15.03/5.11 YES 15.03/5.14 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 15.03/5.14 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 15.03/5.14 15.03/5.14 15.03/5.14 Termination w.r.t. Q of the given QTRS could be proven: 15.03/5.14 15.03/5.14 (0) QTRS 15.03/5.14 (1) QTRSToCSRProof [SOUND, 0 ms] 15.03/5.14 (2) CSR 15.03/5.14 (3) CSDependencyPairsProof [EQUIVALENT, 447 ms] 15.03/5.14 (4) QCSDP 15.03/5.14 (5) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 15.03/5.14 (6) AND 15.03/5.14 (7) QCSDP 15.03/5.14 (8) QCSUsableRulesProof [EQUIVALENT, 0 ms] 15.03/5.14 (9) QCSDP 15.03/5.14 (10) QCSDPMuMonotonicPoloProof [EQUIVALENT, 57 ms] 15.03/5.14 (11) QCSDP 15.03/5.14 (12) QCSDPSubtermProof [EQUIVALENT, 0 ms] 15.03/5.14 (13) QCSDP 15.03/5.14 (14) PIsEmptyProof [EQUIVALENT, 0 ms] 15.03/5.14 (15) YES 15.03/5.14 (16) QCSDP 15.03/5.14 (17) QCSUsableRulesProof [EQUIVALENT, 11 ms] 15.03/5.14 (18) QCSDP 15.03/5.14 (19) QCSDPMuMonotonicPoloProof [EQUIVALENT, 312 ms] 15.03/5.14 (20) QCSDP 15.03/5.14 (21) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 15.03/5.14 (22) TRUE 15.03/5.14 (23) QCSDP 15.03/5.14 (24) QCSDPSubtermProof [EQUIVALENT, 3 ms] 15.03/5.14 (25) QCSDP 15.03/5.14 (26) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 15.03/5.14 (27) TRUE 15.03/5.14 (28) QCSDP 15.03/5.14 (29) QCSDPSubtermProof [EQUIVALENT, 2 ms] 15.03/5.14 (30) QCSDP 15.03/5.14 (31) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 15.03/5.14 (32) TRUE 15.03/5.14 15.03/5.14 15.03/5.14 ---------------------------------------- 15.03/5.14 15.03/5.14 (0) 15.03/5.14 Obligation: 15.03/5.14 Q restricted rewrite system: 15.03/5.14 The TRS R consists of the following rules: 15.03/5.14 15.03/5.14 active(U101(tt, V1, V2)) -> mark(U102(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(U102(tt, V1, V2)) -> mark(U103(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U103(tt, V1, V2)) -> mark(U104(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U104(tt, V1, V2)) -> mark(U105(isNatural(V1), V2)) 15.03/5.14 active(U105(tt, V2)) -> mark(U106(isLNat(V2))) 15.03/5.14 active(U106(tt)) -> mark(tt) 15.03/5.14 active(U11(tt, N, XS)) -> mark(U12(isNaturalKind(N), N, XS)) 15.03/5.14 active(U111(tt, V2)) -> mark(U112(isLNatKind(V2))) 15.03/5.14 active(U112(tt)) -> mark(tt) 15.03/5.14 active(U12(tt, N, XS)) -> mark(U13(isLNat(XS), N, XS)) 15.03/5.14 active(U121(tt, V2)) -> mark(U122(isLNatKind(V2))) 15.03/5.14 active(U122(tt)) -> mark(tt) 15.03/5.14 active(U13(tt, N, XS)) -> mark(U14(isLNatKind(XS), N, XS)) 15.03/5.14 active(U131(tt)) -> mark(tt) 15.03/5.14 active(U14(tt, N, XS)) -> mark(snd(splitAt(N, XS))) 15.03/5.14 active(U141(tt)) -> mark(tt) 15.03/5.14 active(U151(tt)) -> mark(tt) 15.03/5.14 active(U161(tt)) -> mark(tt) 15.03/5.14 active(U171(tt, V2)) -> mark(U172(isLNatKind(V2))) 15.03/5.14 active(U172(tt)) -> mark(tt) 15.03/5.14 active(U181(tt, V1)) -> mark(U182(isLNatKind(V1), V1)) 15.03/5.14 active(U182(tt, V1)) -> mark(U183(isLNat(V1))) 15.03/5.14 active(U183(tt)) -> mark(tt) 15.03/5.14 active(U191(tt, V1)) -> mark(U192(isNaturalKind(V1), V1)) 15.03/5.14 active(U192(tt, V1)) -> mark(U193(isNatural(V1))) 15.03/5.14 active(U193(tt)) -> mark(tt) 15.03/5.14 active(U201(tt, V1, V2)) -> mark(U202(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(U202(tt, V1, V2)) -> mark(U203(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U203(tt, V1, V2)) -> mark(U204(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U204(tt, V1, V2)) -> mark(U205(isNatural(V1), V2)) 15.03/5.14 active(U205(tt, V2)) -> mark(U206(isLNat(V2))) 15.03/5.14 active(U206(tt)) -> mark(tt) 15.03/5.14 active(U21(tt, X, Y)) -> mark(U22(isLNatKind(X), X, Y)) 15.03/5.14 active(U211(tt)) -> mark(tt) 15.03/5.14 active(U22(tt, X, Y)) -> mark(U23(isLNat(Y), X, Y)) 15.03/5.14 active(U221(tt)) -> mark(tt) 15.03/5.14 active(U23(tt, X, Y)) -> mark(U24(isLNatKind(Y), X)) 15.03/5.14 active(U231(tt, V2)) -> mark(U232(isLNatKind(V2))) 15.03/5.14 active(U232(tt)) -> mark(tt) 15.03/5.14 active(U24(tt, X)) -> mark(X) 15.03/5.14 active(U241(tt, V1, V2)) -> mark(U242(isLNatKind(V1), V1, V2)) 15.03/5.14 active(U242(tt, V1, V2)) -> mark(U243(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U243(tt, V1, V2)) -> mark(U244(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U244(tt, V1, V2)) -> mark(U245(isLNat(V1), V2)) 15.03/5.14 active(U245(tt, V2)) -> mark(U246(isLNat(V2))) 15.03/5.14 active(U246(tt)) -> mark(tt) 15.03/5.14 active(U251(tt, V1, V2)) -> mark(U252(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(U252(tt, V1, V2)) -> mark(U253(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U253(tt, V1, V2)) -> mark(U254(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U254(tt, V1, V2)) -> mark(U255(isNatural(V1), V2)) 15.03/5.14 active(U255(tt, V2)) -> mark(U256(isLNat(V2))) 15.03/5.14 active(U256(tt)) -> mark(tt) 15.03/5.14 active(U261(tt, V2)) -> mark(U262(isLNatKind(V2))) 15.03/5.14 active(U262(tt)) -> mark(tt) 15.03/5.14 active(U271(tt, V2)) -> mark(U272(isLNatKind(V2))) 15.03/5.14 active(U272(tt)) -> mark(tt) 15.03/5.14 active(U281(tt, N)) -> mark(U282(isNaturalKind(N), N)) 15.03/5.14 active(U282(tt, N)) -> mark(cons(N, natsFrom(s(N)))) 15.03/5.14 active(U291(tt, N, XS)) -> mark(U292(isNaturalKind(N), N, XS)) 15.03/5.14 active(U292(tt, N, XS)) -> mark(U293(isLNat(XS), N, XS)) 15.03/5.14 active(U293(tt, N, XS)) -> mark(U294(isLNatKind(XS), N, XS)) 15.03/5.14 active(U294(tt, N, XS)) -> mark(head(afterNth(N, XS))) 15.03/5.14 active(U301(tt, X, Y)) -> mark(U302(isLNatKind(X), Y)) 15.03/5.14 active(U302(tt, Y)) -> mark(U303(isLNat(Y), Y)) 15.03/5.14 active(U303(tt, Y)) -> mark(U304(isLNatKind(Y), Y)) 15.03/5.14 active(U304(tt, Y)) -> mark(Y) 15.03/5.14 active(U31(tt, N, XS)) -> mark(U32(isNaturalKind(N), N, XS)) 15.03/5.14 active(U311(tt, XS)) -> mark(U312(isLNatKind(XS), XS)) 15.03/5.14 active(U312(tt, XS)) -> mark(pair(nil, XS)) 15.03/5.14 active(U32(tt, N, XS)) -> mark(U33(isLNat(XS), N, XS)) 15.03/5.14 active(U321(tt, N, X, XS)) -> mark(U322(isNaturalKind(N), N, X, XS)) 15.03/5.14 active(U322(tt, N, X, XS)) -> mark(U323(isNatural(X), N, X, XS)) 15.03/5.14 active(U323(tt, N, X, XS)) -> mark(U324(isNaturalKind(X), N, X, XS)) 15.03/5.14 active(U324(tt, N, X, XS)) -> mark(U325(isLNat(XS), N, X, XS)) 15.03/5.14 active(U325(tt, N, X, XS)) -> mark(U326(isLNatKind(XS), N, X, XS)) 15.03/5.14 active(U326(tt, N, X, XS)) -> mark(U327(splitAt(N, XS), X)) 15.03/5.14 active(U327(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) 15.03/5.14 active(U33(tt, N, XS)) -> mark(U34(isLNatKind(XS), N)) 15.03/5.14 active(U331(tt, N, XS)) -> mark(U332(isNaturalKind(N), XS)) 15.03/5.14 active(U332(tt, XS)) -> mark(U333(isLNat(XS), XS)) 15.03/5.14 active(U333(tt, XS)) -> mark(U334(isLNatKind(XS), XS)) 15.03/5.14 active(U334(tt, XS)) -> mark(XS) 15.03/5.14 active(U34(tt, N)) -> mark(N) 15.03/5.14 active(U341(tt, N, XS)) -> mark(U342(isNaturalKind(N), N, XS)) 15.03/5.14 active(U342(tt, N, XS)) -> mark(U343(isLNat(XS), N, XS)) 15.03/5.14 active(U343(tt, N, XS)) -> mark(U344(isLNatKind(XS), N, XS)) 15.03/5.14 active(U344(tt, N, XS)) -> mark(fst(splitAt(N, XS))) 15.03/5.14 active(U41(tt, V1, V2)) -> mark(U42(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(U42(tt, V1, V2)) -> mark(U43(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U43(tt, V1, V2)) -> mark(U44(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U44(tt, V1, V2)) -> mark(U45(isNatural(V1), V2)) 15.03/5.14 active(U45(tt, V2)) -> mark(U46(isLNat(V2))) 15.03/5.14 active(U46(tt)) -> mark(tt) 15.03/5.14 active(U51(tt, V1, V2)) -> mark(U52(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(U52(tt, V1, V2)) -> mark(U53(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U53(tt, V1, V2)) -> mark(U54(isLNatKind(V2), V1, V2)) 15.03/5.14 active(U54(tt, V1, V2)) -> mark(U55(isNatural(V1), V2)) 15.03/5.14 active(U55(tt, V2)) -> mark(U56(isLNat(V2))) 15.03/5.14 active(U56(tt)) -> mark(tt) 15.03/5.14 active(U61(tt, V1)) -> mark(U62(isPLNatKind(V1), V1)) 15.03/5.14 active(U62(tt, V1)) -> mark(U63(isPLNat(V1))) 15.03/5.14 active(U63(tt)) -> mark(tt) 15.03/5.14 active(U71(tt, V1)) -> mark(U72(isNaturalKind(V1), V1)) 15.03/5.14 active(U72(tt, V1)) -> mark(U73(isNatural(V1))) 15.03/5.14 active(U73(tt)) -> mark(tt) 15.03/5.14 active(U81(tt, V1)) -> mark(U82(isPLNatKind(V1), V1)) 15.03/5.14 active(U82(tt, V1)) -> mark(U83(isPLNat(V1))) 15.03/5.14 active(U83(tt)) -> mark(tt) 15.03/5.14 active(U91(tt, V1)) -> mark(U92(isLNatKind(V1), V1)) 15.03/5.14 active(U92(tt, V1)) -> mark(U93(isLNat(V1))) 15.03/5.14 active(U93(tt)) -> mark(tt) 15.03/5.14 active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS)) 15.03/5.14 active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y)) 15.03/5.14 active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS)) 15.03/5.14 active(isLNat(nil)) -> mark(tt) 15.03/5.14 active(isLNat(afterNth(V1, V2))) -> mark(U41(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(isLNat(cons(V1, V2))) -> mark(U51(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1)) 15.03/5.14 active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1)) 15.03/5.14 active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1)) 15.03/5.14 active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1)) 15.03/5.14 active(isLNat(take(V1, V2))) -> mark(U101(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(isLNatKind(nil)) -> mark(tt) 15.03/5.14 active(isLNatKind(afterNth(V1, V2))) -> mark(U111(isNaturalKind(V1), V2)) 15.03/5.14 active(isLNatKind(cons(V1, V2))) -> mark(U121(isNaturalKind(V1), V2)) 15.03/5.14 active(isLNatKind(fst(V1))) -> mark(U131(isPLNatKind(V1))) 15.03/5.14 active(isLNatKind(natsFrom(V1))) -> mark(U141(isNaturalKind(V1))) 15.03/5.14 active(isLNatKind(snd(V1))) -> mark(U151(isPLNatKind(V1))) 15.03/5.14 active(isLNatKind(tail(V1))) -> mark(U161(isLNatKind(V1))) 15.03/5.14 active(isLNatKind(take(V1, V2))) -> mark(U171(isNaturalKind(V1), V2)) 15.03/5.14 active(isNatural(0)) -> mark(tt) 15.03/5.14 active(isNatural(head(V1))) -> mark(U181(isLNatKind(V1), V1)) 15.03/5.14 active(isNatural(s(V1))) -> mark(U191(isNaturalKind(V1), V1)) 15.03/5.14 active(isNatural(sel(V1, V2))) -> mark(U201(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(isNaturalKind(0)) -> mark(tt) 15.03/5.14 active(isNaturalKind(head(V1))) -> mark(U211(isLNatKind(V1))) 15.03/5.14 active(isNaturalKind(s(V1))) -> mark(U221(isNaturalKind(V1))) 15.03/5.14 active(isNaturalKind(sel(V1, V2))) -> mark(U231(isNaturalKind(V1), V2)) 15.03/5.14 active(isPLNat(pair(V1, V2))) -> mark(U241(isLNatKind(V1), V1, V2)) 15.03/5.14 active(isPLNat(splitAt(V1, V2))) -> mark(U251(isNaturalKind(V1), V1, V2)) 15.03/5.14 active(isPLNatKind(pair(V1, V2))) -> mark(U261(isLNatKind(V1), V2)) 15.03/5.14 active(isPLNatKind(splitAt(V1, V2))) -> mark(U271(isNaturalKind(V1), V2)) 15.03/5.14 active(natsFrom(N)) -> mark(U281(isNatural(N), N)) 15.03/5.14 active(sel(N, XS)) -> mark(U291(isNatural(N), N, XS)) 15.03/5.14 active(snd(pair(X, Y))) -> mark(U301(isLNat(X), X, Y)) 15.03/5.14 active(splitAt(0, XS)) -> mark(U311(isLNat(XS), XS)) 15.03/5.14 active(splitAt(s(N), cons(X, XS))) -> mark(U321(isNatural(N), N, X, XS)) 15.03/5.14 active(tail(cons(N, XS))) -> mark(U331(isNatural(N), N, XS)) 15.03/5.14 active(take(N, XS)) -> mark(U341(isNatural(N), N, XS)) 15.03/5.14 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 15.03/5.14 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 15.03/5.14 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 15.03/5.14 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 15.03/5.14 active(U105(X1, X2)) -> U105(active(X1), X2) 15.03/5.14 active(U106(X)) -> U106(active(X)) 15.03/5.14 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 15.03/5.14 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 15.03/5.14 active(U111(X1, X2)) -> U111(active(X1), X2) 15.03/5.14 active(U112(X)) -> U112(active(X)) 15.03/5.14 active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) 15.03/5.14 active(U121(X1, X2)) -> U121(active(X1), X2) 15.03/5.14 active(U122(X)) -> U122(active(X)) 15.03/5.14 active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) 15.03/5.14 active(U131(X)) -> U131(active(X)) 15.03/5.14 active(snd(X)) -> snd(active(X)) 15.03/5.14 active(splitAt(X1, X2)) -> splitAt(active(X1), X2) 15.03/5.14 active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) 15.03/5.14 active(U141(X)) -> U141(active(X)) 15.03/5.14 active(U151(X)) -> U151(active(X)) 15.03/5.14 active(U161(X)) -> U161(active(X)) 15.03/5.14 active(U171(X1, X2)) -> U171(active(X1), X2) 15.03/5.14 active(U172(X)) -> U172(active(X)) 15.03/5.14 active(U181(X1, X2)) -> U181(active(X1), X2) 15.03/5.14 active(U182(X1, X2)) -> U182(active(X1), X2) 15.03/5.14 active(U183(X)) -> U183(active(X)) 15.03/5.14 active(U191(X1, X2)) -> U191(active(X1), X2) 15.03/5.14 active(U192(X1, X2)) -> U192(active(X1), X2) 15.03/5.14 active(U193(X)) -> U193(active(X)) 15.03/5.14 active(U201(X1, X2, X3)) -> U201(active(X1), X2, X3) 15.03/5.14 active(U202(X1, X2, X3)) -> U202(active(X1), X2, X3) 15.03/5.14 active(U203(X1, X2, X3)) -> U203(active(X1), X2, X3) 15.03/5.14 active(U204(X1, X2, X3)) -> U204(active(X1), X2, X3) 15.03/5.14 active(U205(X1, X2)) -> U205(active(X1), X2) 15.03/5.14 active(U206(X)) -> U206(active(X)) 15.03/5.14 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 15.03/5.14 active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3) 15.03/5.14 active(U211(X)) -> U211(active(X)) 15.03/5.14 active(U23(X1, X2, X3)) -> U23(active(X1), X2, X3) 15.03/5.14 active(U221(X)) -> U221(active(X)) 15.03/5.14 active(U24(X1, X2)) -> U24(active(X1), X2) 15.03/5.14 active(U231(X1, X2)) -> U231(active(X1), X2) 15.03/5.14 active(U232(X)) -> U232(active(X)) 15.03/5.14 active(U241(X1, X2, X3)) -> U241(active(X1), X2, X3) 15.03/5.14 active(U242(X1, X2, X3)) -> U242(active(X1), X2, X3) 15.03/5.14 active(U243(X1, X2, X3)) -> U243(active(X1), X2, X3) 15.03/5.14 active(U244(X1, X2, X3)) -> U244(active(X1), X2, X3) 15.03/5.14 active(U245(X1, X2)) -> U245(active(X1), X2) 15.03/5.14 active(U246(X)) -> U246(active(X)) 15.03/5.14 active(U251(X1, X2, X3)) -> U251(active(X1), X2, X3) 15.03/5.14 active(U252(X1, X2, X3)) -> U252(active(X1), X2, X3) 15.03/5.14 active(U253(X1, X2, X3)) -> U253(active(X1), X2, X3) 15.03/5.14 active(U254(X1, X2, X3)) -> U254(active(X1), X2, X3) 15.03/5.14 active(U255(X1, X2)) -> U255(active(X1), X2) 15.03/5.14 active(U256(X)) -> U256(active(X)) 15.03/5.14 active(U261(X1, X2)) -> U261(active(X1), X2) 15.03/5.14 active(U262(X)) -> U262(active(X)) 15.03/5.14 active(U271(X1, X2)) -> U271(active(X1), X2) 15.03/5.14 active(U272(X)) -> U272(active(X)) 15.03/5.14 active(U281(X1, X2)) -> U281(active(X1), X2) 15.03/5.14 active(U282(X1, X2)) -> U282(active(X1), X2) 15.03/5.14 active(cons(X1, X2)) -> cons(active(X1), X2) 15.03/5.14 active(natsFrom(X)) -> natsFrom(active(X)) 15.03/5.14 active(s(X)) -> s(active(X)) 15.03/5.14 active(U291(X1, X2, X3)) -> U291(active(X1), X2, X3) 15.03/5.14 active(U292(X1, X2, X3)) -> U292(active(X1), X2, X3) 15.03/5.14 active(U293(X1, X2, X3)) -> U293(active(X1), X2, X3) 15.03/5.14 active(U294(X1, X2, X3)) -> U294(active(X1), X2, X3) 15.03/5.14 active(head(X)) -> head(active(X)) 15.03/5.14 active(afterNth(X1, X2)) -> afterNth(active(X1), X2) 15.03/5.14 active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) 15.03/5.14 active(U301(X1, X2, X3)) -> U301(active(X1), X2, X3) 15.03/5.14 active(U302(X1, X2)) -> U302(active(X1), X2) 15.03/5.14 active(U303(X1, X2)) -> U303(active(X1), X2) 15.03/5.14 active(U304(X1, X2)) -> U304(active(X1), X2) 15.03/5.14 active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) 15.03/5.14 active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) 15.03/5.14 active(U311(X1, X2)) -> U311(active(X1), X2) 15.03/5.14 active(U312(X1, X2)) -> U312(active(X1), X2) 15.03/5.14 active(pair(X1, X2)) -> pair(active(X1), X2) 15.03/5.14 active(pair(X1, X2)) -> pair(X1, active(X2)) 15.03/5.14 active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) 15.03/5.14 active(U321(X1, X2, X3, X4)) -> U321(active(X1), X2, X3, X4) 15.03/5.14 active(U322(X1, X2, X3, X4)) -> U322(active(X1), X2, X3, X4) 15.03/5.14 active(U323(X1, X2, X3, X4)) -> U323(active(X1), X2, X3, X4) 15.03/5.14 active(U324(X1, X2, X3, X4)) -> U324(active(X1), X2, X3, X4) 15.03/5.14 active(U325(X1, X2, X3, X4)) -> U325(active(X1), X2, X3, X4) 15.03/5.14 active(U326(X1, X2, X3, X4)) -> U326(active(X1), X2, X3, X4) 15.03/5.14 active(U327(X1, X2)) -> U327(active(X1), X2) 15.03/5.14 active(U34(X1, X2)) -> U34(active(X1), X2) 15.03/5.14 active(U331(X1, X2, X3)) -> U331(active(X1), X2, X3) 15.03/5.14 active(U332(X1, X2)) -> U332(active(X1), X2) 15.03/5.14 active(U333(X1, X2)) -> U333(active(X1), X2) 15.03/5.14 active(U334(X1, X2)) -> U334(active(X1), X2) 15.03/5.14 active(U341(X1, X2, X3)) -> U341(active(X1), X2, X3) 15.03/5.14 active(U342(X1, X2, X3)) -> U342(active(X1), X2, X3) 15.03/5.14 active(U343(X1, X2, X3)) -> U343(active(X1), X2, X3) 15.03/5.14 active(U344(X1, X2, X3)) -> U344(active(X1), X2, X3) 15.03/5.14 active(fst(X)) -> fst(active(X)) 15.03/5.14 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 15.03/5.14 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 15.03/5.14 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 15.03/5.14 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 15.03/5.14 active(U45(X1, X2)) -> U45(active(X1), X2) 15.03/5.14 active(U46(X)) -> U46(active(X)) 15.03/5.14 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 15.03/5.14 active(U52(X1, X2, X3)) -> U52(active(X1), X2, X3) 15.03/5.14 active(U53(X1, X2, X3)) -> U53(active(X1), X2, X3) 15.03/5.14 active(U54(X1, X2, X3)) -> U54(active(X1), X2, X3) 15.03/5.14 active(U55(X1, X2)) -> U55(active(X1), X2) 15.03/5.14 active(U56(X)) -> U56(active(X)) 15.03/5.14 active(U61(X1, X2)) -> U61(active(X1), X2) 15.03/5.14 active(U62(X1, X2)) -> U62(active(X1), X2) 15.03/5.14 active(U63(X)) -> U63(active(X)) 15.03/5.14 active(U71(X1, X2)) -> U71(active(X1), X2) 15.03/5.14 active(U72(X1, X2)) -> U72(active(X1), X2) 15.03/5.14 active(U73(X)) -> U73(active(X)) 15.03/5.14 active(U81(X1, X2)) -> U81(active(X1), X2) 15.03/5.14 active(U82(X1, X2)) -> U82(active(X1), X2) 15.03/5.14 active(U83(X)) -> U83(active(X)) 15.03/5.14 active(U91(X1, X2)) -> U91(active(X1), X2) 15.03/5.14 active(U92(X1, X2)) -> U92(active(X1), X2) 15.03/5.14 active(U93(X)) -> U93(active(X)) 15.03/5.14 active(tail(X)) -> tail(active(X)) 15.03/5.14 active(take(X1, X2)) -> take(active(X1), X2) 15.03/5.14 active(take(X1, X2)) -> take(X1, active(X2)) 15.03/5.14 active(sel(X1, X2)) -> sel(active(X1), X2) 15.03/5.14 active(sel(X1, X2)) -> sel(X1, active(X2)) 15.03/5.14 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 15.03/5.14 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 15.03/5.14 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 15.03/5.14 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 15.03/5.14 U105(mark(X1), X2) -> mark(U105(X1, X2)) 15.03/5.14 U106(mark(X)) -> mark(U106(X)) 15.03/5.14 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 15.03/5.14 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 15.03/5.14 U111(mark(X1), X2) -> mark(U111(X1, X2)) 15.03/5.14 U112(mark(X)) -> mark(U112(X)) 15.03/5.14 U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) 15.03/5.14 U121(mark(X1), X2) -> mark(U121(X1, X2)) 15.03/5.15 U122(mark(X)) -> mark(U122(X)) 15.03/5.15 U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) 15.03/5.15 U131(mark(X)) -> mark(U131(X)) 15.03/5.15 snd(mark(X)) -> mark(snd(X)) 15.03/5.15 splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) 15.03/5.15 splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) 15.03/5.15 U141(mark(X)) -> mark(U141(X)) 15.03/5.15 U151(mark(X)) -> mark(U151(X)) 15.03/5.15 U161(mark(X)) -> mark(U161(X)) 15.03/5.15 U171(mark(X1), X2) -> mark(U171(X1, X2)) 15.03/5.15 U172(mark(X)) -> mark(U172(X)) 15.03/5.15 U181(mark(X1), X2) -> mark(U181(X1, X2)) 15.03/5.15 U182(mark(X1), X2) -> mark(U182(X1, X2)) 15.03/5.15 U183(mark(X)) -> mark(U183(X)) 15.03/5.15 U191(mark(X1), X2) -> mark(U191(X1, X2)) 15.03/5.15 U192(mark(X1), X2) -> mark(U192(X1, X2)) 15.03/5.15 U193(mark(X)) -> mark(U193(X)) 15.03/5.15 U201(mark(X1), X2, X3) -> mark(U201(X1, X2, X3)) 15.03/5.15 U202(mark(X1), X2, X3) -> mark(U202(X1, X2, X3)) 15.03/5.15 U203(mark(X1), X2, X3) -> mark(U203(X1, X2, X3)) 15.03/5.15 U204(mark(X1), X2, X3) -> mark(U204(X1, X2, X3)) 15.03/5.15 U205(mark(X1), X2) -> mark(U205(X1, X2)) 15.03/5.15 U206(mark(X)) -> mark(U206(X)) 15.03/5.15 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 15.03/5.15 U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3)) 15.03/5.15 U211(mark(X)) -> mark(U211(X)) 15.03/5.15 U23(mark(X1), X2, X3) -> mark(U23(X1, X2, X3)) 15.03/5.15 U221(mark(X)) -> mark(U221(X)) 15.03/5.15 U24(mark(X1), X2) -> mark(U24(X1, X2)) 15.03/5.15 U231(mark(X1), X2) -> mark(U231(X1, X2)) 15.03/5.15 U232(mark(X)) -> mark(U232(X)) 15.03/5.15 U241(mark(X1), X2, X3) -> mark(U241(X1, X2, X3)) 15.03/5.15 U242(mark(X1), X2, X3) -> mark(U242(X1, X2, X3)) 15.03/5.15 U243(mark(X1), X2, X3) -> mark(U243(X1, X2, X3)) 15.03/5.15 U244(mark(X1), X2, X3) -> mark(U244(X1, X2, X3)) 15.03/5.15 U245(mark(X1), X2) -> mark(U245(X1, X2)) 15.03/5.15 U246(mark(X)) -> mark(U246(X)) 15.03/5.15 U251(mark(X1), X2, X3) -> mark(U251(X1, X2, X3)) 15.03/5.15 U252(mark(X1), X2, X3) -> mark(U252(X1, X2, X3)) 15.03/5.15 U253(mark(X1), X2, X3) -> mark(U253(X1, X2, X3)) 15.03/5.15 U254(mark(X1), X2, X3) -> mark(U254(X1, X2, X3)) 15.03/5.15 U255(mark(X1), X2) -> mark(U255(X1, X2)) 15.03/5.15 U256(mark(X)) -> mark(U256(X)) 15.03/5.15 U261(mark(X1), X2) -> mark(U261(X1, X2)) 15.03/5.15 U262(mark(X)) -> mark(U262(X)) 15.03/5.15 U271(mark(X1), X2) -> mark(U271(X1, X2)) 15.03/5.15 U272(mark(X)) -> mark(U272(X)) 15.03/5.15 U281(mark(X1), X2) -> mark(U281(X1, X2)) 15.03/5.15 U282(mark(X1), X2) -> mark(U282(X1, X2)) 15.03/5.15 cons(mark(X1), X2) -> mark(cons(X1, X2)) 15.03/5.15 natsFrom(mark(X)) -> mark(natsFrom(X)) 15.03/5.15 s(mark(X)) -> mark(s(X)) 15.03/5.15 U291(mark(X1), X2, X3) -> mark(U291(X1, X2, X3)) 15.03/5.15 U292(mark(X1), X2, X3) -> mark(U292(X1, X2, X3)) 15.03/5.15 U293(mark(X1), X2, X3) -> mark(U293(X1, X2, X3)) 15.03/5.15 U294(mark(X1), X2, X3) -> mark(U294(X1, X2, X3)) 15.03/5.15 head(mark(X)) -> mark(head(X)) 15.03/5.15 afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) 15.03/5.15 afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) 15.03/5.15 U301(mark(X1), X2, X3) -> mark(U301(X1, X2, X3)) 15.03/5.15 U302(mark(X1), X2) -> mark(U302(X1, X2)) 15.03/5.15 U303(mark(X1), X2) -> mark(U303(X1, X2)) 15.03/5.15 U304(mark(X1), X2) -> mark(U304(X1, X2)) 15.03/5.15 U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) 15.03/5.15 U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) 15.03/5.15 U311(mark(X1), X2) -> mark(U311(X1, X2)) 15.03/5.15 U312(mark(X1), X2) -> mark(U312(X1, X2)) 15.03/5.15 pair(mark(X1), X2) -> mark(pair(X1, X2)) 15.03/5.15 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 15.03/5.15 U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) 15.03/5.15 U321(mark(X1), X2, X3, X4) -> mark(U321(X1, X2, X3, X4)) 15.03/5.15 U322(mark(X1), X2, X3, X4) -> mark(U322(X1, X2, X3, X4)) 15.03/5.15 U323(mark(X1), X2, X3, X4) -> mark(U323(X1, X2, X3, X4)) 15.03/5.15 U324(mark(X1), X2, X3, X4) -> mark(U324(X1, X2, X3, X4)) 15.03/5.15 U325(mark(X1), X2, X3, X4) -> mark(U325(X1, X2, X3, X4)) 15.03/5.15 U326(mark(X1), X2, X3, X4) -> mark(U326(X1, X2, X3, X4)) 15.03/5.15 U327(mark(X1), X2) -> mark(U327(X1, X2)) 15.03/5.15 U34(mark(X1), X2) -> mark(U34(X1, X2)) 15.03/5.15 U331(mark(X1), X2, X3) -> mark(U331(X1, X2, X3)) 15.03/5.15 U332(mark(X1), X2) -> mark(U332(X1, X2)) 15.03/5.15 U333(mark(X1), X2) -> mark(U333(X1, X2)) 15.03/5.15 U334(mark(X1), X2) -> mark(U334(X1, X2)) 15.03/5.15 U341(mark(X1), X2, X3) -> mark(U341(X1, X2, X3)) 15.03/5.15 U342(mark(X1), X2, X3) -> mark(U342(X1, X2, X3)) 15.03/5.15 U343(mark(X1), X2, X3) -> mark(U343(X1, X2, X3)) 15.03/5.15 U344(mark(X1), X2, X3) -> mark(U344(X1, X2, X3)) 15.03/5.15 fst(mark(X)) -> mark(fst(X)) 15.03/5.15 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 15.03/5.15 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 15.03/5.15 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 15.03/5.15 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 15.03/5.15 U45(mark(X1), X2) -> mark(U45(X1, X2)) 15.03/5.15 U46(mark(X)) -> mark(U46(X)) 15.03/5.15 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 15.03/5.15 U52(mark(X1), X2, X3) -> mark(U52(X1, X2, X3)) 15.03/5.15 U53(mark(X1), X2, X3) -> mark(U53(X1, X2, X3)) 15.03/5.15 U54(mark(X1), X2, X3) -> mark(U54(X1, X2, X3)) 15.03/5.15 U55(mark(X1), X2) -> mark(U55(X1, X2)) 15.03/5.15 U56(mark(X)) -> mark(U56(X)) 15.03/5.15 U61(mark(X1), X2) -> mark(U61(X1, X2)) 15.03/5.15 U62(mark(X1), X2) -> mark(U62(X1, X2)) 15.03/5.15 U63(mark(X)) -> mark(U63(X)) 15.03/5.15 U71(mark(X1), X2) -> mark(U71(X1, X2)) 15.03/5.15 U72(mark(X1), X2) -> mark(U72(X1, X2)) 15.03/5.15 U73(mark(X)) -> mark(U73(X)) 15.03/5.15 U81(mark(X1), X2) -> mark(U81(X1, X2)) 15.03/5.15 U82(mark(X1), X2) -> mark(U82(X1, X2)) 15.03/5.15 U83(mark(X)) -> mark(U83(X)) 15.03/5.15 U91(mark(X1), X2) -> mark(U91(X1, X2)) 15.03/5.15 U92(mark(X1), X2) -> mark(U92(X1, X2)) 15.03/5.15 U93(mark(X)) -> mark(U93(X)) 15.03/5.15 tail(mark(X)) -> mark(tail(X)) 15.03/5.15 take(mark(X1), X2) -> mark(take(X1, X2)) 15.03/5.15 take(X1, mark(X2)) -> mark(take(X1, X2)) 15.03/5.15 sel(mark(X1), X2) -> mark(sel(X1, X2)) 15.03/5.15 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 15.03/5.15 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(tt) -> ok(tt) 15.03/5.15 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(isNaturalKind(X)) -> isNaturalKind(proper(X)) 15.03/5.15 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(isLNatKind(X)) -> isLNatKind(proper(X)) 15.03/5.15 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U105(X1, X2)) -> U105(proper(X1), proper(X2)) 15.03/5.15 proper(isNatural(X)) -> isNatural(proper(X)) 15.03/5.15 proper(U106(X)) -> U106(proper(X)) 15.03/5.15 proper(isLNat(X)) -> isLNat(proper(X)) 15.03/5.15 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U111(X1, X2)) -> U111(proper(X1), proper(X2)) 15.03/5.15 proper(U112(X)) -> U112(proper(X)) 15.03/5.15 proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)) 15.03/5.15 proper(U122(X)) -> U122(proper(X)) 15.03/5.15 proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U131(X)) -> U131(proper(X)) 15.03/5.15 proper(snd(X)) -> snd(proper(X)) 15.03/5.15 proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) 15.03/5.15 proper(U141(X)) -> U141(proper(X)) 15.03/5.15 proper(U151(X)) -> U151(proper(X)) 15.03/5.15 proper(U161(X)) -> U161(proper(X)) 15.03/5.15 proper(U171(X1, X2)) -> U171(proper(X1), proper(X2)) 15.03/5.15 proper(U172(X)) -> U172(proper(X)) 15.03/5.15 proper(U181(X1, X2)) -> U181(proper(X1), proper(X2)) 15.03/5.15 proper(U182(X1, X2)) -> U182(proper(X1), proper(X2)) 15.03/5.15 proper(U183(X)) -> U183(proper(X)) 15.03/5.15 proper(U191(X1, X2)) -> U191(proper(X1), proper(X2)) 15.03/5.15 proper(U192(X1, X2)) -> U192(proper(X1), proper(X2)) 15.03/5.15 proper(U193(X)) -> U193(proper(X)) 15.03/5.15 proper(U201(X1, X2, X3)) -> U201(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U202(X1, X2, X3)) -> U202(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U203(X1, X2, X3)) -> U203(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U204(X1, X2, X3)) -> U204(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U205(X1, X2)) -> U205(proper(X1), proper(X2)) 15.03/5.15 proper(U206(X)) -> U206(proper(X)) 15.03/5.15 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U211(X)) -> U211(proper(X)) 15.03/5.15 proper(U23(X1, X2, X3)) -> U23(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U221(X)) -> U221(proper(X)) 15.03/5.15 proper(U24(X1, X2)) -> U24(proper(X1), proper(X2)) 15.03/5.15 proper(U231(X1, X2)) -> U231(proper(X1), proper(X2)) 15.03/5.15 proper(U232(X)) -> U232(proper(X)) 15.03/5.15 proper(U241(X1, X2, X3)) -> U241(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U242(X1, X2, X3)) -> U242(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U243(X1, X2, X3)) -> U243(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U244(X1, X2, X3)) -> U244(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U245(X1, X2)) -> U245(proper(X1), proper(X2)) 15.03/5.15 proper(U246(X)) -> U246(proper(X)) 15.03/5.15 proper(U251(X1, X2, X3)) -> U251(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U252(X1, X2, X3)) -> U252(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U253(X1, X2, X3)) -> U253(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U254(X1, X2, X3)) -> U254(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U255(X1, X2)) -> U255(proper(X1), proper(X2)) 15.03/5.15 proper(U256(X)) -> U256(proper(X)) 15.03/5.15 proper(U261(X1, X2)) -> U261(proper(X1), proper(X2)) 15.03/5.15 proper(U262(X)) -> U262(proper(X)) 15.03/5.15 proper(U271(X1, X2)) -> U271(proper(X1), proper(X2)) 15.03/5.15 proper(U272(X)) -> U272(proper(X)) 15.03/5.15 proper(U281(X1, X2)) -> U281(proper(X1), proper(X2)) 15.03/5.15 proper(U282(X1, X2)) -> U282(proper(X1), proper(X2)) 15.03/5.15 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 15.03/5.15 proper(natsFrom(X)) -> natsFrom(proper(X)) 15.03/5.15 proper(s(X)) -> s(proper(X)) 15.03/5.15 proper(U291(X1, X2, X3)) -> U291(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U292(X1, X2, X3)) -> U292(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U293(X1, X2, X3)) -> U293(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U294(X1, X2, X3)) -> U294(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(head(X)) -> head(proper(X)) 15.03/5.15 proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) 15.03/5.15 proper(U301(X1, X2, X3)) -> U301(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U302(X1, X2)) -> U302(proper(X1), proper(X2)) 15.03/5.15 proper(U303(X1, X2)) -> U303(proper(X1), proper(X2)) 15.03/5.15 proper(U304(X1, X2)) -> U304(proper(X1), proper(X2)) 15.03/5.15 proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U311(X1, X2)) -> U311(proper(X1), proper(X2)) 15.03/5.15 proper(U312(X1, X2)) -> U312(proper(X1), proper(X2)) 15.03/5.15 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 15.03/5.15 proper(nil) -> ok(nil) 15.03/5.15 proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U321(X1, X2, X3, X4)) -> U321(proper(X1), proper(X2), proper(X3), proper(X4)) 15.03/5.15 proper(U322(X1, X2, X3, X4)) -> U322(proper(X1), proper(X2), proper(X3), proper(X4)) 15.03/5.15 proper(U323(X1, X2, X3, X4)) -> U323(proper(X1), proper(X2), proper(X3), proper(X4)) 15.03/5.15 proper(U324(X1, X2, X3, X4)) -> U324(proper(X1), proper(X2), proper(X3), proper(X4)) 15.03/5.15 proper(U325(X1, X2, X3, X4)) -> U325(proper(X1), proper(X2), proper(X3), proper(X4)) 15.03/5.15 proper(U326(X1, X2, X3, X4)) -> U326(proper(X1), proper(X2), proper(X3), proper(X4)) 15.03/5.15 proper(U327(X1, X2)) -> U327(proper(X1), proper(X2)) 15.03/5.15 proper(U34(X1, X2)) -> U34(proper(X1), proper(X2)) 15.03/5.15 proper(U331(X1, X2, X3)) -> U331(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U332(X1, X2)) -> U332(proper(X1), proper(X2)) 15.03/5.15 proper(U333(X1, X2)) -> U333(proper(X1), proper(X2)) 15.03/5.15 proper(U334(X1, X2)) -> U334(proper(X1), proper(X2)) 15.03/5.15 proper(U341(X1, X2, X3)) -> U341(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U342(X1, X2, X3)) -> U342(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U343(X1, X2, X3)) -> U343(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U344(X1, X2, X3)) -> U344(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(fst(X)) -> fst(proper(X)) 15.03/5.15 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 15.03/5.15 proper(U46(X)) -> U46(proper(X)) 15.03/5.15 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U52(X1, X2, X3)) -> U52(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U53(X1, X2, X3)) -> U53(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U54(X1, X2, X3)) -> U54(proper(X1), proper(X2), proper(X3)) 15.03/5.15 proper(U55(X1, X2)) -> U55(proper(X1), proper(X2)) 15.03/5.15 proper(U56(X)) -> U56(proper(X)) 15.03/5.15 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 15.03/5.15 proper(U62(X1, X2)) -> U62(proper(X1), proper(X2)) 15.03/5.15 proper(isPLNatKind(X)) -> isPLNatKind(proper(X)) 15.03/5.15 proper(U63(X)) -> U63(proper(X)) 15.03/5.15 proper(isPLNat(X)) -> isPLNat(proper(X)) 15.03/5.15 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 15.03/5.15 proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) 15.03/5.15 proper(U73(X)) -> U73(proper(X)) 15.03/5.15 proper(U81(X1, X2)) -> U81(proper(X1), proper(X2)) 15.03/5.15 proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) 15.03/5.15 proper(U83(X)) -> U83(proper(X)) 15.03/5.15 proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) 15.03/5.15 proper(U92(X1, X2)) -> U92(proper(X1), proper(X2)) 15.03/5.15 proper(U93(X)) -> U93(proper(X)) 15.03/5.15 proper(tail(X)) -> tail(proper(X)) 15.03/5.15 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 15.03/5.15 proper(0) -> ok(0) 15.03/5.15 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 15.03/5.15 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 15.03/5.15 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 15.03/5.15 isNaturalKind(ok(X)) -> ok(isNaturalKind(X)) 15.03/5.15 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 15.03/5.15 isLNatKind(ok(X)) -> ok(isLNatKind(X)) 15.03/5.15 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 15.03/5.15 U105(ok(X1), ok(X2)) -> ok(U105(X1, X2)) 15.03/5.15 isNatural(ok(X)) -> ok(isNatural(X)) 15.03/5.15 U106(ok(X)) -> ok(U106(X)) 15.03/5.15 isLNat(ok(X)) -> ok(isLNat(X)) 15.03/5.15 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 15.03/5.15 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 15.03/5.15 U111(ok(X1), ok(X2)) -> ok(U111(X1, X2)) 15.03/5.15 U112(ok(X)) -> ok(U112(X)) 15.03/5.15 U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) 15.03/5.15 U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)) 15.03/5.15 U122(ok(X)) -> ok(U122(X)) 15.03/5.15 U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) 15.03/5.15 U131(ok(X)) -> ok(U131(X)) 15.03/5.15 snd(ok(X)) -> ok(snd(X)) 15.03/5.15 splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) 15.03/5.15 U141(ok(X)) -> ok(U141(X)) 15.03/5.15 U151(ok(X)) -> ok(U151(X)) 15.03/5.15 U161(ok(X)) -> ok(U161(X)) 15.03/5.15 U171(ok(X1), ok(X2)) -> ok(U171(X1, X2)) 15.03/5.15 U172(ok(X)) -> ok(U172(X)) 15.03/5.15 U181(ok(X1), ok(X2)) -> ok(U181(X1, X2)) 15.03/5.15 U182(ok(X1), ok(X2)) -> ok(U182(X1, X2)) 15.03/5.15 U183(ok(X)) -> ok(U183(X)) 15.03/5.15 U191(ok(X1), ok(X2)) -> ok(U191(X1, X2)) 15.03/5.15 U192(ok(X1), ok(X2)) -> ok(U192(X1, X2)) 15.03/5.15 U193(ok(X)) -> ok(U193(X)) 15.03/5.15 U201(ok(X1), ok(X2), ok(X3)) -> ok(U201(X1, X2, X3)) 15.03/5.15 U202(ok(X1), ok(X2), ok(X3)) -> ok(U202(X1, X2, X3)) 15.03/5.15 U203(ok(X1), ok(X2), ok(X3)) -> ok(U203(X1, X2, X3)) 15.03/5.15 U204(ok(X1), ok(X2), ok(X3)) -> ok(U204(X1, X2, X3)) 15.03/5.15 U205(ok(X1), ok(X2)) -> ok(U205(X1, X2)) 15.03/5.15 U206(ok(X)) -> ok(U206(X)) 15.03/5.15 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 15.03/5.15 U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3)) 15.03/5.15 U211(ok(X)) -> ok(U211(X)) 15.03/5.15 U23(ok(X1), ok(X2), ok(X3)) -> ok(U23(X1, X2, X3)) 15.03/5.15 U221(ok(X)) -> ok(U221(X)) 15.03/5.15 U24(ok(X1), ok(X2)) -> ok(U24(X1, X2)) 15.03/5.15 U231(ok(X1), ok(X2)) -> ok(U231(X1, X2)) 15.03/5.15 U232(ok(X)) -> ok(U232(X)) 15.03/5.15 U241(ok(X1), ok(X2), ok(X3)) -> ok(U241(X1, X2, X3)) 15.03/5.15 U242(ok(X1), ok(X2), ok(X3)) -> ok(U242(X1, X2, X3)) 15.03/5.15 U243(ok(X1), ok(X2), ok(X3)) -> ok(U243(X1, X2, X3)) 15.03/5.15 U244(ok(X1), ok(X2), ok(X3)) -> ok(U244(X1, X2, X3)) 15.03/5.15 U245(ok(X1), ok(X2)) -> ok(U245(X1, X2)) 15.03/5.15 U246(ok(X)) -> ok(U246(X)) 15.03/5.15 U251(ok(X1), ok(X2), ok(X3)) -> ok(U251(X1, X2, X3)) 15.03/5.15 U252(ok(X1), ok(X2), ok(X3)) -> ok(U252(X1, X2, X3)) 15.03/5.15 U253(ok(X1), ok(X2), ok(X3)) -> ok(U253(X1, X2, X3)) 15.03/5.15 U254(ok(X1), ok(X2), ok(X3)) -> ok(U254(X1, X2, X3)) 15.03/5.15 U255(ok(X1), ok(X2)) -> ok(U255(X1, X2)) 15.03/5.15 U256(ok(X)) -> ok(U256(X)) 15.03/5.15 U261(ok(X1), ok(X2)) -> ok(U261(X1, X2)) 15.03/5.15 U262(ok(X)) -> ok(U262(X)) 15.03/5.15 U271(ok(X1), ok(X2)) -> ok(U271(X1, X2)) 15.03/5.15 U272(ok(X)) -> ok(U272(X)) 15.03/5.15 U281(ok(X1), ok(X2)) -> ok(U281(X1, X2)) 15.03/5.15 U282(ok(X1), ok(X2)) -> ok(U282(X1, X2)) 15.03/5.15 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 15.03/5.15 natsFrom(ok(X)) -> ok(natsFrom(X)) 15.03/5.15 s(ok(X)) -> ok(s(X)) 15.03/5.15 U291(ok(X1), ok(X2), ok(X3)) -> ok(U291(X1, X2, X3)) 15.03/5.15 U292(ok(X1), ok(X2), ok(X3)) -> ok(U292(X1, X2, X3)) 15.03/5.15 U293(ok(X1), ok(X2), ok(X3)) -> ok(U293(X1, X2, X3)) 15.03/5.15 U294(ok(X1), ok(X2), ok(X3)) -> ok(U294(X1, X2, X3)) 15.03/5.15 head(ok(X)) -> ok(head(X)) 15.03/5.15 afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) 15.03/5.15 U301(ok(X1), ok(X2), ok(X3)) -> ok(U301(X1, X2, X3)) 15.03/5.15 U302(ok(X1), ok(X2)) -> ok(U302(X1, X2)) 15.03/5.15 U303(ok(X1), ok(X2)) -> ok(U303(X1, X2)) 15.03/5.15 U304(ok(X1), ok(X2)) -> ok(U304(X1, X2)) 15.03/5.15 U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) 15.03/5.15 U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) 15.03/5.15 U311(ok(X1), ok(X2)) -> ok(U311(X1, X2)) 15.03/5.15 U312(ok(X1), ok(X2)) -> ok(U312(X1, X2)) 15.03/5.15 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 15.03/5.15 U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) 15.32/5.15 U321(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U321(X1, X2, X3, X4)) 15.32/5.15 U322(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U322(X1, X2, X3, X4)) 15.32/5.15 U323(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U323(X1, X2, X3, X4)) 15.32/5.15 U324(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U324(X1, X2, X3, X4)) 15.32/5.15 U325(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U325(X1, X2, X3, X4)) 15.32/5.15 U326(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U326(X1, X2, X3, X4)) 15.32/5.15 U327(ok(X1), ok(X2)) -> ok(U327(X1, X2)) 15.32/5.15 U34(ok(X1), ok(X2)) -> ok(U34(X1, X2)) 15.32/5.15 U331(ok(X1), ok(X2), ok(X3)) -> ok(U331(X1, X2, X3)) 15.32/5.15 U332(ok(X1), ok(X2)) -> ok(U332(X1, X2)) 15.32/5.15 U333(ok(X1), ok(X2)) -> ok(U333(X1, X2)) 15.32/5.15 U334(ok(X1), ok(X2)) -> ok(U334(X1, X2)) 15.32/5.15 U341(ok(X1), ok(X2), ok(X3)) -> ok(U341(X1, X2, X3)) 15.32/5.15 U342(ok(X1), ok(X2), ok(X3)) -> ok(U342(X1, X2, X3)) 15.32/5.15 U343(ok(X1), ok(X2), ok(X3)) -> ok(U343(X1, X2, X3)) 15.32/5.15 U344(ok(X1), ok(X2), ok(X3)) -> ok(U344(X1, X2, X3)) 15.32/5.15 fst(ok(X)) -> ok(fst(X)) 15.32/5.15 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 15.32/5.15 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 15.32/5.15 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 15.32/5.15 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 15.32/5.15 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 15.32/5.15 U46(ok(X)) -> ok(U46(X)) 15.32/5.15 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 15.32/5.15 U52(ok(X1), ok(X2), ok(X3)) -> ok(U52(X1, X2, X3)) 15.32/5.15 U53(ok(X1), ok(X2), ok(X3)) -> ok(U53(X1, X2, X3)) 15.32/5.15 U54(ok(X1), ok(X2), ok(X3)) -> ok(U54(X1, X2, X3)) 15.32/5.15 U55(ok(X1), ok(X2)) -> ok(U55(X1, X2)) 15.32/5.15 U56(ok(X)) -> ok(U56(X)) 15.32/5.15 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 15.32/5.15 U62(ok(X1), ok(X2)) -> ok(U62(X1, X2)) 15.32/5.15 isPLNatKind(ok(X)) -> ok(isPLNatKind(X)) 15.32/5.15 U63(ok(X)) -> ok(U63(X)) 15.32/5.15 isPLNat(ok(X)) -> ok(isPLNat(X)) 15.32/5.15 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 15.32/5.15 U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) 15.32/5.15 U73(ok(X)) -> ok(U73(X)) 15.32/5.15 U81(ok(X1), ok(X2)) -> ok(U81(X1, X2)) 15.32/5.15 U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) 15.32/5.15 U83(ok(X)) -> ok(U83(X)) 15.32/5.15 U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) 15.32/5.15 U92(ok(X1), ok(X2)) -> ok(U92(X1, X2)) 15.32/5.15 U93(ok(X)) -> ok(U93(X)) 15.32/5.15 tail(ok(X)) -> ok(tail(X)) 15.32/5.15 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 15.32/5.15 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 15.32/5.15 top(mark(X)) -> top(proper(X)) 15.32/5.15 top(ok(X)) -> top(active(X)) 15.32/5.15 15.32/5.15 The set Q consists of the following terms: 15.32/5.15 15.32/5.15 active(afterNth(x0, x1)) 15.32/5.15 active(isLNat(nil)) 15.32/5.15 active(isLNat(afterNth(x0, x1))) 15.32/5.15 active(isLNat(cons(x0, x1))) 15.32/5.15 active(isLNat(fst(x0))) 15.32/5.15 active(isLNat(natsFrom(x0))) 15.32/5.15 active(isLNat(snd(x0))) 15.32/5.15 active(isLNat(tail(x0))) 15.32/5.15 active(isLNat(take(x0, x1))) 15.32/5.15 active(isLNatKind(nil)) 15.32/5.15 active(isLNatKind(afterNth(x0, x1))) 15.32/5.15 active(isLNatKind(cons(x0, x1))) 15.32/5.15 active(isLNatKind(fst(x0))) 15.32/5.15 active(isLNatKind(natsFrom(x0))) 15.32/5.15 active(isLNatKind(snd(x0))) 15.32/5.15 active(isLNatKind(tail(x0))) 15.32/5.15 active(isLNatKind(take(x0, x1))) 15.32/5.15 active(isNatural(0)) 15.32/5.15 active(isNatural(head(x0))) 15.32/5.15 active(isNatural(s(x0))) 15.32/5.15 active(isNatural(sel(x0, x1))) 15.32/5.15 active(isNaturalKind(0)) 15.32/5.15 active(isNaturalKind(head(x0))) 15.32/5.15 active(isNaturalKind(s(x0))) 15.32/5.15 active(isNaturalKind(sel(x0, x1))) 15.32/5.15 active(isPLNat(pair(x0, x1))) 15.32/5.15 active(isPLNat(splitAt(x0, x1))) 15.32/5.15 active(isPLNatKind(pair(x0, x1))) 15.32/5.15 active(isPLNatKind(splitAt(x0, x1))) 15.32/5.15 active(natsFrom(x0)) 15.32/5.15 active(sel(x0, x1)) 15.32/5.15 active(take(x0, x1)) 15.32/5.15 active(U101(x0, x1, x2)) 15.32/5.15 active(U102(x0, x1, x2)) 15.32/5.15 active(U103(x0, x1, x2)) 15.32/5.15 active(U104(x0, x1, x2)) 15.32/5.15 active(U105(x0, x1)) 15.32/5.15 active(U106(x0)) 15.32/5.15 active(U11(x0, x1, x2)) 15.32/5.15 active(U12(x0, x1, x2)) 15.32/5.15 active(U111(x0, x1)) 15.32/5.15 active(U112(x0)) 15.32/5.15 active(U13(x0, x1, x2)) 15.32/5.15 active(U121(x0, x1)) 15.32/5.15 active(U122(x0)) 15.32/5.15 active(U14(x0, x1, x2)) 15.32/5.15 active(U131(x0)) 15.32/5.15 active(snd(x0)) 15.32/5.15 active(splitAt(x0, x1)) 15.32/5.15 active(U141(x0)) 15.32/5.15 active(U151(x0)) 15.32/5.15 active(U161(x0)) 15.32/5.15 active(U171(x0, x1)) 15.32/5.15 active(U172(x0)) 15.32/5.15 active(U181(x0, x1)) 15.32/5.15 active(U182(x0, x1)) 15.32/5.15 active(U183(x0)) 15.32/5.15 active(U191(x0, x1)) 15.32/5.15 active(U192(x0, x1)) 15.32/5.15 active(U193(x0)) 15.32/5.15 active(U201(x0, x1, x2)) 15.32/5.15 active(U202(x0, x1, x2)) 15.32/5.15 active(U203(x0, x1, x2)) 15.32/5.15 active(U204(x0, x1, x2)) 15.32/5.15 active(U205(x0, x1)) 15.32/5.15 active(U206(x0)) 15.32/5.15 active(U21(x0, x1, x2)) 15.32/5.15 active(U22(x0, x1, x2)) 15.32/5.15 active(U211(x0)) 15.32/5.15 active(U23(x0, x1, x2)) 15.32/5.15 active(U221(x0)) 15.32/5.15 active(U24(x0, x1)) 15.32/5.15 active(U231(x0, x1)) 15.32/5.15 active(U232(x0)) 15.32/5.15 active(U241(x0, x1, x2)) 15.32/5.15 active(U242(x0, x1, x2)) 15.32/5.15 active(U243(x0, x1, x2)) 15.32/5.15 active(U244(x0, x1, x2)) 15.32/5.15 active(U245(x0, x1)) 15.32/5.15 active(U246(x0)) 15.32/5.15 active(U251(x0, x1, x2)) 15.32/5.15 active(U252(x0, x1, x2)) 15.32/5.15 active(U253(x0, x1, x2)) 15.32/5.15 active(U254(x0, x1, x2)) 15.32/5.15 active(U255(x0, x1)) 15.32/5.15 active(U256(x0)) 15.32/5.15 active(U261(x0, x1)) 15.32/5.15 active(U262(x0)) 15.32/5.15 active(U271(x0, x1)) 15.32/5.15 active(U272(x0)) 15.32/5.15 active(U281(x0, x1)) 15.32/5.15 active(U282(x0, x1)) 15.32/5.15 active(cons(x0, x1)) 15.32/5.15 active(s(x0)) 15.32/5.15 active(U291(x0, x1, x2)) 15.32/5.15 active(U292(x0, x1, x2)) 15.32/5.15 active(U293(x0, x1, x2)) 15.32/5.15 active(U294(x0, x1, x2)) 15.32/5.15 active(head(x0)) 15.32/5.15 active(U301(x0, x1, x2)) 15.32/5.15 active(U302(x0, x1)) 15.32/5.15 active(U303(x0, x1)) 15.32/5.15 active(U304(x0, x1)) 15.32/5.15 active(U31(x0, x1, x2)) 15.32/5.15 active(U32(x0, x1, x2)) 15.32/5.15 active(U311(x0, x1)) 15.32/5.15 active(U312(x0, x1)) 15.32/5.15 active(pair(x0, x1)) 15.32/5.15 active(U33(x0, x1, x2)) 15.32/5.15 active(U321(x0, x1, x2, x3)) 15.32/5.15 active(U322(x0, x1, x2, x3)) 15.32/5.15 active(U323(x0, x1, x2, x3)) 15.32/5.15 active(U324(x0, x1, x2, x3)) 15.32/5.15 active(U325(x0, x1, x2, x3)) 15.32/5.15 active(U326(x0, x1, x2, x3)) 15.32/5.15 active(U327(x0, x1)) 15.32/5.15 active(U34(x0, x1)) 15.32/5.15 active(U331(x0, x1, x2)) 15.32/5.15 active(U332(x0, x1)) 15.32/5.15 active(U333(x0, x1)) 15.32/5.15 active(U334(x0, x1)) 15.32/5.15 active(U341(x0, x1, x2)) 15.32/5.15 active(U342(x0, x1, x2)) 15.32/5.15 active(U343(x0, x1, x2)) 15.32/5.15 active(U344(x0, x1, x2)) 15.32/5.15 active(fst(x0)) 15.32/5.15 active(U41(x0, x1, x2)) 15.32/5.15 active(U42(x0, x1, x2)) 15.32/5.15 active(U43(x0, x1, x2)) 15.32/5.15 active(U44(x0, x1, x2)) 15.32/5.15 active(U45(x0, x1)) 15.32/5.15 active(U46(x0)) 15.32/5.15 active(U51(x0, x1, x2)) 15.32/5.15 active(U52(x0, x1, x2)) 15.32/5.15 active(U53(x0, x1, x2)) 15.32/5.15 active(U54(x0, x1, x2)) 15.32/5.15 active(U55(x0, x1)) 15.32/5.15 active(U56(x0)) 15.32/5.15 active(U61(x0, x1)) 15.32/5.15 active(U62(x0, x1)) 15.32/5.15 active(U63(x0)) 15.32/5.15 active(U71(x0, x1)) 15.32/5.15 active(U72(x0, x1)) 15.32/5.15 active(U73(x0)) 15.32/5.15 active(U81(x0, x1)) 15.32/5.15 active(U82(x0, x1)) 15.32/5.15 active(U83(x0)) 15.32/5.15 active(U91(x0, x1)) 15.32/5.15 active(U92(x0, x1)) 15.32/5.15 active(U93(x0)) 15.32/5.15 active(tail(x0)) 15.32/5.15 U101(mark(x0), x1, x2) 15.32/5.15 U102(mark(x0), x1, x2) 15.32/5.15 U103(mark(x0), x1, x2) 15.32/5.15 U104(mark(x0), x1, x2) 15.32/5.15 U105(mark(x0), x1) 15.32/5.15 U106(mark(x0)) 15.32/5.15 U11(mark(x0), x1, x2) 15.32/5.15 U12(mark(x0), x1, x2) 15.32/5.15 U111(mark(x0), x1) 15.32/5.15 U112(mark(x0)) 15.32/5.15 U13(mark(x0), x1, x2) 15.32/5.15 U121(mark(x0), x1) 15.32/5.15 U122(mark(x0)) 15.32/5.15 U14(mark(x0), x1, x2) 15.32/5.15 U131(mark(x0)) 15.32/5.15 snd(mark(x0)) 15.32/5.15 splitAt(mark(x0), x1) 15.32/5.15 splitAt(x0, mark(x1)) 15.32/5.15 U141(mark(x0)) 15.32/5.15 U151(mark(x0)) 15.32/5.15 U161(mark(x0)) 15.32/5.15 U171(mark(x0), x1) 15.32/5.15 U172(mark(x0)) 15.32/5.15 U181(mark(x0), x1) 15.32/5.15 U182(mark(x0), x1) 15.32/5.15 U183(mark(x0)) 15.32/5.15 U191(mark(x0), x1) 15.32/5.15 U192(mark(x0), x1) 15.32/5.15 U193(mark(x0)) 15.32/5.15 U201(mark(x0), x1, x2) 15.32/5.15 U202(mark(x0), x1, x2) 15.32/5.15 U203(mark(x0), x1, x2) 15.32/5.15 U204(mark(x0), x1, x2) 15.32/5.15 U205(mark(x0), x1) 15.32/5.15 U206(mark(x0)) 15.32/5.15 U21(mark(x0), x1, x2) 15.32/5.15 U22(mark(x0), x1, x2) 15.32/5.15 U211(mark(x0)) 15.32/5.15 U23(mark(x0), x1, x2) 15.32/5.15 U221(mark(x0)) 15.32/5.15 U24(mark(x0), x1) 15.32/5.15 U231(mark(x0), x1) 15.32/5.15 U232(mark(x0)) 15.32/5.15 U241(mark(x0), x1, x2) 15.32/5.15 U242(mark(x0), x1, x2) 15.32/5.15 U243(mark(x0), x1, x2) 15.32/5.15 U244(mark(x0), x1, x2) 15.32/5.15 U245(mark(x0), x1) 15.32/5.15 U246(mark(x0)) 15.32/5.15 U251(mark(x0), x1, x2) 15.32/5.15 U252(mark(x0), x1, x2) 15.32/5.15 U253(mark(x0), x1, x2) 15.32/5.15 U254(mark(x0), x1, x2) 15.32/5.15 U255(mark(x0), x1) 15.32/5.15 U256(mark(x0)) 15.32/5.15 U261(mark(x0), x1) 15.32/5.15 U262(mark(x0)) 15.32/5.15 U271(mark(x0), x1) 15.32/5.15 U272(mark(x0)) 15.32/5.15 U281(mark(x0), x1) 15.32/5.15 U282(mark(x0), x1) 15.32/5.15 cons(mark(x0), x1) 15.32/5.15 natsFrom(mark(x0)) 15.32/5.15 s(mark(x0)) 15.32/5.15 U291(mark(x0), x1, x2) 15.32/5.15 U292(mark(x0), x1, x2) 15.32/5.15 U293(mark(x0), x1, x2) 15.32/5.15 U294(mark(x0), x1, x2) 15.32/5.15 head(mark(x0)) 15.32/5.15 afterNth(mark(x0), x1) 15.32/5.15 afterNth(x0, mark(x1)) 15.32/5.15 U301(mark(x0), x1, x2) 15.32/5.15 U302(mark(x0), x1) 15.32/5.15 U303(mark(x0), x1) 15.32/5.15 U304(mark(x0), x1) 15.32/5.15 U31(mark(x0), x1, x2) 15.32/5.15 U32(mark(x0), x1, x2) 15.32/5.15 U311(mark(x0), x1) 15.32/5.15 U312(mark(x0), x1) 15.32/5.15 pair(mark(x0), x1) 15.32/5.15 pair(x0, mark(x1)) 15.32/5.15 U33(mark(x0), x1, x2) 15.32/5.15 U321(mark(x0), x1, x2, x3) 15.32/5.15 U322(mark(x0), x1, x2, x3) 15.32/5.15 U323(mark(x0), x1, x2, x3) 15.32/5.15 U324(mark(x0), x1, x2, x3) 15.32/5.15 U325(mark(x0), x1, x2, x3) 15.32/5.15 U326(mark(x0), x1, x2, x3) 15.32/5.15 U327(mark(x0), x1) 15.32/5.15 U34(mark(x0), x1) 15.32/5.15 U331(mark(x0), x1, x2) 15.32/5.15 U332(mark(x0), x1) 15.32/5.15 U333(mark(x0), x1) 15.32/5.15 U334(mark(x0), x1) 15.32/5.15 U341(mark(x0), x1, x2) 15.32/5.15 U342(mark(x0), x1, x2) 15.32/5.15 U343(mark(x0), x1, x2) 15.32/5.15 U344(mark(x0), x1, x2) 15.32/5.15 fst(mark(x0)) 15.32/5.15 U41(mark(x0), x1, x2) 15.32/5.15 U42(mark(x0), x1, x2) 15.32/5.15 U43(mark(x0), x1, x2) 15.32/5.15 U44(mark(x0), x1, x2) 15.32/5.15 U45(mark(x0), x1) 15.32/5.15 U46(mark(x0)) 15.32/5.15 U51(mark(x0), x1, x2) 15.32/5.15 U52(mark(x0), x1, x2) 15.32/5.15 U53(mark(x0), x1, x2) 15.32/5.15 U54(mark(x0), x1, x2) 15.32/5.15 U55(mark(x0), x1) 15.32/5.15 U56(mark(x0)) 15.32/5.15 U61(mark(x0), x1) 15.32/5.15 U62(mark(x0), x1) 15.32/5.15 U63(mark(x0)) 15.32/5.15 U71(mark(x0), x1) 15.32/5.15 U72(mark(x0), x1) 15.32/5.15 U73(mark(x0)) 15.32/5.15 U81(mark(x0), x1) 15.32/5.15 U82(mark(x0), x1) 15.32/5.15 U83(mark(x0)) 15.32/5.15 U91(mark(x0), x1) 15.32/5.15 U92(mark(x0), x1) 15.32/5.15 U93(mark(x0)) 15.32/5.15 tail(mark(x0)) 15.32/5.15 take(mark(x0), x1) 15.32/5.15 take(x0, mark(x1)) 15.32/5.15 sel(mark(x0), x1) 15.32/5.15 sel(x0, mark(x1)) 15.32/5.15 proper(U101(x0, x1, x2)) 15.32/5.15 proper(tt) 15.32/5.15 proper(U102(x0, x1, x2)) 15.32/5.15 proper(isNaturalKind(x0)) 15.32/5.15 proper(U103(x0, x1, x2)) 15.32/5.15 proper(isLNatKind(x0)) 15.32/5.15 proper(U104(x0, x1, x2)) 15.32/5.15 proper(U105(x0, x1)) 15.32/5.15 proper(isNatural(x0)) 15.32/5.15 proper(U106(x0)) 15.32/5.15 proper(isLNat(x0)) 15.32/5.15 proper(U11(x0, x1, x2)) 15.32/5.15 proper(U12(x0, x1, x2)) 15.32/5.15 proper(U111(x0, x1)) 15.32/5.15 proper(U112(x0)) 15.32/5.15 proper(U13(x0, x1, x2)) 15.32/5.15 proper(U121(x0, x1)) 15.32/5.15 proper(U122(x0)) 15.32/5.15 proper(U14(x0, x1, x2)) 15.32/5.15 proper(U131(x0)) 15.32/5.15 proper(snd(x0)) 15.32/5.15 proper(splitAt(x0, x1)) 15.32/5.15 proper(U141(x0)) 15.32/5.15 proper(U151(x0)) 15.32/5.15 proper(U161(x0)) 15.32/5.15 proper(U171(x0, x1)) 15.32/5.15 proper(U172(x0)) 15.32/5.15 proper(U181(x0, x1)) 15.32/5.15 proper(U182(x0, x1)) 15.32/5.15 proper(U183(x0)) 15.32/5.15 proper(U191(x0, x1)) 15.32/5.15 proper(U192(x0, x1)) 15.32/5.15 proper(U193(x0)) 15.32/5.15 proper(U201(x0, x1, x2)) 15.32/5.15 proper(U202(x0, x1, x2)) 15.32/5.15 proper(U203(x0, x1, x2)) 15.32/5.15 proper(U204(x0, x1, x2)) 15.32/5.15 proper(U205(x0, x1)) 15.32/5.15 proper(U206(x0)) 15.32/5.15 proper(U21(x0, x1, x2)) 15.32/5.15 proper(U22(x0, x1, x2)) 15.32/5.15 proper(U211(x0)) 15.32/5.15 proper(U23(x0, x1, x2)) 15.32/5.15 proper(U221(x0)) 15.32/5.15 proper(U24(x0, x1)) 15.32/5.15 proper(U231(x0, x1)) 15.32/5.15 proper(U232(x0)) 15.32/5.15 proper(U241(x0, x1, x2)) 15.32/5.15 proper(U242(x0, x1, x2)) 15.32/5.15 proper(U243(x0, x1, x2)) 15.32/5.15 proper(U244(x0, x1, x2)) 15.32/5.15 proper(U245(x0, x1)) 15.32/5.15 proper(U246(x0)) 15.32/5.15 proper(U251(x0, x1, x2)) 15.32/5.15 proper(U252(x0, x1, x2)) 15.32/5.15 proper(U253(x0, x1, x2)) 15.32/5.15 proper(U254(x0, x1, x2)) 15.32/5.15 proper(U255(x0, x1)) 15.32/5.15 proper(U256(x0)) 15.32/5.15 proper(U261(x0, x1)) 15.32/5.15 proper(U262(x0)) 15.32/5.15 proper(U271(x0, x1)) 15.32/5.15 proper(U272(x0)) 15.32/5.15 proper(U281(x0, x1)) 15.32/5.15 proper(U282(x0, x1)) 15.32/5.15 proper(cons(x0, x1)) 15.32/5.15 proper(natsFrom(x0)) 15.32/5.15 proper(s(x0)) 15.32/5.15 proper(U291(x0, x1, x2)) 15.32/5.15 proper(U292(x0, x1, x2)) 15.32/5.15 proper(U293(x0, x1, x2)) 15.32/5.15 proper(U294(x0, x1, x2)) 15.32/5.15 proper(head(x0)) 15.32/5.15 proper(afterNth(x0, x1)) 15.32/5.15 proper(U301(x0, x1, x2)) 15.32/5.15 proper(U302(x0, x1)) 15.32/5.15 proper(U303(x0, x1)) 15.32/5.15 proper(U304(x0, x1)) 15.32/5.15 proper(U31(x0, x1, x2)) 15.32/5.15 proper(U32(x0, x1, x2)) 15.32/5.15 proper(U311(x0, x1)) 15.32/5.15 proper(U312(x0, x1)) 15.32/5.15 proper(pair(x0, x1)) 15.32/5.15 proper(nil) 15.32/5.15 proper(U33(x0, x1, x2)) 15.32/5.15 proper(U321(x0, x1, x2, x3)) 15.32/5.15 proper(U322(x0, x1, x2, x3)) 15.32/5.15 proper(U323(x0, x1, x2, x3)) 15.32/5.15 proper(U324(x0, x1, x2, x3)) 15.32/5.15 proper(U325(x0, x1, x2, x3)) 15.32/5.15 proper(U326(x0, x1, x2, x3)) 15.32/5.15 proper(U327(x0, x1)) 15.32/5.15 proper(U34(x0, x1)) 15.32/5.15 proper(U331(x0, x1, x2)) 15.32/5.15 proper(U332(x0, x1)) 15.32/5.15 proper(U333(x0, x1)) 15.32/5.15 proper(U334(x0, x1)) 15.32/5.15 proper(U341(x0, x1, x2)) 15.32/5.15 proper(U342(x0, x1, x2)) 15.32/5.15 proper(U343(x0, x1, x2)) 15.32/5.15 proper(U344(x0, x1, x2)) 15.32/5.15 proper(fst(x0)) 15.32/5.15 proper(U41(x0, x1, x2)) 15.32/5.15 proper(U42(x0, x1, x2)) 15.32/5.15 proper(U43(x0, x1, x2)) 15.32/5.15 proper(U44(x0, x1, x2)) 15.32/5.15 proper(U45(x0, x1)) 15.32/5.15 proper(U46(x0)) 15.32/5.15 proper(U51(x0, x1, x2)) 15.32/5.15 proper(U52(x0, x1, x2)) 15.32/5.15 proper(U53(x0, x1, x2)) 15.32/5.15 proper(U54(x0, x1, x2)) 15.32/5.15 proper(U55(x0, x1)) 15.32/5.15 proper(U56(x0)) 15.32/5.15 proper(U61(x0, x1)) 15.32/5.15 proper(U62(x0, x1)) 15.32/5.15 proper(isPLNatKind(x0)) 15.32/5.15 proper(U63(x0)) 15.32/5.15 proper(isPLNat(x0)) 15.32/5.15 proper(U71(x0, x1)) 15.32/5.15 proper(U72(x0, x1)) 15.32/5.15 proper(U73(x0)) 15.32/5.15 proper(U81(x0, x1)) 15.32/5.15 proper(U82(x0, x1)) 15.32/5.15 proper(U83(x0)) 15.32/5.15 proper(U91(x0, x1)) 15.32/5.15 proper(U92(x0, x1)) 15.32/5.15 proper(U93(x0)) 15.32/5.15 proper(tail(x0)) 15.32/5.15 proper(take(x0, x1)) 15.32/5.15 proper(0) 15.32/5.15 proper(sel(x0, x1)) 15.32/5.15 U101(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U102(ok(x0), ok(x1), ok(x2)) 15.32/5.15 isNaturalKind(ok(x0)) 15.32/5.15 U103(ok(x0), ok(x1), ok(x2)) 15.32/5.15 isLNatKind(ok(x0)) 15.32/5.15 U104(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U105(ok(x0), ok(x1)) 15.32/5.15 isNatural(ok(x0)) 15.32/5.15 U106(ok(x0)) 15.32/5.15 isLNat(ok(x0)) 15.32/5.15 U11(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U12(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U111(ok(x0), ok(x1)) 15.32/5.15 U112(ok(x0)) 15.32/5.15 U13(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U121(ok(x0), ok(x1)) 15.32/5.15 U122(ok(x0)) 15.32/5.15 U14(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U131(ok(x0)) 15.32/5.15 snd(ok(x0)) 15.32/5.15 splitAt(ok(x0), ok(x1)) 15.32/5.15 U141(ok(x0)) 15.32/5.15 U151(ok(x0)) 15.32/5.15 U161(ok(x0)) 15.32/5.15 U171(ok(x0), ok(x1)) 15.32/5.15 U172(ok(x0)) 15.32/5.15 U181(ok(x0), ok(x1)) 15.32/5.15 U182(ok(x0), ok(x1)) 15.32/5.15 U183(ok(x0)) 15.32/5.15 U191(ok(x0), ok(x1)) 15.32/5.15 U192(ok(x0), ok(x1)) 15.32/5.15 U193(ok(x0)) 15.32/5.15 U201(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U202(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U203(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U204(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U205(ok(x0), ok(x1)) 15.32/5.15 U206(ok(x0)) 15.32/5.15 U21(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U22(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U211(ok(x0)) 15.32/5.15 U23(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U221(ok(x0)) 15.32/5.15 U24(ok(x0), ok(x1)) 15.32/5.15 U231(ok(x0), ok(x1)) 15.32/5.15 U232(ok(x0)) 15.32/5.15 U241(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U242(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U243(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U244(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U245(ok(x0), ok(x1)) 15.32/5.15 U246(ok(x0)) 15.32/5.15 U251(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U252(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U253(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U254(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U255(ok(x0), ok(x1)) 15.32/5.15 U256(ok(x0)) 15.32/5.15 U261(ok(x0), ok(x1)) 15.32/5.15 U262(ok(x0)) 15.32/5.15 U271(ok(x0), ok(x1)) 15.32/5.15 U272(ok(x0)) 15.32/5.15 U281(ok(x0), ok(x1)) 15.32/5.15 U282(ok(x0), ok(x1)) 15.32/5.15 cons(ok(x0), ok(x1)) 15.32/5.15 natsFrom(ok(x0)) 15.32/5.15 s(ok(x0)) 15.32/5.15 U291(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U292(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U293(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U294(ok(x0), ok(x1), ok(x2)) 15.32/5.15 head(ok(x0)) 15.32/5.15 afterNth(ok(x0), ok(x1)) 15.32/5.15 U301(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U302(ok(x0), ok(x1)) 15.32/5.15 U303(ok(x0), ok(x1)) 15.32/5.15 U304(ok(x0), ok(x1)) 15.32/5.15 U31(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U32(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U311(ok(x0), ok(x1)) 15.32/5.15 U312(ok(x0), ok(x1)) 15.32/5.15 pair(ok(x0), ok(x1)) 15.32/5.15 U33(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U321(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.15 U322(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.15 U323(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.15 U324(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.15 U325(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.15 U326(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.15 U327(ok(x0), ok(x1)) 15.32/5.15 U34(ok(x0), ok(x1)) 15.32/5.15 U331(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U332(ok(x0), ok(x1)) 15.32/5.15 U333(ok(x0), ok(x1)) 15.32/5.15 U334(ok(x0), ok(x1)) 15.32/5.15 U341(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U342(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U343(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U344(ok(x0), ok(x1), ok(x2)) 15.32/5.15 fst(ok(x0)) 15.32/5.15 U41(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U42(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U43(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U44(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U45(ok(x0), ok(x1)) 15.32/5.15 U46(ok(x0)) 15.32/5.15 U51(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U52(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U53(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U54(ok(x0), ok(x1), ok(x2)) 15.32/5.15 U55(ok(x0), ok(x1)) 15.32/5.15 U56(ok(x0)) 15.32/5.15 U61(ok(x0), ok(x1)) 15.32/5.15 U62(ok(x0), ok(x1)) 15.32/5.15 isPLNatKind(ok(x0)) 15.32/5.15 U63(ok(x0)) 15.32/5.15 isPLNat(ok(x0)) 15.32/5.15 U71(ok(x0), ok(x1)) 15.32/5.15 U72(ok(x0), ok(x1)) 15.32/5.15 U73(ok(x0)) 15.32/5.15 U81(ok(x0), ok(x1)) 15.32/5.15 U82(ok(x0), ok(x1)) 15.32/5.15 U83(ok(x0)) 15.32/5.15 U91(ok(x0), ok(x1)) 15.32/5.15 U92(ok(x0), ok(x1)) 15.32/5.15 U93(ok(x0)) 15.32/5.15 tail(ok(x0)) 15.32/5.15 take(ok(x0), ok(x1)) 15.32/5.15 sel(ok(x0), ok(x1)) 15.32/5.15 top(mark(x0)) 15.32/5.15 top(ok(x0)) 15.32/5.15 15.32/5.15 15.32/5.15 ---------------------------------------- 15.32/5.15 15.32/5.15 (1) QTRSToCSRProof (SOUND) 15.32/5.15 The following Q TRS is given: Q restricted rewrite system: 15.32/5.15 The TRS R consists of the following rules: 15.32/5.15 15.32/5.15 active(U101(tt, V1, V2)) -> mark(U102(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(U102(tt, V1, V2)) -> mark(U103(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U103(tt, V1, V2)) -> mark(U104(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U104(tt, V1, V2)) -> mark(U105(isNatural(V1), V2)) 15.32/5.15 active(U105(tt, V2)) -> mark(U106(isLNat(V2))) 15.32/5.15 active(U106(tt)) -> mark(tt) 15.32/5.15 active(U11(tt, N, XS)) -> mark(U12(isNaturalKind(N), N, XS)) 15.32/5.15 active(U111(tt, V2)) -> mark(U112(isLNatKind(V2))) 15.32/5.15 active(U112(tt)) -> mark(tt) 15.32/5.15 active(U12(tt, N, XS)) -> mark(U13(isLNat(XS), N, XS)) 15.32/5.15 active(U121(tt, V2)) -> mark(U122(isLNatKind(V2))) 15.32/5.15 active(U122(tt)) -> mark(tt) 15.32/5.15 active(U13(tt, N, XS)) -> mark(U14(isLNatKind(XS), N, XS)) 15.32/5.15 active(U131(tt)) -> mark(tt) 15.32/5.15 active(U14(tt, N, XS)) -> mark(snd(splitAt(N, XS))) 15.32/5.15 active(U141(tt)) -> mark(tt) 15.32/5.15 active(U151(tt)) -> mark(tt) 15.32/5.15 active(U161(tt)) -> mark(tt) 15.32/5.15 active(U171(tt, V2)) -> mark(U172(isLNatKind(V2))) 15.32/5.15 active(U172(tt)) -> mark(tt) 15.32/5.15 active(U181(tt, V1)) -> mark(U182(isLNatKind(V1), V1)) 15.32/5.15 active(U182(tt, V1)) -> mark(U183(isLNat(V1))) 15.32/5.15 active(U183(tt)) -> mark(tt) 15.32/5.15 active(U191(tt, V1)) -> mark(U192(isNaturalKind(V1), V1)) 15.32/5.15 active(U192(tt, V1)) -> mark(U193(isNatural(V1))) 15.32/5.15 active(U193(tt)) -> mark(tt) 15.32/5.15 active(U201(tt, V1, V2)) -> mark(U202(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(U202(tt, V1, V2)) -> mark(U203(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U203(tt, V1, V2)) -> mark(U204(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U204(tt, V1, V2)) -> mark(U205(isNatural(V1), V2)) 15.32/5.15 active(U205(tt, V2)) -> mark(U206(isLNat(V2))) 15.32/5.15 active(U206(tt)) -> mark(tt) 15.32/5.15 active(U21(tt, X, Y)) -> mark(U22(isLNatKind(X), X, Y)) 15.32/5.15 active(U211(tt)) -> mark(tt) 15.32/5.15 active(U22(tt, X, Y)) -> mark(U23(isLNat(Y), X, Y)) 15.32/5.15 active(U221(tt)) -> mark(tt) 15.32/5.15 active(U23(tt, X, Y)) -> mark(U24(isLNatKind(Y), X)) 15.32/5.15 active(U231(tt, V2)) -> mark(U232(isLNatKind(V2))) 15.32/5.15 active(U232(tt)) -> mark(tt) 15.32/5.15 active(U24(tt, X)) -> mark(X) 15.32/5.15 active(U241(tt, V1, V2)) -> mark(U242(isLNatKind(V1), V1, V2)) 15.32/5.15 active(U242(tt, V1, V2)) -> mark(U243(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U243(tt, V1, V2)) -> mark(U244(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U244(tt, V1, V2)) -> mark(U245(isLNat(V1), V2)) 15.32/5.15 active(U245(tt, V2)) -> mark(U246(isLNat(V2))) 15.32/5.15 active(U246(tt)) -> mark(tt) 15.32/5.15 active(U251(tt, V1, V2)) -> mark(U252(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(U252(tt, V1, V2)) -> mark(U253(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U253(tt, V1, V2)) -> mark(U254(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U254(tt, V1, V2)) -> mark(U255(isNatural(V1), V2)) 15.32/5.15 active(U255(tt, V2)) -> mark(U256(isLNat(V2))) 15.32/5.15 active(U256(tt)) -> mark(tt) 15.32/5.15 active(U261(tt, V2)) -> mark(U262(isLNatKind(V2))) 15.32/5.15 active(U262(tt)) -> mark(tt) 15.32/5.15 active(U271(tt, V2)) -> mark(U272(isLNatKind(V2))) 15.32/5.15 active(U272(tt)) -> mark(tt) 15.32/5.15 active(U281(tt, N)) -> mark(U282(isNaturalKind(N), N)) 15.32/5.15 active(U282(tt, N)) -> mark(cons(N, natsFrom(s(N)))) 15.32/5.15 active(U291(tt, N, XS)) -> mark(U292(isNaturalKind(N), N, XS)) 15.32/5.15 active(U292(tt, N, XS)) -> mark(U293(isLNat(XS), N, XS)) 15.32/5.15 active(U293(tt, N, XS)) -> mark(U294(isLNatKind(XS), N, XS)) 15.32/5.15 active(U294(tt, N, XS)) -> mark(head(afterNth(N, XS))) 15.32/5.15 active(U301(tt, X, Y)) -> mark(U302(isLNatKind(X), Y)) 15.32/5.15 active(U302(tt, Y)) -> mark(U303(isLNat(Y), Y)) 15.32/5.15 active(U303(tt, Y)) -> mark(U304(isLNatKind(Y), Y)) 15.32/5.15 active(U304(tt, Y)) -> mark(Y) 15.32/5.15 active(U31(tt, N, XS)) -> mark(U32(isNaturalKind(N), N, XS)) 15.32/5.15 active(U311(tt, XS)) -> mark(U312(isLNatKind(XS), XS)) 15.32/5.15 active(U312(tt, XS)) -> mark(pair(nil, XS)) 15.32/5.15 active(U32(tt, N, XS)) -> mark(U33(isLNat(XS), N, XS)) 15.32/5.15 active(U321(tt, N, X, XS)) -> mark(U322(isNaturalKind(N), N, X, XS)) 15.32/5.15 active(U322(tt, N, X, XS)) -> mark(U323(isNatural(X), N, X, XS)) 15.32/5.15 active(U323(tt, N, X, XS)) -> mark(U324(isNaturalKind(X), N, X, XS)) 15.32/5.15 active(U324(tt, N, X, XS)) -> mark(U325(isLNat(XS), N, X, XS)) 15.32/5.15 active(U325(tt, N, X, XS)) -> mark(U326(isLNatKind(XS), N, X, XS)) 15.32/5.15 active(U326(tt, N, X, XS)) -> mark(U327(splitAt(N, XS), X)) 15.32/5.15 active(U327(pair(YS, ZS), X)) -> mark(pair(cons(X, YS), ZS)) 15.32/5.15 active(U33(tt, N, XS)) -> mark(U34(isLNatKind(XS), N)) 15.32/5.15 active(U331(tt, N, XS)) -> mark(U332(isNaturalKind(N), XS)) 15.32/5.15 active(U332(tt, XS)) -> mark(U333(isLNat(XS), XS)) 15.32/5.15 active(U333(tt, XS)) -> mark(U334(isLNatKind(XS), XS)) 15.32/5.15 active(U334(tt, XS)) -> mark(XS) 15.32/5.15 active(U34(tt, N)) -> mark(N) 15.32/5.15 active(U341(tt, N, XS)) -> mark(U342(isNaturalKind(N), N, XS)) 15.32/5.15 active(U342(tt, N, XS)) -> mark(U343(isLNat(XS), N, XS)) 15.32/5.15 active(U343(tt, N, XS)) -> mark(U344(isLNatKind(XS), N, XS)) 15.32/5.15 active(U344(tt, N, XS)) -> mark(fst(splitAt(N, XS))) 15.32/5.15 active(U41(tt, V1, V2)) -> mark(U42(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(U42(tt, V1, V2)) -> mark(U43(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U43(tt, V1, V2)) -> mark(U44(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U44(tt, V1, V2)) -> mark(U45(isNatural(V1), V2)) 15.32/5.15 active(U45(tt, V2)) -> mark(U46(isLNat(V2))) 15.32/5.15 active(U46(tt)) -> mark(tt) 15.32/5.15 active(U51(tt, V1, V2)) -> mark(U52(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(U52(tt, V1, V2)) -> mark(U53(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U53(tt, V1, V2)) -> mark(U54(isLNatKind(V2), V1, V2)) 15.32/5.15 active(U54(tt, V1, V2)) -> mark(U55(isNatural(V1), V2)) 15.32/5.15 active(U55(tt, V2)) -> mark(U56(isLNat(V2))) 15.32/5.15 active(U56(tt)) -> mark(tt) 15.32/5.15 active(U61(tt, V1)) -> mark(U62(isPLNatKind(V1), V1)) 15.32/5.15 active(U62(tt, V1)) -> mark(U63(isPLNat(V1))) 15.32/5.15 active(U63(tt)) -> mark(tt) 15.32/5.15 active(U71(tt, V1)) -> mark(U72(isNaturalKind(V1), V1)) 15.32/5.15 active(U72(tt, V1)) -> mark(U73(isNatural(V1))) 15.32/5.15 active(U73(tt)) -> mark(tt) 15.32/5.15 active(U81(tt, V1)) -> mark(U82(isPLNatKind(V1), V1)) 15.32/5.15 active(U82(tt, V1)) -> mark(U83(isPLNat(V1))) 15.32/5.15 active(U83(tt)) -> mark(tt) 15.32/5.15 active(U91(tt, V1)) -> mark(U92(isLNatKind(V1), V1)) 15.32/5.15 active(U92(tt, V1)) -> mark(U93(isLNat(V1))) 15.32/5.15 active(U93(tt)) -> mark(tt) 15.32/5.15 active(afterNth(N, XS)) -> mark(U11(isNatural(N), N, XS)) 15.32/5.15 active(fst(pair(X, Y))) -> mark(U21(isLNat(X), X, Y)) 15.32/5.15 active(head(cons(N, XS))) -> mark(U31(isNatural(N), N, XS)) 15.32/5.15 active(isLNat(nil)) -> mark(tt) 15.32/5.15 active(isLNat(afterNth(V1, V2))) -> mark(U41(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(isLNat(cons(V1, V2))) -> mark(U51(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(isLNat(fst(V1))) -> mark(U61(isPLNatKind(V1), V1)) 15.32/5.15 active(isLNat(natsFrom(V1))) -> mark(U71(isNaturalKind(V1), V1)) 15.32/5.15 active(isLNat(snd(V1))) -> mark(U81(isPLNatKind(V1), V1)) 15.32/5.15 active(isLNat(tail(V1))) -> mark(U91(isLNatKind(V1), V1)) 15.32/5.15 active(isLNat(take(V1, V2))) -> mark(U101(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(isLNatKind(nil)) -> mark(tt) 15.32/5.15 active(isLNatKind(afterNth(V1, V2))) -> mark(U111(isNaturalKind(V1), V2)) 15.32/5.15 active(isLNatKind(cons(V1, V2))) -> mark(U121(isNaturalKind(V1), V2)) 15.32/5.15 active(isLNatKind(fst(V1))) -> mark(U131(isPLNatKind(V1))) 15.32/5.15 active(isLNatKind(natsFrom(V1))) -> mark(U141(isNaturalKind(V1))) 15.32/5.15 active(isLNatKind(snd(V1))) -> mark(U151(isPLNatKind(V1))) 15.32/5.15 active(isLNatKind(tail(V1))) -> mark(U161(isLNatKind(V1))) 15.32/5.15 active(isLNatKind(take(V1, V2))) -> mark(U171(isNaturalKind(V1), V2)) 15.32/5.15 active(isNatural(0)) -> mark(tt) 15.32/5.15 active(isNatural(head(V1))) -> mark(U181(isLNatKind(V1), V1)) 15.32/5.15 active(isNatural(s(V1))) -> mark(U191(isNaturalKind(V1), V1)) 15.32/5.15 active(isNatural(sel(V1, V2))) -> mark(U201(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(isNaturalKind(0)) -> mark(tt) 15.32/5.15 active(isNaturalKind(head(V1))) -> mark(U211(isLNatKind(V1))) 15.32/5.15 active(isNaturalKind(s(V1))) -> mark(U221(isNaturalKind(V1))) 15.32/5.15 active(isNaturalKind(sel(V1, V2))) -> mark(U231(isNaturalKind(V1), V2)) 15.32/5.15 active(isPLNat(pair(V1, V2))) -> mark(U241(isLNatKind(V1), V1, V2)) 15.32/5.15 active(isPLNat(splitAt(V1, V2))) -> mark(U251(isNaturalKind(V1), V1, V2)) 15.32/5.15 active(isPLNatKind(pair(V1, V2))) -> mark(U261(isLNatKind(V1), V2)) 15.32/5.15 active(isPLNatKind(splitAt(V1, V2))) -> mark(U271(isNaturalKind(V1), V2)) 15.32/5.15 active(natsFrom(N)) -> mark(U281(isNatural(N), N)) 15.32/5.15 active(sel(N, XS)) -> mark(U291(isNatural(N), N, XS)) 15.32/5.15 active(snd(pair(X, Y))) -> mark(U301(isLNat(X), X, Y)) 15.32/5.15 active(splitAt(0, XS)) -> mark(U311(isLNat(XS), XS)) 15.32/5.15 active(splitAt(s(N), cons(X, XS))) -> mark(U321(isNatural(N), N, X, XS)) 15.32/5.15 active(tail(cons(N, XS))) -> mark(U331(isNatural(N), N, XS)) 15.32/5.15 active(take(N, XS)) -> mark(U341(isNatural(N), N, XS)) 15.32/5.15 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 15.32/5.15 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 15.32/5.15 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 15.32/5.15 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 15.32/5.15 active(U105(X1, X2)) -> U105(active(X1), X2) 15.32/5.15 active(U106(X)) -> U106(active(X)) 15.32/5.15 active(U11(X1, X2, X3)) -> U11(active(X1), X2, X3) 15.32/5.15 active(U12(X1, X2, X3)) -> U12(active(X1), X2, X3) 15.32/5.15 active(U111(X1, X2)) -> U111(active(X1), X2) 15.32/5.15 active(U112(X)) -> U112(active(X)) 15.32/5.15 active(U13(X1, X2, X3)) -> U13(active(X1), X2, X3) 15.32/5.15 active(U121(X1, X2)) -> U121(active(X1), X2) 15.32/5.15 active(U122(X)) -> U122(active(X)) 15.32/5.15 active(U14(X1, X2, X3)) -> U14(active(X1), X2, X3) 15.32/5.15 active(U131(X)) -> U131(active(X)) 15.32/5.15 active(snd(X)) -> snd(active(X)) 15.32/5.15 active(splitAt(X1, X2)) -> splitAt(active(X1), X2) 15.32/5.15 active(splitAt(X1, X2)) -> splitAt(X1, active(X2)) 15.32/5.15 active(U141(X)) -> U141(active(X)) 15.32/5.15 active(U151(X)) -> U151(active(X)) 15.32/5.15 active(U161(X)) -> U161(active(X)) 15.32/5.15 active(U171(X1, X2)) -> U171(active(X1), X2) 15.32/5.15 active(U172(X)) -> U172(active(X)) 15.32/5.15 active(U181(X1, X2)) -> U181(active(X1), X2) 15.32/5.15 active(U182(X1, X2)) -> U182(active(X1), X2) 15.32/5.15 active(U183(X)) -> U183(active(X)) 15.32/5.15 active(U191(X1, X2)) -> U191(active(X1), X2) 15.32/5.15 active(U192(X1, X2)) -> U192(active(X1), X2) 15.32/5.15 active(U193(X)) -> U193(active(X)) 15.32/5.15 active(U201(X1, X2, X3)) -> U201(active(X1), X2, X3) 15.32/5.15 active(U202(X1, X2, X3)) -> U202(active(X1), X2, X3) 15.32/5.15 active(U203(X1, X2, X3)) -> U203(active(X1), X2, X3) 15.32/5.15 active(U204(X1, X2, X3)) -> U204(active(X1), X2, X3) 15.32/5.15 active(U205(X1, X2)) -> U205(active(X1), X2) 15.32/5.15 active(U206(X)) -> U206(active(X)) 15.32/5.15 active(U21(X1, X2, X3)) -> U21(active(X1), X2, X3) 15.32/5.15 active(U22(X1, X2, X3)) -> U22(active(X1), X2, X3) 15.32/5.15 active(U211(X)) -> U211(active(X)) 15.32/5.15 active(U23(X1, X2, X3)) -> U23(active(X1), X2, X3) 15.32/5.15 active(U221(X)) -> U221(active(X)) 15.32/5.15 active(U24(X1, X2)) -> U24(active(X1), X2) 15.32/5.15 active(U231(X1, X2)) -> U231(active(X1), X2) 15.32/5.15 active(U232(X)) -> U232(active(X)) 15.32/5.15 active(U241(X1, X2, X3)) -> U241(active(X1), X2, X3) 15.32/5.15 active(U242(X1, X2, X3)) -> U242(active(X1), X2, X3) 15.32/5.15 active(U243(X1, X2, X3)) -> U243(active(X1), X2, X3) 15.32/5.15 active(U244(X1, X2, X3)) -> U244(active(X1), X2, X3) 15.32/5.15 active(U245(X1, X2)) -> U245(active(X1), X2) 15.32/5.15 active(U246(X)) -> U246(active(X)) 15.32/5.15 active(U251(X1, X2, X3)) -> U251(active(X1), X2, X3) 15.32/5.15 active(U252(X1, X2, X3)) -> U252(active(X1), X2, X3) 15.32/5.15 active(U253(X1, X2, X3)) -> U253(active(X1), X2, X3) 15.32/5.15 active(U254(X1, X2, X3)) -> U254(active(X1), X2, X3) 15.32/5.15 active(U255(X1, X2)) -> U255(active(X1), X2) 15.32/5.15 active(U256(X)) -> U256(active(X)) 15.32/5.15 active(U261(X1, X2)) -> U261(active(X1), X2) 15.32/5.15 active(U262(X)) -> U262(active(X)) 15.32/5.15 active(U271(X1, X2)) -> U271(active(X1), X2) 15.32/5.15 active(U272(X)) -> U272(active(X)) 15.32/5.15 active(U281(X1, X2)) -> U281(active(X1), X2) 15.32/5.15 active(U282(X1, X2)) -> U282(active(X1), X2) 15.32/5.15 active(cons(X1, X2)) -> cons(active(X1), X2) 15.32/5.15 active(natsFrom(X)) -> natsFrom(active(X)) 15.32/5.15 active(s(X)) -> s(active(X)) 15.32/5.15 active(U291(X1, X2, X3)) -> U291(active(X1), X2, X3) 15.32/5.15 active(U292(X1, X2, X3)) -> U292(active(X1), X2, X3) 15.32/5.15 active(U293(X1, X2, X3)) -> U293(active(X1), X2, X3) 15.32/5.15 active(U294(X1, X2, X3)) -> U294(active(X1), X2, X3) 15.32/5.15 active(head(X)) -> head(active(X)) 15.32/5.15 active(afterNth(X1, X2)) -> afterNth(active(X1), X2) 15.32/5.15 active(afterNth(X1, X2)) -> afterNth(X1, active(X2)) 15.32/5.15 active(U301(X1, X2, X3)) -> U301(active(X1), X2, X3) 15.32/5.15 active(U302(X1, X2)) -> U302(active(X1), X2) 15.32/5.15 active(U303(X1, X2)) -> U303(active(X1), X2) 15.32/5.15 active(U304(X1, X2)) -> U304(active(X1), X2) 15.32/5.15 active(U31(X1, X2, X3)) -> U31(active(X1), X2, X3) 15.32/5.15 active(U32(X1, X2, X3)) -> U32(active(X1), X2, X3) 15.32/5.15 active(U311(X1, X2)) -> U311(active(X1), X2) 15.32/5.15 active(U312(X1, X2)) -> U312(active(X1), X2) 15.32/5.15 active(pair(X1, X2)) -> pair(active(X1), X2) 15.32/5.15 active(pair(X1, X2)) -> pair(X1, active(X2)) 15.32/5.15 active(U33(X1, X2, X3)) -> U33(active(X1), X2, X3) 15.32/5.15 active(U321(X1, X2, X3, X4)) -> U321(active(X1), X2, X3, X4) 15.32/5.15 active(U322(X1, X2, X3, X4)) -> U322(active(X1), X2, X3, X4) 15.32/5.15 active(U323(X1, X2, X3, X4)) -> U323(active(X1), X2, X3, X4) 15.32/5.15 active(U324(X1, X2, X3, X4)) -> U324(active(X1), X2, X3, X4) 15.32/5.15 active(U325(X1, X2, X3, X4)) -> U325(active(X1), X2, X3, X4) 15.32/5.15 active(U326(X1, X2, X3, X4)) -> U326(active(X1), X2, X3, X4) 15.32/5.15 active(U327(X1, X2)) -> U327(active(X1), X2) 15.32/5.15 active(U34(X1, X2)) -> U34(active(X1), X2) 15.32/5.15 active(U331(X1, X2, X3)) -> U331(active(X1), X2, X3) 15.32/5.15 active(U332(X1, X2)) -> U332(active(X1), X2) 15.32/5.15 active(U333(X1, X2)) -> U333(active(X1), X2) 15.32/5.15 active(U334(X1, X2)) -> U334(active(X1), X2) 15.32/5.15 active(U341(X1, X2, X3)) -> U341(active(X1), X2, X3) 15.32/5.15 active(U342(X1, X2, X3)) -> U342(active(X1), X2, X3) 15.32/5.15 active(U343(X1, X2, X3)) -> U343(active(X1), X2, X3) 15.32/5.15 active(U344(X1, X2, X3)) -> U344(active(X1), X2, X3) 15.32/5.15 active(fst(X)) -> fst(active(X)) 15.32/5.15 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 15.32/5.15 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 15.32/5.15 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 15.32/5.15 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 15.32/5.15 active(U45(X1, X2)) -> U45(active(X1), X2) 15.32/5.15 active(U46(X)) -> U46(active(X)) 15.32/5.15 active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3) 15.32/5.15 active(U52(X1, X2, X3)) -> U52(active(X1), X2, X3) 15.32/5.15 active(U53(X1, X2, X3)) -> U53(active(X1), X2, X3) 15.32/5.15 active(U54(X1, X2, X3)) -> U54(active(X1), X2, X3) 15.32/5.15 active(U55(X1, X2)) -> U55(active(X1), X2) 15.32/5.15 active(U56(X)) -> U56(active(X)) 15.32/5.15 active(U61(X1, X2)) -> U61(active(X1), X2) 15.32/5.15 active(U62(X1, X2)) -> U62(active(X1), X2) 15.32/5.15 active(U63(X)) -> U63(active(X)) 15.32/5.15 active(U71(X1, X2)) -> U71(active(X1), X2) 15.32/5.15 active(U72(X1, X2)) -> U72(active(X1), X2) 15.32/5.15 active(U73(X)) -> U73(active(X)) 15.32/5.15 active(U81(X1, X2)) -> U81(active(X1), X2) 15.32/5.15 active(U82(X1, X2)) -> U82(active(X1), X2) 15.32/5.15 active(U83(X)) -> U83(active(X)) 15.32/5.15 active(U91(X1, X2)) -> U91(active(X1), X2) 15.32/5.15 active(U92(X1, X2)) -> U92(active(X1), X2) 15.32/5.15 active(U93(X)) -> U93(active(X)) 15.32/5.15 active(tail(X)) -> tail(active(X)) 15.32/5.15 active(take(X1, X2)) -> take(active(X1), X2) 15.32/5.15 active(take(X1, X2)) -> take(X1, active(X2)) 15.32/5.15 active(sel(X1, X2)) -> sel(active(X1), X2) 15.32/5.15 active(sel(X1, X2)) -> sel(X1, active(X2)) 15.32/5.15 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 15.32/5.15 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 15.32/5.15 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 15.32/5.15 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 15.32/5.15 U105(mark(X1), X2) -> mark(U105(X1, X2)) 15.32/5.15 U106(mark(X)) -> mark(U106(X)) 15.32/5.15 U11(mark(X1), X2, X3) -> mark(U11(X1, X2, X3)) 15.32/5.15 U12(mark(X1), X2, X3) -> mark(U12(X1, X2, X3)) 15.32/5.15 U111(mark(X1), X2) -> mark(U111(X1, X2)) 15.32/5.15 U112(mark(X)) -> mark(U112(X)) 15.32/5.15 U13(mark(X1), X2, X3) -> mark(U13(X1, X2, X3)) 15.32/5.15 U121(mark(X1), X2) -> mark(U121(X1, X2)) 15.32/5.15 U122(mark(X)) -> mark(U122(X)) 15.32/5.15 U14(mark(X1), X2, X3) -> mark(U14(X1, X2, X3)) 15.32/5.15 U131(mark(X)) -> mark(U131(X)) 15.32/5.15 snd(mark(X)) -> mark(snd(X)) 15.32/5.15 splitAt(mark(X1), X2) -> mark(splitAt(X1, X2)) 15.32/5.15 splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2)) 15.32/5.15 U141(mark(X)) -> mark(U141(X)) 15.32/5.15 U151(mark(X)) -> mark(U151(X)) 15.32/5.15 U161(mark(X)) -> mark(U161(X)) 15.32/5.15 U171(mark(X1), X2) -> mark(U171(X1, X2)) 15.32/5.15 U172(mark(X)) -> mark(U172(X)) 15.32/5.15 U181(mark(X1), X2) -> mark(U181(X1, X2)) 15.32/5.15 U182(mark(X1), X2) -> mark(U182(X1, X2)) 15.32/5.15 U183(mark(X)) -> mark(U183(X)) 15.32/5.15 U191(mark(X1), X2) -> mark(U191(X1, X2)) 15.32/5.15 U192(mark(X1), X2) -> mark(U192(X1, X2)) 15.32/5.15 U193(mark(X)) -> mark(U193(X)) 15.32/5.15 U201(mark(X1), X2, X3) -> mark(U201(X1, X2, X3)) 15.32/5.15 U202(mark(X1), X2, X3) -> mark(U202(X1, X2, X3)) 15.32/5.15 U203(mark(X1), X2, X3) -> mark(U203(X1, X2, X3)) 15.32/5.15 U204(mark(X1), X2, X3) -> mark(U204(X1, X2, X3)) 15.32/5.15 U205(mark(X1), X2) -> mark(U205(X1, X2)) 15.32/5.15 U206(mark(X)) -> mark(U206(X)) 15.32/5.15 U21(mark(X1), X2, X3) -> mark(U21(X1, X2, X3)) 15.32/5.15 U22(mark(X1), X2, X3) -> mark(U22(X1, X2, X3)) 15.32/5.15 U211(mark(X)) -> mark(U211(X)) 15.32/5.15 U23(mark(X1), X2, X3) -> mark(U23(X1, X2, X3)) 15.32/5.15 U221(mark(X)) -> mark(U221(X)) 15.32/5.15 U24(mark(X1), X2) -> mark(U24(X1, X2)) 15.32/5.15 U231(mark(X1), X2) -> mark(U231(X1, X2)) 15.32/5.15 U232(mark(X)) -> mark(U232(X)) 15.32/5.15 U241(mark(X1), X2, X3) -> mark(U241(X1, X2, X3)) 15.32/5.15 U242(mark(X1), X2, X3) -> mark(U242(X1, X2, X3)) 15.32/5.15 U243(mark(X1), X2, X3) -> mark(U243(X1, X2, X3)) 15.32/5.15 U244(mark(X1), X2, X3) -> mark(U244(X1, X2, X3)) 15.32/5.15 U245(mark(X1), X2) -> mark(U245(X1, X2)) 15.32/5.15 U246(mark(X)) -> mark(U246(X)) 15.32/5.15 U251(mark(X1), X2, X3) -> mark(U251(X1, X2, X3)) 15.32/5.15 U252(mark(X1), X2, X3) -> mark(U252(X1, X2, X3)) 15.32/5.15 U253(mark(X1), X2, X3) -> mark(U253(X1, X2, X3)) 15.32/5.15 U254(mark(X1), X2, X3) -> mark(U254(X1, X2, X3)) 15.32/5.15 U255(mark(X1), X2) -> mark(U255(X1, X2)) 15.32/5.15 U256(mark(X)) -> mark(U256(X)) 15.32/5.15 U261(mark(X1), X2) -> mark(U261(X1, X2)) 15.32/5.15 U262(mark(X)) -> mark(U262(X)) 15.32/5.15 U271(mark(X1), X2) -> mark(U271(X1, X2)) 15.32/5.15 U272(mark(X)) -> mark(U272(X)) 15.32/5.15 U281(mark(X1), X2) -> mark(U281(X1, X2)) 15.32/5.15 U282(mark(X1), X2) -> mark(U282(X1, X2)) 15.32/5.15 cons(mark(X1), X2) -> mark(cons(X1, X2)) 15.32/5.15 natsFrom(mark(X)) -> mark(natsFrom(X)) 15.32/5.15 s(mark(X)) -> mark(s(X)) 15.32/5.15 U291(mark(X1), X2, X3) -> mark(U291(X1, X2, X3)) 15.32/5.15 U292(mark(X1), X2, X3) -> mark(U292(X1, X2, X3)) 15.32/5.15 U293(mark(X1), X2, X3) -> mark(U293(X1, X2, X3)) 15.32/5.15 U294(mark(X1), X2, X3) -> mark(U294(X1, X2, X3)) 15.32/5.15 head(mark(X)) -> mark(head(X)) 15.32/5.15 afterNth(mark(X1), X2) -> mark(afterNth(X1, X2)) 15.32/5.15 afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2)) 15.32/5.15 U301(mark(X1), X2, X3) -> mark(U301(X1, X2, X3)) 15.32/5.15 U302(mark(X1), X2) -> mark(U302(X1, X2)) 15.32/5.15 U303(mark(X1), X2) -> mark(U303(X1, X2)) 15.32/5.15 U304(mark(X1), X2) -> mark(U304(X1, X2)) 15.32/5.15 U31(mark(X1), X2, X3) -> mark(U31(X1, X2, X3)) 15.32/5.15 U32(mark(X1), X2, X3) -> mark(U32(X1, X2, X3)) 15.32/5.15 U311(mark(X1), X2) -> mark(U311(X1, X2)) 15.32/5.15 U312(mark(X1), X2) -> mark(U312(X1, X2)) 15.32/5.15 pair(mark(X1), X2) -> mark(pair(X1, X2)) 15.32/5.15 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 15.32/5.15 U33(mark(X1), X2, X3) -> mark(U33(X1, X2, X3)) 15.32/5.15 U321(mark(X1), X2, X3, X4) -> mark(U321(X1, X2, X3, X4)) 15.32/5.15 U322(mark(X1), X2, X3, X4) -> mark(U322(X1, X2, X3, X4)) 15.32/5.15 U323(mark(X1), X2, X3, X4) -> mark(U323(X1, X2, X3, X4)) 15.32/5.15 U324(mark(X1), X2, X3, X4) -> mark(U324(X1, X2, X3, X4)) 15.32/5.15 U325(mark(X1), X2, X3, X4) -> mark(U325(X1, X2, X3, X4)) 15.32/5.15 U326(mark(X1), X2, X3, X4) -> mark(U326(X1, X2, X3, X4)) 15.32/5.15 U327(mark(X1), X2) -> mark(U327(X1, X2)) 15.32/5.15 U34(mark(X1), X2) -> mark(U34(X1, X2)) 15.32/5.15 U331(mark(X1), X2, X3) -> mark(U331(X1, X2, X3)) 15.32/5.15 U332(mark(X1), X2) -> mark(U332(X1, X2)) 15.32/5.15 U333(mark(X1), X2) -> mark(U333(X1, X2)) 15.32/5.15 U334(mark(X1), X2) -> mark(U334(X1, X2)) 15.32/5.15 U341(mark(X1), X2, X3) -> mark(U341(X1, X2, X3)) 15.32/5.15 U342(mark(X1), X2, X3) -> mark(U342(X1, X2, X3)) 15.32/5.15 U343(mark(X1), X2, X3) -> mark(U343(X1, X2, X3)) 15.32/5.15 U344(mark(X1), X2, X3) -> mark(U344(X1, X2, X3)) 15.32/5.15 fst(mark(X)) -> mark(fst(X)) 15.32/5.15 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 15.32/5.15 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 15.32/5.15 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 15.32/5.15 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 15.32/5.15 U45(mark(X1), X2) -> mark(U45(X1, X2)) 15.32/5.15 U46(mark(X)) -> mark(U46(X)) 15.32/5.15 U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3)) 15.32/5.15 U52(mark(X1), X2, X3) -> mark(U52(X1, X2, X3)) 15.32/5.15 U53(mark(X1), X2, X3) -> mark(U53(X1, X2, X3)) 15.32/5.15 U54(mark(X1), X2, X3) -> mark(U54(X1, X2, X3)) 15.32/5.15 U55(mark(X1), X2) -> mark(U55(X1, X2)) 15.32/5.15 U56(mark(X)) -> mark(U56(X)) 15.32/5.15 U61(mark(X1), X2) -> mark(U61(X1, X2)) 15.32/5.15 U62(mark(X1), X2) -> mark(U62(X1, X2)) 15.32/5.15 U63(mark(X)) -> mark(U63(X)) 15.32/5.15 U71(mark(X1), X2) -> mark(U71(X1, X2)) 15.32/5.15 U72(mark(X1), X2) -> mark(U72(X1, X2)) 15.32/5.15 U73(mark(X)) -> mark(U73(X)) 15.32/5.15 U81(mark(X1), X2) -> mark(U81(X1, X2)) 15.32/5.15 U82(mark(X1), X2) -> mark(U82(X1, X2)) 15.32/5.15 U83(mark(X)) -> mark(U83(X)) 15.32/5.15 U91(mark(X1), X2) -> mark(U91(X1, X2)) 15.32/5.15 U92(mark(X1), X2) -> mark(U92(X1, X2)) 15.32/5.15 U93(mark(X)) -> mark(U93(X)) 15.32/5.15 tail(mark(X)) -> mark(tail(X)) 15.32/5.15 take(mark(X1), X2) -> mark(take(X1, X2)) 15.32/5.15 take(X1, mark(X2)) -> mark(take(X1, X2)) 15.32/5.15 sel(mark(X1), X2) -> mark(sel(X1, X2)) 15.32/5.15 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 15.32/5.15 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 15.32/5.15 proper(tt) -> ok(tt) 15.32/5.15 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 15.32/5.15 proper(isNaturalKind(X)) -> isNaturalKind(proper(X)) 15.32/5.15 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 15.32/5.15 proper(isLNatKind(X)) -> isLNatKind(proper(X)) 15.32/5.15 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 15.32/5.15 proper(U105(X1, X2)) -> U105(proper(X1), proper(X2)) 15.32/5.15 proper(isNatural(X)) -> isNatural(proper(X)) 15.32/5.15 proper(U106(X)) -> U106(proper(X)) 15.32/5.15 proper(isLNat(X)) -> isLNat(proper(X)) 15.32/5.15 proper(U11(X1, X2, X3)) -> U11(proper(X1), proper(X2), proper(X3)) 15.32/5.15 proper(U12(X1, X2, X3)) -> U12(proper(X1), proper(X2), proper(X3)) 15.32/5.15 proper(U111(X1, X2)) -> U111(proper(X1), proper(X2)) 15.32/5.15 proper(U112(X)) -> U112(proper(X)) 15.32/5.15 proper(U13(X1, X2, X3)) -> U13(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)) 15.32/5.16 proper(U122(X)) -> U122(proper(X)) 15.32/5.16 proper(U14(X1, X2, X3)) -> U14(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U131(X)) -> U131(proper(X)) 15.32/5.16 proper(snd(X)) -> snd(proper(X)) 15.32/5.16 proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2)) 15.32/5.16 proper(U141(X)) -> U141(proper(X)) 15.32/5.16 proper(U151(X)) -> U151(proper(X)) 15.32/5.16 proper(U161(X)) -> U161(proper(X)) 15.32/5.16 proper(U171(X1, X2)) -> U171(proper(X1), proper(X2)) 15.32/5.16 proper(U172(X)) -> U172(proper(X)) 15.32/5.16 proper(U181(X1, X2)) -> U181(proper(X1), proper(X2)) 15.32/5.16 proper(U182(X1, X2)) -> U182(proper(X1), proper(X2)) 15.32/5.16 proper(U183(X)) -> U183(proper(X)) 15.32/5.16 proper(U191(X1, X2)) -> U191(proper(X1), proper(X2)) 15.32/5.16 proper(U192(X1, X2)) -> U192(proper(X1), proper(X2)) 15.32/5.16 proper(U193(X)) -> U193(proper(X)) 15.32/5.16 proper(U201(X1, X2, X3)) -> U201(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U202(X1, X2, X3)) -> U202(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U203(X1, X2, X3)) -> U203(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U204(X1, X2, X3)) -> U204(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U205(X1, X2)) -> U205(proper(X1), proper(X2)) 15.32/5.16 proper(U206(X)) -> U206(proper(X)) 15.32/5.16 proper(U21(X1, X2, X3)) -> U21(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U22(X1, X2, X3)) -> U22(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U211(X)) -> U211(proper(X)) 15.32/5.16 proper(U23(X1, X2, X3)) -> U23(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U221(X)) -> U221(proper(X)) 15.32/5.16 proper(U24(X1, X2)) -> U24(proper(X1), proper(X2)) 15.32/5.16 proper(U231(X1, X2)) -> U231(proper(X1), proper(X2)) 15.32/5.16 proper(U232(X)) -> U232(proper(X)) 15.32/5.16 proper(U241(X1, X2, X3)) -> U241(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U242(X1, X2, X3)) -> U242(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U243(X1, X2, X3)) -> U243(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U244(X1, X2, X3)) -> U244(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U245(X1, X2)) -> U245(proper(X1), proper(X2)) 15.32/5.16 proper(U246(X)) -> U246(proper(X)) 15.32/5.16 proper(U251(X1, X2, X3)) -> U251(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U252(X1, X2, X3)) -> U252(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U253(X1, X2, X3)) -> U253(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U254(X1, X2, X3)) -> U254(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U255(X1, X2)) -> U255(proper(X1), proper(X2)) 15.32/5.16 proper(U256(X)) -> U256(proper(X)) 15.32/5.16 proper(U261(X1, X2)) -> U261(proper(X1), proper(X2)) 15.32/5.16 proper(U262(X)) -> U262(proper(X)) 15.32/5.16 proper(U271(X1, X2)) -> U271(proper(X1), proper(X2)) 15.32/5.16 proper(U272(X)) -> U272(proper(X)) 15.32/5.16 proper(U281(X1, X2)) -> U281(proper(X1), proper(X2)) 15.32/5.16 proper(U282(X1, X2)) -> U282(proper(X1), proper(X2)) 15.32/5.16 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 15.32/5.16 proper(natsFrom(X)) -> natsFrom(proper(X)) 15.32/5.16 proper(s(X)) -> s(proper(X)) 15.32/5.16 proper(U291(X1, X2, X3)) -> U291(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U292(X1, X2, X3)) -> U292(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U293(X1, X2, X3)) -> U293(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U294(X1, X2, X3)) -> U294(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(head(X)) -> head(proper(X)) 15.32/5.16 proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2)) 15.32/5.16 proper(U301(X1, X2, X3)) -> U301(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U302(X1, X2)) -> U302(proper(X1), proper(X2)) 15.32/5.16 proper(U303(X1, X2)) -> U303(proper(X1), proper(X2)) 15.32/5.16 proper(U304(X1, X2)) -> U304(proper(X1), proper(X2)) 15.32/5.16 proper(U31(X1, X2, X3)) -> U31(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U32(X1, X2, X3)) -> U32(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U311(X1, X2)) -> U311(proper(X1), proper(X2)) 15.32/5.16 proper(U312(X1, X2)) -> U312(proper(X1), proper(X2)) 15.32/5.16 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 15.32/5.16 proper(nil) -> ok(nil) 15.32/5.16 proper(U33(X1, X2, X3)) -> U33(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U321(X1, X2, X3, X4)) -> U321(proper(X1), proper(X2), proper(X3), proper(X4)) 15.32/5.16 proper(U322(X1, X2, X3, X4)) -> U322(proper(X1), proper(X2), proper(X3), proper(X4)) 15.32/5.16 proper(U323(X1, X2, X3, X4)) -> U323(proper(X1), proper(X2), proper(X3), proper(X4)) 15.32/5.16 proper(U324(X1, X2, X3, X4)) -> U324(proper(X1), proper(X2), proper(X3), proper(X4)) 15.32/5.16 proper(U325(X1, X2, X3, X4)) -> U325(proper(X1), proper(X2), proper(X3), proper(X4)) 15.32/5.16 proper(U326(X1, X2, X3, X4)) -> U326(proper(X1), proper(X2), proper(X3), proper(X4)) 15.32/5.16 proper(U327(X1, X2)) -> U327(proper(X1), proper(X2)) 15.32/5.16 proper(U34(X1, X2)) -> U34(proper(X1), proper(X2)) 15.32/5.16 proper(U331(X1, X2, X3)) -> U331(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U332(X1, X2)) -> U332(proper(X1), proper(X2)) 15.32/5.16 proper(U333(X1, X2)) -> U333(proper(X1), proper(X2)) 15.32/5.16 proper(U334(X1, X2)) -> U334(proper(X1), proper(X2)) 15.32/5.16 proper(U341(X1, X2, X3)) -> U341(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U342(X1, X2, X3)) -> U342(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U343(X1, X2, X3)) -> U343(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U344(X1, X2, X3)) -> U344(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(fst(X)) -> fst(proper(X)) 15.32/5.16 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 15.32/5.16 proper(U46(X)) -> U46(proper(X)) 15.32/5.16 proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U52(X1, X2, X3)) -> U52(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U53(X1, X2, X3)) -> U53(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U54(X1, X2, X3)) -> U54(proper(X1), proper(X2), proper(X3)) 15.32/5.16 proper(U55(X1, X2)) -> U55(proper(X1), proper(X2)) 15.32/5.16 proper(U56(X)) -> U56(proper(X)) 15.32/5.16 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 15.32/5.16 proper(U62(X1, X2)) -> U62(proper(X1), proper(X2)) 15.32/5.16 proper(isPLNatKind(X)) -> isPLNatKind(proper(X)) 15.32/5.16 proper(U63(X)) -> U63(proper(X)) 15.32/5.16 proper(isPLNat(X)) -> isPLNat(proper(X)) 15.32/5.16 proper(U71(X1, X2)) -> U71(proper(X1), proper(X2)) 15.32/5.16 proper(U72(X1, X2)) -> U72(proper(X1), proper(X2)) 15.32/5.16 proper(U73(X)) -> U73(proper(X)) 15.32/5.16 proper(U81(X1, X2)) -> U81(proper(X1), proper(X2)) 15.32/5.16 proper(U82(X1, X2)) -> U82(proper(X1), proper(X2)) 15.32/5.16 proper(U83(X)) -> U83(proper(X)) 15.32/5.16 proper(U91(X1, X2)) -> U91(proper(X1), proper(X2)) 15.32/5.16 proper(U92(X1, X2)) -> U92(proper(X1), proper(X2)) 15.32/5.16 proper(U93(X)) -> U93(proper(X)) 15.32/5.16 proper(tail(X)) -> tail(proper(X)) 15.32/5.16 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 15.32/5.16 proper(0) -> ok(0) 15.32/5.16 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 15.32/5.16 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 15.32/5.16 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 15.32/5.16 isNaturalKind(ok(X)) -> ok(isNaturalKind(X)) 15.32/5.16 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 15.32/5.16 isLNatKind(ok(X)) -> ok(isLNatKind(X)) 15.32/5.16 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 15.32/5.16 U105(ok(X1), ok(X2)) -> ok(U105(X1, X2)) 15.32/5.16 isNatural(ok(X)) -> ok(isNatural(X)) 15.32/5.16 U106(ok(X)) -> ok(U106(X)) 15.32/5.16 isLNat(ok(X)) -> ok(isLNat(X)) 15.32/5.16 U11(ok(X1), ok(X2), ok(X3)) -> ok(U11(X1, X2, X3)) 15.32/5.16 U12(ok(X1), ok(X2), ok(X3)) -> ok(U12(X1, X2, X3)) 15.32/5.16 U111(ok(X1), ok(X2)) -> ok(U111(X1, X2)) 15.32/5.16 U112(ok(X)) -> ok(U112(X)) 15.32/5.16 U13(ok(X1), ok(X2), ok(X3)) -> ok(U13(X1, X2, X3)) 15.32/5.16 U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)) 15.32/5.16 U122(ok(X)) -> ok(U122(X)) 15.32/5.16 U14(ok(X1), ok(X2), ok(X3)) -> ok(U14(X1, X2, X3)) 15.32/5.16 U131(ok(X)) -> ok(U131(X)) 15.32/5.16 snd(ok(X)) -> ok(snd(X)) 15.32/5.16 splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2)) 15.32/5.16 U141(ok(X)) -> ok(U141(X)) 15.32/5.16 U151(ok(X)) -> ok(U151(X)) 15.32/5.16 U161(ok(X)) -> ok(U161(X)) 15.32/5.16 U171(ok(X1), ok(X2)) -> ok(U171(X1, X2)) 15.32/5.16 U172(ok(X)) -> ok(U172(X)) 15.32/5.16 U181(ok(X1), ok(X2)) -> ok(U181(X1, X2)) 15.32/5.16 U182(ok(X1), ok(X2)) -> ok(U182(X1, X2)) 15.32/5.16 U183(ok(X)) -> ok(U183(X)) 15.32/5.16 U191(ok(X1), ok(X2)) -> ok(U191(X1, X2)) 15.32/5.16 U192(ok(X1), ok(X2)) -> ok(U192(X1, X2)) 15.32/5.16 U193(ok(X)) -> ok(U193(X)) 15.32/5.16 U201(ok(X1), ok(X2), ok(X3)) -> ok(U201(X1, X2, X3)) 15.32/5.16 U202(ok(X1), ok(X2), ok(X3)) -> ok(U202(X1, X2, X3)) 15.32/5.16 U203(ok(X1), ok(X2), ok(X3)) -> ok(U203(X1, X2, X3)) 15.32/5.16 U204(ok(X1), ok(X2), ok(X3)) -> ok(U204(X1, X2, X3)) 15.32/5.16 U205(ok(X1), ok(X2)) -> ok(U205(X1, X2)) 15.32/5.16 U206(ok(X)) -> ok(U206(X)) 15.32/5.16 U21(ok(X1), ok(X2), ok(X3)) -> ok(U21(X1, X2, X3)) 15.32/5.16 U22(ok(X1), ok(X2), ok(X3)) -> ok(U22(X1, X2, X3)) 15.32/5.16 U211(ok(X)) -> ok(U211(X)) 15.32/5.16 U23(ok(X1), ok(X2), ok(X3)) -> ok(U23(X1, X2, X3)) 15.32/5.16 U221(ok(X)) -> ok(U221(X)) 15.32/5.16 U24(ok(X1), ok(X2)) -> ok(U24(X1, X2)) 15.32/5.16 U231(ok(X1), ok(X2)) -> ok(U231(X1, X2)) 15.32/5.16 U232(ok(X)) -> ok(U232(X)) 15.32/5.16 U241(ok(X1), ok(X2), ok(X3)) -> ok(U241(X1, X2, X3)) 15.32/5.16 U242(ok(X1), ok(X2), ok(X3)) -> ok(U242(X1, X2, X3)) 15.32/5.16 U243(ok(X1), ok(X2), ok(X3)) -> ok(U243(X1, X2, X3)) 15.32/5.16 U244(ok(X1), ok(X2), ok(X3)) -> ok(U244(X1, X2, X3)) 15.32/5.16 U245(ok(X1), ok(X2)) -> ok(U245(X1, X2)) 15.32/5.16 U246(ok(X)) -> ok(U246(X)) 15.32/5.16 U251(ok(X1), ok(X2), ok(X3)) -> ok(U251(X1, X2, X3)) 15.32/5.16 U252(ok(X1), ok(X2), ok(X3)) -> ok(U252(X1, X2, X3)) 15.32/5.16 U253(ok(X1), ok(X2), ok(X3)) -> ok(U253(X1, X2, X3)) 15.32/5.16 U254(ok(X1), ok(X2), ok(X3)) -> ok(U254(X1, X2, X3)) 15.32/5.16 U255(ok(X1), ok(X2)) -> ok(U255(X1, X2)) 15.32/5.16 U256(ok(X)) -> ok(U256(X)) 15.32/5.16 U261(ok(X1), ok(X2)) -> ok(U261(X1, X2)) 15.32/5.16 U262(ok(X)) -> ok(U262(X)) 15.32/5.16 U271(ok(X1), ok(X2)) -> ok(U271(X1, X2)) 15.32/5.16 U272(ok(X)) -> ok(U272(X)) 15.32/5.16 U281(ok(X1), ok(X2)) -> ok(U281(X1, X2)) 15.32/5.16 U282(ok(X1), ok(X2)) -> ok(U282(X1, X2)) 15.32/5.16 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 15.32/5.16 natsFrom(ok(X)) -> ok(natsFrom(X)) 15.32/5.16 s(ok(X)) -> ok(s(X)) 15.32/5.16 U291(ok(X1), ok(X2), ok(X3)) -> ok(U291(X1, X2, X3)) 15.32/5.16 U292(ok(X1), ok(X2), ok(X3)) -> ok(U292(X1, X2, X3)) 15.32/5.16 U293(ok(X1), ok(X2), ok(X3)) -> ok(U293(X1, X2, X3)) 15.32/5.16 U294(ok(X1), ok(X2), ok(X3)) -> ok(U294(X1, X2, X3)) 15.32/5.16 head(ok(X)) -> ok(head(X)) 15.32/5.16 afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2)) 15.32/5.16 U301(ok(X1), ok(X2), ok(X3)) -> ok(U301(X1, X2, X3)) 15.32/5.16 U302(ok(X1), ok(X2)) -> ok(U302(X1, X2)) 15.32/5.16 U303(ok(X1), ok(X2)) -> ok(U303(X1, X2)) 15.32/5.16 U304(ok(X1), ok(X2)) -> ok(U304(X1, X2)) 15.32/5.16 U31(ok(X1), ok(X2), ok(X3)) -> ok(U31(X1, X2, X3)) 15.32/5.16 U32(ok(X1), ok(X2), ok(X3)) -> ok(U32(X1, X2, X3)) 15.32/5.16 U311(ok(X1), ok(X2)) -> ok(U311(X1, X2)) 15.32/5.16 U312(ok(X1), ok(X2)) -> ok(U312(X1, X2)) 15.32/5.16 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 15.32/5.16 U33(ok(X1), ok(X2), ok(X3)) -> ok(U33(X1, X2, X3)) 15.32/5.16 U321(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U321(X1, X2, X3, X4)) 15.32/5.16 U322(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U322(X1, X2, X3, X4)) 15.32/5.16 U323(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U323(X1, X2, X3, X4)) 15.32/5.16 U324(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U324(X1, X2, X3, X4)) 15.32/5.16 U325(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U325(X1, X2, X3, X4)) 15.32/5.16 U326(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U326(X1, X2, X3, X4)) 15.32/5.16 U327(ok(X1), ok(X2)) -> ok(U327(X1, X2)) 15.32/5.16 U34(ok(X1), ok(X2)) -> ok(U34(X1, X2)) 15.32/5.16 U331(ok(X1), ok(X2), ok(X3)) -> ok(U331(X1, X2, X3)) 15.32/5.16 U332(ok(X1), ok(X2)) -> ok(U332(X1, X2)) 15.32/5.16 U333(ok(X1), ok(X2)) -> ok(U333(X1, X2)) 15.32/5.16 U334(ok(X1), ok(X2)) -> ok(U334(X1, X2)) 15.32/5.16 U341(ok(X1), ok(X2), ok(X3)) -> ok(U341(X1, X2, X3)) 15.32/5.16 U342(ok(X1), ok(X2), ok(X3)) -> ok(U342(X1, X2, X3)) 15.32/5.16 U343(ok(X1), ok(X2), ok(X3)) -> ok(U343(X1, X2, X3)) 15.32/5.16 U344(ok(X1), ok(X2), ok(X3)) -> ok(U344(X1, X2, X3)) 15.32/5.16 fst(ok(X)) -> ok(fst(X)) 15.32/5.16 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 15.32/5.16 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 15.32/5.16 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 15.32/5.16 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 15.32/5.16 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 15.32/5.16 U46(ok(X)) -> ok(U46(X)) 15.32/5.16 U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3)) 15.32/5.16 U52(ok(X1), ok(X2), ok(X3)) -> ok(U52(X1, X2, X3)) 15.32/5.16 U53(ok(X1), ok(X2), ok(X3)) -> ok(U53(X1, X2, X3)) 15.32/5.16 U54(ok(X1), ok(X2), ok(X3)) -> ok(U54(X1, X2, X3)) 15.32/5.16 U55(ok(X1), ok(X2)) -> ok(U55(X1, X2)) 15.32/5.16 U56(ok(X)) -> ok(U56(X)) 15.32/5.16 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 15.32/5.16 U62(ok(X1), ok(X2)) -> ok(U62(X1, X2)) 15.32/5.16 isPLNatKind(ok(X)) -> ok(isPLNatKind(X)) 15.32/5.16 U63(ok(X)) -> ok(U63(X)) 15.32/5.16 isPLNat(ok(X)) -> ok(isPLNat(X)) 15.32/5.16 U71(ok(X1), ok(X2)) -> ok(U71(X1, X2)) 15.32/5.16 U72(ok(X1), ok(X2)) -> ok(U72(X1, X2)) 15.32/5.16 U73(ok(X)) -> ok(U73(X)) 15.32/5.16 U81(ok(X1), ok(X2)) -> ok(U81(X1, X2)) 15.32/5.16 U82(ok(X1), ok(X2)) -> ok(U82(X1, X2)) 15.32/5.16 U83(ok(X)) -> ok(U83(X)) 15.32/5.16 U91(ok(X1), ok(X2)) -> ok(U91(X1, X2)) 15.32/5.16 U92(ok(X1), ok(X2)) -> ok(U92(X1, X2)) 15.32/5.16 U93(ok(X)) -> ok(U93(X)) 15.32/5.16 tail(ok(X)) -> ok(tail(X)) 15.32/5.16 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 15.32/5.16 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 15.32/5.16 top(mark(X)) -> top(proper(X)) 15.32/5.16 top(ok(X)) -> top(active(X)) 15.32/5.16 15.32/5.16 The set Q consists of the following terms: 15.32/5.16 15.32/5.16 active(afterNth(x0, x1)) 15.32/5.16 active(isLNat(nil)) 15.32/5.16 active(isLNat(afterNth(x0, x1))) 15.32/5.16 active(isLNat(cons(x0, x1))) 15.32/5.16 active(isLNat(fst(x0))) 15.32/5.16 active(isLNat(natsFrom(x0))) 15.32/5.16 active(isLNat(snd(x0))) 15.32/5.16 active(isLNat(tail(x0))) 15.32/5.16 active(isLNat(take(x0, x1))) 15.32/5.16 active(isLNatKind(nil)) 15.32/5.16 active(isLNatKind(afterNth(x0, x1))) 15.32/5.16 active(isLNatKind(cons(x0, x1))) 15.32/5.16 active(isLNatKind(fst(x0))) 15.32/5.16 active(isLNatKind(natsFrom(x0))) 15.32/5.16 active(isLNatKind(snd(x0))) 15.32/5.16 active(isLNatKind(tail(x0))) 15.32/5.16 active(isLNatKind(take(x0, x1))) 15.32/5.16 active(isNatural(0)) 15.32/5.16 active(isNatural(head(x0))) 15.32/5.16 active(isNatural(s(x0))) 15.32/5.16 active(isNatural(sel(x0, x1))) 15.32/5.16 active(isNaturalKind(0)) 15.32/5.16 active(isNaturalKind(head(x0))) 15.32/5.16 active(isNaturalKind(s(x0))) 15.32/5.16 active(isNaturalKind(sel(x0, x1))) 15.32/5.16 active(isPLNat(pair(x0, x1))) 15.32/5.16 active(isPLNat(splitAt(x0, x1))) 15.32/5.16 active(isPLNatKind(pair(x0, x1))) 15.32/5.16 active(isPLNatKind(splitAt(x0, x1))) 15.32/5.16 active(natsFrom(x0)) 15.32/5.16 active(sel(x0, x1)) 15.32/5.16 active(take(x0, x1)) 15.32/5.16 active(U101(x0, x1, x2)) 15.32/5.16 active(U102(x0, x1, x2)) 15.32/5.16 active(U103(x0, x1, x2)) 15.32/5.16 active(U104(x0, x1, x2)) 15.32/5.16 active(U105(x0, x1)) 15.32/5.16 active(U106(x0)) 15.32/5.16 active(U11(x0, x1, x2)) 15.32/5.16 active(U12(x0, x1, x2)) 15.32/5.16 active(U111(x0, x1)) 15.32/5.16 active(U112(x0)) 15.32/5.16 active(U13(x0, x1, x2)) 15.32/5.16 active(U121(x0, x1)) 15.32/5.16 active(U122(x0)) 15.32/5.16 active(U14(x0, x1, x2)) 15.32/5.16 active(U131(x0)) 15.32/5.16 active(snd(x0)) 15.32/5.16 active(splitAt(x0, x1)) 15.32/5.16 active(U141(x0)) 15.32/5.16 active(U151(x0)) 15.32/5.16 active(U161(x0)) 15.32/5.16 active(U171(x0, x1)) 15.32/5.16 active(U172(x0)) 15.32/5.16 active(U181(x0, x1)) 15.32/5.16 active(U182(x0, x1)) 15.32/5.16 active(U183(x0)) 15.32/5.16 active(U191(x0, x1)) 15.32/5.16 active(U192(x0, x1)) 15.32/5.16 active(U193(x0)) 15.32/5.16 active(U201(x0, x1, x2)) 15.32/5.16 active(U202(x0, x1, x2)) 15.32/5.16 active(U203(x0, x1, x2)) 15.32/5.16 active(U204(x0, x1, x2)) 15.32/5.16 active(U205(x0, x1)) 15.32/5.16 active(U206(x0)) 15.32/5.16 active(U21(x0, x1, x2)) 15.32/5.16 active(U22(x0, x1, x2)) 15.32/5.16 active(U211(x0)) 15.32/5.16 active(U23(x0, x1, x2)) 15.32/5.16 active(U221(x0)) 15.32/5.16 active(U24(x0, x1)) 15.32/5.16 active(U231(x0, x1)) 15.32/5.16 active(U232(x0)) 15.32/5.16 active(U241(x0, x1, x2)) 15.32/5.16 active(U242(x0, x1, x2)) 15.32/5.16 active(U243(x0, x1, x2)) 15.32/5.16 active(U244(x0, x1, x2)) 15.32/5.16 active(U245(x0, x1)) 15.32/5.16 active(U246(x0)) 15.32/5.16 active(U251(x0, x1, x2)) 15.32/5.16 active(U252(x0, x1, x2)) 15.32/5.16 active(U253(x0, x1, x2)) 15.32/5.16 active(U254(x0, x1, x2)) 15.32/5.16 active(U255(x0, x1)) 15.32/5.16 active(U256(x0)) 15.32/5.16 active(U261(x0, x1)) 15.32/5.16 active(U262(x0)) 15.32/5.16 active(U271(x0, x1)) 15.32/5.16 active(U272(x0)) 15.32/5.16 active(U281(x0, x1)) 15.32/5.16 active(U282(x0, x1)) 15.32/5.16 active(cons(x0, x1)) 15.32/5.16 active(s(x0)) 15.32/5.16 active(U291(x0, x1, x2)) 15.32/5.16 active(U292(x0, x1, x2)) 15.32/5.16 active(U293(x0, x1, x2)) 15.32/5.16 active(U294(x0, x1, x2)) 15.32/5.16 active(head(x0)) 15.32/5.16 active(U301(x0, x1, x2)) 15.32/5.16 active(U302(x0, x1)) 15.32/5.16 active(U303(x0, x1)) 15.32/5.16 active(U304(x0, x1)) 15.32/5.16 active(U31(x0, x1, x2)) 15.32/5.16 active(U32(x0, x1, x2)) 15.32/5.16 active(U311(x0, x1)) 15.32/5.16 active(U312(x0, x1)) 15.32/5.16 active(pair(x0, x1)) 15.32/5.16 active(U33(x0, x1, x2)) 15.32/5.16 active(U321(x0, x1, x2, x3)) 15.32/5.16 active(U322(x0, x1, x2, x3)) 15.32/5.16 active(U323(x0, x1, x2, x3)) 15.32/5.16 active(U324(x0, x1, x2, x3)) 15.32/5.16 active(U325(x0, x1, x2, x3)) 15.32/5.16 active(U326(x0, x1, x2, x3)) 15.32/5.16 active(U327(x0, x1)) 15.32/5.16 active(U34(x0, x1)) 15.32/5.16 active(U331(x0, x1, x2)) 15.32/5.16 active(U332(x0, x1)) 15.32/5.16 active(U333(x0, x1)) 15.32/5.16 active(U334(x0, x1)) 15.32/5.16 active(U341(x0, x1, x2)) 15.32/5.16 active(U342(x0, x1, x2)) 15.32/5.16 active(U343(x0, x1, x2)) 15.32/5.16 active(U344(x0, x1, x2)) 15.32/5.16 active(fst(x0)) 15.32/5.16 active(U41(x0, x1, x2)) 15.32/5.16 active(U42(x0, x1, x2)) 15.32/5.16 active(U43(x0, x1, x2)) 15.32/5.16 active(U44(x0, x1, x2)) 15.32/5.16 active(U45(x0, x1)) 15.32/5.16 active(U46(x0)) 15.32/5.16 active(U51(x0, x1, x2)) 15.32/5.16 active(U52(x0, x1, x2)) 15.32/5.16 active(U53(x0, x1, x2)) 15.32/5.16 active(U54(x0, x1, x2)) 15.32/5.16 active(U55(x0, x1)) 15.32/5.16 active(U56(x0)) 15.32/5.16 active(U61(x0, x1)) 15.32/5.16 active(U62(x0, x1)) 15.32/5.16 active(U63(x0)) 15.32/5.16 active(U71(x0, x1)) 15.32/5.16 active(U72(x0, x1)) 15.32/5.16 active(U73(x0)) 15.32/5.16 active(U81(x0, x1)) 15.32/5.16 active(U82(x0, x1)) 15.32/5.16 active(U83(x0)) 15.32/5.16 active(U91(x0, x1)) 15.32/5.16 active(U92(x0, x1)) 15.32/5.16 active(U93(x0)) 15.32/5.16 active(tail(x0)) 15.32/5.16 U101(mark(x0), x1, x2) 15.32/5.16 U102(mark(x0), x1, x2) 15.32/5.16 U103(mark(x0), x1, x2) 15.32/5.16 U104(mark(x0), x1, x2) 15.32/5.16 U105(mark(x0), x1) 15.32/5.16 U106(mark(x0)) 15.32/5.16 U11(mark(x0), x1, x2) 15.32/5.16 U12(mark(x0), x1, x2) 15.32/5.16 U111(mark(x0), x1) 15.32/5.16 U112(mark(x0)) 15.32/5.16 U13(mark(x0), x1, x2) 15.32/5.16 U121(mark(x0), x1) 15.32/5.16 U122(mark(x0)) 15.32/5.16 U14(mark(x0), x1, x2) 15.32/5.16 U131(mark(x0)) 15.32/5.16 snd(mark(x0)) 15.32/5.16 splitAt(mark(x0), x1) 15.32/5.16 splitAt(x0, mark(x1)) 15.32/5.16 U141(mark(x0)) 15.32/5.16 U151(mark(x0)) 15.32/5.16 U161(mark(x0)) 15.32/5.16 U171(mark(x0), x1) 15.32/5.16 U172(mark(x0)) 15.32/5.16 U181(mark(x0), x1) 15.32/5.16 U182(mark(x0), x1) 15.32/5.16 U183(mark(x0)) 15.32/5.16 U191(mark(x0), x1) 15.32/5.16 U192(mark(x0), x1) 15.32/5.16 U193(mark(x0)) 15.32/5.16 U201(mark(x0), x1, x2) 15.32/5.16 U202(mark(x0), x1, x2) 15.32/5.16 U203(mark(x0), x1, x2) 15.32/5.16 U204(mark(x0), x1, x2) 15.32/5.16 U205(mark(x0), x1) 15.32/5.16 U206(mark(x0)) 15.32/5.16 U21(mark(x0), x1, x2) 15.32/5.16 U22(mark(x0), x1, x2) 15.32/5.16 U211(mark(x0)) 15.32/5.16 U23(mark(x0), x1, x2) 15.32/5.16 U221(mark(x0)) 15.32/5.16 U24(mark(x0), x1) 15.32/5.16 U231(mark(x0), x1) 15.32/5.16 U232(mark(x0)) 15.32/5.16 U241(mark(x0), x1, x2) 15.32/5.16 U242(mark(x0), x1, x2) 15.32/5.16 U243(mark(x0), x1, x2) 15.32/5.16 U244(mark(x0), x1, x2) 15.32/5.16 U245(mark(x0), x1) 15.32/5.16 U246(mark(x0)) 15.32/5.16 U251(mark(x0), x1, x2) 15.32/5.16 U252(mark(x0), x1, x2) 15.32/5.16 U253(mark(x0), x1, x2) 15.32/5.16 U254(mark(x0), x1, x2) 15.32/5.16 U255(mark(x0), x1) 15.32/5.16 U256(mark(x0)) 15.32/5.16 U261(mark(x0), x1) 15.32/5.16 U262(mark(x0)) 15.32/5.16 U271(mark(x0), x1) 15.32/5.16 U272(mark(x0)) 15.32/5.16 U281(mark(x0), x1) 15.32/5.16 U282(mark(x0), x1) 15.32/5.16 cons(mark(x0), x1) 15.32/5.16 natsFrom(mark(x0)) 15.32/5.16 s(mark(x0)) 15.32/5.16 U291(mark(x0), x1, x2) 15.32/5.16 U292(mark(x0), x1, x2) 15.32/5.16 U293(mark(x0), x1, x2) 15.32/5.16 U294(mark(x0), x1, x2) 15.32/5.16 head(mark(x0)) 15.32/5.16 afterNth(mark(x0), x1) 15.32/5.16 afterNth(x0, mark(x1)) 15.32/5.16 U301(mark(x0), x1, x2) 15.32/5.16 U302(mark(x0), x1) 15.32/5.16 U303(mark(x0), x1) 15.32/5.16 U304(mark(x0), x1) 15.32/5.16 U31(mark(x0), x1, x2) 15.32/5.16 U32(mark(x0), x1, x2) 15.32/5.16 U311(mark(x0), x1) 15.32/5.16 U312(mark(x0), x1) 15.32/5.16 pair(mark(x0), x1) 15.32/5.16 pair(x0, mark(x1)) 15.32/5.16 U33(mark(x0), x1, x2) 15.32/5.16 U321(mark(x0), x1, x2, x3) 15.32/5.16 U322(mark(x0), x1, x2, x3) 15.32/5.16 U323(mark(x0), x1, x2, x3) 15.32/5.16 U324(mark(x0), x1, x2, x3) 15.32/5.16 U325(mark(x0), x1, x2, x3) 15.32/5.16 U326(mark(x0), x1, x2, x3) 15.32/5.16 U327(mark(x0), x1) 15.32/5.16 U34(mark(x0), x1) 15.32/5.16 U331(mark(x0), x1, x2) 15.32/5.16 U332(mark(x0), x1) 15.32/5.16 U333(mark(x0), x1) 15.32/5.16 U334(mark(x0), x1) 15.32/5.16 U341(mark(x0), x1, x2) 15.32/5.16 U342(mark(x0), x1, x2) 15.32/5.16 U343(mark(x0), x1, x2) 15.32/5.16 U344(mark(x0), x1, x2) 15.32/5.16 fst(mark(x0)) 15.32/5.16 U41(mark(x0), x1, x2) 15.32/5.16 U42(mark(x0), x1, x2) 15.32/5.16 U43(mark(x0), x1, x2) 15.32/5.16 U44(mark(x0), x1, x2) 15.32/5.16 U45(mark(x0), x1) 15.32/5.16 U46(mark(x0)) 15.32/5.16 U51(mark(x0), x1, x2) 15.32/5.16 U52(mark(x0), x1, x2) 15.32/5.16 U53(mark(x0), x1, x2) 15.32/5.16 U54(mark(x0), x1, x2) 15.32/5.16 U55(mark(x0), x1) 15.32/5.16 U56(mark(x0)) 15.32/5.16 U61(mark(x0), x1) 15.32/5.16 U62(mark(x0), x1) 15.32/5.16 U63(mark(x0)) 15.32/5.16 U71(mark(x0), x1) 15.32/5.16 U72(mark(x0), x1) 15.32/5.16 U73(mark(x0)) 15.32/5.16 U81(mark(x0), x1) 15.32/5.16 U82(mark(x0), x1) 15.32/5.16 U83(mark(x0)) 15.32/5.16 U91(mark(x0), x1) 15.32/5.16 U92(mark(x0), x1) 15.32/5.16 U93(mark(x0)) 15.32/5.16 tail(mark(x0)) 15.32/5.16 take(mark(x0), x1) 15.32/5.16 take(x0, mark(x1)) 15.32/5.16 sel(mark(x0), x1) 15.32/5.16 sel(x0, mark(x1)) 15.32/5.16 proper(U101(x0, x1, x2)) 15.32/5.16 proper(tt) 15.32/5.16 proper(U102(x0, x1, x2)) 15.32/5.16 proper(isNaturalKind(x0)) 15.32/5.16 proper(U103(x0, x1, x2)) 15.32/5.16 proper(isLNatKind(x0)) 15.32/5.16 proper(U104(x0, x1, x2)) 15.32/5.16 proper(U105(x0, x1)) 15.32/5.16 proper(isNatural(x0)) 15.32/5.16 proper(U106(x0)) 15.32/5.16 proper(isLNat(x0)) 15.32/5.16 proper(U11(x0, x1, x2)) 15.32/5.16 proper(U12(x0, x1, x2)) 15.32/5.16 proper(U111(x0, x1)) 15.32/5.16 proper(U112(x0)) 15.32/5.16 proper(U13(x0, x1, x2)) 15.32/5.16 proper(U121(x0, x1)) 15.32/5.16 proper(U122(x0)) 15.32/5.16 proper(U14(x0, x1, x2)) 15.32/5.16 proper(U131(x0)) 15.32/5.16 proper(snd(x0)) 15.32/5.16 proper(splitAt(x0, x1)) 15.32/5.16 proper(U141(x0)) 15.32/5.16 proper(U151(x0)) 15.32/5.16 proper(U161(x0)) 15.32/5.16 proper(U171(x0, x1)) 15.32/5.16 proper(U172(x0)) 15.32/5.16 proper(U181(x0, x1)) 15.32/5.16 proper(U182(x0, x1)) 15.32/5.16 proper(U183(x0)) 15.32/5.16 proper(U191(x0, x1)) 15.32/5.16 proper(U192(x0, x1)) 15.32/5.16 proper(U193(x0)) 15.32/5.16 proper(U201(x0, x1, x2)) 15.32/5.16 proper(U202(x0, x1, x2)) 15.32/5.16 proper(U203(x0, x1, x2)) 15.32/5.16 proper(U204(x0, x1, x2)) 15.32/5.16 proper(U205(x0, x1)) 15.32/5.16 proper(U206(x0)) 15.32/5.16 proper(U21(x0, x1, x2)) 15.32/5.16 proper(U22(x0, x1, x2)) 15.32/5.16 proper(U211(x0)) 15.32/5.16 proper(U23(x0, x1, x2)) 15.32/5.16 proper(U221(x0)) 15.32/5.16 proper(U24(x0, x1)) 15.32/5.16 proper(U231(x0, x1)) 15.32/5.16 proper(U232(x0)) 15.32/5.16 proper(U241(x0, x1, x2)) 15.32/5.16 proper(U242(x0, x1, x2)) 15.32/5.16 proper(U243(x0, x1, x2)) 15.32/5.16 proper(U244(x0, x1, x2)) 15.32/5.16 proper(U245(x0, x1)) 15.32/5.16 proper(U246(x0)) 15.32/5.16 proper(U251(x0, x1, x2)) 15.32/5.16 proper(U252(x0, x1, x2)) 15.32/5.16 proper(U253(x0, x1, x2)) 15.32/5.16 proper(U254(x0, x1, x2)) 15.32/5.16 proper(U255(x0, x1)) 15.32/5.16 proper(U256(x0)) 15.32/5.16 proper(U261(x0, x1)) 15.32/5.16 proper(U262(x0)) 15.32/5.16 proper(U271(x0, x1)) 15.32/5.16 proper(U272(x0)) 15.32/5.16 proper(U281(x0, x1)) 15.32/5.16 proper(U282(x0, x1)) 15.32/5.16 proper(cons(x0, x1)) 15.32/5.16 proper(natsFrom(x0)) 15.32/5.16 proper(s(x0)) 15.32/5.16 proper(U291(x0, x1, x2)) 15.32/5.16 proper(U292(x0, x1, x2)) 15.32/5.16 proper(U293(x0, x1, x2)) 15.32/5.16 proper(U294(x0, x1, x2)) 15.32/5.16 proper(head(x0)) 15.32/5.16 proper(afterNth(x0, x1)) 15.32/5.16 proper(U301(x0, x1, x2)) 15.32/5.16 proper(U302(x0, x1)) 15.32/5.16 proper(U303(x0, x1)) 15.32/5.16 proper(U304(x0, x1)) 15.32/5.16 proper(U31(x0, x1, x2)) 15.32/5.16 proper(U32(x0, x1, x2)) 15.32/5.16 proper(U311(x0, x1)) 15.32/5.16 proper(U312(x0, x1)) 15.32/5.16 proper(pair(x0, x1)) 15.32/5.16 proper(nil) 15.32/5.16 proper(U33(x0, x1, x2)) 15.32/5.16 proper(U321(x0, x1, x2, x3)) 15.32/5.16 proper(U322(x0, x1, x2, x3)) 15.32/5.16 proper(U323(x0, x1, x2, x3)) 15.32/5.16 proper(U324(x0, x1, x2, x3)) 15.32/5.16 proper(U325(x0, x1, x2, x3)) 15.32/5.16 proper(U326(x0, x1, x2, x3)) 15.32/5.16 proper(U327(x0, x1)) 15.32/5.16 proper(U34(x0, x1)) 15.32/5.16 proper(U331(x0, x1, x2)) 15.32/5.16 proper(U332(x0, x1)) 15.32/5.16 proper(U333(x0, x1)) 15.32/5.16 proper(U334(x0, x1)) 15.32/5.16 proper(U341(x0, x1, x2)) 15.32/5.16 proper(U342(x0, x1, x2)) 15.32/5.16 proper(U343(x0, x1, x2)) 15.32/5.16 proper(U344(x0, x1, x2)) 15.32/5.16 proper(fst(x0)) 15.32/5.16 proper(U41(x0, x1, x2)) 15.32/5.16 proper(U42(x0, x1, x2)) 15.32/5.16 proper(U43(x0, x1, x2)) 15.32/5.16 proper(U44(x0, x1, x2)) 15.32/5.16 proper(U45(x0, x1)) 15.32/5.16 proper(U46(x0)) 15.32/5.16 proper(U51(x0, x1, x2)) 15.32/5.16 proper(U52(x0, x1, x2)) 15.32/5.16 proper(U53(x0, x1, x2)) 15.32/5.16 proper(U54(x0, x1, x2)) 15.32/5.16 proper(U55(x0, x1)) 15.32/5.16 proper(U56(x0)) 15.32/5.16 proper(U61(x0, x1)) 15.32/5.16 proper(U62(x0, x1)) 15.32/5.16 proper(isPLNatKind(x0)) 15.32/5.16 proper(U63(x0)) 15.32/5.16 proper(isPLNat(x0)) 15.32/5.16 proper(U71(x0, x1)) 15.32/5.16 proper(U72(x0, x1)) 15.32/5.16 proper(U73(x0)) 15.32/5.16 proper(U81(x0, x1)) 15.32/5.16 proper(U82(x0, x1)) 15.32/5.16 proper(U83(x0)) 15.32/5.16 proper(U91(x0, x1)) 15.32/5.16 proper(U92(x0, x1)) 15.32/5.16 proper(U93(x0)) 15.32/5.16 proper(tail(x0)) 15.32/5.16 proper(take(x0, x1)) 15.32/5.16 proper(0) 15.32/5.16 proper(sel(x0, x1)) 15.32/5.16 U101(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U102(ok(x0), ok(x1), ok(x2)) 15.32/5.16 isNaturalKind(ok(x0)) 15.32/5.16 U103(ok(x0), ok(x1), ok(x2)) 15.32/5.16 isLNatKind(ok(x0)) 15.32/5.16 U104(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U105(ok(x0), ok(x1)) 15.32/5.16 isNatural(ok(x0)) 15.32/5.16 U106(ok(x0)) 15.32/5.16 isLNat(ok(x0)) 15.32/5.16 U11(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U12(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U111(ok(x0), ok(x1)) 15.32/5.16 U112(ok(x0)) 15.32/5.16 U13(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U121(ok(x0), ok(x1)) 15.32/5.16 U122(ok(x0)) 15.32/5.16 U14(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U131(ok(x0)) 15.32/5.16 snd(ok(x0)) 15.32/5.16 splitAt(ok(x0), ok(x1)) 15.32/5.16 U141(ok(x0)) 15.32/5.16 U151(ok(x0)) 15.32/5.16 U161(ok(x0)) 15.32/5.16 U171(ok(x0), ok(x1)) 15.32/5.16 U172(ok(x0)) 15.32/5.16 U181(ok(x0), ok(x1)) 15.32/5.16 U182(ok(x0), ok(x1)) 15.32/5.16 U183(ok(x0)) 15.32/5.16 U191(ok(x0), ok(x1)) 15.32/5.16 U192(ok(x0), ok(x1)) 15.32/5.16 U193(ok(x0)) 15.32/5.16 U201(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U202(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U203(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U204(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U205(ok(x0), ok(x1)) 15.32/5.16 U206(ok(x0)) 15.32/5.16 U21(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U22(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U211(ok(x0)) 15.32/5.16 U23(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U221(ok(x0)) 15.32/5.16 U24(ok(x0), ok(x1)) 15.32/5.16 U231(ok(x0), ok(x1)) 15.32/5.16 U232(ok(x0)) 15.32/5.16 U241(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U242(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U243(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U244(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U245(ok(x0), ok(x1)) 15.32/5.16 U246(ok(x0)) 15.32/5.16 U251(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U252(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U253(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U254(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U255(ok(x0), ok(x1)) 15.32/5.16 U256(ok(x0)) 15.32/5.16 U261(ok(x0), ok(x1)) 15.32/5.16 U262(ok(x0)) 15.32/5.16 U271(ok(x0), ok(x1)) 15.32/5.16 U272(ok(x0)) 15.32/5.16 U281(ok(x0), ok(x1)) 15.32/5.16 U282(ok(x0), ok(x1)) 15.32/5.16 cons(ok(x0), ok(x1)) 15.32/5.16 natsFrom(ok(x0)) 15.32/5.16 s(ok(x0)) 15.32/5.16 U291(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U292(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U293(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U294(ok(x0), ok(x1), ok(x2)) 15.32/5.16 head(ok(x0)) 15.32/5.16 afterNth(ok(x0), ok(x1)) 15.32/5.16 U301(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U302(ok(x0), ok(x1)) 15.32/5.16 U303(ok(x0), ok(x1)) 15.32/5.16 U304(ok(x0), ok(x1)) 15.32/5.16 U31(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U32(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U311(ok(x0), ok(x1)) 15.32/5.16 U312(ok(x0), ok(x1)) 15.32/5.16 pair(ok(x0), ok(x1)) 15.32/5.16 U33(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U321(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.16 U322(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.16 U323(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.16 U324(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.16 U325(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.16 U326(ok(x0), ok(x1), ok(x2), ok(x3)) 15.32/5.16 U327(ok(x0), ok(x1)) 15.32/5.16 U34(ok(x0), ok(x1)) 15.32/5.16 U331(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U332(ok(x0), ok(x1)) 15.32/5.16 U333(ok(x0), ok(x1)) 15.32/5.16 U334(ok(x0), ok(x1)) 15.32/5.16 U341(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U342(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U343(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U344(ok(x0), ok(x1), ok(x2)) 15.32/5.16 fst(ok(x0)) 15.32/5.16 U41(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U42(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U43(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U44(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U45(ok(x0), ok(x1)) 15.32/5.16 U46(ok(x0)) 15.32/5.16 U51(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U52(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U53(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U54(ok(x0), ok(x1), ok(x2)) 15.32/5.16 U55(ok(x0), ok(x1)) 15.32/5.16 U56(ok(x0)) 15.32/5.16 U61(ok(x0), ok(x1)) 15.32/5.16 U62(ok(x0), ok(x1)) 15.32/5.16 isPLNatKind(ok(x0)) 15.32/5.16 U63(ok(x0)) 15.32/5.16 isPLNat(ok(x0)) 15.32/5.16 U71(ok(x0), ok(x1)) 15.32/5.16 U72(ok(x0), ok(x1)) 15.32/5.16 U73(ok(x0)) 15.32/5.16 U81(ok(x0), ok(x1)) 15.32/5.16 U82(ok(x0), ok(x1)) 15.32/5.16 U83(ok(x0)) 15.32/5.16 U91(ok(x0), ok(x1)) 15.32/5.16 U92(ok(x0), ok(x1)) 15.32/5.16 U93(ok(x0)) 15.32/5.16 tail(ok(x0)) 15.32/5.16 take(ok(x0), ok(x1)) 15.32/5.16 sel(ok(x0), ok(x1)) 15.32/5.16 top(mark(x0)) 15.32/5.16 top(ok(x0)) 15.32/5.16 15.32/5.16 Special symbols used for the transformation (see [GM04]): 15.32/5.16 top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1 15.32/5.16 The replacement map contains the following entries: 15.32/5.16 15.32/5.16 U101: {1} 15.32/5.16 tt: empty set 15.32/5.16 U102: {1} 15.32/5.16 isNaturalKind: empty set 15.32/5.16 U103: {1} 15.32/5.16 isLNatKind: empty set 15.32/5.16 U104: {1} 15.32/5.16 U105: {1} 15.32/5.16 isNatural: empty set 15.32/5.16 U106: {1} 15.32/5.16 isLNat: empty set 15.32/5.16 U11: {1} 15.32/5.16 U12: {1} 15.32/5.16 U111: {1} 15.32/5.16 U112: {1} 15.32/5.16 U13: {1} 15.32/5.16 U121: {1} 15.32/5.16 U122: {1} 15.32/5.16 U14: {1} 15.32/5.16 U131: {1} 15.32/5.16 snd: {1} 15.32/5.16 splitAt: {1, 2} 15.32/5.16 U141: {1} 15.32/5.16 U151: {1} 15.32/5.16 U161: {1} 15.32/5.16 U171: {1} 15.32/5.16 U172: {1} 15.32/5.16 U181: {1} 15.32/5.16 U182: {1} 15.32/5.16 U183: {1} 15.32/5.16 U191: {1} 15.32/5.16 U192: {1} 15.32/5.16 U193: {1} 15.32/5.16 U201: {1} 15.32/5.16 U202: {1} 15.32/5.16 U203: {1} 15.32/5.16 U204: {1} 15.32/5.16 U205: {1} 15.32/5.16 U206: {1} 15.32/5.16 U21: {1} 15.32/5.16 U22: {1} 15.32/5.16 U211: {1} 15.32/5.16 U23: {1} 15.32/5.16 U221: {1} 15.32/5.16 U24: {1} 15.32/5.16 U231: {1} 15.32/5.16 U232: {1} 15.32/5.16 U241: {1} 15.32/5.16 U242: {1} 15.32/5.16 U243: {1} 15.32/5.16 U244: {1} 15.32/5.16 U245: {1} 15.32/5.16 U246: {1} 15.32/5.16 U251: {1} 15.32/5.16 U252: {1} 15.32/5.16 U253: {1} 15.32/5.16 U254: {1} 15.32/5.16 U255: {1} 15.32/5.16 U256: {1} 15.32/5.16 U261: {1} 15.32/5.16 U262: {1} 15.32/5.16 U271: {1} 15.32/5.16 U272: {1} 15.32/5.16 U281: {1} 15.32/5.16 U282: {1} 15.32/5.16 cons: {1} 15.32/5.16 natsFrom: {1} 15.32/5.16 s: {1} 15.32/5.16 U291: {1} 15.32/5.16 U292: {1} 15.32/5.16 U293: {1} 15.32/5.16 U294: {1} 15.32/5.16 head: {1} 15.32/5.16 afterNth: {1, 2} 15.32/5.16 U301: {1} 15.32/5.16 U302: {1} 15.32/5.16 U303: {1} 15.32/5.16 U304: {1} 15.32/5.16 U31: {1} 15.32/5.16 U32: {1} 15.32/5.16 U311: {1} 15.32/5.16 U312: {1} 15.32/5.16 pair: {1, 2} 15.32/5.16 nil: empty set 15.32/5.16 U33: {1} 15.32/5.16 U321: {1} 15.32/5.16 U322: {1} 15.32/5.16 U323: {1} 15.32/5.16 U324: {1} 15.32/5.16 U325: {1} 15.32/5.16 U326: {1} 15.32/5.16 U327: {1} 15.32/5.16 U34: {1} 15.32/5.16 U331: {1} 15.32/5.16 U332: {1} 15.32/5.16 U333: {1} 15.32/5.16 U334: {1} 15.32/5.16 U341: {1} 15.32/5.16 U342: {1} 15.32/5.16 U343: {1} 15.32/5.16 U344: {1} 15.32/5.16 fst: {1} 15.32/5.16 U41: {1} 15.32/5.16 U42: {1} 15.32/5.16 U43: {1} 15.32/5.16 U44: {1} 15.32/5.16 U45: {1} 15.32/5.16 U46: {1} 15.32/5.16 U51: {1} 15.32/5.16 U52: {1} 15.32/5.16 U53: {1} 15.32/5.16 U54: {1} 15.32/5.16 U55: {1} 15.32/5.16 U56: {1} 15.32/5.16 U61: {1} 15.32/5.16 U62: {1} 15.32/5.16 isPLNatKind: empty set 15.32/5.16 U63: {1} 15.32/5.16 isPLNat: empty set 15.32/5.16 U71: {1} 15.32/5.16 U72: {1} 15.32/5.16 U73: {1} 15.32/5.16 U81: {1} 15.32/5.16 U82: {1} 15.32/5.16 U83: {1} 15.32/5.16 U91: {1} 15.32/5.16 U92: {1} 15.32/5.16 U93: {1} 15.32/5.16 tail: {1} 15.32/5.16 take: {1, 2} 15.32/5.16 0: empty set 15.32/5.16 sel: {1, 2} 15.32/5.16 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. 15.32/5.16 ---------------------------------------- 15.32/5.16 15.32/5.16 (2) 15.32/5.16 Obligation: 15.32/5.16 Context-sensitive rewrite system: 15.32/5.16 The TRS R consists of the following rules: 15.32/5.16 15.32/5.16 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.16 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.16 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.16 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.16 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.16 U106(tt) -> tt 15.32/5.16 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 15.32/5.16 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.16 U112(tt) -> tt 15.32/5.16 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 15.32/5.16 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.16 U122(tt) -> tt 15.32/5.16 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 15.32/5.16 U131(tt) -> tt 15.32/5.16 U14(tt, N, XS) -> snd(splitAt(N, XS)) 15.32/5.16 U141(tt) -> tt 15.32/5.16 U151(tt) -> tt 15.32/5.16 U161(tt) -> tt 15.32/5.16 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.16 U172(tt) -> tt 15.32/5.16 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.16 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.16 U183(tt) -> tt 15.32/5.16 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.16 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.16 U193(tt) -> tt 15.32/5.16 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.16 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.16 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.16 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.16 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.16 U206(tt) -> tt 15.32/5.16 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 15.32/5.16 U211(tt) -> tt 15.32/5.16 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 15.32/5.16 U221(tt) -> tt 15.32/5.16 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 15.32/5.16 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.16 U232(tt) -> tt 15.32/5.16 U24(tt, X) -> X 15.32/5.16 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.16 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.16 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.16 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.16 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.16 U246(tt) -> tt 15.32/5.16 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.16 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.16 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.16 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.16 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.16 U256(tt) -> tt 15.32/5.16 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.16 U262(tt) -> tt 15.32/5.16 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.16 U272(tt) -> tt 15.32/5.16 U281(tt, N) -> U282(isNaturalKind(N), N) 15.32/5.16 U282(tt, N) -> cons(N, natsFrom(s(N))) 15.32/5.16 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 15.32/5.16 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 15.32/5.16 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 15.32/5.16 U294(tt, N, XS) -> head(afterNth(N, XS)) 15.32/5.16 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 15.32/5.16 U302(tt, Y) -> U303(isLNat(Y), Y) 15.32/5.16 U303(tt, Y) -> U304(isLNatKind(Y), Y) 15.32/5.16 U304(tt, Y) -> Y 15.32/5.16 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 15.32/5.16 U311(tt, XS) -> U312(isLNatKind(XS), XS) 15.32/5.16 U312(tt, XS) -> pair(nil, XS) 15.32/5.16 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 15.32/5.16 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 15.32/5.16 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 15.32/5.16 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 15.32/5.16 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 15.32/5.16 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 15.32/5.16 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 15.32/5.16 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 15.32/5.16 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 15.32/5.16 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 15.32/5.16 U332(tt, XS) -> U333(isLNat(XS), XS) 15.32/5.16 U333(tt, XS) -> U334(isLNatKind(XS), XS) 15.32/5.16 U334(tt, XS) -> XS 15.32/5.16 U34(tt, N) -> N 15.32/5.16 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 15.32/5.16 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 15.32/5.16 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 15.32/5.16 U344(tt, N, XS) -> fst(splitAt(N, XS)) 15.32/5.16 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.16 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.16 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.16 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.16 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.16 U46(tt) -> tt 15.32/5.16 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.16 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.16 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.16 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.16 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.16 U56(tt) -> tt 15.32/5.16 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.16 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.16 U63(tt) -> tt 15.32/5.16 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.16 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.16 U73(tt) -> tt 15.32/5.16 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.16 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.16 U83(tt) -> tt 15.32/5.16 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.16 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.16 U93(tt) -> tt 15.32/5.16 afterNth(N, XS) -> U11(isNatural(N), N, XS) 15.32/5.16 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 15.32/5.16 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 15.32/5.16 isLNat(nil) -> tt 15.32/5.16 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.16 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.16 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.16 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.16 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.16 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.16 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.16 isLNatKind(nil) -> tt 15.32/5.16 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.16 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.16 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.16 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.16 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.16 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.16 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.16 isNatural(0) -> tt 15.32/5.16 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.16 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.16 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.16 isNaturalKind(0) -> tt 15.32/5.16 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.16 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.16 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.16 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.16 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.16 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.16 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.16 natsFrom(N) -> U281(isNatural(N), N) 15.32/5.16 sel(N, XS) -> U291(isNatural(N), N, XS) 15.32/5.16 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 15.32/5.16 splitAt(0, XS) -> U311(isLNat(XS), XS) 15.32/5.16 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 15.32/5.16 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 15.32/5.16 take(N, XS) -> U341(isNatural(N), N, XS) 15.32/5.16 15.32/5.16 The replacement map contains the following entries: 15.32/5.16 15.32/5.16 U101: {1} 15.32/5.16 tt: empty set 15.32/5.16 U102: {1} 15.32/5.16 isNaturalKind: empty set 15.32/5.16 U103: {1} 15.32/5.16 isLNatKind: empty set 15.32/5.16 U104: {1} 15.32/5.16 U105: {1} 15.32/5.16 isNatural: empty set 15.32/5.16 U106: {1} 15.32/5.16 isLNat: empty set 15.32/5.16 U11: {1} 15.32/5.16 U12: {1} 15.32/5.16 U111: {1} 15.32/5.16 U112: {1} 15.32/5.16 U13: {1} 15.32/5.16 U121: {1} 15.32/5.16 U122: {1} 15.32/5.16 U14: {1} 15.32/5.16 U131: {1} 15.32/5.16 snd: {1} 15.32/5.16 splitAt: {1, 2} 15.32/5.16 U141: {1} 15.32/5.16 U151: {1} 15.32/5.16 U161: {1} 15.32/5.16 U171: {1} 15.32/5.16 U172: {1} 15.32/5.16 U181: {1} 15.32/5.16 U182: {1} 15.32/5.16 U183: {1} 15.32/5.16 U191: {1} 15.32/5.16 U192: {1} 15.32/5.16 U193: {1} 15.32/5.16 U201: {1} 15.32/5.16 U202: {1} 15.32/5.16 U203: {1} 15.32/5.16 U204: {1} 15.32/5.16 U205: {1} 15.32/5.16 U206: {1} 15.32/5.16 U21: {1} 15.32/5.16 U22: {1} 15.32/5.16 U211: {1} 15.32/5.16 U23: {1} 15.32/5.16 U221: {1} 15.32/5.16 U24: {1} 15.32/5.16 U231: {1} 15.32/5.16 U232: {1} 15.32/5.16 U241: {1} 15.32/5.16 U242: {1} 15.32/5.16 U243: {1} 15.32/5.16 U244: {1} 15.32/5.16 U245: {1} 15.32/5.16 U246: {1} 15.32/5.16 U251: {1} 15.32/5.16 U252: {1} 15.32/5.16 U253: {1} 15.32/5.16 U254: {1} 15.32/5.16 U255: {1} 15.32/5.16 U256: {1} 15.32/5.16 U261: {1} 15.32/5.16 U262: {1} 15.32/5.16 U271: {1} 15.32/5.16 U272: {1} 15.32/5.16 U281: {1} 15.32/5.16 U282: {1} 15.32/5.16 cons: {1} 15.32/5.16 natsFrom: {1} 15.32/5.16 s: {1} 15.32/5.16 U291: {1} 15.32/5.16 U292: {1} 15.32/5.16 U293: {1} 15.32/5.16 U294: {1} 15.32/5.16 head: {1} 15.32/5.16 afterNth: {1, 2} 15.32/5.16 U301: {1} 15.32/5.16 U302: {1} 15.32/5.16 U303: {1} 15.32/5.16 U304: {1} 15.32/5.16 U31: {1} 15.32/5.16 U32: {1} 15.32/5.16 U311: {1} 15.32/5.16 U312: {1} 15.32/5.16 pair: {1, 2} 15.32/5.16 nil: empty set 15.32/5.16 U33: {1} 15.32/5.16 U321: {1} 15.32/5.16 U322: {1} 15.32/5.16 U323: {1} 15.32/5.16 U324: {1} 15.32/5.16 U325: {1} 15.32/5.16 U326: {1} 15.32/5.16 U327: {1} 15.32/5.16 U34: {1} 15.32/5.16 U331: {1} 15.32/5.16 U332: {1} 15.32/5.16 U333: {1} 15.32/5.16 U334: {1} 15.32/5.16 U341: {1} 15.32/5.16 U342: {1} 15.32/5.16 U343: {1} 15.32/5.16 U344: {1} 15.32/5.16 fst: {1} 15.32/5.16 U41: {1} 15.32/5.16 U42: {1} 15.32/5.16 U43: {1} 15.32/5.16 U44: {1} 15.32/5.16 U45: {1} 15.32/5.16 U46: {1} 15.32/5.16 U51: {1} 15.32/5.16 U52: {1} 15.32/5.16 U53: {1} 15.32/5.16 U54: {1} 15.32/5.16 U55: {1} 15.32/5.16 U56: {1} 15.32/5.16 U61: {1} 15.32/5.16 U62: {1} 15.32/5.16 isPLNatKind: empty set 15.32/5.16 U63: {1} 15.32/5.16 isPLNat: empty set 15.32/5.16 U71: {1} 15.32/5.16 U72: {1} 15.32/5.16 U73: {1} 15.32/5.16 U81: {1} 15.32/5.16 U82: {1} 15.32/5.16 U83: {1} 15.32/5.16 U91: {1} 15.32/5.16 U92: {1} 15.32/5.16 U93: {1} 15.32/5.16 tail: {1} 15.32/5.16 take: {1, 2} 15.32/5.16 0: empty set 15.32/5.16 sel: {1, 2} 15.32/5.16 15.32/5.16 ---------------------------------------- 15.32/5.16 15.32/5.16 (3) CSDependencyPairsProof (EQUIVALENT) 15.32/5.16 Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem. 15.32/5.16 ---------------------------------------- 15.32/5.16 15.32/5.16 (4) 15.32/5.16 Obligation: 15.32/5.16 Q-restricted context-sensitive dependency pair problem: 15.32/5.16 The symbols in {U106_1, U112_1, U122_1, U131_1, snd_1, splitAt_2, U141_1, U151_1, U161_1, U172_1, U183_1, U193_1, U206_1, U211_1, U221_1, U232_1, U246_1, U256_1, U262_1, U272_1, natsFrom_1, s_1, head_1, afterNth_2, pair_2, fst_1, U46_1, U56_1, U63_1, U73_1, U83_1, U93_1, tail_1, take_2, sel_2, U106'_1, U112'_1, U122'_1, SND_1, SPLITAT_2, U172'_1, U183'_1, U193'_1, U206'_1, U232'_1, U246'_1, U256'_1, U262'_1, U272'_1, HEAD_1, AFTERNTH_2, FST_1, U46'_1, U56'_1, U63'_1, U73'_1, U83'_1, U93'_1, U131'_1, U141'_1, U151'_1, U161'_1, U211'_1, U221'_1, NATSFROM_1, SEL_2, TAIL_1, TAKE_2} are replacing on all positions. 15.32/5.16 For all symbols f in {U101_3, U102_3, U103_3, U104_3, U105_2, U11_3, U12_3, U111_2, U13_3, U121_2, U14_3, U171_2, U181_2, U182_2, U191_2, U192_2, U201_3, U202_3, U203_3, U204_3, U205_2, U21_3, U22_3, U23_3, U24_2, U231_2, U241_3, U242_3, U243_3, U244_3, U245_2, U251_3, U252_3, U253_3, U254_3, U255_2, U261_2, U271_2, U281_2, U282_2, cons_2, U291_3, U292_3, U293_3, U294_3, U301_3, U302_2, U303_2, U304_2, U31_3, U32_3, U311_2, U312_2, U33_3, U321_4, U322_4, U323_4, U324_4, U325_4, U326_4, U327_2, U34_2, U331_3, U332_2, U333_2, U334_2, U341_3, U342_3, U343_3, U344_3, U41_3, U42_3, U43_3, U44_3, U45_2, U51_3, U52_3, U53_3, U54_3, U55_2, U61_2, U62_2, U71_2, U72_2, U81_2, U82_2, U91_2, U92_2, U102'_3, U101'_3, U103'_3, U104'_3, U105'_2, U12'_3, U11'_3, U111'_2, U13'_3, U121'_2, U14'_3, U171'_2, U182'_2, U181'_2, U192'_2, U191'_2, U202'_3, U201'_3, U203'_3, U204'_3, U205'_2, U22'_3, U21'_3, U23'_3, U24'_2, U231'_2, U242'_3, U241'_3, U243'_3, U244'_3, U245'_2, U252'_3, U251'_3, U253'_3, U254'_3, U255'_2, U261'_2, U271'_2, U282'_2, U281'_2, U292'_3, U291'_3, U293'_3, U294'_3, U302'_2, U301'_3, U303'_2, U304'_2, U32'_3, U31'_3, U312'_2, U311'_2, U33'_3, U322'_4, U321'_4, U323'_4, U324'_4, U325'_4, U326'_4, U327'_2, U34'_2, U332'_2, U331'_3, U333'_2, U334'_2, U342'_3, U341'_3, U343'_3, U344'_3, U42'_3, U41'_3, U43'_3, U44'_3, U45'_2, U52'_3, U51'_3, U53'_3, U54'_3, U55'_2, U62'_2, U61'_2, U72'_2, U71'_2, U82'_2, U81'_2, U92'_2, U91'_2} we have mu(f) = {1}. 15.32/5.16 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1, ISNATURALKIND_1, ISLNATKIND_1, ISNATURAL_1, ISLNAT_1, ISPLNATKIND_1, ISPLNAT_1, U_1} are not replacing on any position. 15.32/5.16 15.32/5.16 The ordinary context-sensitive dependency pairs DP_o are: 15.32/5.16 U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) 15.32/5.16 U101'(tt, V1, V2) -> ISNATURALKIND(V1) 15.32/5.16 U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) 15.32/5.16 U102'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) 15.32/5.16 U103'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) 15.32/5.16 U104'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.16 U105'(tt, V2) -> U106'(isLNat(V2)) 15.32/5.16 U105'(tt, V2) -> ISLNAT(V2) 15.32/5.16 U11'(tt, N, XS) -> U12'(isNaturalKind(N), N, XS) 15.32/5.16 U11'(tt, N, XS) -> ISNATURALKIND(N) 15.32/5.16 U111'(tt, V2) -> U112'(isLNatKind(V2)) 15.32/5.16 U111'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 U12'(tt, N, XS) -> U13'(isLNat(XS), N, XS) 15.32/5.16 U12'(tt, N, XS) -> ISLNAT(XS) 15.32/5.16 U121'(tt, V2) -> U122'(isLNatKind(V2)) 15.32/5.16 U121'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 U13'(tt, N, XS) -> U14'(isLNatKind(XS), N, XS) 15.32/5.16 U13'(tt, N, XS) -> ISLNATKIND(XS) 15.32/5.16 U14'(tt, N, XS) -> SND(splitAt(N, XS)) 15.32/5.16 U14'(tt, N, XS) -> SPLITAT(N, XS) 15.32/5.16 U171'(tt, V2) -> U172'(isLNatKind(V2)) 15.32/5.16 U171'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 U181'(tt, V1) -> U182'(isLNatKind(V1), V1) 15.32/5.16 U181'(tt, V1) -> ISLNATKIND(V1) 15.32/5.16 U182'(tt, V1) -> U183'(isLNat(V1)) 15.32/5.16 U182'(tt, V1) -> ISLNAT(V1) 15.32/5.16 U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) 15.32/5.16 U191'(tt, V1) -> ISNATURALKIND(V1) 15.32/5.16 U192'(tt, V1) -> U193'(isNatural(V1)) 15.32/5.16 U192'(tt, V1) -> ISNATURAL(V1) 15.32/5.16 U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) 15.32/5.16 U201'(tt, V1, V2) -> ISNATURALKIND(V1) 15.32/5.16 U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) 15.32/5.16 U202'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) 15.32/5.16 U203'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) 15.32/5.16 U204'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.16 U205'(tt, V2) -> U206'(isLNat(V2)) 15.32/5.16 U205'(tt, V2) -> ISLNAT(V2) 15.32/5.16 U21'(tt, X, Y) -> U22'(isLNatKind(X), X, Y) 15.32/5.16 U21'(tt, X, Y) -> ISLNATKIND(X) 15.32/5.16 U22'(tt, X, Y) -> U23'(isLNat(Y), X, Y) 15.32/5.16 U22'(tt, X, Y) -> ISLNAT(Y) 15.32/5.16 U23'(tt, X, Y) -> U24'(isLNatKind(Y), X) 15.32/5.16 U23'(tt, X, Y) -> ISLNATKIND(Y) 15.32/5.16 U231'(tt, V2) -> U232'(isLNatKind(V2)) 15.32/5.16 U231'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) 15.32/5.16 U241'(tt, V1, V2) -> ISLNATKIND(V1) 15.32/5.16 U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) 15.32/5.16 U242'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) 15.32/5.16 U243'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) 15.32/5.16 U244'(tt, V1, V2) -> ISLNAT(V1) 15.32/5.16 U245'(tt, V2) -> U246'(isLNat(V2)) 15.32/5.16 U245'(tt, V2) -> ISLNAT(V2) 15.32/5.16 U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) 15.32/5.16 U251'(tt, V1, V2) -> ISNATURALKIND(V1) 15.32/5.16 U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) 15.32/5.16 U252'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) 15.32/5.16 U253'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) 15.32/5.16 U254'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.16 U255'(tt, V2) -> U256'(isLNat(V2)) 15.32/5.16 U255'(tt, V2) -> ISLNAT(V2) 15.32/5.16 U261'(tt, V2) -> U262'(isLNatKind(V2)) 15.32/5.16 U261'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 U271'(tt, V2) -> U272'(isLNatKind(V2)) 15.32/5.16 U271'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 U281'(tt, N) -> U282'(isNaturalKind(N), N) 15.32/5.16 U281'(tt, N) -> ISNATURALKIND(N) 15.32/5.16 U291'(tt, N, XS) -> U292'(isNaturalKind(N), N, XS) 15.32/5.16 U291'(tt, N, XS) -> ISNATURALKIND(N) 15.32/5.16 U292'(tt, N, XS) -> U293'(isLNat(XS), N, XS) 15.32/5.16 U292'(tt, N, XS) -> ISLNAT(XS) 15.32/5.16 U293'(tt, N, XS) -> U294'(isLNatKind(XS), N, XS) 15.32/5.16 U293'(tt, N, XS) -> ISLNATKIND(XS) 15.32/5.16 U294'(tt, N, XS) -> HEAD(afterNth(N, XS)) 15.32/5.16 U294'(tt, N, XS) -> AFTERNTH(N, XS) 15.32/5.16 U301'(tt, X, Y) -> U302'(isLNatKind(X), Y) 15.32/5.16 U301'(tt, X, Y) -> ISLNATKIND(X) 15.32/5.16 U302'(tt, Y) -> U303'(isLNat(Y), Y) 15.32/5.16 U302'(tt, Y) -> ISLNAT(Y) 15.32/5.16 U303'(tt, Y) -> U304'(isLNatKind(Y), Y) 15.32/5.16 U303'(tt, Y) -> ISLNATKIND(Y) 15.32/5.16 U31'(tt, N, XS) -> U32'(isNaturalKind(N), N, XS) 15.32/5.16 U31'(tt, N, XS) -> ISNATURALKIND(N) 15.32/5.16 U311'(tt, XS) -> U312'(isLNatKind(XS), XS) 15.32/5.16 U311'(tt, XS) -> ISLNATKIND(XS) 15.32/5.16 U32'(tt, N, XS) -> U33'(isLNat(XS), N, XS) 15.32/5.16 U32'(tt, N, XS) -> ISLNAT(XS) 15.32/5.16 U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) 15.32/5.16 U321'(tt, N, X, XS) -> ISNATURALKIND(N) 15.32/5.16 U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) 15.32/5.16 U322'(tt, N, X, XS) -> ISNATURAL(X) 15.32/5.16 U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) 15.32/5.16 U323'(tt, N, X, XS) -> ISNATURALKIND(X) 15.32/5.16 U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) 15.32/5.16 U324'(tt, N, X, XS) -> ISLNAT(XS) 15.32/5.16 U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) 15.32/5.16 U325'(tt, N, X, XS) -> ISLNATKIND(XS) 15.32/5.16 U326'(tt, N, X, XS) -> U327'(splitAt(N, XS), X) 15.32/5.16 U326'(tt, N, X, XS) -> SPLITAT(N, XS) 15.32/5.16 U33'(tt, N, XS) -> U34'(isLNatKind(XS), N) 15.32/5.16 U33'(tt, N, XS) -> ISLNATKIND(XS) 15.32/5.16 U331'(tt, N, XS) -> U332'(isNaturalKind(N), XS) 15.32/5.16 U331'(tt, N, XS) -> ISNATURALKIND(N) 15.32/5.16 U332'(tt, XS) -> U333'(isLNat(XS), XS) 15.32/5.16 U332'(tt, XS) -> ISLNAT(XS) 15.32/5.16 U333'(tt, XS) -> U334'(isLNatKind(XS), XS) 15.32/5.16 U333'(tt, XS) -> ISLNATKIND(XS) 15.32/5.16 U341'(tt, N, XS) -> U342'(isNaturalKind(N), N, XS) 15.32/5.16 U341'(tt, N, XS) -> ISNATURALKIND(N) 15.32/5.16 U342'(tt, N, XS) -> U343'(isLNat(XS), N, XS) 15.32/5.16 U342'(tt, N, XS) -> ISLNAT(XS) 15.32/5.16 U343'(tt, N, XS) -> U344'(isLNatKind(XS), N, XS) 15.32/5.16 U343'(tt, N, XS) -> ISLNATKIND(XS) 15.32/5.16 U344'(tt, N, XS) -> FST(splitAt(N, XS)) 15.32/5.16 U344'(tt, N, XS) -> SPLITAT(N, XS) 15.32/5.16 U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) 15.32/5.16 U41'(tt, V1, V2) -> ISNATURALKIND(V1) 15.32/5.16 U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) 15.32/5.16 U42'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) 15.32/5.16 U43'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) 15.32/5.16 U44'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.16 U45'(tt, V2) -> U46'(isLNat(V2)) 15.32/5.16 U45'(tt, V2) -> ISLNAT(V2) 15.32/5.16 U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) 15.32/5.16 U51'(tt, V1, V2) -> ISNATURALKIND(V1) 15.32/5.16 U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) 15.32/5.16 U52'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) 15.32/5.16 U53'(tt, V1, V2) -> ISLNATKIND(V2) 15.32/5.16 U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) 15.32/5.16 U54'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.16 U55'(tt, V2) -> U56'(isLNat(V2)) 15.32/5.16 U55'(tt, V2) -> ISLNAT(V2) 15.32/5.16 U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) 15.32/5.16 U61'(tt, V1) -> ISPLNATKIND(V1) 15.32/5.16 U62'(tt, V1) -> U63'(isPLNat(V1)) 15.32/5.16 U62'(tt, V1) -> ISPLNAT(V1) 15.32/5.16 U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) 15.32/5.16 U71'(tt, V1) -> ISNATURALKIND(V1) 15.32/5.16 U72'(tt, V1) -> U73'(isNatural(V1)) 15.32/5.16 U72'(tt, V1) -> ISNATURAL(V1) 15.32/5.16 U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) 15.32/5.16 U81'(tt, V1) -> ISPLNATKIND(V1) 15.32/5.16 U82'(tt, V1) -> U83'(isPLNat(V1)) 15.32/5.16 U82'(tt, V1) -> ISPLNAT(V1) 15.32/5.16 U91'(tt, V1) -> U92'(isLNatKind(V1), V1) 15.32/5.16 U91'(tt, V1) -> ISLNATKIND(V1) 15.32/5.16 U92'(tt, V1) -> U93'(isLNat(V1)) 15.32/5.16 U92'(tt, V1) -> ISLNAT(V1) 15.32/5.16 AFTERNTH(N, XS) -> U11'(isNatural(N), N, XS) 15.32/5.16 AFTERNTH(N, XS) -> ISNATURAL(N) 15.32/5.16 FST(pair(X, Y)) -> U21'(isLNat(X), X, Y) 15.32/5.16 FST(pair(X, Y)) -> ISLNAT(X) 15.32/5.16 HEAD(cons(N, XS)) -> U31'(isNatural(N), N, XS) 15.32/5.16 HEAD(cons(N, XS)) -> ISNATURAL(N) 15.32/5.16 ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) 15.32/5.16 ISLNAT(afterNth(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) 15.32/5.16 ISLNAT(cons(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) 15.32/5.16 ISLNAT(fst(V1)) -> ISPLNATKIND(V1) 15.32/5.16 ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) 15.32/5.16 ISLNAT(natsFrom(V1)) -> ISNATURALKIND(V1) 15.32/5.16 ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) 15.32/5.16 ISLNAT(snd(V1)) -> ISPLNATKIND(V1) 15.32/5.16 ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) 15.32/5.16 ISLNAT(tail(V1)) -> ISLNATKIND(V1) 15.32/5.16 ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) 15.32/5.16 ISLNAT(take(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) 15.32/5.16 ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) 15.32/5.16 ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISLNATKIND(fst(V1)) -> U131'(isPLNatKind(V1)) 15.32/5.16 ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) 15.32/5.16 ISLNATKIND(natsFrom(V1)) -> U141'(isNaturalKind(V1)) 15.32/5.16 ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) 15.32/5.16 ISLNATKIND(snd(V1)) -> U151'(isPLNatKind(V1)) 15.32/5.16 ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) 15.32/5.16 ISLNATKIND(tail(V1)) -> U161'(isLNatKind(V1)) 15.32/5.16 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 15.32/5.16 ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) 15.32/5.16 ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) 15.32/5.16 ISNATURAL(head(V1)) -> ISLNATKIND(V1) 15.32/5.16 ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) 15.32/5.16 ISNATURAL(s(V1)) -> ISNATURALKIND(V1) 15.32/5.16 ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) 15.32/5.16 ISNATURAL(sel(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISNATURALKIND(head(V1)) -> U211'(isLNatKind(V1)) 15.32/5.16 ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) 15.32/5.16 ISNATURALKIND(s(V1)) -> U221'(isNaturalKind(V1)) 15.32/5.16 ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) 15.32/5.16 ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) 15.32/5.16 ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) 15.32/5.16 ISPLNAT(pair(V1, V2)) -> ISLNATKIND(V1) 15.32/5.16 ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) 15.32/5.16 ISPLNAT(splitAt(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) 15.32/5.16 ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) 15.32/5.16 ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) 15.32/5.16 ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 NATSFROM(N) -> U281'(isNatural(N), N) 15.32/5.16 NATSFROM(N) -> ISNATURAL(N) 15.32/5.16 SEL(N, XS) -> U291'(isNatural(N), N, XS) 15.32/5.16 SEL(N, XS) -> ISNATURAL(N) 15.32/5.16 SND(pair(X, Y)) -> U301'(isLNat(X), X, Y) 15.32/5.16 SND(pair(X, Y)) -> ISLNAT(X) 15.32/5.16 SPLITAT(0, XS) -> U311'(isLNat(XS), XS) 15.32/5.16 SPLITAT(0, XS) -> ISLNAT(XS) 15.32/5.16 SPLITAT(s(N), cons(X, XS)) -> U321'(isNatural(N), N, X, XS) 15.32/5.16 SPLITAT(s(N), cons(X, XS)) -> ISNATURAL(N) 15.32/5.16 TAIL(cons(N, XS)) -> U331'(isNatural(N), N, XS) 15.32/5.16 TAIL(cons(N, XS)) -> ISNATURAL(N) 15.32/5.16 TAKE(N, XS) -> U341'(isNatural(N), N, XS) 15.32/5.16 TAKE(N, XS) -> ISNATURAL(N) 15.32/5.16 15.32/5.16 The collapsing dependency pairs are DP_c: 15.32/5.16 U14'(tt, N, XS) -> N 15.32/5.16 U14'(tt, N, XS) -> XS 15.32/5.16 U24'(tt, X) -> X 15.32/5.16 U282'(tt, N) -> N 15.32/5.16 U294'(tt, N, XS) -> N 15.32/5.16 U294'(tt, N, XS) -> XS 15.32/5.16 U304'(tt, Y) -> Y 15.32/5.16 U312'(tt, XS) -> XS 15.32/5.16 U326'(tt, N, X, XS) -> N 15.32/5.16 U326'(tt, N, X, XS) -> XS 15.32/5.16 U327'(pair(YS, ZS), X) -> X 15.32/5.16 U334'(tt, XS) -> XS 15.32/5.16 U34'(tt, N) -> N 15.32/5.16 U344'(tt, N, XS) -> N 15.32/5.16 U344'(tt, N, XS) -> XS 15.32/5.16 15.32/5.16 15.32/5.16 The hidden terms of R are: 15.32/5.16 15.32/5.16 natsFrom(s(x0)) 15.32/5.16 15.32/5.16 Every hiding context is built from: 15.32/5.16 aprove.DPFramework.CSDPProblem.QCSDPProblem$1@4408733c 15.32/5.16 aprove.DPFramework.CSDPProblem.QCSDPProblem$1@6178b6fd 15.32/5.16 15.32/5.16 Hence, the new unhiding pairs DP_u are : 15.32/5.16 U14'(tt, N, XS) -> U(N) 15.32/5.16 U14'(tt, N, XS) -> U(XS) 15.32/5.16 U24'(tt, X) -> U(X) 15.32/5.16 U282'(tt, N) -> U(N) 15.32/5.16 U294'(tt, N, XS) -> U(N) 15.32/5.16 U294'(tt, N, XS) -> U(XS) 15.32/5.16 U304'(tt, Y) -> U(Y) 15.32/5.16 U312'(tt, XS) -> U(XS) 15.32/5.16 U326'(tt, N, X, XS) -> U(N) 15.32/5.16 U326'(tt, N, X, XS) -> U(XS) 15.32/5.16 U327'(pair(YS, ZS), X) -> U(X) 15.32/5.16 U334'(tt, XS) -> U(XS) 15.32/5.16 U34'(tt, N) -> U(N) 15.32/5.16 U344'(tt, N, XS) -> U(N) 15.32/5.16 U344'(tt, N, XS) -> U(XS) 15.32/5.16 U(s(x_0)) -> U(x_0) 15.32/5.16 U(natsFrom(x_0)) -> U(x_0) 15.32/5.16 U(natsFrom(s(x0))) -> NATSFROM(s(x0)) 15.32/5.16 15.32/5.16 The TRS R consists of the following rules: 15.32/5.16 15.32/5.16 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.16 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.16 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.16 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.16 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.16 U106(tt) -> tt 15.32/5.16 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 15.32/5.16 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.16 U112(tt) -> tt 15.32/5.16 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 15.32/5.16 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.16 U122(tt) -> tt 15.32/5.16 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 15.32/5.16 U131(tt) -> tt 15.32/5.16 U14(tt, N, XS) -> snd(splitAt(N, XS)) 15.32/5.16 U141(tt) -> tt 15.32/5.16 U151(tt) -> tt 15.32/5.16 U161(tt) -> tt 15.32/5.16 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.16 U172(tt) -> tt 15.32/5.16 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.16 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.16 U183(tt) -> tt 15.32/5.16 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.16 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.16 U193(tt) -> tt 15.32/5.16 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.16 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.16 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.16 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.16 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.16 U206(tt) -> tt 15.32/5.16 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 15.32/5.16 U211(tt) -> tt 15.32/5.16 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 15.32/5.16 U221(tt) -> tt 15.32/5.16 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 15.32/5.16 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.16 U232(tt) -> tt 15.32/5.16 U24(tt, X) -> X 15.32/5.16 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.16 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.16 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.16 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.16 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.16 U246(tt) -> tt 15.32/5.16 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.16 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.16 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.16 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.16 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.16 U256(tt) -> tt 15.32/5.16 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.16 U262(tt) -> tt 15.32/5.16 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.16 U272(tt) -> tt 15.32/5.16 U281(tt, N) -> U282(isNaturalKind(N), N) 15.32/5.16 U282(tt, N) -> cons(N, natsFrom(s(N))) 15.32/5.16 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 15.32/5.16 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 15.32/5.16 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 15.32/5.16 U294(tt, N, XS) -> head(afterNth(N, XS)) 15.32/5.16 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 15.32/5.16 U302(tt, Y) -> U303(isLNat(Y), Y) 15.32/5.16 U303(tt, Y) -> U304(isLNatKind(Y), Y) 15.32/5.16 U304(tt, Y) -> Y 15.32/5.16 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 15.32/5.16 U311(tt, XS) -> U312(isLNatKind(XS), XS) 15.32/5.16 U312(tt, XS) -> pair(nil, XS) 15.32/5.16 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 15.32/5.16 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 15.32/5.16 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 15.32/5.16 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 15.32/5.16 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 15.32/5.16 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 15.32/5.16 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 15.32/5.16 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 15.32/5.16 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 15.32/5.16 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 15.32/5.16 U332(tt, XS) -> U333(isLNat(XS), XS) 15.32/5.16 U333(tt, XS) -> U334(isLNatKind(XS), XS) 15.32/5.16 U334(tt, XS) -> XS 15.32/5.16 U34(tt, N) -> N 15.32/5.16 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 15.32/5.16 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 15.32/5.16 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 15.32/5.16 U344(tt, N, XS) -> fst(splitAt(N, XS)) 15.32/5.16 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.16 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.16 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.16 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.16 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.16 U46(tt) -> tt 15.32/5.16 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.16 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.16 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.16 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.16 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.16 U56(tt) -> tt 15.32/5.16 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.16 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.16 U63(tt) -> tt 15.32/5.16 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.16 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.16 U73(tt) -> tt 15.32/5.16 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.16 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.16 U83(tt) -> tt 15.32/5.16 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.16 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.16 U93(tt) -> tt 15.32/5.16 afterNth(N, XS) -> U11(isNatural(N), N, XS) 15.32/5.16 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 15.32/5.16 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 15.32/5.16 isLNat(nil) -> tt 15.32/5.16 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.16 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.16 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.16 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.16 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.16 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.16 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.16 isLNatKind(nil) -> tt 15.32/5.16 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.16 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.16 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.16 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.16 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.16 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.16 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.16 isNatural(0) -> tt 15.32/5.16 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.16 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.16 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.16 isNaturalKind(0) -> tt 15.32/5.16 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.16 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.16 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.16 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.16 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.16 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.16 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.16 natsFrom(N) -> U281(isNatural(N), N) 15.32/5.16 sel(N, XS) -> U291(isNatural(N), N, XS) 15.32/5.16 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 15.32/5.16 splitAt(0, XS) -> U311(isLNat(XS), XS) 15.32/5.16 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 15.32/5.16 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 15.32/5.16 take(N, XS) -> U341(isNatural(N), N, XS) 15.32/5.16 15.32/5.16 Q is empty. 15.32/5.16 15.32/5.16 ---------------------------------------- 15.32/5.16 15.32/5.16 (5) QCSDependencyGraphProof (EQUIVALENT) 15.32/5.16 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 149 less nodes. 15.32/5.16 15.32/5.16 ---------------------------------------- 15.32/5.16 15.32/5.16 (6) 15.32/5.16 Complex Obligation (AND) 15.32/5.16 15.32/5.16 ---------------------------------------- 15.32/5.16 15.32/5.16 (7) 15.32/5.16 Obligation: 15.32/5.16 Q-restricted context-sensitive dependency pair problem: 15.32/5.16 The symbols in {U106_1, U112_1, U122_1, U131_1, snd_1, splitAt_2, U141_1, U151_1, U161_1, U172_1, U183_1, U193_1, U206_1, U211_1, U221_1, U232_1, U246_1, U256_1, U262_1, U272_1, natsFrom_1, s_1, head_1, afterNth_2, pair_2, fst_1, U46_1, U56_1, U63_1, U73_1, U83_1, U93_1, tail_1, take_2, sel_2} are replacing on all positions. 15.32/5.16 For all symbols f in {U101_3, U102_3, U103_3, U104_3, U105_2, U11_3, U12_3, U111_2, U13_3, U121_2, U14_3, U171_2, U181_2, U182_2, U191_2, U192_2, U201_3, U202_3, U203_3, U204_3, U205_2, U21_3, U22_3, U23_3, U24_2, U231_2, U241_3, U242_3, U243_3, U244_3, U245_2, U251_3, U252_3, U253_3, U254_3, U255_2, U261_2, U271_2, U281_2, U282_2, cons_2, U291_3, U292_3, U293_3, U294_3, U301_3, U302_2, U303_2, U304_2, U31_3, U32_3, U311_2, U312_2, U33_3, U321_4, U322_4, U323_4, U324_4, U325_4, U326_4, U327_2, U34_2, U331_3, U332_2, U333_2, U334_2, U341_3, U342_3, U343_3, U344_3, U41_3, U42_3, U43_3, U44_3, U45_2, U51_3, U52_3, U53_3, U54_3, U55_2, U61_2, U62_2, U71_2, U72_2, U81_2, U82_2, U91_2, U92_2, U111'_2, U231'_2, U121'_2, U261'_2, U171'_2, U271'_2} we have mu(f) = {1}. 15.32/5.16 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1, ISLNATKIND_1, ISNATURALKIND_1, ISPLNATKIND_1} are not replacing on any position. 15.32/5.16 15.32/5.16 The TRS P consists of the following rules: 15.32/5.16 15.32/5.16 ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) 15.32/5.16 ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) 15.32/5.16 U111'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) 15.32/5.16 ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) 15.32/5.16 U231'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) 15.32/5.16 U121'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) 15.32/5.16 ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) 15.32/5.16 U261'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) 15.32/5.16 ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) 15.32/5.16 ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) 15.32/5.16 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 15.32/5.16 ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) 15.32/5.16 U171'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) 15.32/5.16 U271'(tt, V2) -> ISLNATKIND(V2) 15.32/5.16 ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.16 15.32/5.16 The TRS R consists of the following rules: 15.32/5.16 15.32/5.16 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.16 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.16 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.16 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.16 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.16 U106(tt) -> tt 15.32/5.16 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 15.32/5.16 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.16 U112(tt) -> tt 15.32/5.16 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 15.32/5.16 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.16 U122(tt) -> tt 15.32/5.16 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 15.32/5.16 U131(tt) -> tt 15.32/5.16 U14(tt, N, XS) -> snd(splitAt(N, XS)) 15.32/5.16 U141(tt) -> tt 15.32/5.16 U151(tt) -> tt 15.32/5.16 U161(tt) -> tt 15.32/5.16 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.16 U172(tt) -> tt 15.32/5.16 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.16 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.16 U183(tt) -> tt 15.32/5.16 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.16 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.16 U193(tt) -> tt 15.32/5.16 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.16 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.16 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.16 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.16 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.16 U206(tt) -> tt 15.32/5.16 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 15.32/5.16 U211(tt) -> tt 15.32/5.16 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 15.32/5.16 U221(tt) -> tt 15.32/5.16 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 15.32/5.16 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.16 U232(tt) -> tt 15.32/5.16 U24(tt, X) -> X 15.32/5.16 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.16 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.16 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.16 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.16 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.16 U246(tt) -> tt 15.32/5.16 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.16 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.16 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.16 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.16 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.16 U256(tt) -> tt 15.32/5.16 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.16 U262(tt) -> tt 15.32/5.16 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.16 U272(tt) -> tt 15.32/5.16 U281(tt, N) -> U282(isNaturalKind(N), N) 15.32/5.16 U282(tt, N) -> cons(N, natsFrom(s(N))) 15.32/5.16 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 15.32/5.16 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 15.32/5.16 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 15.32/5.16 U294(tt, N, XS) -> head(afterNth(N, XS)) 15.32/5.16 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 15.32/5.16 U302(tt, Y) -> U303(isLNat(Y), Y) 15.32/5.16 U303(tt, Y) -> U304(isLNatKind(Y), Y) 15.32/5.16 U304(tt, Y) -> Y 15.32/5.16 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 15.32/5.16 U311(tt, XS) -> U312(isLNatKind(XS), XS) 15.32/5.16 U312(tt, XS) -> pair(nil, XS) 15.32/5.16 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 15.32/5.16 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 15.32/5.16 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 15.32/5.16 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 15.32/5.16 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 15.32/5.16 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 15.32/5.16 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 15.32/5.16 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 15.32/5.16 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 15.32/5.16 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 15.32/5.16 U332(tt, XS) -> U333(isLNat(XS), XS) 15.32/5.16 U333(tt, XS) -> U334(isLNatKind(XS), XS) 15.32/5.16 U334(tt, XS) -> XS 15.32/5.16 U34(tt, N) -> N 15.32/5.16 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 15.32/5.16 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 15.32/5.16 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 15.32/5.16 U344(tt, N, XS) -> fst(splitAt(N, XS)) 15.32/5.16 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.16 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.16 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.16 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.16 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.16 U46(tt) -> tt 15.32/5.16 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.16 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.16 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.16 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.16 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.16 U56(tt) -> tt 15.32/5.16 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.16 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.16 U63(tt) -> tt 15.32/5.16 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.16 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.16 U73(tt) -> tt 15.32/5.16 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.16 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.16 U83(tt) -> tt 15.32/5.16 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.16 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.16 U93(tt) -> tt 15.32/5.16 afterNth(N, XS) -> U11(isNatural(N), N, XS) 15.32/5.16 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 15.32/5.16 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 15.32/5.16 isLNat(nil) -> tt 15.32/5.16 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.16 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.16 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.16 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.16 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.16 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.16 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.16 isLNatKind(nil) -> tt 15.32/5.16 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.16 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.16 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.16 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.16 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.16 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.16 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.16 isNatural(0) -> tt 15.32/5.16 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.16 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.16 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.16 isNaturalKind(0) -> tt 15.32/5.16 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.16 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.16 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.16 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.16 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.16 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.16 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.16 natsFrom(N) -> U281(isNatural(N), N) 15.32/5.16 sel(N, XS) -> U291(isNatural(N), N, XS) 15.32/5.16 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 15.32/5.16 splitAt(0, XS) -> U311(isLNat(XS), XS) 15.32/5.16 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 15.32/5.16 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 15.32/5.16 take(N, XS) -> U341(isNatural(N), N, XS) 15.32/5.16 15.32/5.16 Q is empty. 15.32/5.16 15.32/5.16 ---------------------------------------- 15.32/5.16 15.32/5.16 (8) QCSUsableRulesProof (EQUIVALENT) 15.32/5.16 The following rules are not useable [DA_EMMES] and can be deleted: 15.32/5.16 15.32/5.16 U101(tt, x0, x1) -> U102(isNaturalKind(x0), x0, x1) 15.32/5.16 U102(tt, x0, x1) -> U103(isLNatKind(x1), x0, x1) 15.32/5.16 U103(tt, x0, x1) -> U104(isLNatKind(x1), x0, x1) 15.32/5.16 U104(tt, x0, x1) -> U105(isNatural(x0), x1) 15.32/5.16 U105(tt, x0) -> U106(isLNat(x0)) 15.32/5.16 U106(tt) -> tt 15.32/5.16 U11(tt, x0, x1) -> U12(isNaturalKind(x0), x0, x1) 15.32/5.16 U12(tt, x0, x1) -> U13(isLNat(x1), x0, x1) 15.32/5.16 U13(tt, x0, x1) -> U14(isLNatKind(x1), x0, x1) 15.32/5.16 U14(tt, x0, x1) -> snd(splitAt(x0, x1)) 15.32/5.16 U181(tt, x0) -> U182(isLNatKind(x0), x0) 15.32/5.16 U182(tt, x0) -> U183(isLNat(x0)) 15.32/5.16 U183(tt) -> tt 15.32/5.16 U191(tt, x0) -> U192(isNaturalKind(x0), x0) 15.32/5.16 U192(tt, x0) -> U193(isNatural(x0)) 15.32/5.16 U193(tt) -> tt 15.32/5.16 U201(tt, x0, x1) -> U202(isNaturalKind(x0), x0, x1) 15.32/5.16 U202(tt, x0, x1) -> U203(isLNatKind(x1), x0, x1) 15.32/5.16 U203(tt, x0, x1) -> U204(isLNatKind(x1), x0, x1) 15.32/5.16 U204(tt, x0, x1) -> U205(isNatural(x0), x1) 15.32/5.16 U205(tt, x0) -> U206(isLNat(x0)) 15.32/5.16 U206(tt) -> tt 15.32/5.16 U21(tt, x0, x1) -> U22(isLNatKind(x0), x0, x1) 15.32/5.16 U22(tt, x0, x1) -> U23(isLNat(x1), x0, x1) 15.32/5.16 U23(tt, x0, x1) -> U24(isLNatKind(x1), x0) 15.32/5.16 U24(tt, x0) -> x0 15.32/5.16 U241(tt, x0, x1) -> U242(isLNatKind(x0), x0, x1) 15.32/5.16 U242(tt, x0, x1) -> U243(isLNatKind(x1), x0, x1) 15.32/5.16 U243(tt, x0, x1) -> U244(isLNatKind(x1), x0, x1) 15.32/5.16 U244(tt, x0, x1) -> U245(isLNat(x0), x1) 15.32/5.16 U245(tt, x0) -> U246(isLNat(x0)) 15.32/5.16 U246(tt) -> tt 15.32/5.16 U251(tt, x0, x1) -> U252(isNaturalKind(x0), x0, x1) 15.32/5.16 U252(tt, x0, x1) -> U253(isLNatKind(x1), x0, x1) 15.32/5.16 U253(tt, x0, x1) -> U254(isLNatKind(x1), x0, x1) 15.32/5.16 U254(tt, x0, x1) -> U255(isNatural(x0), x1) 15.32/5.16 U255(tt, x0) -> U256(isLNat(x0)) 15.32/5.16 U256(tt) -> tt 15.32/5.16 U281(tt, x0) -> U282(isNaturalKind(x0), x0) 15.32/5.16 U282(tt, x0) -> cons(x0, natsFrom(s(x0))) 15.32/5.16 U291(tt, x0, x1) -> U292(isNaturalKind(x0), x0, x1) 15.32/5.16 U292(tt, x0, x1) -> U293(isLNat(x1), x0, x1) 15.32/5.16 U293(tt, x0, x1) -> U294(isLNatKind(x1), x0, x1) 15.32/5.16 U294(tt, x0, x1) -> head(afterNth(x0, x1)) 15.32/5.16 U301(tt, x0, x1) -> U302(isLNatKind(x0), x1) 15.32/5.16 U302(tt, x0) -> U303(isLNat(x0), x0) 15.32/5.16 U303(tt, x0) -> U304(isLNatKind(x0), x0) 15.32/5.16 U304(tt, x0) -> x0 15.32/5.16 U31(tt, x0, x1) -> U32(isNaturalKind(x0), x0, x1) 15.32/5.16 U311(tt, x0) -> U312(isLNatKind(x0), x0) 15.32/5.16 U312(tt, x0) -> pair(nil, x0) 15.32/5.16 U32(tt, x0, x1) -> U33(isLNat(x1), x0, x1) 15.32/5.16 U321(tt, x0, x1, x2) -> U322(isNaturalKind(x0), x0, x1, x2) 15.32/5.16 U322(tt, x0, x1, x2) -> U323(isNatural(x1), x0, x1, x2) 15.32/5.16 U323(tt, x0, x1, x2) -> U324(isNaturalKind(x1), x0, x1, x2) 15.32/5.16 U324(tt, x0, x1, x2) -> U325(isLNat(x2), x0, x1, x2) 15.32/5.16 U325(tt, x0, x1, x2) -> U326(isLNatKind(x2), x0, x1, x2) 15.32/5.16 U326(tt, x0, x1, x2) -> U327(splitAt(x0, x2), x1) 15.32/5.16 U327(pair(x0, x1), x2) -> pair(cons(x2, x0), x1) 15.32/5.16 U33(tt, x0, x1) -> U34(isLNatKind(x1), x0) 15.32/5.16 U331(tt, x0, x1) -> U332(isNaturalKind(x0), x1) 15.32/5.16 U332(tt, x0) -> U333(isLNat(x0), x0) 15.32/5.16 U333(tt, x0) -> U334(isLNatKind(x0), x0) 15.32/5.16 U334(tt, x0) -> x0 15.32/5.16 U34(tt, x0) -> x0 15.32/5.16 U341(tt, x0, x1) -> U342(isNaturalKind(x0), x0, x1) 15.32/5.16 U342(tt, x0, x1) -> U343(isLNat(x1), x0, x1) 15.32/5.16 U343(tt, x0, x1) -> U344(isLNatKind(x1), x0, x1) 15.32/5.16 U344(tt, x0, x1) -> fst(splitAt(x0, x1)) 15.32/5.16 U41(tt, x0, x1) -> U42(isNaturalKind(x0), x0, x1) 15.32/5.16 U42(tt, x0, x1) -> U43(isLNatKind(x1), x0, x1) 15.32/5.16 U43(tt, x0, x1) -> U44(isLNatKind(x1), x0, x1) 15.32/5.16 U44(tt, x0, x1) -> U45(isNatural(x0), x1) 15.32/5.16 U45(tt, x0) -> U46(isLNat(x0)) 15.32/5.16 U46(tt) -> tt 15.32/5.16 U51(tt, x0, x1) -> U52(isNaturalKind(x0), x0, x1) 15.32/5.16 U52(tt, x0, x1) -> U53(isLNatKind(x1), x0, x1) 15.32/5.16 U53(tt, x0, x1) -> U54(isLNatKind(x1), x0, x1) 15.32/5.16 U54(tt, x0, x1) -> U55(isNatural(x0), x1) 15.32/5.16 U55(tt, x0) -> U56(isLNat(x0)) 15.32/5.16 U56(tt) -> tt 15.32/5.16 U61(tt, x0) -> U62(isPLNatKind(x0), x0) 15.32/5.16 U62(tt, x0) -> U63(isPLNat(x0)) 15.32/5.16 U63(tt) -> tt 15.32/5.16 U71(tt, x0) -> U72(isNaturalKind(x0), x0) 15.32/5.16 U72(tt, x0) -> U73(isNatural(x0)) 15.32/5.16 U73(tt) -> tt 15.32/5.16 U81(tt, x0) -> U82(isPLNatKind(x0), x0) 15.32/5.16 U82(tt, x0) -> U83(isPLNat(x0)) 15.32/5.16 U83(tt) -> tt 15.32/5.16 U91(tt, x0) -> U92(isLNatKind(x0), x0) 15.32/5.16 U92(tt, x0) -> U93(isLNat(x0)) 15.32/5.16 U93(tt) -> tt 15.32/5.16 afterNth(x0, x1) -> U11(isNatural(x0), x0, x1) 15.32/5.17 fst(pair(x0, x1)) -> U21(isLNat(x0), x0, x1) 15.32/5.17 head(cons(x0, x1)) -> U31(isNatural(x0), x0, x1) 15.32/5.17 isLNat(nil) -> tt 15.32/5.17 isLNat(afterNth(x0, x1)) -> U41(isNaturalKind(x0), x0, x1) 15.32/5.17 isLNat(cons(x0, x1)) -> U51(isNaturalKind(x0), x0, x1) 15.32/5.17 isLNat(fst(x0)) -> U61(isPLNatKind(x0), x0) 15.32/5.17 isLNat(natsFrom(x0)) -> U71(isNaturalKind(x0), x0) 15.32/5.17 isLNat(snd(x0)) -> U81(isPLNatKind(x0), x0) 15.32/5.17 isLNat(tail(x0)) -> U91(isLNatKind(x0), x0) 15.32/5.17 isLNat(take(x0, x1)) -> U101(isNaturalKind(x0), x0, x1) 15.32/5.17 isNatural(0) -> tt 15.32/5.17 isNatural(head(x0)) -> U181(isLNatKind(x0), x0) 15.32/5.17 isNatural(s(x0)) -> U191(isNaturalKind(x0), x0) 15.32/5.17 isNatural(sel(x0, x1)) -> U201(isNaturalKind(x0), x0, x1) 15.32/5.17 isPLNat(pair(x0, x1)) -> U241(isLNatKind(x0), x0, x1) 15.32/5.17 isPLNat(splitAt(x0, x1)) -> U251(isNaturalKind(x0), x0, x1) 15.32/5.17 natsFrom(x0) -> U281(isNatural(x0), x0) 15.32/5.17 sel(x0, x1) -> U291(isNatural(x0), x0, x1) 15.32/5.17 snd(pair(x0, x1)) -> U301(isLNat(x0), x0, x1) 15.32/5.17 splitAt(0, x0) -> U311(isLNat(x0), x0) 15.32/5.17 splitAt(s(x0), cons(x1, x2)) -> U321(isNatural(x0), x0, x1, x2) 15.32/5.17 tail(cons(x0, x1)) -> U331(isNatural(x0), x0, x1) 15.32/5.17 take(x0, x1) -> U341(isNatural(x0), x0, x1) 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (9) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {head_1, U211_1, afterNth_2, s_1, U221_1, sel_2, U232_1, U122_1, fst_1, U131_1, pair_2, natsFrom_1, U141_1, snd_1, U151_1, splitAt_2, U272_1, tail_1, U161_1, take_2, U172_1, U262_1, U112_1} are replacing on all positions. 15.32/5.17 For all symbols f in {U111_2, U231_2, cons_2, U121_2, U261_2, U271_2, U171_2, U111'_2, U231'_2, U121'_2, U261'_2, U171'_2, U271'_2} we have mu(f) = {1}. 15.32/5.17 The symbols in {isNaturalKind_1, isLNatKind_1, isPLNatKind_1, ISLNATKIND_1, ISNATURALKIND_1, ISPLNATKIND_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 15.32/5.17 ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) 15.32/5.17 ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) 15.32/5.17 U111'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) 15.32/5.17 ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) 15.32/5.17 U231'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) 15.32/5.17 U121'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) 15.32/5.17 ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) 15.32/5.17 U261'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) 15.32/5.17 ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) 15.32/5.17 ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) 15.32/5.17 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 15.32/5.17 ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) 15.32/5.17 U171'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) 15.32/5.17 U271'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 isNaturalKind(0) -> tt 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isLNatKind(nil) -> tt 15.32/5.17 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.17 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.17 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.17 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.17 U141(tt) -> tt 15.32/5.17 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.17 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.17 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.17 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.17 U172(tt) -> tt 15.32/5.17 U161(tt) -> tt 15.32/5.17 U272(tt) -> tt 15.32/5.17 U151(tt) -> tt 15.32/5.17 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.17 U262(tt) -> tt 15.32/5.17 U131(tt) -> tt 15.32/5.17 U122(tt) -> tt 15.32/5.17 U232(tt) -> tt 15.32/5.17 U221(tt) -> tt 15.32/5.17 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.17 U112(tt) -> tt 15.32/5.17 U211(tt) -> tt 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (10) QCSDPMuMonotonicPoloProof (EQUIVALENT) 15.32/5.17 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. 15.32/5.17 15.32/5.17 Strictly oriented dependency pairs: 15.32/5.17 15.32/5.17 ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) 15.32/5.17 ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) 15.32/5.17 U111'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) 15.32/5.17 ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) 15.32/5.17 U231'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) 15.32/5.17 U121'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) 15.32/5.17 ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) 15.32/5.17 U261'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) 15.32/5.17 ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) 15.32/5.17 ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) 15.32/5.17 ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) 15.32/5.17 U171'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) 15.32/5.17 U271'(tt, V2) -> ISLNATKIND(V2) 15.32/5.17 ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) 15.32/5.17 15.32/5.17 Strictly oriented rules of the TRS R: 15.32/5.17 15.32/5.17 isNaturalKind(0) -> tt 15.32/5.17 isLNatKind(nil) -> tt 15.32/5.17 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.17 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.17 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.17 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.17 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.17 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.17 U141(tt) -> tt 15.32/5.17 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.17 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.17 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.17 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.17 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.17 U172(tt) -> tt 15.32/5.17 U272(tt) -> tt 15.32/5.17 U151(tt) -> tt 15.32/5.17 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.17 U262(tt) -> tt 15.32/5.17 U131(tt) -> tt 15.32/5.17 U122(tt) -> tt 15.32/5.17 U232(tt) -> tt 15.32/5.17 U221(tt) -> tt 15.32/5.17 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.17 U112(tt) -> tt 15.32/5.17 15.32/5.17 Used ordering: POLO with Polynomial interpretation [POLO]: 15.32/5.17 15.32/5.17 POL(0) = 2 15.32/5.17 POL(ISLNATKIND(x_1)) = 1 + 2*x_1 15.32/5.17 POL(ISNATURALKIND(x_1)) = 2 + x_1 15.32/5.17 POL(ISPLNATKIND(x_1)) = 1 + 2*x_1 15.32/5.17 POL(U111(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U111'(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U112(x_1)) = 1 + x_1 15.32/5.17 POL(U121(x_1, x_2)) = 1 + x_1 + 2*x_2 15.32/5.17 POL(U121'(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U122(x_1)) = 1 + x_1 15.32/5.17 POL(U131(x_1)) = 1 + x_1 15.32/5.17 POL(U141(x_1)) = 1 + x_1 15.32/5.17 POL(U151(x_1)) = 1 + 2*x_1 15.32/5.17 POL(U161(x_1)) = x_1 15.32/5.17 POL(U171(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U171'(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U172(x_1)) = 1 + x_1 15.32/5.17 POL(U211(x_1)) = x_1 15.32/5.17 POL(U221(x_1)) = 1 + 2*x_1 15.32/5.17 POL(U231(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U231'(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U232(x_1)) = 1 + x_1 15.32/5.17 POL(U261(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U261'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 15.32/5.17 POL(U262(x_1)) = 1 + x_1 15.32/5.17 POL(U271(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U271'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 15.32/5.17 POL(U272(x_1)) = 1 + x_1 15.32/5.17 POL(afterNth(x_1, x_2)) = 2 + 2*x_1 + x_2 15.32/5.17 POL(cons(x_1, x_2)) = 2 + 2*x_1 + x_2 15.32/5.17 POL(fst(x_1)) = 1 + 2*x_1 15.32/5.17 POL(head(x_1)) = 2*x_1 15.32/5.17 POL(isLNatKind(x_1)) = 2*x_1 15.32/5.17 POL(isNaturalKind(x_1)) = 2*x_1 15.32/5.17 POL(isPLNatKind(x_1)) = 1 + 2*x_1 15.32/5.17 POL(natsFrom(x_1)) = 2 + 2*x_1 15.32/5.17 POL(nil) = 1 15.32/5.17 POL(pair(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 15.32/5.17 POL(s(x_1)) = 1 + 2*x_1 15.32/5.17 POL(sel(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 15.32/5.17 POL(snd(x_1)) = 2 + 2*x_1 15.32/5.17 POL(splitAt(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 15.32/5.17 POL(tail(x_1)) = x_1 15.32/5.17 POL(take(x_1, x_2)) = 2 + 2*x_1 + x_2 15.32/5.17 POL(tt) = 0 15.32/5.17 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (11) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {head_1, U211_1, sel_2, U122_1, fst_1, U131_1, tail_1, U161_1} are replacing on all positions. 15.32/5.17 For all symbols f in {U231_2, U121_2} we have mu(f) = {1}. 15.32/5.17 The symbols in {isNaturalKind_1, isLNatKind_1, isPLNatKind_1, ISLNATKIND_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 15.32/5.17 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 U161(tt) -> tt 15.32/5.17 U211(tt) -> tt 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (12) QCSDPSubtermProof (EQUIVALENT) 15.32/5.17 We use the subterm processor [DA_EMMES]. 15.32/5.17 15.32/5.17 15.32/5.17 The following pairs can be oriented strictly and are deleted. 15.32/5.17 15.32/5.17 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 15.32/5.17 The remaining pairs can at least be oriented weakly. 15.32/5.17 none 15.32/5.17 Used ordering: Combined order from the following AFS and order. 15.32/5.17 ISLNATKIND(x1) = x1 15.32/5.17 15.32/5.17 15.32/5.17 Subterm Order 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (13) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {head_1, U211_1, sel_2, U122_1, fst_1, U131_1, tail_1, U161_1} are replacing on all positions. 15.32/5.17 For all symbols f in {U231_2, U121_2} we have mu(f) = {1}. 15.32/5.17 The symbols in {isNaturalKind_1, isLNatKind_1, isPLNatKind_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 none 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 U161(tt) -> tt 15.32/5.17 U211(tt) -> tt 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (14) PIsEmptyProof (EQUIVALENT) 15.32/5.17 The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain. 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (15) 15.32/5.17 YES 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (16) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {U106_1, U112_1, U122_1, U131_1, snd_1, splitAt_2, U141_1, U151_1, U161_1, U172_1, U183_1, U193_1, U206_1, U211_1, U221_1, U232_1, U246_1, U256_1, U262_1, U272_1, natsFrom_1, s_1, head_1, afterNth_2, pair_2, fst_1, U46_1, U56_1, U63_1, U73_1, U83_1, U93_1, tail_1, take_2, sel_2} are replacing on all positions. 15.32/5.17 For all symbols f in {U101_3, U102_3, U103_3, U104_3, U105_2, U11_3, U12_3, U111_2, U13_3, U121_2, U14_3, U171_2, U181_2, U182_2, U191_2, U192_2, U201_3, U202_3, U203_3, U204_3, U205_2, U21_3, U22_3, U23_3, U24_2, U231_2, U241_3, U242_3, U243_3, U244_3, U245_2, U251_3, U252_3, U253_3, U254_3, U255_2, U261_2, U271_2, U281_2, U282_2, cons_2, U291_3, U292_3, U293_3, U294_3, U301_3, U302_2, U303_2, U304_2, U31_3, U32_3, U311_2, U312_2, U33_3, U321_4, U322_4, U323_4, U324_4, U325_4, U326_4, U327_2, U34_2, U331_3, U332_2, U333_2, U334_2, U341_3, U342_3, U343_3, U344_3, U41_3, U42_3, U43_3, U44_3, U45_2, U51_3, U52_3, U53_3, U54_3, U55_2, U61_2, U62_2, U71_2, U72_2, U81_2, U82_2, U91_2, U92_2, U103'_3, U102'_3, U104'_3, U105'_2, U41'_3, U42'_3, U43'_3, U44'_3, U45'_2, U51'_3, U52'_3, U53'_3, U54'_3, U55'_2, U61'_2, U62'_2, U241'_3, U242'_3, U243'_3, U244'_3, U245'_2, U71'_2, U72'_2, U181'_2, U182'_2, U81'_2, U82'_2, U251'_3, U252'_3, U253'_3, U254'_3, U255'_2, U91'_2, U92'_2, U101'_3, U191'_2, U192'_2, U201'_3, U202'_3, U203'_3, U204'_3, U205'_2} we have mu(f) = {1}. 15.32/5.17 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1, ISLNAT_1, ISPLNAT_1, ISNATURAL_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 15.32/5.17 U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) 15.32/5.17 U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) 15.32/5.17 U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) 15.32/5.17 U105'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) 15.32/5.17 U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) 15.32/5.17 U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) 15.32/5.17 U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) 15.32/5.17 U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) 15.32/5.17 U45'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) 15.32/5.17 U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) 15.32/5.17 U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) 15.32/5.17 U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) 15.32/5.17 U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) 15.32/5.17 U55'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) 15.32/5.17 U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) 15.32/5.17 U62'(tt, V1) -> ISPLNAT(V1) 15.32/5.17 ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) 15.32/5.17 U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) 15.32/5.17 U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) 15.32/5.17 U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) 15.32/5.17 U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) 15.32/5.17 U245'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) 15.32/5.17 U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) 15.32/5.17 U72'(tt, V1) -> ISNATURAL(V1) 15.32/5.17 ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) 15.32/5.17 U181'(tt, V1) -> U182'(isLNatKind(V1), V1) 15.32/5.17 U182'(tt, V1) -> ISLNAT(V1) 15.32/5.17 ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) 15.32/5.17 U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) 15.32/5.17 U82'(tt, V1) -> ISPLNAT(V1) 15.32/5.17 ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) 15.32/5.17 U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) 15.32/5.17 U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) 15.32/5.17 U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) 15.32/5.17 U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) 15.32/5.17 U255'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) 15.32/5.17 U91'(tt, V1) -> U92'(isLNatKind(V1), V1) 15.32/5.17 U92'(tt, V1) -> ISLNAT(V1) 15.32/5.17 ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) 15.32/5.17 U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) 15.32/5.17 U254'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) 15.32/5.17 U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) 15.32/5.17 U192'(tt, V1) -> ISNATURAL(V1) 15.32/5.17 ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) 15.32/5.17 U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) 15.32/5.17 U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) 15.32/5.17 U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) 15.32/5.17 U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) 15.32/5.17 U205'(tt, V2) -> ISLNAT(V2) 15.32/5.17 U204'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 U244'(tt, V1, V2) -> ISLNAT(V1) 15.32/5.17 U54'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 U44'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 U104'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.17 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.17 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.17 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.17 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.17 U106(tt) -> tt 15.32/5.17 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 15.32/5.17 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.17 U112(tt) -> tt 15.32/5.17 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 U122(tt) -> tt 15.32/5.17 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 15.32/5.17 U131(tt) -> tt 15.32/5.17 U14(tt, N, XS) -> snd(splitAt(N, XS)) 15.32/5.17 U141(tt) -> tt 15.32/5.17 U151(tt) -> tt 15.32/5.17 U161(tt) -> tt 15.32/5.17 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.17 U172(tt) -> tt 15.32/5.17 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.17 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.17 U183(tt) -> tt 15.32/5.17 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.17 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.17 U193(tt) -> tt 15.32/5.17 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.17 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.17 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.17 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.17 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.17 U206(tt) -> tt 15.32/5.17 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 15.32/5.17 U211(tt) -> tt 15.32/5.17 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 15.32/5.17 U221(tt) -> tt 15.32/5.17 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 15.32/5.17 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.17 U232(tt) -> tt 15.32/5.17 U24(tt, X) -> X 15.32/5.17 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.17 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.17 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.17 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.17 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.17 U246(tt) -> tt 15.32/5.17 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.17 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.17 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.17 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.17 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.17 U256(tt) -> tt 15.32/5.17 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.17 U262(tt) -> tt 15.32/5.17 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.17 U272(tt) -> tt 15.32/5.17 U281(tt, N) -> U282(isNaturalKind(N), N) 15.32/5.17 U282(tt, N) -> cons(N, natsFrom(s(N))) 15.32/5.17 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 15.32/5.17 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 15.32/5.17 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 15.32/5.17 U294(tt, N, XS) -> head(afterNth(N, XS)) 15.32/5.17 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 15.32/5.17 U302(tt, Y) -> U303(isLNat(Y), Y) 15.32/5.17 U303(tt, Y) -> U304(isLNatKind(Y), Y) 15.32/5.17 U304(tt, Y) -> Y 15.32/5.17 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 15.32/5.17 U311(tt, XS) -> U312(isLNatKind(XS), XS) 15.32/5.17 U312(tt, XS) -> pair(nil, XS) 15.32/5.17 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 15.32/5.17 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 15.32/5.17 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 15.32/5.17 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 15.32/5.17 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 15.32/5.17 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 15.32/5.17 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 15.32/5.17 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 15.32/5.17 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 15.32/5.17 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 15.32/5.17 U332(tt, XS) -> U333(isLNat(XS), XS) 15.32/5.17 U333(tt, XS) -> U334(isLNatKind(XS), XS) 15.32/5.17 U334(tt, XS) -> XS 15.32/5.17 U34(tt, N) -> N 15.32/5.17 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 15.32/5.17 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 15.32/5.17 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 15.32/5.17 U344(tt, N, XS) -> fst(splitAt(N, XS)) 15.32/5.17 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.17 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.17 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.17 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.17 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.17 U46(tt) -> tt 15.32/5.17 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.17 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.17 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.17 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.17 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.17 U56(tt) -> tt 15.32/5.17 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.17 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.17 U63(tt) -> tt 15.32/5.17 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.17 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.17 U73(tt) -> tt 15.32/5.17 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.17 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.17 U83(tt) -> tt 15.32/5.17 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.17 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.17 U93(tt) -> tt 15.32/5.17 afterNth(N, XS) -> U11(isNatural(N), N, XS) 15.32/5.17 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 15.32/5.17 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 15.32/5.17 isLNat(nil) -> tt 15.32/5.17 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.17 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.17 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.17 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.17 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNatKind(nil) -> tt 15.32/5.17 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.17 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.17 isNatural(0) -> tt 15.32/5.17 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.17 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.17 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.17 isNaturalKind(0) -> tt 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.17 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.17 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.17 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.17 natsFrom(N) -> U281(isNatural(N), N) 15.32/5.17 sel(N, XS) -> U291(isNatural(N), N, XS) 15.32/5.17 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 15.32/5.17 splitAt(0, XS) -> U311(isLNat(XS), XS) 15.32/5.17 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 15.32/5.17 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 15.32/5.17 take(N, XS) -> U341(isNatural(N), N, XS) 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (17) QCSUsableRulesProof (EQUIVALENT) 15.32/5.17 The following rules are not useable [DA_EMMES] and can be deleted: 15.32/5.17 15.32/5.17 U11(tt, x0, x1) -> U12(isNaturalKind(x0), x0, x1) 15.32/5.17 U12(tt, x0, x1) -> U13(isLNat(x1), x0, x1) 15.32/5.17 U13(tt, x0, x1) -> U14(isLNatKind(x1), x0, x1) 15.32/5.17 U14(tt, x0, x1) -> snd(splitAt(x0, x1)) 15.32/5.17 U21(tt, x0, x1) -> U22(isLNatKind(x0), x0, x1) 15.32/5.17 U22(tt, x0, x1) -> U23(isLNat(x1), x0, x1) 15.32/5.17 U23(tt, x0, x1) -> U24(isLNatKind(x1), x0) 15.32/5.17 U24(tt, x0) -> x0 15.32/5.17 U281(tt, x0) -> U282(isNaturalKind(x0), x0) 15.32/5.17 U282(tt, x0) -> cons(x0, natsFrom(s(x0))) 15.32/5.17 U291(tt, x0, x1) -> U292(isNaturalKind(x0), x0, x1) 15.32/5.17 U292(tt, x0, x1) -> U293(isLNat(x1), x0, x1) 15.32/5.17 U293(tt, x0, x1) -> U294(isLNatKind(x1), x0, x1) 15.32/5.17 U294(tt, x0, x1) -> head(afterNth(x0, x1)) 15.32/5.17 U301(tt, x0, x1) -> U302(isLNatKind(x0), x1) 15.32/5.17 U302(tt, x0) -> U303(isLNat(x0), x0) 15.32/5.17 U303(tt, x0) -> U304(isLNatKind(x0), x0) 15.32/5.17 U304(tt, x0) -> x0 15.32/5.17 U31(tt, x0, x1) -> U32(isNaturalKind(x0), x0, x1) 15.32/5.17 U311(tt, x0) -> U312(isLNatKind(x0), x0) 15.32/5.17 U312(tt, x0) -> pair(nil, x0) 15.32/5.17 U32(tt, x0, x1) -> U33(isLNat(x1), x0, x1) 15.32/5.17 U321(tt, x0, x1, x2) -> U322(isNaturalKind(x0), x0, x1, x2) 15.32/5.17 U322(tt, x0, x1, x2) -> U323(isNatural(x1), x0, x1, x2) 15.32/5.17 U323(tt, x0, x1, x2) -> U324(isNaturalKind(x1), x0, x1, x2) 15.32/5.17 U324(tt, x0, x1, x2) -> U325(isLNat(x2), x0, x1, x2) 15.32/5.17 U325(tt, x0, x1, x2) -> U326(isLNatKind(x2), x0, x1, x2) 15.32/5.17 U326(tt, x0, x1, x2) -> U327(splitAt(x0, x2), x1) 15.32/5.17 U327(pair(x0, x1), x2) -> pair(cons(x2, x0), x1) 15.32/5.17 U33(tt, x0, x1) -> U34(isLNatKind(x1), x0) 15.32/5.17 U331(tt, x0, x1) -> U332(isNaturalKind(x0), x1) 15.32/5.17 U332(tt, x0) -> U333(isLNat(x0), x0) 15.32/5.17 U333(tt, x0) -> U334(isLNatKind(x0), x0) 15.32/5.17 U334(tt, x0) -> x0 15.32/5.17 U34(tt, x0) -> x0 15.32/5.17 U341(tt, x0, x1) -> U342(isNaturalKind(x0), x0, x1) 15.32/5.17 U342(tt, x0, x1) -> U343(isLNat(x1), x0, x1) 15.32/5.17 U343(tt, x0, x1) -> U344(isLNatKind(x1), x0, x1) 15.32/5.17 U344(tt, x0, x1) -> fst(splitAt(x0, x1)) 15.32/5.17 afterNth(x0, x1) -> U11(isNatural(x0), x0, x1) 15.32/5.17 fst(pair(x0, x1)) -> U21(isLNat(x0), x0, x1) 15.32/5.17 head(cons(x0, x1)) -> U31(isNatural(x0), x0, x1) 15.32/5.17 natsFrom(x0) -> U281(isNatural(x0), x0) 15.32/5.17 sel(x0, x1) -> U291(isNatural(x0), x0, x1) 15.32/5.17 snd(pair(x0, x1)) -> U301(isLNat(x0), x0, x1) 15.32/5.17 splitAt(0, x0) -> U311(isLNat(x0), x0) 15.32/5.17 splitAt(s(x0), cons(x1, x2)) -> U321(isNatural(x0), x0, x1, x2) 15.32/5.17 tail(cons(x0, x1)) -> U331(isNatural(x0), x0, x1) 15.32/5.17 take(x0, x1) -> U341(isNatural(x0), x0, x1) 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (18) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {afterNth_2, head_1, U211_1, s_1, U221_1, sel_2, U232_1, fst_1, U131_1, pair_2, natsFrom_1, U141_1, snd_1, U151_1, splitAt_2, U272_1, tail_1, U161_1, take_2, U172_1, U262_1, U122_1, U112_1, U183_1, U193_1, U206_1, U56_1, U63_1, U73_1, U83_1, U256_1, U93_1, U106_1, U246_1, U46_1} are replacing on all positions. 15.32/5.17 For all symbols f in {U111_2, cons_2, U121_2, U231_2, U261_2, U271_2, U171_2, U181_2, U182_2, U41_3, U42_3, U43_3, U44_3, U45_2, U191_2, U192_2, U201_3, U202_3, U203_3, U204_3, U205_2, U51_3, U52_3, U53_3, U54_3, U55_2, U61_2, U62_2, U241_3, U242_3, U243_3, U244_3, U245_2, U71_2, U72_2, U81_2, U82_2, U251_3, U252_3, U253_3, U254_3, U255_2, U91_2, U92_2, U101_3, U102_3, U103_3, U104_3, U105_2, U103'_3, U102'_3, U104'_3, U105'_2, U41'_3, U42'_3, U43'_3, U44'_3, U45'_2, U51'_3, U52'_3, U53'_3, U54'_3, U55'_2, U61'_2, U62'_2, U241'_3, U242'_3, U243'_3, U244'_3, U245'_2, U71'_2, U72'_2, U181'_2, U182'_2, U81'_2, U82'_2, U251'_3, U252'_3, U253'_3, U254'_3, U255'_2, U91'_2, U92'_2, U101'_3, U191'_2, U192'_2, U201'_3, U202'_3, U203'_3, U204'_3, U205'_2} we have mu(f) = {1}. 15.32/5.17 The symbols in {isLNatKind_1, isNaturalKind_1, isPLNatKind_1, isNatural_1, isLNat_1, isPLNat_1, ISLNAT_1, ISPLNAT_1, ISNATURAL_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 15.32/5.17 U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) 15.32/5.17 U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) 15.32/5.17 U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) 15.32/5.17 U105'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) 15.32/5.17 U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) 15.32/5.17 U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) 15.32/5.17 U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) 15.32/5.17 U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) 15.32/5.17 U45'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) 15.32/5.17 U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) 15.32/5.17 U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) 15.32/5.17 U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) 15.32/5.17 U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) 15.32/5.17 U55'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) 15.32/5.17 U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) 15.32/5.17 U62'(tt, V1) -> ISPLNAT(V1) 15.32/5.17 ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) 15.32/5.17 U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) 15.32/5.17 U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) 15.32/5.17 U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) 15.32/5.17 U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) 15.32/5.17 U245'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) 15.32/5.17 U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) 15.32/5.17 U72'(tt, V1) -> ISNATURAL(V1) 15.32/5.17 ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) 15.32/5.17 U181'(tt, V1) -> U182'(isLNatKind(V1), V1) 15.32/5.17 U182'(tt, V1) -> ISLNAT(V1) 15.32/5.17 ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) 15.32/5.17 U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) 15.32/5.17 U82'(tt, V1) -> ISPLNAT(V1) 15.32/5.17 ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) 15.32/5.17 U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) 15.32/5.17 U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) 15.32/5.17 U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) 15.32/5.17 U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) 15.32/5.17 U255'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) 15.32/5.17 U91'(tt, V1) -> U92'(isLNatKind(V1), V1) 15.32/5.17 U92'(tt, V1) -> ISLNAT(V1) 15.32/5.17 ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) 15.32/5.17 U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) 15.32/5.17 U254'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) 15.32/5.17 U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) 15.32/5.17 U192'(tt, V1) -> ISNATURAL(V1) 15.32/5.17 ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) 15.32/5.17 U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) 15.32/5.17 U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) 15.32/5.17 U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) 15.32/5.17 U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) 15.32/5.17 U205'(tt, V2) -> ISLNAT(V2) 15.32/5.17 U204'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 U244'(tt, V1, V2) -> ISLNAT(V1) 15.32/5.17 U54'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 U44'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 U104'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 isLNatKind(nil) -> tt 15.32/5.17 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.17 isNaturalKind(0) -> tt 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.17 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.17 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.17 U141(tt) -> tt 15.32/5.17 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.17 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.17 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.17 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.17 U172(tt) -> tt 15.32/5.17 U161(tt) -> tt 15.32/5.17 U272(tt) -> tt 15.32/5.17 U151(tt) -> tt 15.32/5.17 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.17 U262(tt) -> tt 15.32/5.17 U131(tt) -> tt 15.32/5.17 U232(tt) -> tt 15.32/5.17 U221(tt) -> tt 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 U122(tt) -> tt 15.32/5.17 U211(tt) -> tt 15.32/5.17 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.17 U112(tt) -> tt 15.32/5.17 isNatural(0) -> tt 15.32/5.17 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.17 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.17 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.17 isLNat(nil) -> tt 15.32/5.17 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.17 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.17 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.17 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.17 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.17 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.17 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.17 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.17 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.17 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.17 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.17 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.17 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.17 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.17 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.17 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.17 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.17 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.17 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.17 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.17 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.17 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.17 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.17 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.17 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.17 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.17 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.17 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.17 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.17 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.17 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.17 U73(tt) -> tt 15.32/5.17 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.17 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.17 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.17 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.17 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.17 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.17 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.17 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.17 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.17 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.17 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.17 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.17 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.17 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.17 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.17 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.17 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.17 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.17 U106(tt) -> tt 15.32/5.17 U93(tt) -> tt 15.32/5.17 U256(tt) -> tt 15.32/5.17 U83(tt) -> tt 15.32/5.17 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.17 U246(tt) -> tt 15.32/5.17 U63(tt) -> tt 15.32/5.17 U56(tt) -> tt 15.32/5.17 U206(tt) -> tt 15.32/5.17 U193(tt) -> tt 15.32/5.17 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.17 U46(tt) -> tt 15.32/5.17 U183(tt) -> tt 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (19) QCSDPMuMonotonicPoloProof (EQUIVALENT) 15.32/5.17 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. 15.32/5.17 15.32/5.17 Strictly oriented dependency pairs: 15.32/5.17 15.32/5.17 U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) 15.32/5.17 U105'(tt, V2) -> ISLNAT(V2) 15.32/5.17 U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) 15.32/5.17 U45'(tt, V2) -> ISLNAT(V2) 15.32/5.17 U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) 15.32/5.17 U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) 15.32/5.17 ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) 15.32/5.17 U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) 15.32/5.17 ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) 15.32/5.17 U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) 15.32/5.17 U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) 15.32/5.17 U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) 15.32/5.17 U72'(tt, V1) -> ISNATURAL(V1) 15.32/5.17 U181'(tt, V1) -> U182'(isLNatKind(V1), V1) 15.32/5.17 U182'(tt, V1) -> ISLNAT(V1) 15.32/5.17 U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) 15.32/5.17 U82'(tt, V1) -> ISPLNAT(V1) 15.32/5.17 ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) 15.32/5.17 U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) 15.32/5.17 U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) 15.32/5.17 ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) 15.32/5.17 U91'(tt, V1) -> U92'(isLNatKind(V1), V1) 15.32/5.17 U254'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) 15.32/5.17 U192'(tt, V1) -> ISNATURAL(V1) 15.32/5.17 ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) 15.32/5.17 U44'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 U104'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 15.32/5.17 Strictly oriented rules of the TRS R: 15.32/5.17 15.32/5.17 isNatural(0) -> tt 15.32/5.17 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.17 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.17 isLNat(nil) -> tt 15.32/5.17 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.17 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.17 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.17 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.17 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.17 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.17 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.17 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.17 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.17 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.17 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.17 U73(tt) -> tt 15.32/5.17 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.17 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.17 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.17 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.17 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.17 U93(tt) -> tt 15.32/5.17 U83(tt) -> tt 15.32/5.17 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.17 U63(tt) -> tt 15.32/5.17 U183(tt) -> tt 15.32/5.17 15.32/5.17 Used ordering: POLO with Polynomial interpretation [POLO]: 15.32/5.17 15.32/5.17 POL(0) = 2 15.32/5.17 POL(ISLNAT(x_1)) = x_1 15.32/5.17 POL(ISNATURAL(x_1)) = x_1 15.32/5.17 POL(ISPLNAT(x_1)) = 2*x_1 15.32/5.17 POL(U101(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U101'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U102(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 15.32/5.17 POL(U102'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U103(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U103'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U104(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U104'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U105(x_1, x_2)) = x_1 + x_2 15.32/5.17 POL(U105'(x_1, x_2)) = 1 + x_1 + 2*x_2 15.32/5.17 POL(U106(x_1)) = x_1 15.32/5.17 POL(U111(x_1, x_2)) = 2*x_1 15.32/5.17 POL(U112(x_1)) = 2*x_1 15.32/5.17 POL(U121(x_1, x_2)) = 2*x_1 15.32/5.17 POL(U122(x_1)) = x_1 15.32/5.17 POL(U131(x_1)) = 2*x_1 15.32/5.17 POL(U141(x_1)) = 2*x_1 15.32/5.17 POL(U151(x_1)) = 2*x_1 15.32/5.17 POL(U161(x_1)) = 2*x_1 15.32/5.17 POL(U171(x_1, x_2)) = x_1 15.32/5.17 POL(U172(x_1)) = 2*x_1 15.32/5.17 POL(U181(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U181'(x_1, x_2)) = 2 + 2*x_1 + x_2 15.32/5.17 POL(U182(x_1, x_2)) = 2 + 2*x_1 + x_2 15.32/5.17 POL(U182'(x_1, x_2)) = 1 + 2*x_1 + x_2 15.32/5.17 POL(U183(x_1)) = 1 + x_1 15.32/5.17 POL(U191(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U191'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 15.32/5.17 POL(U192(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 15.32/5.17 POL(U192'(x_1, x_2)) = 1 + x_1 + 2*x_2 15.32/5.17 POL(U193(x_1)) = x_1 15.32/5.17 POL(U201(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U201'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U202(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U202'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U203(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U203'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U204(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 15.32/5.17 POL(U204'(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 15.32/5.17 POL(U205(x_1, x_2)) = x_1 + x_2 15.32/5.17 POL(U205'(x_1, x_2)) = x_1 + x_2 15.32/5.17 POL(U206(x_1)) = x_1 15.32/5.17 POL(U211(x_1)) = 2*x_1 15.32/5.17 POL(U221(x_1)) = 2*x_1 15.32/5.17 POL(U231(x_1, x_2)) = x_1 15.32/5.17 POL(U232(x_1)) = 2*x_1 15.32/5.17 POL(U241(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3 15.32/5.17 POL(U241'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U242(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3 15.32/5.17 POL(U242'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U243(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3 15.32/5.17 POL(U243'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U244(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3 15.32/5.17 POL(U244'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U245(x_1, x_2)) = 1 + x_1 + x_2 15.32/5.17 POL(U245'(x_1, x_2)) = 2*x_1 + x_2 15.32/5.17 POL(U246(x_1)) = x_1 15.32/5.17 POL(U251(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 15.32/5.17 POL(U251'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U252(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 15.32/5.17 POL(U252'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U253(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 15.32/5.17 POL(U253'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U254(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U254'(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 15.32/5.17 POL(U255(x_1, x_2)) = x_1 + x_2 15.32/5.17 POL(U255'(x_1, x_2)) = x_1 + x_2 15.32/5.17 POL(U256(x_1)) = x_1 15.32/5.17 POL(U261(x_1, x_2)) = x_1 15.32/5.17 POL(U262(x_1)) = x_1 15.32/5.17 POL(U271(x_1, x_2)) = 2*x_1 15.32/5.17 POL(U272(x_1)) = x_1 15.32/5.17 POL(U41(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U41'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U42(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U42'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U43(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U43'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U44(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U44'(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 15.32/5.17 POL(U45(x_1, x_2)) = x_1 + 2*x_2 15.32/5.17 POL(U45'(x_1, x_2)) = 1 + x_1 + x_2 15.32/5.17 POL(U46(x_1)) = 2*x_1 15.32/5.17 POL(U51(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U51'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U52(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U52'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U53(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U53'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U54(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 15.32/5.17 POL(U54'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 15.32/5.17 POL(U55(x_1, x_2)) = x_1 + 2*x_2 15.32/5.17 POL(U55'(x_1, x_2)) = x_1 + x_2 15.32/5.17 POL(U56(x_1)) = 2*x_1 15.32/5.17 POL(U61(x_1, x_2)) = 1 + 2*x_1 + x_2 15.32/5.17 POL(U61'(x_1, x_2)) = 1 + x_1 + 2*x_2 15.32/5.17 POL(U62(x_1, x_2)) = 1 + 2*x_1 + x_2 15.32/5.17 POL(U62'(x_1, x_2)) = 2*x_1 + 2*x_2 15.32/5.17 POL(U63(x_1)) = 1 + x_1 15.32/5.17 POL(U71(x_1, x_2)) = 2 + x_1 + 2*x_2 15.32/5.17 POL(U71'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 15.32/5.17 POL(U72(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 15.32/5.17 POL(U72'(x_1, x_2)) = 1 + x_1 + 2*x_2 15.32/5.17 POL(U73(x_1)) = 1 + x_1 15.32/5.17 POL(U81(x_1, x_2)) = 2 + x_1 + x_2 15.32/5.17 POL(U81'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 15.32/5.17 POL(U82(x_1, x_2)) = 2 + 2*x_1 + x_2 15.32/5.17 POL(U82'(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 15.32/5.17 POL(U83(x_1)) = 1 + x_1 15.32/5.17 POL(U91(x_1, x_2)) = 1 + 2*x_1 + x_2 15.32/5.17 POL(U91'(x_1, x_2)) = 1 + x_1 + 2*x_2 15.32/5.17 POL(U92(x_1, x_2)) = 1 + 2*x_1 + x_2 15.32/5.17 POL(U92'(x_1, x_2)) = x_1 + 2*x_2 15.32/5.17 POL(U93(x_1)) = 1 + x_1 15.32/5.17 POL(afterNth(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 15.32/5.17 POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 15.32/5.17 POL(fst(x_1)) = 2 + 2*x_1 15.32/5.17 POL(head(x_1)) = 2 + 2*x_1 15.32/5.17 POL(isLNat(x_1)) = x_1 15.32/5.17 POL(isLNatKind(x_1)) = 0 15.32/5.17 POL(isNatural(x_1)) = 2*x_1 15.32/5.17 POL(isNaturalKind(x_1)) = 0 15.32/5.17 POL(isPLNat(x_1)) = x_1 15.32/5.17 POL(isPLNatKind(x_1)) = 0 15.32/5.17 POL(natsFrom(x_1)) = 2 + 2*x_1 15.32/5.17 POL(nil) = 2 15.32/5.17 POL(pair(x_1, x_2)) = 2 + 2*x_1 + x_2 15.32/5.17 POL(s(x_1)) = 2 + 2*x_1 15.32/5.17 POL(sel(x_1, x_2)) = 1 + 2*x_1 + x_2 15.32/5.17 POL(snd(x_1)) = 2 + 2*x_1 15.32/5.17 POL(splitAt(x_1, x_2)) = 2 + 2*x_1 + x_2 15.32/5.17 POL(tail(x_1)) = 2 + 2*x_1 15.32/5.17 POL(take(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 15.32/5.17 POL(tt) = 0 15.32/5.17 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (20) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {afterNth_2, head_1, U211_1, s_1, U221_1, sel_2, U232_1, fst_1, U131_1, pair_2, natsFrom_1, U141_1, snd_1, U151_1, splitAt_2, U272_1, tail_1, U161_1, take_2, U172_1, U262_1, U122_1, U112_1, U206_1, U56_1, U63_1, U73_1, U256_1, U93_1, U106_1, U246_1, U193_1, U46_1} are replacing on all positions. 15.32/5.17 For all symbols f in {U111_2, cons_2, U121_2, U231_2, U261_2, U271_2, U171_2, U181_2, U182_2, U41_3, U42_3, U43_3, U201_3, U202_3, U203_3, U204_3, U205_2, U51_3, U52_3, U53_3, U54_3, U55_2, U61_2, U62_2, U241_3, U242_3, U243_3, U244_3, U71_2, U72_2, U81_2, U82_2, U251_3, U252_3, U253_3, U254_3, U255_2, U91_2, U92_2, U101_3, U102_3, U103_3, U104_3, U105_2, U45_2, U103'_3, U102'_3, U104'_3, U41'_3, U42'_3, U43'_3, U45'_2, U44'_3, U51'_3, U54'_3, U53'_3, U55'_2, U62'_2, U244'_3, U243'_3, U245'_2, U71'_2, U181'_2, U81'_2, U252'_3, U251'_3, U253'_3, U255'_2, U92'_2, U101'_3, U191'_2, U202'_3, U201'_3, U203'_3, U204'_3, U205'_2} we have mu(f) = {1}. 15.32/5.17 The symbols in {isLNatKind_1, isNaturalKind_1, isPLNatKind_1, isLNat_1, isNatural_1, isPLNat_1, ISLNAT_1, ISPLNAT_1, ISNATURAL_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 15.32/5.17 U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) 15.32/5.17 U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) 15.32/5.17 ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) 15.32/5.17 U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) 15.32/5.17 U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) 15.32/5.17 U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) 15.32/5.17 ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) 15.32/5.17 U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) 15.32/5.17 U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) 15.32/5.17 U55'(tt, V2) -> ISLNAT(V2) 15.32/5.17 U62'(tt, V1) -> ISPLNAT(V1) 15.32/5.17 U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) 15.32/5.17 U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) 15.32/5.17 U245'(tt, V2) -> ISLNAT(V2) 15.32/5.17 ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) 15.32/5.17 ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) 15.32/5.17 ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) 15.32/5.17 U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) 15.32/5.17 U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) 15.32/5.17 U255'(tt, V2) -> ISLNAT(V2) 15.32/5.17 U92'(tt, V1) -> ISLNAT(V1) 15.32/5.17 ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) 15.32/5.17 U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) 15.32/5.17 ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) 15.32/5.17 U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) 15.32/5.17 U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) 15.32/5.17 U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) 15.32/5.17 U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) 15.32/5.17 U205'(tt, V2) -> ISLNAT(V2) 15.32/5.17 U204'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 U244'(tt, V1, V2) -> ISLNAT(V1) 15.32/5.17 U54'(tt, V1, V2) -> ISNATURAL(V1) 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 isLNatKind(nil) -> tt 15.32/5.17 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.17 isNaturalKind(0) -> tt 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.17 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.17 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.17 U141(tt) -> tt 15.32/5.17 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.17 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.17 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.17 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.17 U172(tt) -> tt 15.32/5.17 U161(tt) -> tt 15.32/5.17 U272(tt) -> tt 15.32/5.17 U151(tt) -> tt 15.32/5.17 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.17 U262(tt) -> tt 15.32/5.17 U131(tt) -> tt 15.32/5.17 U232(tt) -> tt 15.32/5.17 U221(tt) -> tt 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 U122(tt) -> tt 15.32/5.17 U211(tt) -> tt 15.32/5.17 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.17 U112(tt) -> tt 15.32/5.17 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.17 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.17 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.17 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.17 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.17 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.17 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.17 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.17 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.17 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.17 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.17 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.17 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.17 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.17 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.17 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.17 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.17 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.17 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.17 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.17 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.17 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.17 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.17 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.17 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.17 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.17 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.17 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.17 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.17 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.17 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.17 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.17 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.17 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.17 U106(tt) -> tt 15.32/5.17 U256(tt) -> tt 15.32/5.17 U246(tt) -> tt 15.32/5.17 U56(tt) -> tt 15.32/5.17 U206(tt) -> tt 15.32/5.17 U193(tt) -> tt 15.32/5.17 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.17 U46(tt) -> tt 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (21) QCSDependencyGraphProof (EQUIVALENT) 15.32/5.17 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 32 less nodes. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (22) 15.32/5.17 TRUE 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (23) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {U106_1, U112_1, U122_1, U131_1, snd_1, splitAt_2, U141_1, U151_1, U161_1, U172_1, U183_1, U193_1, U206_1, U211_1, U221_1, U232_1, U246_1, U256_1, U262_1, U272_1, natsFrom_1, s_1, head_1, afterNth_2, pair_2, fst_1, U46_1, U56_1, U63_1, U73_1, U83_1, U93_1, tail_1, take_2, sel_2, NATSFROM_1} are replacing on all positions. 15.32/5.17 For all symbols f in {U101_3, U102_3, U103_3, U104_3, U105_2, U11_3, U12_3, U111_2, U13_3, U121_2, U14_3, U171_2, U181_2, U182_2, U191_2, U192_2, U201_3, U202_3, U203_3, U204_3, U205_2, U21_3, U22_3, U23_3, U24_2, U231_2, U241_3, U242_3, U243_3, U244_3, U245_2, U251_3, U252_3, U253_3, U254_3, U255_2, U261_2, U271_2, U281_2, U282_2, cons_2, U291_3, U292_3, U293_3, U294_3, U301_3, U302_2, U303_2, U304_2, U31_3, U32_3, U311_2, U312_2, U33_3, U321_4, U322_4, U323_4, U324_4, U325_4, U326_4, U327_2, U34_2, U331_3, U332_2, U333_2, U334_2, U341_3, U342_3, U343_3, U344_3, U41_3, U42_3, U43_3, U44_3, U45_2, U51_3, U52_3, U53_3, U54_3, U55_2, U61_2, U62_2, U71_2, U72_2, U81_2, U82_2, U91_2, U92_2, U281'_2, U282'_2} we have mu(f) = {1}. 15.32/5.17 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1, U_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 15.32/5.17 NATSFROM(N) -> U281'(isNatural(N), N) 15.32/5.17 U281'(tt, N) -> U282'(isNaturalKind(N), N) 15.32/5.17 U282'(tt, N) -> U(N) 15.32/5.17 U(s(x_0)) -> U(x_0) 15.32/5.17 U(natsFrom(x_0)) -> U(x_0) 15.32/5.17 U(natsFrom(s(x0))) -> NATSFROM(s(x0)) 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.17 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.17 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.17 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.17 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.17 U106(tt) -> tt 15.32/5.17 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 15.32/5.17 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.17 U112(tt) -> tt 15.32/5.17 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 U122(tt) -> tt 15.32/5.17 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 15.32/5.17 U131(tt) -> tt 15.32/5.17 U14(tt, N, XS) -> snd(splitAt(N, XS)) 15.32/5.17 U141(tt) -> tt 15.32/5.17 U151(tt) -> tt 15.32/5.17 U161(tt) -> tt 15.32/5.17 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.17 U172(tt) -> tt 15.32/5.17 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.17 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.17 U183(tt) -> tt 15.32/5.17 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.17 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.17 U193(tt) -> tt 15.32/5.17 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.17 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.17 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.17 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.17 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.17 U206(tt) -> tt 15.32/5.17 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 15.32/5.17 U211(tt) -> tt 15.32/5.17 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 15.32/5.17 U221(tt) -> tt 15.32/5.17 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 15.32/5.17 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.17 U232(tt) -> tt 15.32/5.17 U24(tt, X) -> X 15.32/5.17 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.17 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.17 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.17 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.17 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.17 U246(tt) -> tt 15.32/5.17 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.17 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.17 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.17 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.17 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.17 U256(tt) -> tt 15.32/5.17 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.17 U262(tt) -> tt 15.32/5.17 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.17 U272(tt) -> tt 15.32/5.17 U281(tt, N) -> U282(isNaturalKind(N), N) 15.32/5.17 U282(tt, N) -> cons(N, natsFrom(s(N))) 15.32/5.17 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 15.32/5.17 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 15.32/5.17 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 15.32/5.17 U294(tt, N, XS) -> head(afterNth(N, XS)) 15.32/5.17 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 15.32/5.17 U302(tt, Y) -> U303(isLNat(Y), Y) 15.32/5.17 U303(tt, Y) -> U304(isLNatKind(Y), Y) 15.32/5.17 U304(tt, Y) -> Y 15.32/5.17 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 15.32/5.17 U311(tt, XS) -> U312(isLNatKind(XS), XS) 15.32/5.17 U312(tt, XS) -> pair(nil, XS) 15.32/5.17 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 15.32/5.17 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 15.32/5.17 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 15.32/5.17 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 15.32/5.17 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 15.32/5.17 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 15.32/5.17 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 15.32/5.17 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 15.32/5.17 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 15.32/5.17 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 15.32/5.17 U332(tt, XS) -> U333(isLNat(XS), XS) 15.32/5.17 U333(tt, XS) -> U334(isLNatKind(XS), XS) 15.32/5.17 U334(tt, XS) -> XS 15.32/5.17 U34(tt, N) -> N 15.32/5.17 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 15.32/5.17 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 15.32/5.17 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 15.32/5.17 U344(tt, N, XS) -> fst(splitAt(N, XS)) 15.32/5.17 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.17 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.17 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.17 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.17 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.17 U46(tt) -> tt 15.32/5.17 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.17 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.17 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.17 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.17 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.17 U56(tt) -> tt 15.32/5.17 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.17 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.17 U63(tt) -> tt 15.32/5.17 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.17 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.17 U73(tt) -> tt 15.32/5.17 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.17 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.17 U83(tt) -> tt 15.32/5.17 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.17 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.17 U93(tt) -> tt 15.32/5.17 afterNth(N, XS) -> U11(isNatural(N), N, XS) 15.32/5.17 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 15.32/5.17 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 15.32/5.17 isLNat(nil) -> tt 15.32/5.17 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.17 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.17 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.17 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.17 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNatKind(nil) -> tt 15.32/5.17 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.17 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.17 isNatural(0) -> tt 15.32/5.17 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.17 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.17 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.17 isNaturalKind(0) -> tt 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.17 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.17 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.17 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.17 natsFrom(N) -> U281(isNatural(N), N) 15.32/5.17 sel(N, XS) -> U291(isNatural(N), N, XS) 15.32/5.17 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 15.32/5.17 splitAt(0, XS) -> U311(isLNat(XS), XS) 15.32/5.17 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 15.32/5.17 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 15.32/5.17 take(N, XS) -> U341(isNatural(N), N, XS) 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (24) QCSDPSubtermProof (EQUIVALENT) 15.32/5.17 We use the subterm processor [DA_EMMES]. 15.32/5.17 15.32/5.17 15.32/5.17 The following pairs can be oriented strictly and are deleted. 15.32/5.17 15.32/5.17 U(s(x_0)) -> U(x_0) 15.32/5.17 U(natsFrom(x_0)) -> U(x_0) 15.32/5.17 U(natsFrom(s(x0))) -> NATSFROM(s(x0)) 15.32/5.17 The remaining pairs can at least be oriented weakly. 15.32/5.17 15.32/5.17 NATSFROM(N) -> U281'(isNatural(N), N) 15.32/5.17 U281'(tt, N) -> U282'(isNaturalKind(N), N) 15.32/5.17 U282'(tt, N) -> U(N) 15.32/5.17 Used ordering: Combined order from the following AFS and order. 15.32/5.17 U281'(x1, x2) = x2 15.32/5.17 15.32/5.17 NATSFROM(x1) = x1 15.32/5.17 15.32/5.17 U282'(x1, x2) = x2 15.32/5.17 15.32/5.17 U(x1) = x1 15.32/5.17 15.32/5.17 15.32/5.17 Subterm Order 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (25) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {U106_1, U112_1, U122_1, U131_1, snd_1, splitAt_2, U141_1, U151_1, U161_1, U172_1, U183_1, U193_1, U206_1, U211_1, U221_1, U232_1, U246_1, U256_1, U262_1, U272_1, natsFrom_1, s_1, head_1, afterNth_2, pair_2, fst_1, U46_1, U56_1, U63_1, U73_1, U83_1, U93_1, tail_1, take_2, sel_2, NATSFROM_1} are replacing on all positions. 15.32/5.17 For all symbols f in {U101_3, U102_3, U103_3, U104_3, U105_2, U11_3, U12_3, U111_2, U13_3, U121_2, U14_3, U171_2, U181_2, U182_2, U191_2, U192_2, U201_3, U202_3, U203_3, U204_3, U205_2, U21_3, U22_3, U23_3, U24_2, U231_2, U241_3, U242_3, U243_3, U244_3, U245_2, U251_3, U252_3, U253_3, U254_3, U255_2, U261_2, U271_2, U281_2, U282_2, cons_2, U291_3, U292_3, U293_3, U294_3, U301_3, U302_2, U303_2, U304_2, U31_3, U32_3, U311_2, U312_2, U33_3, U321_4, U322_4, U323_4, U324_4, U325_4, U326_4, U327_2, U34_2, U331_3, U332_2, U333_2, U334_2, U341_3, U342_3, U343_3, U344_3, U41_3, U42_3, U43_3, U44_3, U45_2, U51_3, U52_3, U53_3, U54_3, U55_2, U61_2, U62_2, U71_2, U72_2, U81_2, U82_2, U91_2, U92_2, U281'_2, U282'_2} we have mu(f) = {1}. 15.32/5.17 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1, U_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 15.32/5.17 NATSFROM(N) -> U281'(isNatural(N), N) 15.32/5.17 U281'(tt, N) -> U282'(isNaturalKind(N), N) 15.32/5.17 U282'(tt, N) -> U(N) 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.17 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.17 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.17 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.17 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.17 U106(tt) -> tt 15.32/5.17 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 15.32/5.17 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.17 U112(tt) -> tt 15.32/5.17 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 U122(tt) -> tt 15.32/5.17 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 15.32/5.17 U131(tt) -> tt 15.32/5.17 U14(tt, N, XS) -> snd(splitAt(N, XS)) 15.32/5.17 U141(tt) -> tt 15.32/5.17 U151(tt) -> tt 15.32/5.17 U161(tt) -> tt 15.32/5.17 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.17 U172(tt) -> tt 15.32/5.17 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.17 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.17 U183(tt) -> tt 15.32/5.17 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.17 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.17 U193(tt) -> tt 15.32/5.17 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.17 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.17 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.17 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.17 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.17 U206(tt) -> tt 15.32/5.17 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 15.32/5.17 U211(tt) -> tt 15.32/5.17 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 15.32/5.17 U221(tt) -> tt 15.32/5.17 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 15.32/5.17 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.17 U232(tt) -> tt 15.32/5.17 U24(tt, X) -> X 15.32/5.17 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.17 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.17 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.17 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.17 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.17 U246(tt) -> tt 15.32/5.17 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.17 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.17 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.17 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.17 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.17 U256(tt) -> tt 15.32/5.17 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.17 U262(tt) -> tt 15.32/5.17 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.17 U272(tt) -> tt 15.32/5.17 U281(tt, N) -> U282(isNaturalKind(N), N) 15.32/5.17 U282(tt, N) -> cons(N, natsFrom(s(N))) 15.32/5.17 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 15.32/5.17 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 15.32/5.17 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 15.32/5.17 U294(tt, N, XS) -> head(afterNth(N, XS)) 15.32/5.17 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 15.32/5.17 U302(tt, Y) -> U303(isLNat(Y), Y) 15.32/5.17 U303(tt, Y) -> U304(isLNatKind(Y), Y) 15.32/5.17 U304(tt, Y) -> Y 15.32/5.17 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 15.32/5.17 U311(tt, XS) -> U312(isLNatKind(XS), XS) 15.32/5.17 U312(tt, XS) -> pair(nil, XS) 15.32/5.17 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 15.32/5.17 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 15.32/5.17 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 15.32/5.17 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 15.32/5.17 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 15.32/5.17 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 15.32/5.17 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 15.32/5.17 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 15.32/5.17 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 15.32/5.17 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 15.32/5.17 U332(tt, XS) -> U333(isLNat(XS), XS) 15.32/5.17 U333(tt, XS) -> U334(isLNatKind(XS), XS) 15.32/5.17 U334(tt, XS) -> XS 15.32/5.17 U34(tt, N) -> N 15.32/5.17 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 15.32/5.17 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 15.32/5.17 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 15.32/5.17 U344(tt, N, XS) -> fst(splitAt(N, XS)) 15.32/5.17 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.17 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.17 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.17 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.17 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.17 U46(tt) -> tt 15.32/5.17 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.17 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.17 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.17 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.17 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.17 U56(tt) -> tt 15.32/5.17 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.17 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.17 U63(tt) -> tt 15.32/5.17 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.17 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.17 U73(tt) -> tt 15.32/5.17 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.17 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.17 U83(tt) -> tt 15.32/5.17 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.17 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.17 U93(tt) -> tt 15.32/5.17 afterNth(N, XS) -> U11(isNatural(N), N, XS) 15.32/5.17 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 15.32/5.17 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 15.32/5.17 isLNat(nil) -> tt 15.32/5.17 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.17 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.17 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.17 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.17 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNatKind(nil) -> tt 15.32/5.17 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.17 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.17 isNatural(0) -> tt 15.32/5.17 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.17 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.17 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.17 isNaturalKind(0) -> tt 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.17 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.17 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.17 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.17 natsFrom(N) -> U281(isNatural(N), N) 15.32/5.17 sel(N, XS) -> U291(isNatural(N), N, XS) 15.32/5.17 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 15.32/5.17 splitAt(0, XS) -> U311(isLNat(XS), XS) 15.32/5.17 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 15.32/5.17 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 15.32/5.17 take(N, XS) -> U341(isNatural(N), N, XS) 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (26) QCSDependencyGraphProof (EQUIVALENT) 15.32/5.17 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (27) 15.32/5.17 TRUE 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (28) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {U106_1, U112_1, U122_1, U131_1, snd_1, splitAt_2, U141_1, U151_1, U161_1, U172_1, U183_1, U193_1, U206_1, U211_1, U221_1, U232_1, U246_1, U256_1, U262_1, U272_1, natsFrom_1, s_1, head_1, afterNth_2, pair_2, fst_1, U46_1, U56_1, U63_1, U73_1, U83_1, U93_1, tail_1, take_2, sel_2, SPLITAT_2} are replacing on all positions. 15.32/5.17 For all symbols f in {U101_3, U102_3, U103_3, U104_3, U105_2, U11_3, U12_3, U111_2, U13_3, U121_2, U14_3, U171_2, U181_2, U182_2, U191_2, U192_2, U201_3, U202_3, U203_3, U204_3, U205_2, U21_3, U22_3, U23_3, U24_2, U231_2, U241_3, U242_3, U243_3, U244_3, U245_2, U251_3, U252_3, U253_3, U254_3, U255_2, U261_2, U271_2, U281_2, U282_2, cons_2, U291_3, U292_3, U293_3, U294_3, U301_3, U302_2, U303_2, U304_2, U31_3, U32_3, U311_2, U312_2, U33_3, U321_4, U322_4, U323_4, U324_4, U325_4, U326_4, U327_2, U34_2, U331_3, U332_2, U333_2, U334_2, U341_3, U342_3, U343_3, U344_3, U41_3, U42_3, U43_3, U44_3, U45_2, U51_3, U52_3, U53_3, U54_3, U55_2, U61_2, U62_2, U71_2, U72_2, U81_2, U82_2, U91_2, U92_2, U325'_4, U324'_4, U326'_4, U321'_4, U322'_4, U323'_4} we have mu(f) = {1}. 15.32/5.17 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 15.32/5.17 U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) 15.32/5.17 U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) 15.32/5.17 U326'(tt, N, X, XS) -> SPLITAT(N, XS) 15.32/5.17 SPLITAT(s(N), cons(X, XS)) -> U321'(isNatural(N), N, X, XS) 15.32/5.17 U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) 15.32/5.17 U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) 15.32/5.17 U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.17 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.17 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.17 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.17 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.17 U106(tt) -> tt 15.32/5.17 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 15.32/5.17 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.17 U112(tt) -> tt 15.32/5.17 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 U122(tt) -> tt 15.32/5.17 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 15.32/5.17 U131(tt) -> tt 15.32/5.17 U14(tt, N, XS) -> snd(splitAt(N, XS)) 15.32/5.17 U141(tt) -> tt 15.32/5.17 U151(tt) -> tt 15.32/5.17 U161(tt) -> tt 15.32/5.17 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.17 U172(tt) -> tt 15.32/5.17 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.17 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.17 U183(tt) -> tt 15.32/5.17 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.17 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.17 U193(tt) -> tt 15.32/5.17 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.17 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.17 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.17 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.17 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.17 U206(tt) -> tt 15.32/5.17 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 15.32/5.17 U211(tt) -> tt 15.32/5.17 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 15.32/5.17 U221(tt) -> tt 15.32/5.17 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 15.32/5.17 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.17 U232(tt) -> tt 15.32/5.17 U24(tt, X) -> X 15.32/5.17 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.17 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.17 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.17 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.17 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.17 U246(tt) -> tt 15.32/5.17 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.17 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.17 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.17 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.17 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.17 U256(tt) -> tt 15.32/5.17 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.17 U262(tt) -> tt 15.32/5.17 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.17 U272(tt) -> tt 15.32/5.17 U281(tt, N) -> U282(isNaturalKind(N), N) 15.32/5.17 U282(tt, N) -> cons(N, natsFrom(s(N))) 15.32/5.17 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 15.32/5.17 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 15.32/5.17 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 15.32/5.17 U294(tt, N, XS) -> head(afterNth(N, XS)) 15.32/5.17 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 15.32/5.17 U302(tt, Y) -> U303(isLNat(Y), Y) 15.32/5.17 U303(tt, Y) -> U304(isLNatKind(Y), Y) 15.32/5.17 U304(tt, Y) -> Y 15.32/5.17 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 15.32/5.17 U311(tt, XS) -> U312(isLNatKind(XS), XS) 15.32/5.17 U312(tt, XS) -> pair(nil, XS) 15.32/5.17 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 15.32/5.17 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 15.32/5.17 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 15.32/5.17 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 15.32/5.17 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 15.32/5.17 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 15.32/5.17 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 15.32/5.17 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 15.32/5.17 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 15.32/5.17 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 15.32/5.17 U332(tt, XS) -> U333(isLNat(XS), XS) 15.32/5.17 U333(tt, XS) -> U334(isLNatKind(XS), XS) 15.32/5.17 U334(tt, XS) -> XS 15.32/5.17 U34(tt, N) -> N 15.32/5.17 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 15.32/5.17 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 15.32/5.17 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 15.32/5.17 U344(tt, N, XS) -> fst(splitAt(N, XS)) 15.32/5.17 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.17 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.17 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.17 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.17 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.17 U46(tt) -> tt 15.32/5.17 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.17 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.17 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.17 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.17 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.17 U56(tt) -> tt 15.32/5.17 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.17 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.17 U63(tt) -> tt 15.32/5.17 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.17 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.17 U73(tt) -> tt 15.32/5.17 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.17 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.17 U83(tt) -> tt 15.32/5.17 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.17 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.17 U93(tt) -> tt 15.32/5.17 afterNth(N, XS) -> U11(isNatural(N), N, XS) 15.32/5.17 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 15.32/5.17 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 15.32/5.17 isLNat(nil) -> tt 15.32/5.17 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.17 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.17 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.17 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.17 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNatKind(nil) -> tt 15.32/5.17 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.17 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.17 isNatural(0) -> tt 15.32/5.17 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.17 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.17 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.17 isNaturalKind(0) -> tt 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.17 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.17 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.17 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.17 natsFrom(N) -> U281(isNatural(N), N) 15.32/5.17 sel(N, XS) -> U291(isNatural(N), N, XS) 15.32/5.17 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 15.32/5.17 splitAt(0, XS) -> U311(isLNat(XS), XS) 15.32/5.17 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 15.32/5.17 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 15.32/5.17 take(N, XS) -> U341(isNatural(N), N, XS) 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (29) QCSDPSubtermProof (EQUIVALENT) 15.32/5.17 We use the subterm processor [DA_EMMES]. 15.32/5.17 15.32/5.17 15.32/5.17 The following pairs can be oriented strictly and are deleted. 15.32/5.17 15.32/5.17 SPLITAT(s(N), cons(X, XS)) -> U321'(isNatural(N), N, X, XS) 15.32/5.17 The remaining pairs can at least be oriented weakly. 15.32/5.17 15.32/5.17 U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) 15.32/5.17 U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) 15.32/5.17 U326'(tt, N, X, XS) -> SPLITAT(N, XS) 15.32/5.17 U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) 15.32/5.17 U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) 15.32/5.17 U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) 15.32/5.17 Used ordering: Combined order from the following AFS and order. 15.32/5.17 U325'(x1, x2, x3, x4) = x2 15.32/5.17 15.32/5.17 U324'(x1, x2, x3, x4) = x2 15.32/5.17 15.32/5.17 U326'(x1, x2, x3, x4) = x2 15.32/5.17 15.32/5.17 SPLITAT(x1, x2) = x1 15.32/5.17 15.32/5.17 U321'(x1, x2, x3, x4) = x2 15.32/5.17 15.32/5.17 U322'(x1, x2, x3, x4) = x2 15.32/5.17 15.32/5.17 U323'(x1, x2, x3, x4) = x2 15.32/5.17 15.32/5.17 15.32/5.17 Subterm Order 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (30) 15.32/5.17 Obligation: 15.32/5.17 Q-restricted context-sensitive dependency pair problem: 15.32/5.17 The symbols in {U106_1, U112_1, U122_1, U131_1, snd_1, splitAt_2, U141_1, U151_1, U161_1, U172_1, U183_1, U193_1, U206_1, U211_1, U221_1, U232_1, U246_1, U256_1, U262_1, U272_1, natsFrom_1, s_1, head_1, afterNth_2, pair_2, fst_1, U46_1, U56_1, U63_1, U73_1, U83_1, U93_1, tail_1, take_2, sel_2, SPLITAT_2} are replacing on all positions. 15.32/5.17 For all symbols f in {U101_3, U102_3, U103_3, U104_3, U105_2, U11_3, U12_3, U111_2, U13_3, U121_2, U14_3, U171_2, U181_2, U182_2, U191_2, U192_2, U201_3, U202_3, U203_3, U204_3, U205_2, U21_3, U22_3, U23_3, U24_2, U231_2, U241_3, U242_3, U243_3, U244_3, U245_2, U251_3, U252_3, U253_3, U254_3, U255_2, U261_2, U271_2, U281_2, U282_2, cons_2, U291_3, U292_3, U293_3, U294_3, U301_3, U302_2, U303_2, U304_2, U31_3, U32_3, U311_2, U312_2, U33_3, U321_4, U322_4, U323_4, U324_4, U325_4, U326_4, U327_2, U34_2, U331_3, U332_2, U333_2, U334_2, U341_3, U342_3, U343_3, U344_3, U41_3, U42_3, U43_3, U44_3, U45_2, U51_3, U52_3, U53_3, U54_3, U55_2, U61_2, U62_2, U71_2, U72_2, U81_2, U82_2, U91_2, U92_2, U325'_4, U324'_4, U326'_4, U322'_4, U321'_4, U323'_4} we have mu(f) = {1}. 15.32/5.17 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1} are not replacing on any position. 15.32/5.17 15.32/5.17 The TRS P consists of the following rules: 15.32/5.17 15.32/5.17 U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) 15.32/5.17 U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) 15.32/5.17 U326'(tt, N, X, XS) -> SPLITAT(N, XS) 15.32/5.17 U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) 15.32/5.17 U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) 15.32/5.17 U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) 15.32/5.17 15.32/5.17 The TRS R consists of the following rules: 15.32/5.17 15.32/5.17 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 15.32/5.17 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 15.32/5.17 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 15.32/5.17 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 15.32/5.17 U105(tt, V2) -> U106(isLNat(V2)) 15.32/5.17 U106(tt) -> tt 15.32/5.17 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 15.32/5.17 U111(tt, V2) -> U112(isLNatKind(V2)) 15.32/5.17 U112(tt) -> tt 15.32/5.17 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 15.32/5.17 U121(tt, V2) -> U122(isLNatKind(V2)) 15.32/5.17 U122(tt) -> tt 15.32/5.17 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 15.32/5.17 U131(tt) -> tt 15.32/5.17 U14(tt, N, XS) -> snd(splitAt(N, XS)) 15.32/5.17 U141(tt) -> tt 15.32/5.17 U151(tt) -> tt 15.32/5.17 U161(tt) -> tt 15.32/5.17 U171(tt, V2) -> U172(isLNatKind(V2)) 15.32/5.17 U172(tt) -> tt 15.32/5.17 U181(tt, V1) -> U182(isLNatKind(V1), V1) 15.32/5.17 U182(tt, V1) -> U183(isLNat(V1)) 15.32/5.17 U183(tt) -> tt 15.32/5.17 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 15.32/5.17 U192(tt, V1) -> U193(isNatural(V1)) 15.32/5.17 U193(tt) -> tt 15.32/5.17 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 15.32/5.17 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 15.32/5.17 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 15.32/5.17 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 15.32/5.17 U205(tt, V2) -> U206(isLNat(V2)) 15.32/5.17 U206(tt) -> tt 15.32/5.17 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 15.32/5.17 U211(tt) -> tt 15.32/5.17 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 15.32/5.17 U221(tt) -> tt 15.32/5.17 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 15.32/5.17 U231(tt, V2) -> U232(isLNatKind(V2)) 15.32/5.17 U232(tt) -> tt 15.32/5.17 U24(tt, X) -> X 15.32/5.17 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 15.32/5.17 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 15.32/5.17 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 15.32/5.17 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 15.32/5.17 U245(tt, V2) -> U246(isLNat(V2)) 15.32/5.17 U246(tt) -> tt 15.32/5.17 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 15.32/5.17 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 15.32/5.17 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 15.32/5.17 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 15.32/5.17 U255(tt, V2) -> U256(isLNat(V2)) 15.32/5.17 U256(tt) -> tt 15.32/5.17 U261(tt, V2) -> U262(isLNatKind(V2)) 15.32/5.17 U262(tt) -> tt 15.32/5.17 U271(tt, V2) -> U272(isLNatKind(V2)) 15.32/5.17 U272(tt) -> tt 15.32/5.17 U281(tt, N) -> U282(isNaturalKind(N), N) 15.32/5.17 U282(tt, N) -> cons(N, natsFrom(s(N))) 15.32/5.17 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 15.32/5.17 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 15.32/5.17 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 15.32/5.17 U294(tt, N, XS) -> head(afterNth(N, XS)) 15.32/5.17 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 15.32/5.17 U302(tt, Y) -> U303(isLNat(Y), Y) 15.32/5.17 U303(tt, Y) -> U304(isLNatKind(Y), Y) 15.32/5.17 U304(tt, Y) -> Y 15.32/5.17 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 15.32/5.17 U311(tt, XS) -> U312(isLNatKind(XS), XS) 15.32/5.17 U312(tt, XS) -> pair(nil, XS) 15.32/5.17 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 15.32/5.17 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 15.32/5.17 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 15.32/5.17 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 15.32/5.17 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 15.32/5.17 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 15.32/5.17 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 15.32/5.17 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 15.32/5.17 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 15.32/5.17 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 15.32/5.17 U332(tt, XS) -> U333(isLNat(XS), XS) 15.32/5.17 U333(tt, XS) -> U334(isLNatKind(XS), XS) 15.32/5.17 U334(tt, XS) -> XS 15.32/5.17 U34(tt, N) -> N 15.32/5.17 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 15.32/5.17 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 15.32/5.17 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 15.32/5.17 U344(tt, N, XS) -> fst(splitAt(N, XS)) 15.32/5.17 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 15.32/5.17 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 15.32/5.17 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 15.32/5.17 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 15.32/5.17 U45(tt, V2) -> U46(isLNat(V2)) 15.32/5.17 U46(tt) -> tt 15.32/5.17 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 15.32/5.17 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 15.32/5.17 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 15.32/5.17 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 15.32/5.17 U55(tt, V2) -> U56(isLNat(V2)) 15.32/5.17 U56(tt) -> tt 15.32/5.17 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 15.32/5.17 U62(tt, V1) -> U63(isPLNat(V1)) 15.32/5.17 U63(tt) -> tt 15.32/5.17 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 15.32/5.17 U72(tt, V1) -> U73(isNatural(V1)) 15.32/5.17 U73(tt) -> tt 15.32/5.17 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 15.32/5.17 U82(tt, V1) -> U83(isPLNat(V1)) 15.32/5.17 U83(tt) -> tt 15.32/5.17 U91(tt, V1) -> U92(isLNatKind(V1), V1) 15.32/5.17 U92(tt, V1) -> U93(isLNat(V1)) 15.32/5.17 U93(tt) -> tt 15.32/5.17 afterNth(N, XS) -> U11(isNatural(N), N, XS) 15.32/5.17 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 15.32/5.17 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 15.32/5.17 isLNat(nil) -> tt 15.32/5.17 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 15.32/5.17 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 15.32/5.17 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 15.32/5.17 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 15.32/5.17 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 15.32/5.17 isLNatKind(nil) -> tt 15.32/5.17 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 15.32/5.17 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 15.32/5.17 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 15.32/5.17 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 15.32/5.17 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 15.32/5.17 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 15.32/5.17 isNatural(0) -> tt 15.32/5.17 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 15.32/5.17 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 15.32/5.17 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 15.32/5.17 isNaturalKind(0) -> tt 15.32/5.17 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 15.32/5.17 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 15.32/5.17 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 15.32/5.17 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 15.32/5.17 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 15.32/5.17 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 15.32/5.17 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 15.32/5.17 natsFrom(N) -> U281(isNatural(N), N) 15.32/5.17 sel(N, XS) -> U291(isNatural(N), N, XS) 15.32/5.17 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 15.32/5.17 splitAt(0, XS) -> U311(isLNat(XS), XS) 15.32/5.17 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 15.32/5.17 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 15.32/5.17 take(N, XS) -> U341(isNatural(N), N, XS) 15.32/5.17 15.32/5.17 Q is empty. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (31) QCSDependencyGraphProof (EQUIVALENT) 15.32/5.17 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 6 less nodes. 15.32/5.17 15.32/5.17 ---------------------------------------- 15.32/5.17 15.32/5.17 (32) 15.32/5.17 TRUE 15.45/5.23 EOF