1)
11.73/3.86	   U12'(tt, V1) -> ISNATLIST(V1)
11.73/3.86	   ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2)
11.73/3.86	   U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U45'(tt, V2) -> ISNATILIST(V2)
11.73/3.86	   U44'(tt, V1, V2) -> ISNAT(V1)
11.73/3.86	
11.73/3.86	
11.73/3.86	Used ordering: POLO with Polynomial interpretation [POLO]:
11.73/3.86	
11.73/3.86	   POL(0) = 0
11.73/3.86	   POL(ISNAT(x_1)) = x_1
11.73/3.86	   POL(ISNATILIST(x_1)) = 2*x_1
11.73/3.86	   POL(ISNATLIST(x_1)) = 2*x_1
11.73/3.86	   POL(U101(x_1, x_2, x_3)) = x_1
11.73/3.86	   POL(U101'(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U102(x_1, x_2, x_3)) = 2*x_1
11.73/3.86	   POL(U102'(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U103(x_1, x_2, x_3)) = 2*x_1
11.73/3.86	   POL(U103'(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U104(x_1, x_2, x_3)) = x_1
11.73/3.86	   POL(U104'(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U105(x_1, x_2)) = 2*x_1
11.73/3.86	   POL(U105'(x_1, x_2)) = 2*x_1 + 2*x_2
11.73/3.86	   POL(U106(x_1)) = x_1
11.73/3.86	   POL(U11(x_1, x_2)) = x_1
11.73/3.86	   POL(U11'(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
11.73/3.86	   POL(U12(x_1, x_2)) = 2*x_1
11.73/3.86	   POL(U12'(x_1, x_2)) = 1 + x_1 + 2*x_2
11.73/3.86	   POL(U13(x_1)) = 2*x_1
11.73/3.86	   POL(U21(x_1, x_2)) = x_1
11.73/3.86	   POL(U21'(x_1, x_2)) = x_1 + x_2
11.73/3.86	   POL(U22(x_1, x_2)) = x_1
11.73/3.86	   POL(U22'(x_1, x_2)) = x_1 + x_2
11.73/3.86	   POL(U23(x_1)) = 2*x_1
11.73/3.86	   POL(U31(x_1, x_2)) = 2*x_1
11.73/3.86	   POL(U31'(x_1, x_2)) = 2*x_1 + 2*x_2
11.73/3.86	   POL(U32(x_1, x_2)) = x_1
11.73/3.86	   POL(U32'(x_1, x_2)) = 2*x_1 + 2*x_2
11.73/3.86	   POL(U33(x_1)) = 2*x_1
11.73/3.86	   POL(U41(x_1, x_2, x_3)) = x_1
11.73/3.86	   POL(U41'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3
11.73/3.86	   POL(U42(x_1, x_2, x_3)) = 2*x_1
11.73/3.86	   POL(U42'(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U43(x_1, x_2, x_3)) = x_1
11.73/3.86	   POL(U43'(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U44(x_1, x_2, x_3)) = 2*x_1
11.73/3.86	   POL(U44'(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U45(x_1, x_2)) = x_1
11.73/3.86	   POL(U45'(x_1, x_2)) = 1 + x_1 + 2*x_2
11.73/3.86	   POL(U46(x_1)) = 2*x_1
11.73/3.86	   POL(U51(x_1, x_2)) = 2*x_1
11.73/3.86	   POL(U52(x_1)) = x_1
11.73/3.86	   POL(U61(x_1, x_2)) = 2*x_1
11.73/3.86	   POL(U62(x_1)) = x_1
11.73/3.86	   POL(U71(x_1)) = 2*x_1
11.73/3.86	   POL(U81(x_1)) = 2*x_1
11.73/3.86	   POL(U91(x_1, x_2, x_3)) = 2*x_1
11.73/3.86	   POL(U91'(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U92(x_1, x_2, x_3)) = x_1
11.73/3.86	   POL(U92'(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U93(x_1, x_2, x_3)) = x_1
11.73/3.86	   POL(U93'(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U94(x_1, x_2, x_3)) = x_1
11.73/3.86	   POL(U94'(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3
11.73/3.86	   POL(U95(x_1, x_2)) = 2*x_1
11.73/3.86	   POL(U95'(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
11.73/3.86	   POL(U96(x_1)) = x_1
11.73/3.86	   POL(cons(x_1, x_2)) = 2 + x_1 + 2*x_2
11.73/3.86	   POL(isNat(x_1)) = 0
11.73/3.86	   POL(isNatIList(x_1)) = 0
11.73/3.86	   POL(isNatIListKind(x_1)) = 0
11.73/3.86	   POL(isNatKind(x_1)) = 0
11.73/3.86	   POL(isNatList(x_1)) = 0
11.73/3.86	   POL(length(x_1)) = 2 + 2*x_1
11.73/3.86	   POL(nil) = 0
11.73/3.86	   POL(s(x_1)) = 2*x_1
11.73/3.86	   POL(take(x_1, x_2)) = 2 + 2*x_1 + x_2
11.73/3.86	   POL(tt) = 0
11.73/3.86	   POL(zeros) = 0
11.73/3.86	
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(24)
11.73/3.86	Obligation:
11.73/3.86	Q-restricted context-sensitive dependency pair problem:
11.73/3.86	The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions.
11.73/3.86	For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U104'_3, U103'_3, U105'_2, U31'_2, U32'_2, U92'_3, U91'_3, U94'_3, U93'_3, U95'_2, U12'_2, U11'_2, U21'_2, U22'_2, U42'_3, U41'_3, U44'_3, U43'_3, U45'_2} we have mu(f) = {1}.
11.73/3.86	The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1} are not replacing on any position.
11.73/3.86	
11.73/3.86	The TRS P consists of the following rules:
11.73/3.86	
11.73/3.86	   U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104'(tt, V1, V2) -> U105'(isNat(V1), V2)
11.73/3.86	   U105'(tt, V2) -> ISNATILIST(V2)
11.73/3.86	   ISNATILIST(V) -> U31'(isNatIListKind(V), V)
11.73/3.86	   U31'(tt, V) -> U32'(isNatIListKind(V), V)
11.73/3.86	   U32'(tt, V) -> ISNATLIST(V)
11.73/3.86	   U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2)
11.73/3.86	   U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94'(tt, V1, V2) -> U95'(isNat(V1), V2)
11.73/3.86	   U11'(tt, V1) -> U12'(isNatIListKind(V1), V1)
11.73/3.86	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.73/3.86	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
11.73/3.86	   U22'(tt, V1) -> ISNAT(V1)
11.73/3.86	   U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2)
11.73/3.86	   U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44'(tt, V1, V2) -> U45'(isNat(V1), V2)
11.73/3.86	   U104'(tt, V1, V2) -> ISNAT(V1)
11.73/3.86	
11.73/3.86	The TRS R consists of the following rules:
11.73/3.86	
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   U13(tt) -> tt
11.73/3.86	
11.73/3.86	Q is empty.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(25) QCSDependencyGraphProof (EQUIVALENT)
11.73/3.86	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 14 less nodes.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(26)
11.73/3.86	Obligation:
11.73/3.86	Q-restricted context-sensitive dependency pair problem:
11.73/3.86	The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions.
11.73/3.86	For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U22'_2, U21'_2} we have mu(f) = {1}.
11.73/3.86	The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNAT_1} are not replacing on any position.
11.73/3.86	
11.73/3.86	The TRS P consists of the following rules:
11.73/3.86	
11.73/3.86	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
11.73/3.86	   U22'(tt, V1) -> ISNAT(V1)
11.73/3.86	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.73/3.86	
11.73/3.86	The TRS R consists of the following rules:
11.73/3.86	
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   U13(tt) -> tt
11.73/3.86	
11.73/3.86	Q is empty.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(27) QCSDPSubtermProof (EQUIVALENT)
11.73/3.86	We use the subterm processor [DA_EMMES].
11.73/3.86	
11.73/3.86	
11.73/3.86	The following pairs can be oriented strictly and are deleted.
11.73/3.86	
11.73/3.86	   ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
11.73/3.86	The remaining pairs can at least be oriented weakly.
11.73/3.86	
11.73/3.86	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
11.73/3.86	   U22'(tt, V1) -> ISNAT(V1)
11.73/3.86	Used ordering:  Combined order from the following AFS and order.
11.73/3.86	U22'(x1, x2)  =  x2
11.73/3.86	
11.73/3.86	U21'(x1, x2)  =  x2
11.73/3.86	
11.73/3.86	ISNAT(x1)  =  x1
11.73/3.86	
11.73/3.86	
11.73/3.86	Subterm Order
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(28)
11.73/3.86	Obligation:
11.73/3.86	Q-restricted context-sensitive dependency pair problem:
11.73/3.86	The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions.
11.73/3.86	For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U22'_2, U21'_2} we have mu(f) = {1}.
11.73/3.86	The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNAT_1} are not replacing on any position.
11.73/3.86	
11.73/3.86	The TRS P consists of the following rules:
11.73/3.86	
11.73/3.86	   U21'(tt, V1) -> U22'(isNatKind(V1), V1)
11.73/3.86	   U22'(tt, V1) -> ISNAT(V1)
11.73/3.86	
11.73/3.86	The TRS R consists of the following rules:
11.73/3.86	
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   U13(tt) -> tt
11.73/3.86	
11.73/3.86	Q is empty.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(29) QCSDependencyGraphProof (EQUIVALENT)
11.73/3.86	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(30)
11.73/3.86	TRUE
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(31)
11.73/3.86	Obligation:
11.73/3.86	Q-restricted context-sensitive dependency pair problem:
11.73/3.86	The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, TAKE_2} are replacing on all positions.
11.73/3.86	For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U133'_4, U132'_4, U134'_4, U135'_4, U136'_4, U131'_4} we have mu(f) = {1}.
11.73/3.86	The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, U_1} are not replacing on any position.
11.73/3.86	
11.73/3.86	The TRS P consists of the following rules:
11.73/3.86	
11.73/3.86	   U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N)
11.73/3.86	   U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N)
11.73/3.86	   U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N)
11.73/3.86	   U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N)
11.73/3.86	   U136'(tt, IL, M, N) -> U(N)
11.73/3.86	   U(take(x_0, x_1)) -> U(x_0)
11.73/3.86	   U(take(x_0, x_1)) -> U(x_1)
11.73/3.86	   U(take(x0, x1)) -> TAKE(x0, x1)
11.73/3.86	   TAKE(s(M), cons(N, IL)) -> U131'(isNatIList(IL), IL, M, N)
11.73/3.86	   U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N)
11.73/3.86	
11.73/3.86	The TRS R consists of the following rules:
11.73/3.86	
11.73/3.86	   zeros -> cons(0, zeros)
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.86	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.86	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.86	   U114(tt, L) -> s(length(L))
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   U13(tt) -> tt
11.73/3.86	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.86	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.86	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.86	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.86	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.86	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.86	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.86	
11.73/3.86	Q is empty.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(32) QCSDPSubtermProof (EQUIVALENT)
11.73/3.86	We use the subterm processor [DA_EMMES].
11.73/3.86	
11.73/3.86	
11.73/3.86	The following pairs can be oriented strictly and are deleted.
11.73/3.86	
11.73/3.86	   U(take(x_0, x_1)) -> U(x_0)
11.73/3.86	   U(take(x_0, x_1)) -> U(x_1)
11.73/3.86	   U(take(x0, x1)) -> TAKE(x0, x1)
11.73/3.86	   TAKE(s(M), cons(N, IL)) -> U131'(isNatIList(IL), IL, M, N)
11.73/3.86	The remaining pairs can at least be oriented weakly.
11.73/3.86	
11.73/3.86	   U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N)
11.73/3.86	   U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N)
11.73/3.86	   U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N)
11.73/3.86	   U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N)
11.73/3.86	   U136'(tt, IL, M, N) -> U(N)
11.73/3.86	   U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N)
11.73/3.86	Used ordering:  Combined order from the following AFS and order.
11.73/3.86	U133'(x1, x2, x3, x4)  =  x4
11.73/3.86	
11.73/3.86	U132'(x1, x2, x3, x4)  =  x4
11.73/3.86	
11.73/3.86	U134'(x1, x2, x3, x4)  =  x4
11.73/3.86	
11.73/3.86	U135'(x1, x2, x3, x4)  =  x4
11.73/3.86	
11.73/3.86	U136'(x1, x2, x3, x4)  =  x4
11.73/3.86	
11.73/3.86	U(x1)  =  x1
11.73/3.86	
11.73/3.86	TAKE(x1, x2)  =  x2
11.73/3.86	
11.73/3.86	U131'(x1, x2, x3, x4)  =  x4
11.73/3.86	
11.73/3.86	
11.73/3.86	Subterm Order
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(33)
11.73/3.86	Obligation:
11.73/3.86	Q-restricted context-sensitive dependency pair problem:
11.73/3.86	The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1} are replacing on all positions.
11.73/3.86	For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U133'_4, U132'_4, U134'_4, U135'_4, U136'_4, U131'_4} we have mu(f) = {1}.
11.73/3.86	The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, U_1} are not replacing on any position.
11.73/3.86	
11.73/3.86	The TRS P consists of the following rules:
11.73/3.86	
11.73/3.86	   U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N)
11.73/3.86	   U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N)
11.73/3.86	   U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N)
11.73/3.86	   U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N)
11.73/3.86	   U136'(tt, IL, M, N) -> U(N)
11.73/3.86	   U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N)
11.73/3.86	
11.73/3.86	The TRS R consists of the following rules:
11.73/3.86	
11.73/3.86	   zeros -> cons(0, zeros)
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.86	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.86	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.86	   U114(tt, L) -> s(length(L))
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   U13(tt) -> tt
11.73/3.86	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.86	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.86	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.86	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.86	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.86	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.86	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.86	
11.73/3.86	Q is empty.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(34) QCSDependencyGraphProof (EQUIVALENT)
11.73/3.86	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 6 less nodes.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(35)
11.73/3.86	TRUE
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(36)
11.73/3.86	Obligation:
11.73/3.86	Q-restricted context-sensitive dependency pair problem:
11.73/3.86	The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, LENGTH_1} are replacing on all positions.
11.73/3.86	For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U113'_3, U112'_3, U114'_2, U111'_3} we have mu(f) = {1}.
11.73/3.86	The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1} are not replacing on any position.
11.73/3.86	
11.73/3.86	The TRS P consists of the following rules:
11.73/3.86	
11.73/3.86	   U112'(tt, L, N) -> U113'(isNat(N), L, N)
11.73/3.86	   U113'(tt, L, N) -> U114'(isNatKind(N), L)
11.73/3.86	   U114'(tt, L) -> LENGTH(L)
11.73/3.86	   LENGTH(cons(N, L)) -> U111'(isNatList(L), L, N)
11.73/3.86	   U111'(tt, L, N) -> U112'(isNatIListKind(L), L, N)
11.73/3.86	
11.73/3.86	The TRS R consists of the following rules:
11.73/3.86	
11.73/3.86	   zeros -> cons(0, zeros)
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.86	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.86	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.86	   U114(tt, L) -> s(length(L))
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   U13(tt) -> tt
11.73/3.86	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.86	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.86	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.86	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.86	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.86	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.86	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.86	
11.73/3.86	Q is empty.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(37) QCSDPReductionPairProof (EQUIVALENT)
11.73/3.86	Using the order
11.73/3.86	
11.73/3.86	Polynomial interpretation [POLO]:
11.73/3.86	
11.73/3.86	   POL(0) = 2
11.73/3.86	   POL(LENGTH(x_1)) = 2 + 2*x_1
11.73/3.86	   POL(U101(x_1, x_2, x_3)) = x_2
11.73/3.86	   POL(U102(x_1, x_2, x_3)) = x_2
11.73/3.86	   POL(U103(x_1, x_2, x_3)) = x_2
11.73/3.86	   POL(U104(x_1, x_2, x_3)) = x_2
11.73/3.86	   POL(U105(x_1, x_2)) = x_1
11.73/3.86	   POL(U106(x_1)) = 2
11.73/3.86	   POL(U11(x_1, x_2)) = x_2
11.73/3.86	   POL(U111(x_1, x_2, x_3)) = 2*x_2
11.73/3.86	   POL(U111'(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2
11.73/3.86	   POL(U112(x_1, x_2, x_3)) = 2*x_2
11.73/3.86	   POL(U112'(x_1, x_2, x_3)) = 2 + 2*x_2
11.73/3.86	   POL(U113(x_1, x_2, x_3)) = 2*x_2
11.73/3.86	   POL(U113'(x_1, x_2, x_3)) = 2 + 2*x_2
11.73/3.86	   POL(U114(x_1, x_2)) = 2*x_2
11.73/3.86	   POL(U114'(x_1, x_2)) = 2 + 2*x_2
11.73/3.86	   POL(U12(x_1, x_2)) = x_2
11.73/3.86	   POL(U13(x_1)) = x_1
11.73/3.86	   POL(U131(x_1, x_2, x_3, x_4)) = 2*x_3
11.73/3.86	   POL(U132(x_1, x_2, x_3, x_4)) = 2*x_3
11.73/3.86	   POL(U133(x_1, x_2, x_3, x_4)) = 2*x_3
11.73/3.86	   POL(U134(x_1, x_2, x_3, x_4)) = 2*x_3
11.73/3.86	   POL(U135(x_1, x_2, x_3, x_4)) = 2*x_3
11.73/3.86	   POL(U136(x_1, x_2, x_3, x_4)) = 2*x_3
11.73/3.86	   POL(U21(x_1, x_2)) = 2*x_2
11.73/3.86	   POL(U22(x_1, x_2)) = 2*x_2
11.73/3.86	   POL(U23(x_1)) = 2*x_1
11.73/3.86	   POL(U31(x_1, x_2)) = 2*x_2
11.73/3.86	   POL(U32(x_1, x_2)) = 2*x_2
11.73/3.86	   POL(U33(x_1)) = 2*x_1
11.73/3.86	   POL(U41(x_1, x_2, x_3)) = 2 + 2*x_3
11.73/3.86	   POL(U42(x_1, x_2, x_3)) = 2
11.73/3.86	   POL(U43(x_1, x_2, x_3)) = x_1
11.73/3.86	   POL(U44(x_1, x_2, x_3)) = x_1
11.73/3.86	   POL(U45(x_1, x_2)) = 2
11.73/3.86	   POL(U46(x_1)) = 2
11.73/3.86	   POL(U51(x_1, x_2)) = 2
11.73/3.86	   POL(U52(x_1)) = x_1
11.73/3.86	   POL(U61(x_1, x_2)) = 2
11.73/3.86	   POL(U62(x_1)) = 2
11.73/3.86	   POL(U71(x_1)) = 2
11.73/3.86	   POL(U81(x_1)) = 2
11.73/3.86	   POL(U91(x_1, x_2, x_3)) = 2*x_3
11.73/3.86	   POL(U92(x_1, x_2, x_3)) = x_3
11.73/3.86	   POL(U93(x_1, x_2, x_3)) = x_3
11.73/3.86	   POL(U94(x_1, x_2, x_3)) = x_3
11.73/3.86	   POL(U95(x_1, x_2)) = x_2
11.73/3.86	   POL(U96(x_1)) = x_1
11.73/3.86	   POL(cons(x_1, x_2)) = 2*x_2
11.73/3.86	   POL(isNat(x_1)) = x_1
11.73/3.86	   POL(isNatIList(x_1)) = 2 + 2*x_1
11.73/3.86	   POL(isNatIListKind(x_1)) = 2
11.73/3.86	   POL(isNatKind(x_1)) = 2
11.73/3.86	   POL(isNatList(x_1)) = x_1
11.73/3.86	   POL(length(x_1)) = x_1
11.73/3.86	   POL(nil) = 2
11.73/3.86	   POL(s(x_1)) = 2*x_1
11.73/3.86	   POL(take(x_1, x_2)) = x_1
11.73/3.86	   POL(tt) = 2
11.73/3.86	   POL(zeros) = 0
11.73/3.86	
11.73/3.86	
11.73/3.86	the following usable rules
11.73/3.86	
11.73/3.86	
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.86	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.86	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.86	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.86	   U114(tt, L) -> s(length(L))
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   zeros -> cons(0, zeros)
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.86	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.86	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.86	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.86	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.86	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.86	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   U13(tt) -> tt
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	
11.73/3.86	
11.73/3.86	could all be oriented weakly.
11.73/3.86	
11.73/3.86	Furthermore, the pairs
11.73/3.86	
11.73/3.86	
11.73/3.86	   LENGTH(cons(N, L)) -> U111'(isNatList(L), L, N)
11.73/3.86	   U111'(tt, L, N) -> U112'(isNatIListKind(L), L, N)
11.73/3.86	
11.73/3.86	
11.73/3.86	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
11.73/3.86	
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(38)
11.73/3.86	Obligation:
11.73/3.86	Q-restricted context-sensitive dependency pair problem:
11.73/3.86	The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, LENGTH_1} are replacing on all positions.
11.73/3.86	For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U113'_3, U112'_3, U114'_2} we have mu(f) = {1}.
11.73/3.86	The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1} are not replacing on any position.
11.73/3.86	
11.73/3.86	The TRS P consists of the following rules:
11.73/3.86	
11.73/3.86	   U112'(tt, L, N) -> U113'(isNat(N), L, N)
11.73/3.86	   U113'(tt, L, N) -> U114'(isNatKind(N), L)
11.73/3.86	   U114'(tt, L) -> LENGTH(L)
11.73/3.86	
11.73/3.86	The TRS R consists of the following rules:
11.73/3.86	
11.73/3.86	   zeros -> cons(0, zeros)
11.73/3.86	   U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2)
11.73/3.86	   U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2)
11.73/3.86	   U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2)
11.73/3.86	   U104(tt, V1, V2) -> U105(isNat(V1), V2)
11.73/3.86	   U105(tt, V2) -> U106(isNatIList(V2))
11.73/3.86	   U106(tt) -> tt
11.73/3.86	   U11(tt, V1) -> U12(isNatIListKind(V1), V1)
11.73/3.86	   U111(tt, L, N) -> U112(isNatIListKind(L), L, N)
11.73/3.86	   U112(tt, L, N) -> U113(isNat(N), L, N)
11.73/3.86	   U113(tt, L, N) -> U114(isNatKind(N), L)
11.73/3.86	   U114(tt, L) -> s(length(L))
11.73/3.86	   U12(tt, V1) -> U13(isNatList(V1))
11.73/3.86	   U13(tt) -> tt
11.73/3.86	   U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N)
11.73/3.86	   U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N)
11.73/3.86	   U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N)
11.73/3.86	   U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N)
11.73/3.86	   U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N)
11.73/3.86	   U136(tt, IL, M, N) -> cons(N, take(M, IL))
11.73/3.86	   U21(tt, V1) -> U22(isNatKind(V1), V1)
11.73/3.86	   U22(tt, V1) -> U23(isNat(V1))
11.73/3.86	   U23(tt) -> tt
11.73/3.86	   U31(tt, V) -> U32(isNatIListKind(V), V)
11.73/3.86	   U32(tt, V) -> U33(isNatList(V))
11.73/3.86	   U33(tt) -> tt
11.73/3.86	   U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2)
11.73/3.86	   U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2)
11.73/3.86	   U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2)
11.73/3.86	   U44(tt, V1, V2) -> U45(isNat(V1), V2)
11.73/3.86	   U45(tt, V2) -> U46(isNatIList(V2))
11.73/3.86	   U46(tt) -> tt
11.73/3.86	   U51(tt, V2) -> U52(isNatIListKind(V2))
11.73/3.86	   U52(tt) -> tt
11.73/3.86	   U61(tt, V2) -> U62(isNatIListKind(V2))
11.73/3.86	   U62(tt) -> tt
11.73/3.86	   U71(tt) -> tt
11.73/3.86	   U81(tt) -> tt
11.73/3.86	   U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2)
11.73/3.86	   U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2)
11.73/3.86	   U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2)
11.73/3.86	   U94(tt, V1, V2) -> U95(isNat(V1), V2)
11.73/3.86	   U95(tt, V2) -> U96(isNatList(V2))
11.73/3.86	   U96(tt) -> tt
11.73/3.86	   isNat(0) -> tt
11.73/3.86	   isNat(length(V1)) -> U11(isNatIListKind(V1), V1)
11.73/3.86	   isNat(s(V1)) -> U21(isNatKind(V1), V1)
11.73/3.86	   isNatIList(V) -> U31(isNatIListKind(V), V)
11.73/3.86	   isNatIList(zeros) -> tt
11.73/3.86	   isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2)
11.73/3.86	   isNatIListKind(nil) -> tt
11.73/3.86	   isNatIListKind(zeros) -> tt
11.73/3.86	   isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2)
11.73/3.86	   isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2)
11.73/3.86	   isNatKind(0) -> tt
11.73/3.86	   isNatKind(length(V1)) -> U71(isNatIListKind(V1))
11.73/3.86	   isNatKind(s(V1)) -> U81(isNatKind(V1))
11.73/3.86	   isNatList(nil) -> tt
11.73/3.86	   isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2)
11.73/3.86	   isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2)
11.73/3.86	   length(cons(N, L)) -> U111(isNatList(L), L, N)
11.73/3.86	   take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N)
11.73/3.86	
11.73/3.86	Q is empty.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(39) QCSDependencyGraphProof (EQUIVALENT)
11.73/3.86	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes.
11.73/3.86	
11.73/3.86	----------------------------------------
11.73/3.86	
11.73/3.86	(40)
11.73/3.86	TRUE
11.73/3.91	EOF