12.43/3.83 YES 12.43/3.85 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 12.43/3.85 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 12.43/3.85 12.43/3.85 12.43/3.85 Termination of the given CSR could be proven: 12.43/3.85 12.43/3.85 (0) CSR 12.43/3.85 (1) CSDependencyPairsProof [EQUIVALENT, 657 ms] 12.43/3.85 (2) QCSDP 12.43/3.85 (3) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 12.43/3.85 (4) AND 12.43/3.85 (5) QCSDP 12.43/3.85 (6) QCSUsableRulesProof [EQUIVALENT, 0 ms] 12.43/3.85 (7) QCSDP 12.43/3.85 (8) QCSDPMuMonotonicPoloProof [EQUIVALENT, 76 ms] 12.43/3.85 (9) QCSDP 12.43/3.85 (10) QCSDPSubtermProof [EQUIVALENT, 0 ms] 12.43/3.85 (11) QCSDP 12.43/3.85 (12) PIsEmptyProof [EQUIVALENT, 0 ms] 12.43/3.85 (13) YES 12.43/3.85 (14) QCSDP 12.43/3.85 (15) QCSUsableRulesProof [EQUIVALENT, 36 ms] 12.43/3.85 (16) QCSDP 12.43/3.85 (17) QCSDPMuMonotonicPoloProof [EQUIVALENT, 368 ms] 12.43/3.85 (18) QCSDP 12.43/3.85 (19) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 12.43/3.85 (20) TRUE 12.43/3.85 (21) QCSDP 12.43/3.85 (22) QCSDPSubtermProof [EQUIVALENT, 0 ms] 12.43/3.85 (23) QCSDP 12.43/3.85 (24) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 12.43/3.85 (25) TRUE 12.43/3.85 (26) QCSDP 12.43/3.85 (27) QCSDPSubtermProof [EQUIVALENT, 0 ms] 12.43/3.85 (28) QCSDP 12.43/3.85 (29) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 12.43/3.85 (30) TRUE 12.43/3.85 12.43/3.85 12.43/3.85 ---------------------------------------- 12.43/3.85 12.43/3.85 (0) 12.43/3.85 Obligation: 12.43/3.85 Context-sensitive rewrite system: 12.43/3.85 The TRS R consists of the following rules: 12.43/3.85 12.43/3.85 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.43/3.85 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.43/3.85 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.43/3.85 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.43/3.85 U105(tt, V2) -> U106(isLNat(V2)) 12.43/3.85 U106(tt) -> tt 12.43/3.85 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 12.43/3.85 U111(tt, V2) -> U112(isLNatKind(V2)) 12.43/3.85 U112(tt) -> tt 12.43/3.85 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 12.43/3.85 U121(tt, V2) -> U122(isLNatKind(V2)) 12.43/3.85 U122(tt) -> tt 12.43/3.85 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 12.43/3.85 U131(tt) -> tt 12.43/3.85 U14(tt, N, XS) -> snd(splitAt(N, XS)) 12.43/3.85 U141(tt) -> tt 12.43/3.85 U151(tt) -> tt 12.43/3.85 U161(tt) -> tt 12.43/3.85 U171(tt, V2) -> U172(isLNatKind(V2)) 12.43/3.85 U172(tt) -> tt 12.43/3.85 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.43/3.85 U182(tt, V1) -> U183(isLNat(V1)) 12.43/3.85 U183(tt) -> tt 12.43/3.85 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.43/3.85 U192(tt, V1) -> U193(isNatural(V1)) 12.43/3.85 U193(tt) -> tt 12.43/3.85 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.43/3.85 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.43/3.85 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.43/3.85 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.43/3.85 U205(tt, V2) -> U206(isLNat(V2)) 12.43/3.85 U206(tt) -> tt 12.43/3.85 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 12.43/3.85 U211(tt) -> tt 12.43/3.85 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 12.43/3.85 U221(tt) -> tt 12.43/3.85 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 12.43/3.85 U231(tt, V2) -> U232(isLNatKind(V2)) 12.43/3.85 U232(tt) -> tt 12.43/3.85 U24(tt, X) -> X 12.43/3.85 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.43/3.85 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.43/3.85 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.43/3.85 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.43/3.85 U245(tt, V2) -> U246(isLNat(V2)) 12.43/3.85 U246(tt) -> tt 12.43/3.85 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.43/3.85 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.43/3.85 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.43/3.85 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.43/3.85 U255(tt, V2) -> U256(isLNat(V2)) 12.43/3.85 U256(tt) -> tt 12.43/3.85 U261(tt, V2) -> U262(isLNatKind(V2)) 12.43/3.85 U262(tt) -> tt 12.43/3.85 U271(tt, V2) -> U272(isLNatKind(V2)) 12.43/3.85 U272(tt) -> tt 12.43/3.85 U281(tt, N) -> U282(isNaturalKind(N), N) 12.43/3.85 U282(tt, N) -> cons(N, natsFrom(s(N))) 12.43/3.85 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 12.43/3.85 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 12.43/3.85 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 12.43/3.85 U294(tt, N, XS) -> head(afterNth(N, XS)) 12.43/3.85 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 12.43/3.85 U302(tt, Y) -> U303(isLNat(Y), Y) 12.43/3.85 U303(tt, Y) -> U304(isLNatKind(Y), Y) 12.43/3.85 U304(tt, Y) -> Y 12.43/3.85 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 12.43/3.85 U311(tt, XS) -> U312(isLNatKind(XS), XS) 12.43/3.85 U312(tt, XS) -> pair(nil, XS) 12.43/3.85 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 12.43/3.85 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 12.43/3.85 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 12.43/3.85 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 12.43/3.85 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 12.43/3.85 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 12.43/3.85 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 12.43/3.85 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 12.43/3.85 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 12.43/3.85 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 12.43/3.85 U332(tt, XS) -> U333(isLNat(XS), XS) 12.43/3.85 U333(tt, XS) -> U334(isLNatKind(XS), XS) 12.43/3.85 U334(tt, XS) -> XS 12.43/3.85 U34(tt, N) -> N 12.43/3.85 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 12.43/3.85 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 12.43/3.85 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 12.43/3.85 U344(tt, N, XS) -> fst(splitAt(N, XS)) 12.43/3.85 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.43/3.85 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.43/3.85 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.43/3.85 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.43/3.85 U45(tt, V2) -> U46(isLNat(V2)) 12.43/3.85 U46(tt) -> tt 12.43/3.85 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.43/3.85 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.43/3.85 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.43/3.85 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.43/3.85 U55(tt, V2) -> U56(isLNat(V2)) 12.43/3.85 U56(tt) -> tt 12.43/3.85 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.43/3.85 U62(tt, V1) -> U63(isPLNat(V1)) 12.43/3.85 U63(tt) -> tt 12.43/3.85 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.43/3.85 U72(tt, V1) -> U73(isNatural(V1)) 12.43/3.85 U73(tt) -> tt 12.43/3.85 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.43/3.85 U82(tt, V1) -> U83(isPLNat(V1)) 12.43/3.85 U83(tt) -> tt 12.43/3.85 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.43/3.85 U92(tt, V1) -> U93(isLNat(V1)) 12.43/3.85 U93(tt) -> tt 12.43/3.85 afterNth(N, XS) -> U11(isNatural(N), N, XS) 12.43/3.85 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 12.43/3.85 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 12.43/3.85 isLNat(nil) -> tt 12.43/3.85 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.43/3.85 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.43/3.85 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.43/3.85 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.43/3.85 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.43/3.85 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.43/3.85 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.43/3.85 isLNatKind(nil) -> tt 12.43/3.85 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.43/3.85 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.43/3.85 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.43/3.85 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.43/3.85 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.43/3.85 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.43/3.85 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.43/3.85 isNatural(0) -> tt 12.43/3.85 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.43/3.85 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.43/3.85 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.43/3.85 isNaturalKind(0) -> tt 12.43/3.85 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.43/3.85 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.43/3.85 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.43/3.85 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.43/3.85 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.43/3.85 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.43/3.85 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.43/3.85 natsFrom(N) -> U281(isNatural(N), N) 12.43/3.86 sel(N, XS) -> U291(isNatural(N), N, XS) 12.43/3.86 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 12.43/3.86 splitAt(0, XS) -> U311(isLNat(XS), XS) 12.43/3.86 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 12.43/3.86 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 12.43/3.86 take(N, XS) -> U341(isNatural(N), N, XS) 12.43/3.86 12.43/3.86 The replacement map contains the following entries: 12.43/3.86 12.43/3.86 U101: {1} 12.43/3.86 tt: empty set 12.43/3.86 U102: {1} 12.43/3.86 isNaturalKind: empty set 12.43/3.86 U103: {1} 12.43/3.86 isLNatKind: empty set 12.43/3.86 U104: {1} 12.43/3.86 U105: {1} 12.43/3.86 isNatural: empty set 12.43/3.86 U106: {1} 12.43/3.86 isLNat: empty set 12.43/3.86 U11: {1} 12.43/3.86 U12: {1} 12.43/3.86 U111: {1} 12.43/3.86 U112: {1} 12.43/3.86 U13: {1} 12.43/3.86 U121: {1} 12.43/3.86 U122: {1} 12.43/3.86 U14: {1} 12.43/3.86 U131: {1} 12.43/3.86 snd: {1} 12.43/3.86 splitAt: {1, 2} 12.43/3.86 U141: {1} 12.43/3.86 U151: {1} 12.43/3.86 U161: {1} 12.43/3.86 U171: {1} 12.43/3.86 U172: {1} 12.43/3.86 U181: {1} 12.43/3.86 U182: {1} 12.43/3.86 U183: {1} 12.43/3.86 U191: {1} 12.43/3.86 U192: {1} 12.43/3.86 U193: {1} 12.43/3.86 U201: {1} 12.43/3.86 U202: {1} 12.43/3.86 U203: {1} 12.43/3.86 U204: {1} 12.43/3.86 U205: {1} 12.43/3.86 U206: {1} 12.43/3.86 U21: {1} 12.43/3.86 U22: {1} 12.43/3.86 U211: {1} 12.43/3.86 U23: {1} 12.43/3.86 U221: {1} 12.43/3.86 U24: {1} 12.43/3.86 U231: {1} 12.43/3.86 U232: {1} 12.43/3.86 U241: {1} 12.43/3.86 U242: {1} 12.43/3.86 U243: {1} 12.43/3.86 U244: {1} 12.43/3.86 U245: {1} 12.43/3.86 U246: {1} 12.43/3.86 U251: {1} 12.43/3.86 U252: {1} 12.43/3.86 U253: {1} 12.43/3.86 U254: {1} 12.43/3.86 U255: {1} 12.43/3.86 U256: {1} 12.43/3.86 U261: {1} 12.43/3.86 U262: {1} 12.43/3.86 U271: {1} 12.43/3.86 U272: {1} 12.43/3.86 U281: {1} 12.43/3.86 U282: {1} 12.43/3.86 cons: {1} 12.43/3.86 natsFrom: {1} 12.43/3.86 s: {1} 12.43/3.86 U291: {1} 12.43/3.86 U292: {1} 12.43/3.86 U293: {1} 12.43/3.86 U294: {1} 12.43/3.86 head: {1} 12.43/3.86 afterNth: {1, 2} 12.43/3.86 U301: {1} 12.43/3.86 U302: {1} 12.43/3.86 U303: {1} 12.43/3.86 U304: {1} 12.43/3.86 U31: {1} 12.43/3.86 U32: {1} 12.43/3.86 U311: {1} 12.43/3.86 U312: {1} 12.43/3.86 pair: {1, 2} 12.43/3.86 nil: empty set 12.43/3.86 U33: {1} 12.43/3.86 U321: {1} 12.43/3.86 U322: {1} 12.43/3.86 U323: {1} 12.43/3.86 U324: {1} 12.43/3.86 U325: {1} 12.43/3.86 U326: {1} 12.43/3.86 U327: {1} 12.43/3.86 U34: {1} 12.43/3.86 U331: {1} 12.43/3.86 U332: {1} 12.43/3.86 U333: {1} 12.43/3.86 U334: {1} 12.43/3.86 U341: {1} 12.43/3.86 U342: {1} 12.43/3.86 U343: {1} 12.43/3.86 U344: {1} 12.43/3.86 fst: {1} 12.43/3.86 U41: {1} 12.43/3.86 U42: {1} 12.43/3.86 U43: {1} 12.43/3.86 U44: {1} 12.43/3.86 U45: {1} 12.43/3.86 U46: {1} 12.43/3.86 U51: {1} 12.43/3.86 U52: {1} 12.43/3.86 U53: {1} 12.43/3.86 U54: {1} 12.43/3.86 U55: {1} 12.43/3.86 U56: {1} 12.43/3.86 U61: {1} 12.43/3.86 U62: {1} 12.43/3.86 isPLNatKind: empty set 12.43/3.86 U63: {1} 12.43/3.86 isPLNat: empty set 12.43/3.86 U71: {1} 12.43/3.86 U72: {1} 12.43/3.86 U73: {1} 12.43/3.86 U81: {1} 12.43/3.86 U82: {1} 12.43/3.86 U83: {1} 12.43/3.86 U91: {1} 12.43/3.86 U92: {1} 12.43/3.86 U93: {1} 12.43/3.86 tail: {1} 12.43/3.86 take: {1, 2} 12.43/3.86 0: empty set 12.43/3.86 sel: {1, 2} 12.43/3.86 12.43/3.86 ---------------------------------------- 12.43/3.86 12.43/3.86 (1) CSDependencyPairsProof (EQUIVALENT) 12.43/3.86 Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem. 12.43/3.86 ---------------------------------------- 12.43/3.86 12.43/3.86 (2) 12.43/3.86 Obligation: 12.43/3.86 Q-restricted context-sensitive dependency pair problem: 12.43/3.86 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. 12.43/3.86 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}. 12.43/3.86 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. 12.43/3.86 12.43/3.86 The ordinary context-sensitive dependency pairs DP_o are: 12.43/3.86 U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) 12.43/3.86 U101'(tt, V1, V2) -> ISNATURALKIND(V1) 12.43/3.86 U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) 12.43/3.86 U102'(tt, V1, V2) -> ISLNATKIND(V2) 12.43/3.86 U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) 12.43/3.86 U103'(tt, V1, V2) -> ISLNATKIND(V2) 12.43/3.86 U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) 12.43/3.86 U104'(tt, V1, V2) -> ISNATURAL(V1) 12.43/3.86 U105'(tt, V2) -> U106'(isLNat(V2)) 12.43/3.86 U105'(tt, V2) -> ISLNAT(V2) 12.43/3.86 U11'(tt, N, XS) -> U12'(isNaturalKind(N), N, XS) 12.43/3.86 U11'(tt, N, XS) -> ISNATURALKIND(N) 12.43/3.86 U111'(tt, V2) -> U112'(isLNatKind(V2)) 12.43/3.86 U111'(tt, V2) -> ISLNATKIND(V2) 12.43/3.86 U12'(tt, N, XS) -> U13'(isLNat(XS), N, XS) 12.43/3.86 U12'(tt, N, XS) -> ISLNAT(XS) 12.43/3.86 U121'(tt, V2) -> U122'(isLNatKind(V2)) 12.43/3.86 U121'(tt, V2) -> ISLNATKIND(V2) 12.43/3.86 U13'(tt, N, XS) -> U14'(isLNatKind(XS), N, XS) 12.43/3.86 U13'(tt, N, XS) -> ISLNATKIND(XS) 12.43/3.86 U14'(tt, N, XS) -> SND(splitAt(N, XS)) 12.43/3.86 U14'(tt, N, XS) -> SPLITAT(N, XS) 12.43/3.86 U171'(tt, V2) -> U172'(isLNatKind(V2)) 12.43/3.86 U171'(tt, V2) -> ISLNATKIND(V2) 12.43/3.86 U181'(tt, V1) -> U182'(isLNatKind(V1), V1) 12.43/3.86 U181'(tt, V1) -> ISLNATKIND(V1) 12.43/3.86 U182'(tt, V1) -> U183'(isLNat(V1)) 12.43/3.86 U182'(tt, V1) -> ISLNAT(V1) 12.43/3.86 U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) 12.43/3.86 U191'(tt, V1) -> ISNATURALKIND(V1) 12.43/3.86 U192'(tt, V1) -> U193'(isNatural(V1)) 12.43/3.86 U192'(tt, V1) -> ISNATURAL(V1) 12.43/3.86 U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) 12.43/3.86 U201'(tt, V1, V2) -> ISNATURALKIND(V1) 12.43/3.86 U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) 12.43/3.86 U202'(tt, V1, V2) -> ISLNATKIND(V2) 12.43/3.86 U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) 12.43/3.86 U203'(tt, V1, V2) -> ISLNATKIND(V2) 12.43/3.86 U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) 12.43/3.86 U204'(tt, V1, V2) -> ISNATURAL(V1) 12.43/3.86 U205'(tt, V2) -> U206'(isLNat(V2)) 12.43/3.86 U205'(tt, V2) -> ISLNAT(V2) 12.43/3.86 U21'(tt, X, Y) -> U22'(isLNatKind(X), X, Y) 12.55/3.86 U21'(tt, X, Y) -> ISLNATKIND(X) 12.55/3.86 U22'(tt, X, Y) -> U23'(isLNat(Y), X, Y) 12.55/3.86 U22'(tt, X, Y) -> ISLNAT(Y) 12.55/3.86 U23'(tt, X, Y) -> U24'(isLNatKind(Y), X) 12.55/3.86 U23'(tt, X, Y) -> ISLNATKIND(Y) 12.55/3.86 U231'(tt, V2) -> U232'(isLNatKind(V2)) 12.55/3.86 U231'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) 12.55/3.86 U241'(tt, V1, V2) -> ISLNATKIND(V1) 12.55/3.86 U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) 12.55/3.86 U242'(tt, V1, V2) -> ISLNATKIND(V2) 12.55/3.86 U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) 12.55/3.86 U243'(tt, V1, V2) -> ISLNATKIND(V2) 12.55/3.86 U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) 12.55/3.86 U244'(tt, V1, V2) -> ISLNAT(V1) 12.55/3.86 U245'(tt, V2) -> U246'(isLNat(V2)) 12.55/3.86 U245'(tt, V2) -> ISLNAT(V2) 12.55/3.86 U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) 12.55/3.86 U251'(tt, V1, V2) -> ISNATURALKIND(V1) 12.55/3.86 U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) 12.55/3.86 U252'(tt, V1, V2) -> ISLNATKIND(V2) 12.55/3.86 U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) 12.55/3.86 U253'(tt, V1, V2) -> ISLNATKIND(V2) 12.55/3.86 U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) 12.55/3.86 U254'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U255'(tt, V2) -> U256'(isLNat(V2)) 12.55/3.86 U255'(tt, V2) -> ISLNAT(V2) 12.55/3.86 U261'(tt, V2) -> U262'(isLNatKind(V2)) 12.55/3.86 U261'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 U271'(tt, V2) -> U272'(isLNatKind(V2)) 12.55/3.86 U271'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 U281'(tt, N) -> U282'(isNaturalKind(N), N) 12.55/3.86 U281'(tt, N) -> ISNATURALKIND(N) 12.55/3.86 U291'(tt, N, XS) -> U292'(isNaturalKind(N), N, XS) 12.55/3.86 U291'(tt, N, XS) -> ISNATURALKIND(N) 12.55/3.86 U292'(tt, N, XS) -> U293'(isLNat(XS), N, XS) 12.55/3.86 U292'(tt, N, XS) -> ISLNAT(XS) 12.55/3.86 U293'(tt, N, XS) -> U294'(isLNatKind(XS), N, XS) 12.55/3.86 U293'(tt, N, XS) -> ISLNATKIND(XS) 12.55/3.86 U294'(tt, N, XS) -> HEAD(afterNth(N, XS)) 12.55/3.86 U294'(tt, N, XS) -> AFTERNTH(N, XS) 12.55/3.86 U301'(tt, X, Y) -> U302'(isLNatKind(X), Y) 12.55/3.86 U301'(tt, X, Y) -> ISLNATKIND(X) 12.55/3.86 U302'(tt, Y) -> U303'(isLNat(Y), Y) 12.55/3.86 U302'(tt, Y) -> ISLNAT(Y) 12.55/3.86 U303'(tt, Y) -> U304'(isLNatKind(Y), Y) 12.55/3.86 U303'(tt, Y) -> ISLNATKIND(Y) 12.55/3.86 U31'(tt, N, XS) -> U32'(isNaturalKind(N), N, XS) 12.55/3.86 U31'(tt, N, XS) -> ISNATURALKIND(N) 12.55/3.86 U311'(tt, XS) -> U312'(isLNatKind(XS), XS) 12.55/3.86 U311'(tt, XS) -> ISLNATKIND(XS) 12.55/3.86 U32'(tt, N, XS) -> U33'(isLNat(XS), N, XS) 12.55/3.86 U32'(tt, N, XS) -> ISLNAT(XS) 12.55/3.86 U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) 12.55/3.86 U321'(tt, N, X, XS) -> ISNATURALKIND(N) 12.55/3.86 U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) 12.55/3.86 U322'(tt, N, X, XS) -> ISNATURAL(X) 12.55/3.86 U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) 12.55/3.86 U323'(tt, N, X, XS) -> ISNATURALKIND(X) 12.55/3.86 U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) 12.55/3.86 U324'(tt, N, X, XS) -> ISLNAT(XS) 12.55/3.86 U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) 12.55/3.86 U325'(tt, N, X, XS) -> ISLNATKIND(XS) 12.55/3.86 U326'(tt, N, X, XS) -> U327'(splitAt(N, XS), X) 12.55/3.86 U326'(tt, N, X, XS) -> SPLITAT(N, XS) 12.55/3.86 U33'(tt, N, XS) -> U34'(isLNatKind(XS), N) 12.55/3.86 U33'(tt, N, XS) -> ISLNATKIND(XS) 12.55/3.86 U331'(tt, N, XS) -> U332'(isNaturalKind(N), XS) 12.55/3.86 U331'(tt, N, XS) -> ISNATURALKIND(N) 12.55/3.86 U332'(tt, XS) -> U333'(isLNat(XS), XS) 12.55/3.86 U332'(tt, XS) -> ISLNAT(XS) 12.55/3.86 U333'(tt, XS) -> U334'(isLNatKind(XS), XS) 12.55/3.86 U333'(tt, XS) -> ISLNATKIND(XS) 12.55/3.86 U341'(tt, N, XS) -> U342'(isNaturalKind(N), N, XS) 12.55/3.86 U341'(tt, N, XS) -> ISNATURALKIND(N) 12.55/3.86 U342'(tt, N, XS) -> U343'(isLNat(XS), N, XS) 12.55/3.86 U342'(tt, N, XS) -> ISLNAT(XS) 12.55/3.86 U343'(tt, N, XS) -> U344'(isLNatKind(XS), N, XS) 12.55/3.86 U343'(tt, N, XS) -> ISLNATKIND(XS) 12.55/3.86 U344'(tt, N, XS) -> FST(splitAt(N, XS)) 12.55/3.86 U344'(tt, N, XS) -> SPLITAT(N, XS) 12.55/3.86 U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) 12.55/3.86 U41'(tt, V1, V2) -> ISNATURALKIND(V1) 12.55/3.86 U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) 12.55/3.86 U42'(tt, V1, V2) -> ISLNATKIND(V2) 12.55/3.86 U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) 12.55/3.86 U43'(tt, V1, V2) -> ISLNATKIND(V2) 12.55/3.86 U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) 12.55/3.86 U44'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U45'(tt, V2) -> U46'(isLNat(V2)) 12.55/3.86 U45'(tt, V2) -> ISLNAT(V2) 12.55/3.86 U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) 12.55/3.86 U51'(tt, V1, V2) -> ISNATURALKIND(V1) 12.55/3.86 U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) 12.55/3.86 U52'(tt, V1, V2) -> ISLNATKIND(V2) 12.55/3.86 U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) 12.55/3.86 U53'(tt, V1, V2) -> ISLNATKIND(V2) 12.55/3.86 U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) 12.55/3.86 U54'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U55'(tt, V2) -> U56'(isLNat(V2)) 12.55/3.86 U55'(tt, V2) -> ISLNAT(V2) 12.55/3.86 U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) 12.55/3.86 U61'(tt, V1) -> ISPLNATKIND(V1) 12.55/3.86 U62'(tt, V1) -> U63'(isPLNat(V1)) 12.55/3.86 U62'(tt, V1) -> ISPLNAT(V1) 12.55/3.86 U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) 12.55/3.86 U71'(tt, V1) -> ISNATURALKIND(V1) 12.55/3.86 U72'(tt, V1) -> U73'(isNatural(V1)) 12.55/3.86 U72'(tt, V1) -> ISNATURAL(V1) 12.55/3.86 U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) 12.55/3.86 U81'(tt, V1) -> ISPLNATKIND(V1) 12.55/3.86 U82'(tt, V1) -> U83'(isPLNat(V1)) 12.55/3.86 U82'(tt, V1) -> ISPLNAT(V1) 12.55/3.86 U91'(tt, V1) -> U92'(isLNatKind(V1), V1) 12.55/3.86 U91'(tt, V1) -> ISLNATKIND(V1) 12.55/3.86 U92'(tt, V1) -> U93'(isLNat(V1)) 12.55/3.86 U92'(tt, V1) -> ISLNAT(V1) 12.55/3.86 AFTERNTH(N, XS) -> U11'(isNatural(N), N, XS) 12.55/3.86 AFTERNTH(N, XS) -> ISNATURAL(N) 12.55/3.86 FST(pair(X, Y)) -> U21'(isLNat(X), X, Y) 12.55/3.86 FST(pair(X, Y)) -> ISLNAT(X) 12.55/3.86 HEAD(cons(N, XS)) -> U31'(isNatural(N), N, XS) 12.55/3.86 HEAD(cons(N, XS)) -> ISNATURAL(N) 12.55/3.86 ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) 12.55/3.86 ISLNAT(afterNth(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) 12.55/3.86 ISLNAT(cons(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) 12.55/3.86 ISLNAT(fst(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) 12.55/3.86 ISLNAT(natsFrom(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) 12.55/3.86 ISLNAT(snd(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) 12.55/3.86 ISLNAT(tail(V1)) -> ISLNATKIND(V1) 12.55/3.86 ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) 12.55/3.86 ISLNAT(take(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) 12.55/3.86 ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) 12.55/3.86 ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(fst(V1)) -> U131'(isPLNatKind(V1)) 12.55/3.86 ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISLNATKIND(natsFrom(V1)) -> U141'(isNaturalKind(V1)) 12.55/3.86 ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(snd(V1)) -> U151'(isPLNatKind(V1)) 12.55/3.86 ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISLNATKIND(tail(V1)) -> U161'(isLNatKind(V1)) 12.55/3.86 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 12.55/3.86 ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) 12.55/3.86 ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) 12.55/3.86 ISNATURAL(head(V1)) -> ISLNATKIND(V1) 12.55/3.86 ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) 12.55/3.86 ISNATURAL(s(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) 12.55/3.86 ISNATURAL(sel(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(head(V1)) -> U211'(isLNatKind(V1)) 12.55/3.86 ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) 12.55/3.86 ISNATURALKIND(s(V1)) -> U221'(isNaturalKind(V1)) 12.55/3.86 ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) 12.55/3.86 ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) 12.55/3.86 ISPLNAT(pair(V1, V2)) -> ISLNATKIND(V1) 12.55/3.86 ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) 12.55/3.86 ISPLNAT(splitAt(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) 12.55/3.86 ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) 12.55/3.86 ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) 12.55/3.86 ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 NATSFROM(N) -> U281'(isNatural(N), N) 12.55/3.86 NATSFROM(N) -> ISNATURAL(N) 12.55/3.86 SEL(N, XS) -> U291'(isNatural(N), N, XS) 12.55/3.86 SEL(N, XS) -> ISNATURAL(N) 12.55/3.86 SND(pair(X, Y)) -> U301'(isLNat(X), X, Y) 12.55/3.86 SND(pair(X, Y)) -> ISLNAT(X) 12.55/3.86 SPLITAT(0, XS) -> U311'(isLNat(XS), XS) 12.55/3.86 SPLITAT(0, XS) -> ISLNAT(XS) 12.55/3.86 SPLITAT(s(N), cons(X, XS)) -> U321'(isNatural(N), N, X, XS) 12.55/3.86 SPLITAT(s(N), cons(X, XS)) -> ISNATURAL(N) 12.55/3.86 TAIL(cons(N, XS)) -> U331'(isNatural(N), N, XS) 12.55/3.86 TAIL(cons(N, XS)) -> ISNATURAL(N) 12.55/3.86 TAKE(N, XS) -> U341'(isNatural(N), N, XS) 12.55/3.86 TAKE(N, XS) -> ISNATURAL(N) 12.55/3.86 12.55/3.86 The collapsing dependency pairs are DP_c: 12.55/3.86 U14'(tt, N, XS) -> N 12.55/3.86 U14'(tt, N, XS) -> XS 12.55/3.86 U24'(tt, X) -> X 12.55/3.86 U282'(tt, N) -> N 12.55/3.86 U294'(tt, N, XS) -> N 12.55/3.86 U294'(tt, N, XS) -> XS 12.55/3.86 U304'(tt, Y) -> Y 12.55/3.86 U312'(tt, XS) -> XS 12.55/3.86 U326'(tt, N, X, XS) -> N 12.55/3.86 U326'(tt, N, X, XS) -> XS 12.55/3.86 U327'(pair(YS, ZS), X) -> X 12.55/3.86 U334'(tt, XS) -> XS 12.55/3.86 U34'(tt, N) -> N 12.55/3.86 U344'(tt, N, XS) -> N 12.55/3.86 U344'(tt, N, XS) -> XS 12.55/3.86 12.55/3.86 12.55/3.86 The hidden terms of R are: 12.55/3.86 12.55/3.86 natsFrom(s(x0)) 12.55/3.86 12.55/3.86 Every hiding context is built from: 12.55/3.86 aprove.DPFramework.CSDPProblem.QCSDPProblem$1@74126c51 12.55/3.86 aprove.DPFramework.CSDPProblem.QCSDPProblem$1@27bcb0b7 12.55/3.86 12.55/3.86 Hence, the new unhiding pairs DP_u are : 12.55/3.86 U14'(tt, N, XS) -> U(N) 12.55/3.86 U14'(tt, N, XS) -> U(XS) 12.55/3.86 U24'(tt, X) -> U(X) 12.55/3.86 U282'(tt, N) -> U(N) 12.55/3.86 U294'(tt, N, XS) -> U(N) 12.55/3.86 U294'(tt, N, XS) -> U(XS) 12.55/3.86 U304'(tt, Y) -> U(Y) 12.55/3.86 U312'(tt, XS) -> U(XS) 12.55/3.86 U326'(tt, N, X, XS) -> U(N) 12.55/3.86 U326'(tt, N, X, XS) -> U(XS) 12.55/3.86 U327'(pair(YS, ZS), X) -> U(X) 12.55/3.86 U334'(tt, XS) -> U(XS) 12.55/3.86 U34'(tt, N) -> U(N) 12.55/3.86 U344'(tt, N, XS) -> U(N) 12.55/3.86 U344'(tt, N, XS) -> U(XS) 12.55/3.86 U(s(x_0)) -> U(x_0) 12.55/3.86 U(natsFrom(x_0)) -> U(x_0) 12.55/3.86 U(natsFrom(s(x0))) -> NATSFROM(s(x0)) 12.55/3.86 12.55/3.86 The TRS R consists of the following rules: 12.55/3.86 12.55/3.86 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.55/3.86 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.55/3.86 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.55/3.86 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.55/3.86 U105(tt, V2) -> U106(isLNat(V2)) 12.55/3.86 U106(tt) -> tt 12.55/3.86 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 12.55/3.86 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.86 U112(tt) -> tt 12.55/3.86 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 12.55/3.86 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.86 U122(tt) -> tt 12.55/3.86 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 12.55/3.86 U131(tt) -> tt 12.55/3.86 U14(tt, N, XS) -> snd(splitAt(N, XS)) 12.55/3.86 U141(tt) -> tt 12.55/3.86 U151(tt) -> tt 12.55/3.86 U161(tt) -> tt 12.55/3.86 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.86 U172(tt) -> tt 12.55/3.86 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.55/3.86 U182(tt, V1) -> U183(isLNat(V1)) 12.55/3.86 U183(tt) -> tt 12.55/3.86 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.55/3.86 U192(tt, V1) -> U193(isNatural(V1)) 12.55/3.86 U193(tt) -> tt 12.55/3.86 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.55/3.86 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.55/3.86 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.55/3.86 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.55/3.86 U205(tt, V2) -> U206(isLNat(V2)) 12.55/3.86 U206(tt) -> tt 12.55/3.86 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 12.55/3.86 U211(tt) -> tt 12.55/3.86 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 12.55/3.86 U221(tt) -> tt 12.55/3.86 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 12.55/3.86 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.86 U232(tt) -> tt 12.55/3.86 U24(tt, X) -> X 12.55/3.86 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.55/3.86 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.55/3.86 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.55/3.86 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.55/3.86 U245(tt, V2) -> U246(isLNat(V2)) 12.55/3.86 U246(tt) -> tt 12.55/3.86 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.55/3.86 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.55/3.86 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.55/3.86 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.55/3.86 U255(tt, V2) -> U256(isLNat(V2)) 12.55/3.86 U256(tt) -> tt 12.55/3.86 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.86 U262(tt) -> tt 12.55/3.86 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.86 U272(tt) -> tt 12.55/3.86 U281(tt, N) -> U282(isNaturalKind(N), N) 12.55/3.86 U282(tt, N) -> cons(N, natsFrom(s(N))) 12.55/3.86 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 12.55/3.86 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 12.55/3.86 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 12.55/3.86 U294(tt, N, XS) -> head(afterNth(N, XS)) 12.55/3.86 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 12.55/3.86 U302(tt, Y) -> U303(isLNat(Y), Y) 12.55/3.86 U303(tt, Y) -> U304(isLNatKind(Y), Y) 12.55/3.86 U304(tt, Y) -> Y 12.55/3.86 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 12.55/3.86 U311(tt, XS) -> U312(isLNatKind(XS), XS) 12.55/3.86 U312(tt, XS) -> pair(nil, XS) 12.55/3.86 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 12.55/3.86 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 12.55/3.86 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 12.55/3.86 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 12.55/3.86 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 12.55/3.86 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 12.55/3.86 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 12.55/3.86 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 12.55/3.86 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 12.55/3.86 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 12.55/3.86 U332(tt, XS) -> U333(isLNat(XS), XS) 12.55/3.86 U333(tt, XS) -> U334(isLNatKind(XS), XS) 12.55/3.86 U334(tt, XS) -> XS 12.55/3.86 U34(tt, N) -> N 12.55/3.86 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 12.55/3.86 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 12.55/3.86 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 12.55/3.86 U344(tt, N, XS) -> fst(splitAt(N, XS)) 12.55/3.86 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.55/3.86 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.55/3.86 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.55/3.86 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.55/3.86 U45(tt, V2) -> U46(isLNat(V2)) 12.55/3.86 U46(tt) -> tt 12.55/3.86 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.55/3.86 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.55/3.86 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.55/3.86 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.55/3.86 U55(tt, V2) -> U56(isLNat(V2)) 12.55/3.86 U56(tt) -> tt 12.55/3.86 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.55/3.86 U62(tt, V1) -> U63(isPLNat(V1)) 12.55/3.86 U63(tt) -> tt 12.55/3.86 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.55/3.86 U72(tt, V1) -> U73(isNatural(V1)) 12.55/3.86 U73(tt) -> tt 12.55/3.86 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.55/3.86 U82(tt, V1) -> U83(isPLNat(V1)) 12.55/3.86 U83(tt) -> tt 12.55/3.86 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.55/3.86 U92(tt, V1) -> U93(isLNat(V1)) 12.55/3.86 U93(tt) -> tt 12.55/3.86 afterNth(N, XS) -> U11(isNatural(N), N, XS) 12.55/3.86 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 12.55/3.86 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 12.55/3.86 isLNat(nil) -> tt 12.55/3.86 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.55/3.86 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.55/3.86 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.55/3.86 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.55/3.86 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.55/3.86 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.55/3.86 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.55/3.86 isLNatKind(nil) -> tt 12.55/3.86 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.86 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.86 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.86 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.86 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.86 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.86 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.86 isNatural(0) -> tt 12.55/3.86 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.55/3.86 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.55/3.86 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.55/3.86 isNaturalKind(0) -> tt 12.55/3.86 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.86 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.86 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.86 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.55/3.86 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.55/3.86 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.86 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.86 natsFrom(N) -> U281(isNatural(N), N) 12.55/3.86 sel(N, XS) -> U291(isNatural(N), N, XS) 12.55/3.86 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 12.55/3.86 splitAt(0, XS) -> U311(isLNat(XS), XS) 12.55/3.86 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 12.55/3.86 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 12.55/3.86 take(N, XS) -> U341(isNatural(N), N, XS) 12.55/3.86 12.55/3.86 Q is empty. 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (3) QCSDependencyGraphProof (EQUIVALENT) 12.55/3.86 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 149 less nodes. 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (4) 12.55/3.86 Complex Obligation (AND) 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (5) 12.55/3.86 Obligation: 12.55/3.86 Q-restricted context-sensitive dependency pair problem: 12.55/3.86 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. 12.55/3.86 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}. 12.55/3.86 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. 12.55/3.86 12.55/3.86 The TRS P consists of the following rules: 12.55/3.86 12.55/3.86 ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) 12.55/3.86 ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) 12.55/3.86 U111'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) 12.55/3.86 U231'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) 12.55/3.86 U121'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) 12.55/3.86 U261'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) 12.55/3.86 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 12.55/3.86 ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) 12.55/3.86 U171'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) 12.55/3.86 U271'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 12.55/3.86 The TRS R consists of the following rules: 12.55/3.86 12.55/3.86 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.55/3.86 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.55/3.86 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.55/3.86 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.55/3.86 U105(tt, V2) -> U106(isLNat(V2)) 12.55/3.86 U106(tt) -> tt 12.55/3.86 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 12.55/3.86 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.86 U112(tt) -> tt 12.55/3.86 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 12.55/3.86 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.86 U122(tt) -> tt 12.55/3.86 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 12.55/3.86 U131(tt) -> tt 12.55/3.86 U14(tt, N, XS) -> snd(splitAt(N, XS)) 12.55/3.86 U141(tt) -> tt 12.55/3.86 U151(tt) -> tt 12.55/3.86 U161(tt) -> tt 12.55/3.86 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.86 U172(tt) -> tt 12.55/3.86 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.55/3.86 U182(tt, V1) -> U183(isLNat(V1)) 12.55/3.86 U183(tt) -> tt 12.55/3.86 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.55/3.86 U192(tt, V1) -> U193(isNatural(V1)) 12.55/3.86 U193(tt) -> tt 12.55/3.86 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.55/3.86 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.55/3.86 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.55/3.86 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.55/3.86 U205(tt, V2) -> U206(isLNat(V2)) 12.55/3.86 U206(tt) -> tt 12.55/3.86 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 12.55/3.86 U211(tt) -> tt 12.55/3.86 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 12.55/3.86 U221(tt) -> tt 12.55/3.86 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 12.55/3.86 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.86 U232(tt) -> tt 12.55/3.86 U24(tt, X) -> X 12.55/3.86 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.55/3.86 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.55/3.86 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.55/3.86 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.55/3.86 U245(tt, V2) -> U246(isLNat(V2)) 12.55/3.86 U246(tt) -> tt 12.55/3.86 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.55/3.86 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.55/3.86 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.55/3.86 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.55/3.86 U255(tt, V2) -> U256(isLNat(V2)) 12.55/3.86 U256(tt) -> tt 12.55/3.86 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.86 U262(tt) -> tt 12.55/3.86 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.86 U272(tt) -> tt 12.55/3.86 U281(tt, N) -> U282(isNaturalKind(N), N) 12.55/3.86 U282(tt, N) -> cons(N, natsFrom(s(N))) 12.55/3.86 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 12.55/3.86 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 12.55/3.86 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 12.55/3.86 U294(tt, N, XS) -> head(afterNth(N, XS)) 12.55/3.86 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 12.55/3.86 U302(tt, Y) -> U303(isLNat(Y), Y) 12.55/3.86 U303(tt, Y) -> U304(isLNatKind(Y), Y) 12.55/3.86 U304(tt, Y) -> Y 12.55/3.86 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 12.55/3.86 U311(tt, XS) -> U312(isLNatKind(XS), XS) 12.55/3.86 U312(tt, XS) -> pair(nil, XS) 12.55/3.86 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 12.55/3.86 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 12.55/3.86 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 12.55/3.86 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 12.55/3.86 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 12.55/3.86 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 12.55/3.86 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 12.55/3.86 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 12.55/3.86 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 12.55/3.86 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 12.55/3.86 U332(tt, XS) -> U333(isLNat(XS), XS) 12.55/3.86 U333(tt, XS) -> U334(isLNatKind(XS), XS) 12.55/3.86 U334(tt, XS) -> XS 12.55/3.86 U34(tt, N) -> N 12.55/3.86 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 12.55/3.86 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 12.55/3.86 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 12.55/3.86 U344(tt, N, XS) -> fst(splitAt(N, XS)) 12.55/3.86 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.55/3.86 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.55/3.86 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.55/3.86 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.55/3.86 U45(tt, V2) -> U46(isLNat(V2)) 12.55/3.86 U46(tt) -> tt 12.55/3.86 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.55/3.86 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.55/3.86 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.55/3.86 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.55/3.86 U55(tt, V2) -> U56(isLNat(V2)) 12.55/3.86 U56(tt) -> tt 12.55/3.86 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.55/3.86 U62(tt, V1) -> U63(isPLNat(V1)) 12.55/3.86 U63(tt) -> tt 12.55/3.86 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.55/3.86 U72(tt, V1) -> U73(isNatural(V1)) 12.55/3.86 U73(tt) -> tt 12.55/3.86 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.55/3.86 U82(tt, V1) -> U83(isPLNat(V1)) 12.55/3.86 U83(tt) -> tt 12.55/3.86 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.55/3.86 U92(tt, V1) -> U93(isLNat(V1)) 12.55/3.86 U93(tt) -> tt 12.55/3.86 afterNth(N, XS) -> U11(isNatural(N), N, XS) 12.55/3.86 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 12.55/3.86 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 12.55/3.86 isLNat(nil) -> tt 12.55/3.86 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.55/3.86 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.55/3.86 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.55/3.86 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.55/3.86 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.55/3.86 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.55/3.86 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.55/3.86 isLNatKind(nil) -> tt 12.55/3.86 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.86 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.86 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.86 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.86 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.86 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.86 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.86 isNatural(0) -> tt 12.55/3.86 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.55/3.86 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.55/3.86 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.55/3.86 isNaturalKind(0) -> tt 12.55/3.86 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.86 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.86 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.86 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.55/3.86 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.55/3.86 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.86 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.86 natsFrom(N) -> U281(isNatural(N), N) 12.55/3.86 sel(N, XS) -> U291(isNatural(N), N, XS) 12.55/3.86 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 12.55/3.86 splitAt(0, XS) -> U311(isLNat(XS), XS) 12.55/3.86 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 12.55/3.86 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 12.55/3.86 take(N, XS) -> U341(isNatural(N), N, XS) 12.55/3.86 12.55/3.86 Q is empty. 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (6) QCSUsableRulesProof (EQUIVALENT) 12.55/3.86 The following rules are not useable [DA_EMMES] and can be deleted: 12.55/3.86 12.55/3.86 U101(tt, x0, x1) -> U102(isNaturalKind(x0), x0, x1) 12.55/3.86 U102(tt, x0, x1) -> U103(isLNatKind(x1), x0, x1) 12.55/3.86 U103(tt, x0, x1) -> U104(isLNatKind(x1), x0, x1) 12.55/3.86 U104(tt, x0, x1) -> U105(isNatural(x0), x1) 12.55/3.86 U105(tt, x0) -> U106(isLNat(x0)) 12.55/3.86 U106(tt) -> tt 12.55/3.86 U11(tt, x0, x1) -> U12(isNaturalKind(x0), x0, x1) 12.55/3.86 U12(tt, x0, x1) -> U13(isLNat(x1), x0, x1) 12.55/3.86 U13(tt, x0, x1) -> U14(isLNatKind(x1), x0, x1) 12.55/3.86 U14(tt, x0, x1) -> snd(splitAt(x0, x1)) 12.55/3.86 U181(tt, x0) -> U182(isLNatKind(x0), x0) 12.55/3.86 U182(tt, x0) -> U183(isLNat(x0)) 12.55/3.86 U183(tt) -> tt 12.55/3.86 U191(tt, x0) -> U192(isNaturalKind(x0), x0) 12.55/3.86 U192(tt, x0) -> U193(isNatural(x0)) 12.55/3.86 U193(tt) -> tt 12.55/3.86 U201(tt, x0, x1) -> U202(isNaturalKind(x0), x0, x1) 12.55/3.86 U202(tt, x0, x1) -> U203(isLNatKind(x1), x0, x1) 12.55/3.86 U203(tt, x0, x1) -> U204(isLNatKind(x1), x0, x1) 12.55/3.86 U204(tt, x0, x1) -> U205(isNatural(x0), x1) 12.55/3.86 U205(tt, x0) -> U206(isLNat(x0)) 12.55/3.86 U206(tt) -> tt 12.55/3.86 U21(tt, x0, x1) -> U22(isLNatKind(x0), x0, x1) 12.55/3.86 U22(tt, x0, x1) -> U23(isLNat(x1), x0, x1) 12.55/3.86 U23(tt, x0, x1) -> U24(isLNatKind(x1), x0) 12.55/3.86 U24(tt, x0) -> x0 12.55/3.86 U241(tt, x0, x1) -> U242(isLNatKind(x0), x0, x1) 12.55/3.86 U242(tt, x0, x1) -> U243(isLNatKind(x1), x0, x1) 12.55/3.86 U243(tt, x0, x1) -> U244(isLNatKind(x1), x0, x1) 12.55/3.86 U244(tt, x0, x1) -> U245(isLNat(x0), x1) 12.55/3.86 U245(tt, x0) -> U246(isLNat(x0)) 12.55/3.86 U246(tt) -> tt 12.55/3.86 U251(tt, x0, x1) -> U252(isNaturalKind(x0), x0, x1) 12.55/3.86 U252(tt, x0, x1) -> U253(isLNatKind(x1), x0, x1) 12.55/3.86 U253(tt, x0, x1) -> U254(isLNatKind(x1), x0, x1) 12.55/3.86 U254(tt, x0, x1) -> U255(isNatural(x0), x1) 12.55/3.86 U255(tt, x0) -> U256(isLNat(x0)) 12.55/3.86 U256(tt) -> tt 12.55/3.86 U281(tt, x0) -> U282(isNaturalKind(x0), x0) 12.55/3.86 U282(tt, x0) -> cons(x0, natsFrom(s(x0))) 12.55/3.86 U291(tt, x0, x1) -> U292(isNaturalKind(x0), x0, x1) 12.55/3.86 U292(tt, x0, x1) -> U293(isLNat(x1), x0, x1) 12.55/3.86 U293(tt, x0, x1) -> U294(isLNatKind(x1), x0, x1) 12.55/3.86 U294(tt, x0, x1) -> head(afterNth(x0, x1)) 12.55/3.86 U301(tt, x0, x1) -> U302(isLNatKind(x0), x1) 12.55/3.86 U302(tt, x0) -> U303(isLNat(x0), x0) 12.55/3.86 U303(tt, x0) -> U304(isLNatKind(x0), x0) 12.55/3.86 U304(tt, x0) -> x0 12.55/3.86 U31(tt, x0, x1) -> U32(isNaturalKind(x0), x0, x1) 12.55/3.86 U311(tt, x0) -> U312(isLNatKind(x0), x0) 12.55/3.86 U312(tt, x0) -> pair(nil, x0) 12.55/3.86 U32(tt, x0, x1) -> U33(isLNat(x1), x0, x1) 12.55/3.86 U321(tt, x0, x1, x2) -> U322(isNaturalKind(x0), x0, x1, x2) 12.55/3.86 U322(tt, x0, x1, x2) -> U323(isNatural(x1), x0, x1, x2) 12.55/3.86 U323(tt, x0, x1, x2) -> U324(isNaturalKind(x1), x0, x1, x2) 12.55/3.86 U324(tt, x0, x1, x2) -> U325(isLNat(x2), x0, x1, x2) 12.55/3.86 U325(tt, x0, x1, x2) -> U326(isLNatKind(x2), x0, x1, x2) 12.55/3.86 U326(tt, x0, x1, x2) -> U327(splitAt(x0, x2), x1) 12.55/3.86 U327(pair(x0, x1), x2) -> pair(cons(x2, x0), x1) 12.55/3.86 U33(tt, x0, x1) -> U34(isLNatKind(x1), x0) 12.55/3.86 U331(tt, x0, x1) -> U332(isNaturalKind(x0), x1) 12.55/3.86 U332(tt, x0) -> U333(isLNat(x0), x0) 12.55/3.86 U333(tt, x0) -> U334(isLNatKind(x0), x0) 12.55/3.86 U334(tt, x0) -> x0 12.55/3.86 U34(tt, x0) -> x0 12.55/3.86 U341(tt, x0, x1) -> U342(isNaturalKind(x0), x0, x1) 12.55/3.86 U342(tt, x0, x1) -> U343(isLNat(x1), x0, x1) 12.55/3.86 U343(tt, x0, x1) -> U344(isLNatKind(x1), x0, x1) 12.55/3.86 U344(tt, x0, x1) -> fst(splitAt(x0, x1)) 12.55/3.86 U41(tt, x0, x1) -> U42(isNaturalKind(x0), x0, x1) 12.55/3.86 U42(tt, x0, x1) -> U43(isLNatKind(x1), x0, x1) 12.55/3.86 U43(tt, x0, x1) -> U44(isLNatKind(x1), x0, x1) 12.55/3.86 U44(tt, x0, x1) -> U45(isNatural(x0), x1) 12.55/3.86 U45(tt, x0) -> U46(isLNat(x0)) 12.55/3.86 U46(tt) -> tt 12.55/3.86 U51(tt, x0, x1) -> U52(isNaturalKind(x0), x0, x1) 12.55/3.86 U52(tt, x0, x1) -> U53(isLNatKind(x1), x0, x1) 12.55/3.86 U53(tt, x0, x1) -> U54(isLNatKind(x1), x0, x1) 12.55/3.86 U54(tt, x0, x1) -> U55(isNatural(x0), x1) 12.55/3.86 U55(tt, x0) -> U56(isLNat(x0)) 12.55/3.86 U56(tt) -> tt 12.55/3.86 U61(tt, x0) -> U62(isPLNatKind(x0), x0) 12.55/3.86 U62(tt, x0) -> U63(isPLNat(x0)) 12.55/3.86 U63(tt) -> tt 12.55/3.86 U71(tt, x0) -> U72(isNaturalKind(x0), x0) 12.55/3.86 U72(tt, x0) -> U73(isNatural(x0)) 12.55/3.86 U73(tt) -> tt 12.55/3.86 U81(tt, x0) -> U82(isPLNatKind(x0), x0) 12.55/3.86 U82(tt, x0) -> U83(isPLNat(x0)) 12.55/3.86 U83(tt) -> tt 12.55/3.86 U91(tt, x0) -> U92(isLNatKind(x0), x0) 12.55/3.86 U92(tt, x0) -> U93(isLNat(x0)) 12.55/3.86 U93(tt) -> tt 12.55/3.86 afterNth(x0, x1) -> U11(isNatural(x0), x0, x1) 12.55/3.86 fst(pair(x0, x1)) -> U21(isLNat(x0), x0, x1) 12.55/3.86 head(cons(x0, x1)) -> U31(isNatural(x0), x0, x1) 12.55/3.86 isLNat(nil) -> tt 12.55/3.86 isLNat(afterNth(x0, x1)) -> U41(isNaturalKind(x0), x0, x1) 12.55/3.86 isLNat(cons(x0, x1)) -> U51(isNaturalKind(x0), x0, x1) 12.55/3.86 isLNat(fst(x0)) -> U61(isPLNatKind(x0), x0) 12.55/3.86 isLNat(natsFrom(x0)) -> U71(isNaturalKind(x0), x0) 12.55/3.86 isLNat(snd(x0)) -> U81(isPLNatKind(x0), x0) 12.55/3.86 isLNat(tail(x0)) -> U91(isLNatKind(x0), x0) 12.55/3.86 isLNat(take(x0, x1)) -> U101(isNaturalKind(x0), x0, x1) 12.55/3.86 isNatural(0) -> tt 12.55/3.86 isNatural(head(x0)) -> U181(isLNatKind(x0), x0) 12.55/3.86 isNatural(s(x0)) -> U191(isNaturalKind(x0), x0) 12.55/3.86 isNatural(sel(x0, x1)) -> U201(isNaturalKind(x0), x0, x1) 12.55/3.86 isPLNat(pair(x0, x1)) -> U241(isLNatKind(x0), x0, x1) 12.55/3.86 isPLNat(splitAt(x0, x1)) -> U251(isNaturalKind(x0), x0, x1) 12.55/3.86 natsFrom(x0) -> U281(isNatural(x0), x0) 12.55/3.86 sel(x0, x1) -> U291(isNatural(x0), x0, x1) 12.55/3.86 snd(pair(x0, x1)) -> U301(isLNat(x0), x0, x1) 12.55/3.86 splitAt(0, x0) -> U311(isLNat(x0), x0) 12.55/3.86 splitAt(s(x0), cons(x1, x2)) -> U321(isNatural(x0), x0, x1, x2) 12.55/3.86 tail(cons(x0, x1)) -> U331(isNatural(x0), x0, x1) 12.55/3.86 take(x0, x1) -> U341(isNatural(x0), x0, x1) 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (7) 12.55/3.86 Obligation: 12.55/3.86 Q-restricted context-sensitive dependency pair problem: 12.55/3.86 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. 12.55/3.86 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}. 12.55/3.86 The symbols in {isNaturalKind_1, isLNatKind_1, isPLNatKind_1, ISLNATKIND_1, ISNATURALKIND_1, ISPLNATKIND_1} are not replacing on any position. 12.55/3.86 12.55/3.86 The TRS P consists of the following rules: 12.55/3.86 12.55/3.86 ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) 12.55/3.86 ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) 12.55/3.86 U111'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) 12.55/3.86 U231'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) 12.55/3.86 U121'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) 12.55/3.86 U261'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) 12.55/3.86 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 12.55/3.86 ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) 12.55/3.86 U171'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) 12.55/3.86 U271'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 12.55/3.86 The TRS R consists of the following rules: 12.55/3.86 12.55/3.86 isNaturalKind(0) -> tt 12.55/3.86 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.86 isLNatKind(nil) -> tt 12.55/3.86 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.86 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.86 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.86 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.86 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.86 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.86 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.86 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.86 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.86 U141(tt) -> tt 12.55/3.86 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.86 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.86 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.86 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.86 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.86 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.86 U172(tt) -> tt 12.55/3.86 U161(tt) -> tt 12.55/3.86 U272(tt) -> tt 12.55/3.86 U151(tt) -> tt 12.55/3.86 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.86 U262(tt) -> tt 12.55/3.86 U131(tt) -> tt 12.55/3.86 U122(tt) -> tt 12.55/3.86 U232(tt) -> tt 12.55/3.86 U221(tt) -> tt 12.55/3.86 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.86 U112(tt) -> tt 12.55/3.86 U211(tt) -> tt 12.55/3.86 12.55/3.86 Q is empty. 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (8) QCSDPMuMonotonicPoloProof (EQUIVALENT) 12.55/3.86 By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted. 12.55/3.86 12.55/3.86 Strictly oriented dependency pairs: 12.55/3.86 12.55/3.86 ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) 12.55/3.86 ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) 12.55/3.86 U111'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) 12.55/3.86 U231'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) 12.55/3.86 U121'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) 12.55/3.86 U261'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) 12.55/3.86 ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) 12.55/3.86 ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) 12.55/3.86 ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) 12.55/3.86 U171'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) 12.55/3.86 U271'(tt, V2) -> ISLNATKIND(V2) 12.55/3.86 ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) 12.55/3.86 12.55/3.86 Strictly oriented rules of the TRS R: 12.55/3.86 12.55/3.86 isNaturalKind(0) -> tt 12.55/3.86 isLNatKind(nil) -> tt 12.55/3.86 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.86 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.86 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.86 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.86 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.86 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.86 U141(tt) -> tt 12.55/3.86 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.86 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.86 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.86 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.86 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.86 U172(tt) -> tt 12.55/3.86 U272(tt) -> tt 12.55/3.86 U151(tt) -> tt 12.55/3.86 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.86 U262(tt) -> tt 12.55/3.86 U131(tt) -> tt 12.55/3.86 U122(tt) -> tt 12.55/3.86 U232(tt) -> tt 12.55/3.86 U221(tt) -> tt 12.55/3.86 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.86 U112(tt) -> tt 12.55/3.86 12.55/3.86 Used ordering: POLO with Polynomial interpretation [POLO]: 12.55/3.86 12.55/3.86 POL(0) = 2 12.55/3.86 POL(ISLNATKIND(x_1)) = 1 + 2*x_1 12.55/3.86 POL(ISNATURALKIND(x_1)) = 2 + x_1 12.55/3.86 POL(ISPLNATKIND(x_1)) = 1 + 2*x_1 12.55/3.86 POL(U111(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.86 POL(U111'(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.86 POL(U112(x_1)) = 1 + x_1 12.55/3.86 POL(U121(x_1, x_2)) = 1 + x_1 + 2*x_2 12.55/3.86 POL(U121'(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.86 POL(U122(x_1)) = 1 + x_1 12.55/3.86 POL(U131(x_1)) = 1 + x_1 12.55/3.86 POL(U141(x_1)) = 1 + x_1 12.55/3.86 POL(U151(x_1)) = 1 + 2*x_1 12.55/3.86 POL(U161(x_1)) = x_1 12.55/3.86 POL(U171(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.86 POL(U171'(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.86 POL(U172(x_1)) = 1 + x_1 12.55/3.86 POL(U211(x_1)) = x_1 12.55/3.86 POL(U221(x_1)) = 1 + 2*x_1 12.55/3.86 POL(U231(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.86 POL(U231'(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.86 POL(U232(x_1)) = 1 + x_1 12.55/3.86 POL(U261(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.86 POL(U261'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 12.55/3.86 POL(U262(x_1)) = 1 + x_1 12.55/3.86 POL(U271(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.86 POL(U271'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 12.55/3.86 POL(U272(x_1)) = 1 + x_1 12.55/3.86 POL(afterNth(x_1, x_2)) = 2 + 2*x_1 + x_2 12.55/3.86 POL(cons(x_1, x_2)) = 2 + 2*x_1 + x_2 12.55/3.86 POL(fst(x_1)) = 1 + 2*x_1 12.55/3.86 POL(head(x_1)) = 2*x_1 12.55/3.86 POL(isLNatKind(x_1)) = 2*x_1 12.55/3.86 POL(isNaturalKind(x_1)) = 2*x_1 12.55/3.86 POL(isPLNatKind(x_1)) = 1 + 2*x_1 12.55/3.86 POL(natsFrom(x_1)) = 2 + 2*x_1 12.55/3.86 POL(nil) = 1 12.55/3.86 POL(pair(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 12.55/3.86 POL(s(x_1)) = 1 + 2*x_1 12.55/3.86 POL(sel(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 12.55/3.86 POL(snd(x_1)) = 2 + 2*x_1 12.55/3.86 POL(splitAt(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 12.55/3.86 POL(tail(x_1)) = x_1 12.55/3.86 POL(take(x_1, x_2)) = 2 + 2*x_1 + x_2 12.55/3.86 POL(tt) = 0 12.55/3.86 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (9) 12.55/3.86 Obligation: 12.55/3.86 Q-restricted context-sensitive dependency pair problem: 12.55/3.86 The symbols in {head_1, U211_1, sel_2, U122_1, fst_1, U131_1, tail_1, U161_1} are replacing on all positions. 12.55/3.86 For all symbols f in {U231_2, U121_2} we have mu(f) = {1}. 12.55/3.86 The symbols in {isNaturalKind_1, isLNatKind_1, isPLNatKind_1, ISLNATKIND_1} are not replacing on any position. 12.55/3.86 12.55/3.86 The TRS P consists of the following rules: 12.55/3.86 12.55/3.86 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 12.55/3.86 12.55/3.86 The TRS R consists of the following rules: 12.55/3.86 12.55/3.86 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.86 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.86 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.86 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.86 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.86 U161(tt) -> tt 12.55/3.86 U211(tt) -> tt 12.55/3.86 12.55/3.86 Q is empty. 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (10) QCSDPSubtermProof (EQUIVALENT) 12.55/3.86 We use the subterm processor [DA_EMMES]. 12.55/3.86 12.55/3.86 12.55/3.86 The following pairs can be oriented strictly and are deleted. 12.55/3.86 12.55/3.86 ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) 12.55/3.86 The remaining pairs can at least be oriented weakly. 12.55/3.86 none 12.55/3.86 Used ordering: Combined order from the following AFS and order. 12.55/3.86 ISLNATKIND(x1) = x1 12.55/3.86 12.55/3.86 12.55/3.86 Subterm Order 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (11) 12.55/3.86 Obligation: 12.55/3.86 Q-restricted context-sensitive dependency pair problem: 12.55/3.86 The symbols in {head_1, U211_1, sel_2, U122_1, fst_1, U131_1, tail_1, U161_1} are replacing on all positions. 12.55/3.86 For all symbols f in {U231_2, U121_2} we have mu(f) = {1}. 12.55/3.86 The symbols in {isNaturalKind_1, isLNatKind_1, isPLNatKind_1} are not replacing on any position. 12.55/3.86 12.55/3.86 The TRS P consists of the following rules: 12.55/3.86 none 12.55/3.86 12.55/3.86 The TRS R consists of the following rules: 12.55/3.86 12.55/3.86 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.86 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.86 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.86 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.86 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.86 U161(tt) -> tt 12.55/3.86 U211(tt) -> tt 12.55/3.86 12.55/3.86 Q is empty. 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (12) PIsEmptyProof (EQUIVALENT) 12.55/3.86 The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain. 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (13) 12.55/3.86 YES 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (14) 12.55/3.86 Obligation: 12.55/3.86 Q-restricted context-sensitive dependency pair problem: 12.55/3.86 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. 12.55/3.86 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}. 12.55/3.86 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. 12.55/3.86 12.55/3.86 The TRS P consists of the following rules: 12.55/3.86 12.55/3.86 U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) 12.55/3.86 U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) 12.55/3.86 U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) 12.55/3.86 U105'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) 12.55/3.86 U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) 12.55/3.86 U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) 12.55/3.86 U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) 12.55/3.86 U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) 12.55/3.86 U45'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) 12.55/3.86 U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) 12.55/3.86 U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) 12.55/3.86 U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) 12.55/3.86 U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) 12.55/3.86 U55'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) 12.55/3.86 U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) 12.55/3.86 U62'(tt, V1) -> ISPLNAT(V1) 12.55/3.86 ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) 12.55/3.86 U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) 12.55/3.86 U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) 12.55/3.86 U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) 12.55/3.86 U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) 12.55/3.86 U245'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) 12.55/3.86 U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) 12.55/3.86 U72'(tt, V1) -> ISNATURAL(V1) 12.55/3.86 ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) 12.55/3.86 U181'(tt, V1) -> U182'(isLNatKind(V1), V1) 12.55/3.86 U182'(tt, V1) -> ISLNAT(V1) 12.55/3.86 ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) 12.55/3.86 U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) 12.55/3.86 U82'(tt, V1) -> ISPLNAT(V1) 12.55/3.86 ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) 12.55/3.86 U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) 12.55/3.86 U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) 12.55/3.86 U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) 12.55/3.86 U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) 12.55/3.86 U255'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) 12.55/3.86 U91'(tt, V1) -> U92'(isLNatKind(V1), V1) 12.55/3.86 U92'(tt, V1) -> ISLNAT(V1) 12.55/3.86 ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) 12.55/3.86 U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) 12.55/3.86 U254'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) 12.55/3.86 U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) 12.55/3.86 U192'(tt, V1) -> ISNATURAL(V1) 12.55/3.86 ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) 12.55/3.86 U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) 12.55/3.86 U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) 12.55/3.86 U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) 12.55/3.86 U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) 12.55/3.86 U205'(tt, V2) -> ISLNAT(V2) 12.55/3.86 U204'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U244'(tt, V1, V2) -> ISLNAT(V1) 12.55/3.86 U54'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U44'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U104'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 12.55/3.86 The TRS R consists of the following rules: 12.55/3.86 12.55/3.86 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.55/3.86 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.55/3.86 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.55/3.86 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.55/3.86 U105(tt, V2) -> U106(isLNat(V2)) 12.55/3.86 U106(tt) -> tt 12.55/3.86 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 12.55/3.86 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.86 U112(tt) -> tt 12.55/3.86 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 12.55/3.86 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.86 U122(tt) -> tt 12.55/3.86 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 12.55/3.86 U131(tt) -> tt 12.55/3.86 U14(tt, N, XS) -> snd(splitAt(N, XS)) 12.55/3.86 U141(tt) -> tt 12.55/3.86 U151(tt) -> tt 12.55/3.86 U161(tt) -> tt 12.55/3.86 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.86 U172(tt) -> tt 12.55/3.86 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.55/3.86 U182(tt, V1) -> U183(isLNat(V1)) 12.55/3.86 U183(tt) -> tt 12.55/3.86 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.55/3.86 U192(tt, V1) -> U193(isNatural(V1)) 12.55/3.86 U193(tt) -> tt 12.55/3.86 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.55/3.86 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.55/3.86 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.55/3.86 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.55/3.86 U205(tt, V2) -> U206(isLNat(V2)) 12.55/3.86 U206(tt) -> tt 12.55/3.86 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 12.55/3.86 U211(tt) -> tt 12.55/3.86 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 12.55/3.86 U221(tt) -> tt 12.55/3.86 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 12.55/3.86 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.86 U232(tt) -> tt 12.55/3.86 U24(tt, X) -> X 12.55/3.86 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.55/3.86 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.55/3.86 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.55/3.86 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.55/3.86 U245(tt, V2) -> U246(isLNat(V2)) 12.55/3.86 U246(tt) -> tt 12.55/3.86 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.55/3.86 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.55/3.86 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.55/3.86 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.55/3.86 U255(tt, V2) -> U256(isLNat(V2)) 12.55/3.86 U256(tt) -> tt 12.55/3.86 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.86 U262(tt) -> tt 12.55/3.86 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.86 U272(tt) -> tt 12.55/3.86 U281(tt, N) -> U282(isNaturalKind(N), N) 12.55/3.86 U282(tt, N) -> cons(N, natsFrom(s(N))) 12.55/3.86 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 12.55/3.86 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 12.55/3.86 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 12.55/3.86 U294(tt, N, XS) -> head(afterNth(N, XS)) 12.55/3.86 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 12.55/3.86 U302(tt, Y) -> U303(isLNat(Y), Y) 12.55/3.86 U303(tt, Y) -> U304(isLNatKind(Y), Y) 12.55/3.86 U304(tt, Y) -> Y 12.55/3.86 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 12.55/3.86 U311(tt, XS) -> U312(isLNatKind(XS), XS) 12.55/3.86 U312(tt, XS) -> pair(nil, XS) 12.55/3.86 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 12.55/3.86 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 12.55/3.86 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 12.55/3.86 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 12.55/3.86 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 12.55/3.86 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 12.55/3.86 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 12.55/3.86 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 12.55/3.86 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 12.55/3.86 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 12.55/3.86 U332(tt, XS) -> U333(isLNat(XS), XS) 12.55/3.86 U333(tt, XS) -> U334(isLNatKind(XS), XS) 12.55/3.86 U334(tt, XS) -> XS 12.55/3.86 U34(tt, N) -> N 12.55/3.86 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 12.55/3.86 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 12.55/3.86 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 12.55/3.86 U344(tt, N, XS) -> fst(splitAt(N, XS)) 12.55/3.86 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.55/3.86 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.55/3.86 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.55/3.86 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.55/3.86 U45(tt, V2) -> U46(isLNat(V2)) 12.55/3.86 U46(tt) -> tt 12.55/3.86 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.55/3.86 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.55/3.86 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.55/3.86 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.55/3.86 U55(tt, V2) -> U56(isLNat(V2)) 12.55/3.86 U56(tt) -> tt 12.55/3.86 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.55/3.86 U62(tt, V1) -> U63(isPLNat(V1)) 12.55/3.86 U63(tt) -> tt 12.55/3.86 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.55/3.86 U72(tt, V1) -> U73(isNatural(V1)) 12.55/3.86 U73(tt) -> tt 12.55/3.86 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.55/3.86 U82(tt, V1) -> U83(isPLNat(V1)) 12.55/3.86 U83(tt) -> tt 12.55/3.86 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.55/3.86 U92(tt, V1) -> U93(isLNat(V1)) 12.55/3.86 U93(tt) -> tt 12.55/3.86 afterNth(N, XS) -> U11(isNatural(N), N, XS) 12.55/3.86 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 12.55/3.86 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 12.55/3.86 isLNat(nil) -> tt 12.55/3.86 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.55/3.86 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.55/3.86 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.55/3.86 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.55/3.86 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.55/3.86 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.55/3.86 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.55/3.86 isLNatKind(nil) -> tt 12.55/3.86 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.86 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.86 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.86 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.86 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.86 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.86 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.86 isNatural(0) -> tt 12.55/3.86 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.55/3.86 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.55/3.86 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.55/3.86 isNaturalKind(0) -> tt 12.55/3.86 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.86 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.86 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.86 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.55/3.86 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.55/3.86 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.86 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.86 natsFrom(N) -> U281(isNatural(N), N) 12.55/3.86 sel(N, XS) -> U291(isNatural(N), N, XS) 12.55/3.86 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 12.55/3.86 splitAt(0, XS) -> U311(isLNat(XS), XS) 12.55/3.86 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 12.55/3.86 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 12.55/3.86 take(N, XS) -> U341(isNatural(N), N, XS) 12.55/3.86 12.55/3.86 Q is empty. 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (15) QCSUsableRulesProof (EQUIVALENT) 12.55/3.86 The following rules are not useable [DA_EMMES] and can be deleted: 12.55/3.86 12.55/3.86 U11(tt, x0, x1) -> U12(isNaturalKind(x0), x0, x1) 12.55/3.86 U12(tt, x0, x1) -> U13(isLNat(x1), x0, x1) 12.55/3.86 U13(tt, x0, x1) -> U14(isLNatKind(x1), x0, x1) 12.55/3.86 U14(tt, x0, x1) -> snd(splitAt(x0, x1)) 12.55/3.86 U21(tt, x0, x1) -> U22(isLNatKind(x0), x0, x1) 12.55/3.86 U22(tt, x0, x1) -> U23(isLNat(x1), x0, x1) 12.55/3.86 U23(tt, x0, x1) -> U24(isLNatKind(x1), x0) 12.55/3.86 U24(tt, x0) -> x0 12.55/3.86 U281(tt, x0) -> U282(isNaturalKind(x0), x0) 12.55/3.86 U282(tt, x0) -> cons(x0, natsFrom(s(x0))) 12.55/3.86 U291(tt, x0, x1) -> U292(isNaturalKind(x0), x0, x1) 12.55/3.86 U292(tt, x0, x1) -> U293(isLNat(x1), x0, x1) 12.55/3.86 U293(tt, x0, x1) -> U294(isLNatKind(x1), x0, x1) 12.55/3.86 U294(tt, x0, x1) -> head(afterNth(x0, x1)) 12.55/3.86 U301(tt, x0, x1) -> U302(isLNatKind(x0), x1) 12.55/3.86 U302(tt, x0) -> U303(isLNat(x0), x0) 12.55/3.86 U303(tt, x0) -> U304(isLNatKind(x0), x0) 12.55/3.86 U304(tt, x0) -> x0 12.55/3.86 U31(tt, x0, x1) -> U32(isNaturalKind(x0), x0, x1) 12.55/3.86 U311(tt, x0) -> U312(isLNatKind(x0), x0) 12.55/3.86 U312(tt, x0) -> pair(nil, x0) 12.55/3.86 U32(tt, x0, x1) -> U33(isLNat(x1), x0, x1) 12.55/3.86 U321(tt, x0, x1, x2) -> U322(isNaturalKind(x0), x0, x1, x2) 12.55/3.86 U322(tt, x0, x1, x2) -> U323(isNatural(x1), x0, x1, x2) 12.55/3.86 U323(tt, x0, x1, x2) -> U324(isNaturalKind(x1), x0, x1, x2) 12.55/3.86 U324(tt, x0, x1, x2) -> U325(isLNat(x2), x0, x1, x2) 12.55/3.86 U325(tt, x0, x1, x2) -> U326(isLNatKind(x2), x0, x1, x2) 12.55/3.86 U326(tt, x0, x1, x2) -> U327(splitAt(x0, x2), x1) 12.55/3.86 U327(pair(x0, x1), x2) -> pair(cons(x2, x0), x1) 12.55/3.86 U33(tt, x0, x1) -> U34(isLNatKind(x1), x0) 12.55/3.86 U331(tt, x0, x1) -> U332(isNaturalKind(x0), x1) 12.55/3.86 U332(tt, x0) -> U333(isLNat(x0), x0) 12.55/3.86 U333(tt, x0) -> U334(isLNatKind(x0), x0) 12.55/3.86 U334(tt, x0) -> x0 12.55/3.86 U34(tt, x0) -> x0 12.55/3.86 U341(tt, x0, x1) -> U342(isNaturalKind(x0), x0, x1) 12.55/3.86 U342(tt, x0, x1) -> U343(isLNat(x1), x0, x1) 12.55/3.86 U343(tt, x0, x1) -> U344(isLNatKind(x1), x0, x1) 12.55/3.86 U344(tt, x0, x1) -> fst(splitAt(x0, x1)) 12.55/3.86 afterNth(x0, x1) -> U11(isNatural(x0), x0, x1) 12.55/3.86 fst(pair(x0, x1)) -> U21(isLNat(x0), x0, x1) 12.55/3.86 head(cons(x0, x1)) -> U31(isNatural(x0), x0, x1) 12.55/3.86 natsFrom(x0) -> U281(isNatural(x0), x0) 12.55/3.86 sel(x0, x1) -> U291(isNatural(x0), x0, x1) 12.55/3.86 snd(pair(x0, x1)) -> U301(isLNat(x0), x0, x1) 12.55/3.86 splitAt(0, x0) -> U311(isLNat(x0), x0) 12.55/3.86 splitAt(s(x0), cons(x1, x2)) -> U321(isNatural(x0), x0, x1, x2) 12.55/3.86 tail(cons(x0, x1)) -> U331(isNatural(x0), x0, x1) 12.55/3.86 take(x0, x1) -> U341(isNatural(x0), x0, x1) 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (16) 12.55/3.86 Obligation: 12.55/3.86 Q-restricted context-sensitive dependency pair problem: 12.55/3.86 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. 12.55/3.86 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}. 12.55/3.86 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. 12.55/3.86 12.55/3.86 The TRS P consists of the following rules: 12.55/3.86 12.55/3.86 U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) 12.55/3.86 U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) 12.55/3.86 U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) 12.55/3.86 U105'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) 12.55/3.86 U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) 12.55/3.86 U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) 12.55/3.86 U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) 12.55/3.86 U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) 12.55/3.86 U45'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) 12.55/3.86 U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) 12.55/3.86 U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) 12.55/3.86 U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) 12.55/3.86 U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) 12.55/3.86 U55'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) 12.55/3.86 U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) 12.55/3.86 U62'(tt, V1) -> ISPLNAT(V1) 12.55/3.86 ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) 12.55/3.86 U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) 12.55/3.86 U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) 12.55/3.86 U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) 12.55/3.86 U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) 12.55/3.86 U245'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) 12.55/3.86 U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) 12.55/3.86 U72'(tt, V1) -> ISNATURAL(V1) 12.55/3.86 ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) 12.55/3.86 U181'(tt, V1) -> U182'(isLNatKind(V1), V1) 12.55/3.86 U182'(tt, V1) -> ISLNAT(V1) 12.55/3.86 ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) 12.55/3.86 U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) 12.55/3.86 U82'(tt, V1) -> ISPLNAT(V1) 12.55/3.86 ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) 12.55/3.86 U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) 12.55/3.86 U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) 12.55/3.86 U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) 12.55/3.86 U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) 12.55/3.86 U255'(tt, V2) -> ISLNAT(V2) 12.55/3.86 ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) 12.55/3.86 U91'(tt, V1) -> U92'(isLNatKind(V1), V1) 12.55/3.86 U92'(tt, V1) -> ISLNAT(V1) 12.55/3.86 ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) 12.55/3.86 U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) 12.55/3.86 U254'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) 12.55/3.86 U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) 12.55/3.86 U192'(tt, V1) -> ISNATURAL(V1) 12.55/3.86 ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) 12.55/3.86 U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) 12.55/3.86 U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) 12.55/3.86 U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) 12.55/3.86 U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) 12.55/3.86 U205'(tt, V2) -> ISLNAT(V2) 12.55/3.86 U204'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U244'(tt, V1, V2) -> ISLNAT(V1) 12.55/3.86 U54'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U44'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U104'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 12.55/3.86 The TRS R consists of the following rules: 12.55/3.86 12.55/3.86 isLNatKind(nil) -> tt 12.55/3.86 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.86 isNaturalKind(0) -> tt 12.55/3.86 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.86 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.86 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.86 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.86 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.86 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.86 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.86 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.86 U141(tt) -> tt 12.55/3.86 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.86 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.86 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.86 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.86 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.86 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.86 U172(tt) -> tt 12.55/3.86 U161(tt) -> tt 12.55/3.86 U272(tt) -> tt 12.55/3.86 U151(tt) -> tt 12.55/3.86 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.86 U262(tt) -> tt 12.55/3.86 U131(tt) -> tt 12.55/3.86 U232(tt) -> tt 12.55/3.86 U221(tt) -> tt 12.55/3.86 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.86 U122(tt) -> tt 12.55/3.86 U211(tt) -> tt 12.55/3.86 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.86 U112(tt) -> tt 12.55/3.86 isNatural(0) -> tt 12.55/3.86 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.55/3.86 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.55/3.86 U182(tt, V1) -> U183(isLNat(V1)) 12.55/3.86 isLNat(nil) -> tt 12.55/3.86 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.55/3.86 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.55/3.86 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.55/3.86 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.55/3.86 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.55/3.86 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.55/3.86 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.55/3.86 U192(tt, V1) -> U193(isNatural(V1)) 12.55/3.86 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.55/3.86 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.55/3.86 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.55/3.86 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.55/3.86 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.55/3.86 U205(tt, V2) -> U206(isLNat(V2)) 12.55/3.86 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.55/3.86 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.55/3.86 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.55/3.86 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.55/3.86 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.55/3.86 U55(tt, V2) -> U56(isLNat(V2)) 12.55/3.86 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.55/3.86 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.55/3.86 U62(tt, V1) -> U63(isPLNat(V1)) 12.55/3.86 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.55/3.86 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.55/3.86 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.55/3.86 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.55/3.86 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.55/3.86 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.55/3.86 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.55/3.86 U72(tt, V1) -> U73(isNatural(V1)) 12.55/3.86 U73(tt) -> tt 12.55/3.86 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.55/3.86 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.55/3.86 U82(tt, V1) -> U83(isPLNat(V1)) 12.55/3.86 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.55/3.86 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.55/3.86 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.55/3.86 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.55/3.86 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.55/3.86 U255(tt, V2) -> U256(isLNat(V2)) 12.55/3.86 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.55/3.86 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.55/3.86 U92(tt, V1) -> U93(isLNat(V1)) 12.55/3.86 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.55/3.86 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.55/3.86 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.55/3.86 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.55/3.86 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.55/3.86 U105(tt, V2) -> U106(isLNat(V2)) 12.55/3.86 U106(tt) -> tt 12.55/3.86 U93(tt) -> tt 12.55/3.86 U256(tt) -> tt 12.55/3.86 U83(tt) -> tt 12.55/3.86 U245(tt, V2) -> U246(isLNat(V2)) 12.55/3.86 U246(tt) -> tt 12.55/3.86 U63(tt) -> tt 12.55/3.86 U56(tt) -> tt 12.55/3.86 U206(tt) -> tt 12.55/3.86 U193(tt) -> tt 12.55/3.86 U45(tt, V2) -> U46(isLNat(V2)) 12.55/3.86 U46(tt) -> tt 12.55/3.86 U183(tt) -> tt 12.55/3.86 12.55/3.86 Q is empty. 12.55/3.86 12.55/3.86 ---------------------------------------- 12.55/3.86 12.55/3.86 (17) QCSDPMuMonotonicPoloProof (EQUIVALENT) 12.55/3.86 By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted. 12.55/3.86 12.55/3.86 Strictly oriented dependency pairs: 12.55/3.86 12.55/3.86 U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) 12.55/3.86 U105'(tt, V2) -> ISLNAT(V2) 12.55/3.86 U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) 12.55/3.86 U45'(tt, V2) -> ISLNAT(V2) 12.55/3.86 U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) 12.55/3.86 U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) 12.55/3.86 ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) 12.55/3.86 U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) 12.55/3.86 ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) 12.55/3.86 U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) 12.55/3.86 U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) 12.55/3.86 U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) 12.55/3.86 U72'(tt, V1) -> ISNATURAL(V1) 12.55/3.86 U181'(tt, V1) -> U182'(isLNatKind(V1), V1) 12.55/3.86 U182'(tt, V1) -> ISLNAT(V1) 12.55/3.86 U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) 12.55/3.86 U82'(tt, V1) -> ISPLNAT(V1) 12.55/3.86 ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) 12.55/3.86 U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) 12.55/3.86 U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) 12.55/3.86 ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) 12.55/3.86 U91'(tt, V1) -> U92'(isLNatKind(V1), V1) 12.55/3.86 U254'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) 12.55/3.86 U192'(tt, V1) -> ISNATURAL(V1) 12.55/3.86 ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) 12.55/3.86 U44'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 U104'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.86 12.55/3.86 Strictly oriented rules of the TRS R: 12.55/3.86 12.55/3.86 isNatural(0) -> tt 12.55/3.86 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.55/3.86 U182(tt, V1) -> U183(isLNat(V1)) 12.55/3.86 isLNat(nil) -> tt 12.55/3.86 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.55/3.86 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.55/3.86 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.55/3.86 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.55/3.87 U192(tt, V1) -> U193(isNatural(V1)) 12.55/3.87 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.55/3.87 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.55/3.87 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.55/3.87 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.55/3.87 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.55/3.87 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.55/3.87 U73(tt) -> tt 12.55/3.87 U82(tt, V1) -> U83(isPLNat(V1)) 12.55/3.87 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.55/3.87 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.55/3.87 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.55/3.87 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.55/3.87 U93(tt) -> tt 12.55/3.87 U83(tt) -> tt 12.55/3.87 U245(tt, V2) -> U246(isLNat(V2)) 12.55/3.87 U63(tt) -> tt 12.55/3.87 U183(tt) -> tt 12.55/3.87 12.55/3.87 Used ordering: POLO with Polynomial interpretation [POLO]: 12.55/3.87 12.55/3.87 POL(0) = 2 12.55/3.87 POL(ISLNAT(x_1)) = x_1 12.55/3.87 POL(ISNATURAL(x_1)) = x_1 12.55/3.87 POL(ISPLNAT(x_1)) = 2*x_1 12.55/3.87 POL(U101(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U101'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U102(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 12.55/3.87 POL(U102'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U103(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U103'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U104(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U104'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U105(x_1, x_2)) = x_1 + x_2 12.55/3.87 POL(U105'(x_1, x_2)) = 1 + x_1 + 2*x_2 12.55/3.87 POL(U106(x_1)) = x_1 12.55/3.87 POL(U111(x_1, x_2)) = 2*x_1 12.55/3.87 POL(U112(x_1)) = 2*x_1 12.55/3.87 POL(U121(x_1, x_2)) = 2*x_1 12.55/3.87 POL(U122(x_1)) = x_1 12.55/3.87 POL(U131(x_1)) = 2*x_1 12.55/3.87 POL(U141(x_1)) = 2*x_1 12.55/3.87 POL(U151(x_1)) = 2*x_1 12.55/3.87 POL(U161(x_1)) = 2*x_1 12.55/3.87 POL(U171(x_1, x_2)) = x_1 12.55/3.87 POL(U172(x_1)) = 2*x_1 12.55/3.87 POL(U181(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.87 POL(U181'(x_1, x_2)) = 2 + 2*x_1 + x_2 12.55/3.87 POL(U182(x_1, x_2)) = 2 + 2*x_1 + x_2 12.55/3.87 POL(U182'(x_1, x_2)) = 1 + 2*x_1 + x_2 12.55/3.87 POL(U183(x_1)) = 1 + x_1 12.55/3.87 POL(U191(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.87 POL(U191'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 12.55/3.87 POL(U192(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 12.55/3.87 POL(U192'(x_1, x_2)) = 1 + x_1 + 2*x_2 12.55/3.87 POL(U193(x_1)) = x_1 12.55/3.87 POL(U201(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U201'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U202(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U202'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U203(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U203'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U204(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 12.55/3.87 POL(U204'(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 12.55/3.87 POL(U205(x_1, x_2)) = x_1 + x_2 12.55/3.87 POL(U205'(x_1, x_2)) = x_1 + x_2 12.55/3.87 POL(U206(x_1)) = x_1 12.55/3.87 POL(U211(x_1)) = 2*x_1 12.55/3.87 POL(U221(x_1)) = 2*x_1 12.55/3.87 POL(U231(x_1, x_2)) = x_1 12.55/3.87 POL(U232(x_1)) = 2*x_1 12.55/3.87 POL(U241(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3 12.55/3.87 POL(U241'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U242(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3 12.55/3.87 POL(U242'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U243(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3 12.55/3.87 POL(U243'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U244(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3 12.55/3.87 POL(U244'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U245(x_1, x_2)) = 1 + x_1 + x_2 12.55/3.87 POL(U245'(x_1, x_2)) = 2*x_1 + x_2 12.55/3.87 POL(U246(x_1)) = x_1 12.55/3.87 POL(U251(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 12.55/3.87 POL(U251'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U252(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 12.55/3.87 POL(U252'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U253(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 12.55/3.87 POL(U253'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U254(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U254'(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 12.55/3.87 POL(U255(x_1, x_2)) = x_1 + x_2 12.55/3.87 POL(U255'(x_1, x_2)) = x_1 + x_2 12.55/3.87 POL(U256(x_1)) = x_1 12.55/3.87 POL(U261(x_1, x_2)) = x_1 12.55/3.87 POL(U262(x_1)) = x_1 12.55/3.87 POL(U271(x_1, x_2)) = 2*x_1 12.55/3.87 POL(U272(x_1)) = x_1 12.55/3.87 POL(U41(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U41'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U42(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U42'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U43(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U43'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U44(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U44'(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 12.55/3.87 POL(U45(x_1, x_2)) = x_1 + 2*x_2 12.55/3.87 POL(U45'(x_1, x_2)) = 1 + x_1 + x_2 12.55/3.87 POL(U46(x_1)) = 2*x_1 12.55/3.87 POL(U51(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U51'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U52(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U52'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U53(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U53'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U54(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 12.55/3.87 POL(U54'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 12.55/3.87 POL(U55(x_1, x_2)) = x_1 + 2*x_2 12.55/3.87 POL(U55'(x_1, x_2)) = x_1 + x_2 12.55/3.87 POL(U56(x_1)) = 2*x_1 12.55/3.87 POL(U61(x_1, x_2)) = 1 + 2*x_1 + x_2 12.55/3.87 POL(U61'(x_1, x_2)) = 1 + x_1 + 2*x_2 12.55/3.87 POL(U62(x_1, x_2)) = 1 + 2*x_1 + x_2 12.55/3.87 POL(U62'(x_1, x_2)) = 2*x_1 + 2*x_2 12.55/3.87 POL(U63(x_1)) = 1 + x_1 12.55/3.87 POL(U71(x_1, x_2)) = 2 + x_1 + 2*x_2 12.55/3.87 POL(U71'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 12.55/3.87 POL(U72(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 12.55/3.87 POL(U72'(x_1, x_2)) = 1 + x_1 + 2*x_2 12.55/3.87 POL(U73(x_1)) = 1 + x_1 12.55/3.87 POL(U81(x_1, x_2)) = 2 + x_1 + x_2 12.55/3.87 POL(U81'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 12.55/3.87 POL(U82(x_1, x_2)) = 2 + 2*x_1 + x_2 12.55/3.87 POL(U82'(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 12.55/3.87 POL(U83(x_1)) = 1 + x_1 12.55/3.87 POL(U91(x_1, x_2)) = 1 + 2*x_1 + x_2 12.55/3.87 POL(U91'(x_1, x_2)) = 1 + x_1 + 2*x_2 12.55/3.87 POL(U92(x_1, x_2)) = 1 + 2*x_1 + x_2 12.55/3.87 POL(U92'(x_1, x_2)) = x_1 + 2*x_2 12.55/3.87 POL(U93(x_1)) = 1 + x_1 12.55/3.87 POL(afterNth(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 12.55/3.87 POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 12.55/3.87 POL(fst(x_1)) = 2 + 2*x_1 12.55/3.87 POL(head(x_1)) = 2 + 2*x_1 12.55/3.87 POL(isLNat(x_1)) = x_1 12.55/3.87 POL(isLNatKind(x_1)) = 0 12.55/3.87 POL(isNatural(x_1)) = 2*x_1 12.55/3.87 POL(isNaturalKind(x_1)) = 0 12.55/3.87 POL(isPLNat(x_1)) = x_1 12.55/3.87 POL(isPLNatKind(x_1)) = 0 12.55/3.87 POL(natsFrom(x_1)) = 2 + 2*x_1 12.55/3.87 POL(nil) = 2 12.55/3.87 POL(pair(x_1, x_2)) = 2 + 2*x_1 + x_2 12.55/3.87 POL(s(x_1)) = 2 + 2*x_1 12.55/3.87 POL(sel(x_1, x_2)) = 1 + 2*x_1 + x_2 12.55/3.87 POL(snd(x_1)) = 2 + 2*x_1 12.55/3.87 POL(splitAt(x_1, x_2)) = 2 + 2*x_1 + x_2 12.55/3.87 POL(tail(x_1)) = 2 + 2*x_1 12.55/3.87 POL(take(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 12.55/3.87 POL(tt) = 0 12.55/3.87 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (18) 12.55/3.87 Obligation: 12.55/3.87 Q-restricted context-sensitive dependency pair problem: 12.55/3.87 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. 12.55/3.87 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}. 12.55/3.87 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. 12.55/3.87 12.55/3.87 The TRS P consists of the following rules: 12.55/3.87 12.55/3.87 U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) 12.55/3.87 U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) 12.55/3.87 ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) 12.55/3.87 U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) 12.55/3.87 U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) 12.55/3.87 U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) 12.55/3.87 ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) 12.55/3.87 U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) 12.55/3.87 U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) 12.55/3.87 U55'(tt, V2) -> ISLNAT(V2) 12.55/3.87 U62'(tt, V1) -> ISPLNAT(V1) 12.55/3.87 U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) 12.55/3.87 U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) 12.55/3.87 U245'(tt, V2) -> ISLNAT(V2) 12.55/3.87 ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) 12.55/3.87 ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) 12.55/3.87 ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) 12.55/3.87 U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) 12.55/3.87 U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) 12.55/3.87 U255'(tt, V2) -> ISLNAT(V2) 12.55/3.87 U92'(tt, V1) -> ISLNAT(V1) 12.55/3.87 ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) 12.55/3.87 U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) 12.55/3.87 ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) 12.55/3.87 U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) 12.55/3.87 U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) 12.55/3.87 U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) 12.55/3.87 U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) 12.55/3.87 U205'(tt, V2) -> ISLNAT(V2) 12.55/3.87 U204'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.87 U244'(tt, V1, V2) -> ISLNAT(V1) 12.55/3.87 U54'(tt, V1, V2) -> ISNATURAL(V1) 12.55/3.87 12.55/3.87 The TRS R consists of the following rules: 12.55/3.87 12.55/3.87 isLNatKind(nil) -> tt 12.55/3.87 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.87 isNaturalKind(0) -> tt 12.55/3.87 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.87 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.87 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.87 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.87 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.87 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.87 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.87 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.87 U141(tt) -> tt 12.55/3.87 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.87 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.87 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.87 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.87 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.87 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.87 U172(tt) -> tt 12.55/3.87 U161(tt) -> tt 12.55/3.87 U272(tt) -> tt 12.55/3.87 U151(tt) -> tt 12.55/3.87 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.87 U262(tt) -> tt 12.55/3.87 U131(tt) -> tt 12.55/3.87 U232(tt) -> tt 12.55/3.87 U221(tt) -> tt 12.55/3.87 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.87 U122(tt) -> tt 12.55/3.87 U211(tt) -> tt 12.55/3.87 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.87 U112(tt) -> tt 12.55/3.87 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.55/3.87 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.55/3.87 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.55/3.87 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.55/3.87 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.55/3.87 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.55/3.87 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.55/3.87 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.55/3.87 U205(tt, V2) -> U206(isLNat(V2)) 12.55/3.87 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.55/3.87 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.55/3.87 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.55/3.87 U55(tt, V2) -> U56(isLNat(V2)) 12.55/3.87 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.55/3.87 U62(tt, V1) -> U63(isPLNat(V1)) 12.55/3.87 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.55/3.87 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.55/3.87 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.55/3.87 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.55/3.87 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.55/3.87 U72(tt, V1) -> U73(isNatural(V1)) 12.55/3.87 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.55/3.87 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.55/3.87 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.55/3.87 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.55/3.87 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.55/3.87 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.55/3.87 U255(tt, V2) -> U256(isLNat(V2)) 12.55/3.87 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.55/3.87 U92(tt, V1) -> U93(isLNat(V1)) 12.55/3.87 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.55/3.87 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.55/3.87 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.55/3.87 U105(tt, V2) -> U106(isLNat(V2)) 12.55/3.87 U106(tt) -> tt 12.55/3.87 U256(tt) -> tt 12.55/3.87 U246(tt) -> tt 12.55/3.87 U56(tt) -> tt 12.55/3.87 U206(tt) -> tt 12.55/3.87 U193(tt) -> tt 12.55/3.87 U45(tt, V2) -> U46(isLNat(V2)) 12.55/3.87 U46(tt) -> tt 12.55/3.87 12.55/3.87 Q is empty. 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (19) QCSDependencyGraphProof (EQUIVALENT) 12.55/3.87 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 32 less nodes. 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (20) 12.55/3.87 TRUE 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (21) 12.55/3.87 Obligation: 12.55/3.87 Q-restricted context-sensitive dependency pair problem: 12.55/3.87 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. 12.55/3.87 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}. 12.55/3.87 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1, U_1} are not replacing on any position. 12.55/3.87 12.55/3.87 The TRS P consists of the following rules: 12.55/3.87 12.55/3.87 NATSFROM(N) -> U281'(isNatural(N), N) 12.55/3.87 U281'(tt, N) -> U282'(isNaturalKind(N), N) 12.55/3.87 U282'(tt, N) -> U(N) 12.55/3.87 U(s(x_0)) -> U(x_0) 12.55/3.87 U(natsFrom(x_0)) -> U(x_0) 12.55/3.87 U(natsFrom(s(x0))) -> NATSFROM(s(x0)) 12.55/3.87 12.55/3.87 The TRS R consists of the following rules: 12.55/3.87 12.55/3.87 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.55/3.87 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.55/3.87 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.55/3.87 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.55/3.87 U105(tt, V2) -> U106(isLNat(V2)) 12.55/3.87 U106(tt) -> tt 12.55/3.87 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 12.55/3.87 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.87 U112(tt) -> tt 12.55/3.87 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 12.55/3.87 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.87 U122(tt) -> tt 12.55/3.87 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 12.55/3.87 U131(tt) -> tt 12.55/3.87 U14(tt, N, XS) -> snd(splitAt(N, XS)) 12.55/3.87 U141(tt) -> tt 12.55/3.87 U151(tt) -> tt 12.55/3.87 U161(tt) -> tt 12.55/3.87 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.87 U172(tt) -> tt 12.55/3.87 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.55/3.87 U182(tt, V1) -> U183(isLNat(V1)) 12.55/3.87 U183(tt) -> tt 12.55/3.87 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.55/3.87 U192(tt, V1) -> U193(isNatural(V1)) 12.55/3.87 U193(tt) -> tt 12.55/3.87 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.55/3.87 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.55/3.87 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.55/3.87 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.55/3.87 U205(tt, V2) -> U206(isLNat(V2)) 12.55/3.87 U206(tt) -> tt 12.55/3.87 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 12.55/3.87 U211(tt) -> tt 12.55/3.87 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 12.55/3.87 U221(tt) -> tt 12.55/3.87 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 12.55/3.87 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.87 U232(tt) -> tt 12.55/3.87 U24(tt, X) -> X 12.55/3.87 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.55/3.87 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.55/3.87 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.55/3.87 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.55/3.87 U245(tt, V2) -> U246(isLNat(V2)) 12.55/3.87 U246(tt) -> tt 12.55/3.87 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.55/3.87 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.55/3.87 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.55/3.87 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.55/3.87 U255(tt, V2) -> U256(isLNat(V2)) 12.55/3.87 U256(tt) -> tt 12.55/3.87 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.87 U262(tt) -> tt 12.55/3.87 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.87 U272(tt) -> tt 12.55/3.87 U281(tt, N) -> U282(isNaturalKind(N), N) 12.55/3.87 U282(tt, N) -> cons(N, natsFrom(s(N))) 12.55/3.87 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 12.55/3.87 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 12.55/3.87 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 12.55/3.87 U294(tt, N, XS) -> head(afterNth(N, XS)) 12.55/3.87 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 12.55/3.87 U302(tt, Y) -> U303(isLNat(Y), Y) 12.55/3.87 U303(tt, Y) -> U304(isLNatKind(Y), Y) 12.55/3.87 U304(tt, Y) -> Y 12.55/3.87 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 12.55/3.87 U311(tt, XS) -> U312(isLNatKind(XS), XS) 12.55/3.87 U312(tt, XS) -> pair(nil, XS) 12.55/3.87 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 12.55/3.87 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 12.55/3.87 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 12.55/3.87 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 12.55/3.87 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 12.55/3.87 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 12.55/3.87 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 12.55/3.87 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 12.55/3.87 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 12.55/3.87 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 12.55/3.87 U332(tt, XS) -> U333(isLNat(XS), XS) 12.55/3.87 U333(tt, XS) -> U334(isLNatKind(XS), XS) 12.55/3.87 U334(tt, XS) -> XS 12.55/3.87 U34(tt, N) -> N 12.55/3.87 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 12.55/3.87 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 12.55/3.87 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 12.55/3.87 U344(tt, N, XS) -> fst(splitAt(N, XS)) 12.55/3.87 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.55/3.87 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.55/3.87 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.55/3.87 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.55/3.87 U45(tt, V2) -> U46(isLNat(V2)) 12.55/3.87 U46(tt) -> tt 12.55/3.87 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.55/3.87 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.55/3.87 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.55/3.87 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.55/3.87 U55(tt, V2) -> U56(isLNat(V2)) 12.55/3.87 U56(tt) -> tt 12.55/3.87 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.55/3.87 U62(tt, V1) -> U63(isPLNat(V1)) 12.55/3.87 U63(tt) -> tt 12.55/3.87 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.55/3.87 U72(tt, V1) -> U73(isNatural(V1)) 12.55/3.87 U73(tt) -> tt 12.55/3.87 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.55/3.87 U82(tt, V1) -> U83(isPLNat(V1)) 12.55/3.87 U83(tt) -> tt 12.55/3.87 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.55/3.87 U92(tt, V1) -> U93(isLNat(V1)) 12.55/3.87 U93(tt) -> tt 12.55/3.87 afterNth(N, XS) -> U11(isNatural(N), N, XS) 12.55/3.87 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 12.55/3.87 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 12.55/3.87 isLNat(nil) -> tt 12.55/3.87 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.55/3.87 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.55/3.87 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.55/3.87 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.55/3.87 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNatKind(nil) -> tt 12.55/3.87 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.87 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.87 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.87 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.87 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.87 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.87 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.87 isNatural(0) -> tt 12.55/3.87 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.55/3.87 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.55/3.87 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.55/3.87 isNaturalKind(0) -> tt 12.55/3.87 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.87 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.87 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.87 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.55/3.87 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.55/3.87 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.87 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.87 natsFrom(N) -> U281(isNatural(N), N) 12.55/3.87 sel(N, XS) -> U291(isNatural(N), N, XS) 12.55/3.87 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 12.55/3.87 splitAt(0, XS) -> U311(isLNat(XS), XS) 12.55/3.87 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 12.55/3.87 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 12.55/3.87 take(N, XS) -> U341(isNatural(N), N, XS) 12.55/3.87 12.55/3.87 Q is empty. 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (22) QCSDPSubtermProof (EQUIVALENT) 12.55/3.87 We use the subterm processor [DA_EMMES]. 12.55/3.87 12.55/3.87 12.55/3.87 The following pairs can be oriented strictly and are deleted. 12.55/3.87 12.55/3.87 U(s(x_0)) -> U(x_0) 12.55/3.87 U(natsFrom(x_0)) -> U(x_0) 12.55/3.87 U(natsFrom(s(x0))) -> NATSFROM(s(x0)) 12.55/3.87 The remaining pairs can at least be oriented weakly. 12.55/3.87 12.55/3.87 NATSFROM(N) -> U281'(isNatural(N), N) 12.55/3.87 U281'(tt, N) -> U282'(isNaturalKind(N), N) 12.55/3.87 U282'(tt, N) -> U(N) 12.55/3.87 Used ordering: Combined order from the following AFS and order. 12.55/3.87 U281'(x1, x2) = x2 12.55/3.87 12.55/3.87 NATSFROM(x1) = x1 12.55/3.87 12.55/3.87 U282'(x1, x2) = x2 12.55/3.87 12.55/3.87 U(x1) = x1 12.55/3.87 12.55/3.87 12.55/3.87 Subterm Order 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (23) 12.55/3.87 Obligation: 12.55/3.87 Q-restricted context-sensitive dependency pair problem: 12.55/3.87 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. 12.55/3.87 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}. 12.55/3.87 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1, U_1} are not replacing on any position. 12.55/3.87 12.55/3.87 The TRS P consists of the following rules: 12.55/3.87 12.55/3.87 NATSFROM(N) -> U281'(isNatural(N), N) 12.55/3.87 U281'(tt, N) -> U282'(isNaturalKind(N), N) 12.55/3.87 U282'(tt, N) -> U(N) 12.55/3.87 12.55/3.87 The TRS R consists of the following rules: 12.55/3.87 12.55/3.87 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.55/3.87 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.55/3.87 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.55/3.87 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.55/3.87 U105(tt, V2) -> U106(isLNat(V2)) 12.55/3.87 U106(tt) -> tt 12.55/3.87 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 12.55/3.87 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.87 U112(tt) -> tt 12.55/3.87 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 12.55/3.87 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.87 U122(tt) -> tt 12.55/3.87 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 12.55/3.87 U131(tt) -> tt 12.55/3.87 U14(tt, N, XS) -> snd(splitAt(N, XS)) 12.55/3.87 U141(tt) -> tt 12.55/3.87 U151(tt) -> tt 12.55/3.87 U161(tt) -> tt 12.55/3.87 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.87 U172(tt) -> tt 12.55/3.87 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.55/3.87 U182(tt, V1) -> U183(isLNat(V1)) 12.55/3.87 U183(tt) -> tt 12.55/3.87 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.55/3.87 U192(tt, V1) -> U193(isNatural(V1)) 12.55/3.87 U193(tt) -> tt 12.55/3.87 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.55/3.87 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.55/3.87 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.55/3.87 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.55/3.87 U205(tt, V2) -> U206(isLNat(V2)) 12.55/3.87 U206(tt) -> tt 12.55/3.87 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 12.55/3.87 U211(tt) -> tt 12.55/3.87 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 12.55/3.87 U221(tt) -> tt 12.55/3.87 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 12.55/3.87 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.87 U232(tt) -> tt 12.55/3.87 U24(tt, X) -> X 12.55/3.87 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.55/3.87 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.55/3.87 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.55/3.87 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.55/3.87 U245(tt, V2) -> U246(isLNat(V2)) 12.55/3.87 U246(tt) -> tt 12.55/3.87 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.55/3.87 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.55/3.87 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.55/3.87 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.55/3.87 U255(tt, V2) -> U256(isLNat(V2)) 12.55/3.87 U256(tt) -> tt 12.55/3.87 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.87 U262(tt) -> tt 12.55/3.87 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.87 U272(tt) -> tt 12.55/3.87 U281(tt, N) -> U282(isNaturalKind(N), N) 12.55/3.87 U282(tt, N) -> cons(N, natsFrom(s(N))) 12.55/3.87 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 12.55/3.87 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 12.55/3.87 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 12.55/3.87 U294(tt, N, XS) -> head(afterNth(N, XS)) 12.55/3.87 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 12.55/3.87 U302(tt, Y) -> U303(isLNat(Y), Y) 12.55/3.87 U303(tt, Y) -> U304(isLNatKind(Y), Y) 12.55/3.87 U304(tt, Y) -> Y 12.55/3.87 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 12.55/3.87 U311(tt, XS) -> U312(isLNatKind(XS), XS) 12.55/3.87 U312(tt, XS) -> pair(nil, XS) 12.55/3.87 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 12.55/3.87 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 12.55/3.87 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 12.55/3.87 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 12.55/3.87 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 12.55/3.87 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 12.55/3.87 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 12.55/3.87 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 12.55/3.87 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 12.55/3.87 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 12.55/3.87 U332(tt, XS) -> U333(isLNat(XS), XS) 12.55/3.87 U333(tt, XS) -> U334(isLNatKind(XS), XS) 12.55/3.87 U334(tt, XS) -> XS 12.55/3.87 U34(tt, N) -> N 12.55/3.87 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 12.55/3.87 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 12.55/3.87 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 12.55/3.87 U344(tt, N, XS) -> fst(splitAt(N, XS)) 12.55/3.87 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.55/3.87 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.55/3.87 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.55/3.87 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.55/3.87 U45(tt, V2) -> U46(isLNat(V2)) 12.55/3.87 U46(tt) -> tt 12.55/3.87 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.55/3.87 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.55/3.87 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.55/3.87 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.55/3.87 U55(tt, V2) -> U56(isLNat(V2)) 12.55/3.87 U56(tt) -> tt 12.55/3.87 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.55/3.87 U62(tt, V1) -> U63(isPLNat(V1)) 12.55/3.87 U63(tt) -> tt 12.55/3.87 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.55/3.87 U72(tt, V1) -> U73(isNatural(V1)) 12.55/3.87 U73(tt) -> tt 12.55/3.87 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.55/3.87 U82(tt, V1) -> U83(isPLNat(V1)) 12.55/3.87 U83(tt) -> tt 12.55/3.87 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.55/3.87 U92(tt, V1) -> U93(isLNat(V1)) 12.55/3.87 U93(tt) -> tt 12.55/3.87 afterNth(N, XS) -> U11(isNatural(N), N, XS) 12.55/3.87 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 12.55/3.87 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 12.55/3.87 isLNat(nil) -> tt 12.55/3.87 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.55/3.87 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.55/3.87 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.55/3.87 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.55/3.87 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNatKind(nil) -> tt 12.55/3.87 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.87 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.87 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.87 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.87 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.87 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.87 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.87 isNatural(0) -> tt 12.55/3.87 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.55/3.87 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.55/3.87 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.55/3.87 isNaturalKind(0) -> tt 12.55/3.87 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.87 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.87 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.87 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.55/3.87 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.55/3.87 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.87 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.87 natsFrom(N) -> U281(isNatural(N), N) 12.55/3.87 sel(N, XS) -> U291(isNatural(N), N, XS) 12.55/3.87 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 12.55/3.87 splitAt(0, XS) -> U311(isLNat(XS), XS) 12.55/3.87 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 12.55/3.87 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 12.55/3.87 take(N, XS) -> U341(isNatural(N), N, XS) 12.55/3.87 12.55/3.87 Q is empty. 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (24) QCSDependencyGraphProof (EQUIVALENT) 12.55/3.87 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes. 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (25) 12.55/3.87 TRUE 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (26) 12.55/3.87 Obligation: 12.55/3.87 Q-restricted context-sensitive dependency pair problem: 12.55/3.87 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. 12.55/3.87 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}. 12.55/3.87 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1} are not replacing on any position. 12.55/3.87 12.55/3.87 The TRS P consists of the following rules: 12.55/3.87 12.55/3.87 U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) 12.55/3.87 U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) 12.55/3.87 U326'(tt, N, X, XS) -> SPLITAT(N, XS) 12.55/3.87 SPLITAT(s(N), cons(X, XS)) -> U321'(isNatural(N), N, X, XS) 12.55/3.87 U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) 12.55/3.87 U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) 12.55/3.87 U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) 12.55/3.87 12.55/3.87 The TRS R consists of the following rules: 12.55/3.87 12.55/3.87 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.55/3.87 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.55/3.87 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.55/3.87 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.55/3.87 U105(tt, V2) -> U106(isLNat(V2)) 12.55/3.87 U106(tt) -> tt 12.55/3.87 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 12.55/3.87 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.87 U112(tt) -> tt 12.55/3.87 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 12.55/3.87 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.87 U122(tt) -> tt 12.55/3.87 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 12.55/3.87 U131(tt) -> tt 12.55/3.87 U14(tt, N, XS) -> snd(splitAt(N, XS)) 12.55/3.87 U141(tt) -> tt 12.55/3.87 U151(tt) -> tt 12.55/3.87 U161(tt) -> tt 12.55/3.87 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.87 U172(tt) -> tt 12.55/3.87 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.55/3.87 U182(tt, V1) -> U183(isLNat(V1)) 12.55/3.87 U183(tt) -> tt 12.55/3.87 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.55/3.87 U192(tt, V1) -> U193(isNatural(V1)) 12.55/3.87 U193(tt) -> tt 12.55/3.87 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.55/3.87 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.55/3.87 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.55/3.87 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.55/3.87 U205(tt, V2) -> U206(isLNat(V2)) 12.55/3.87 U206(tt) -> tt 12.55/3.87 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 12.55/3.87 U211(tt) -> tt 12.55/3.87 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 12.55/3.87 U221(tt) -> tt 12.55/3.87 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 12.55/3.87 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.87 U232(tt) -> tt 12.55/3.87 U24(tt, X) -> X 12.55/3.87 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.55/3.87 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.55/3.87 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.55/3.87 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.55/3.87 U245(tt, V2) -> U246(isLNat(V2)) 12.55/3.87 U246(tt) -> tt 12.55/3.87 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.55/3.87 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.55/3.87 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.55/3.87 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.55/3.87 U255(tt, V2) -> U256(isLNat(V2)) 12.55/3.87 U256(tt) -> tt 12.55/3.87 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.87 U262(tt) -> tt 12.55/3.87 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.87 U272(tt) -> tt 12.55/3.87 U281(tt, N) -> U282(isNaturalKind(N), N) 12.55/3.87 U282(tt, N) -> cons(N, natsFrom(s(N))) 12.55/3.87 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 12.55/3.87 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 12.55/3.87 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 12.55/3.87 U294(tt, N, XS) -> head(afterNth(N, XS)) 12.55/3.87 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 12.55/3.87 U302(tt, Y) -> U303(isLNat(Y), Y) 12.55/3.87 U303(tt, Y) -> U304(isLNatKind(Y), Y) 12.55/3.87 U304(tt, Y) -> Y 12.55/3.87 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 12.55/3.87 U311(tt, XS) -> U312(isLNatKind(XS), XS) 12.55/3.87 U312(tt, XS) -> pair(nil, XS) 12.55/3.87 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 12.55/3.87 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 12.55/3.87 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 12.55/3.87 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 12.55/3.87 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 12.55/3.87 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 12.55/3.87 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 12.55/3.87 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 12.55/3.87 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 12.55/3.87 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 12.55/3.87 U332(tt, XS) -> U333(isLNat(XS), XS) 12.55/3.87 U333(tt, XS) -> U334(isLNatKind(XS), XS) 12.55/3.87 U334(tt, XS) -> XS 12.55/3.87 U34(tt, N) -> N 12.55/3.87 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 12.55/3.87 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 12.55/3.87 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 12.55/3.87 U344(tt, N, XS) -> fst(splitAt(N, XS)) 12.55/3.87 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.55/3.87 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.55/3.87 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.55/3.87 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.55/3.87 U45(tt, V2) -> U46(isLNat(V2)) 12.55/3.87 U46(tt) -> tt 12.55/3.87 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.55/3.87 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.55/3.87 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.55/3.87 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.55/3.87 U55(tt, V2) -> U56(isLNat(V2)) 12.55/3.87 U56(tt) -> tt 12.55/3.87 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.55/3.87 U62(tt, V1) -> U63(isPLNat(V1)) 12.55/3.87 U63(tt) -> tt 12.55/3.87 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.55/3.87 U72(tt, V1) -> U73(isNatural(V1)) 12.55/3.87 U73(tt) -> tt 12.55/3.87 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.55/3.87 U82(tt, V1) -> U83(isPLNat(V1)) 12.55/3.87 U83(tt) -> tt 12.55/3.87 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.55/3.87 U92(tt, V1) -> U93(isLNat(V1)) 12.55/3.87 U93(tt) -> tt 12.55/3.87 afterNth(N, XS) -> U11(isNatural(N), N, XS) 12.55/3.87 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 12.55/3.87 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 12.55/3.87 isLNat(nil) -> tt 12.55/3.87 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.55/3.87 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.55/3.87 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.55/3.87 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.55/3.87 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNatKind(nil) -> tt 12.55/3.87 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.87 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.87 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.87 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.87 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.87 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.87 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.87 isNatural(0) -> tt 12.55/3.87 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.55/3.87 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.55/3.87 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.55/3.87 isNaturalKind(0) -> tt 12.55/3.87 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.87 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.87 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.87 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.55/3.87 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.55/3.87 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.87 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.87 natsFrom(N) -> U281(isNatural(N), N) 12.55/3.87 sel(N, XS) -> U291(isNatural(N), N, XS) 12.55/3.87 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 12.55/3.87 splitAt(0, XS) -> U311(isLNat(XS), XS) 12.55/3.87 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 12.55/3.87 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 12.55/3.87 take(N, XS) -> U341(isNatural(N), N, XS) 12.55/3.87 12.55/3.87 Q is empty. 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (27) QCSDPSubtermProof (EQUIVALENT) 12.55/3.87 We use the subterm processor [DA_EMMES]. 12.55/3.87 12.55/3.87 12.55/3.87 The following pairs can be oriented strictly and are deleted. 12.55/3.87 12.55/3.87 SPLITAT(s(N), cons(X, XS)) -> U321'(isNatural(N), N, X, XS) 12.55/3.87 The remaining pairs can at least be oriented weakly. 12.55/3.87 12.55/3.87 U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) 12.55/3.87 U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) 12.55/3.87 U326'(tt, N, X, XS) -> SPLITAT(N, XS) 12.55/3.87 U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) 12.55/3.87 U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) 12.55/3.87 U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) 12.55/3.87 Used ordering: Combined order from the following AFS and order. 12.55/3.87 U325'(x1, x2, x3, x4) = x2 12.55/3.87 12.55/3.87 U324'(x1, x2, x3, x4) = x2 12.55/3.87 12.55/3.87 U326'(x1, x2, x3, x4) = x2 12.55/3.87 12.55/3.87 SPLITAT(x1, x2) = x1 12.55/3.87 12.55/3.87 U321'(x1, x2, x3, x4) = x2 12.55/3.87 12.55/3.87 U322'(x1, x2, x3, x4) = x2 12.55/3.87 12.55/3.87 U323'(x1, x2, x3, x4) = x2 12.55/3.87 12.55/3.87 12.55/3.87 Subterm Order 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (28) 12.55/3.87 Obligation: 12.55/3.87 Q-restricted context-sensitive dependency pair problem: 12.55/3.87 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. 12.55/3.87 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}. 12.55/3.87 The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1} are not replacing on any position. 12.55/3.87 12.55/3.87 The TRS P consists of the following rules: 12.55/3.87 12.55/3.87 U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) 12.55/3.87 U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) 12.55/3.87 U326'(tt, N, X, XS) -> SPLITAT(N, XS) 12.55/3.87 U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) 12.55/3.87 U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) 12.55/3.87 U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) 12.55/3.87 12.55/3.87 The TRS R consists of the following rules: 12.55/3.87 12.55/3.87 U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) 12.55/3.87 U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) 12.55/3.87 U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) 12.55/3.87 U104(tt, V1, V2) -> U105(isNatural(V1), V2) 12.55/3.87 U105(tt, V2) -> U106(isLNat(V2)) 12.55/3.87 U106(tt) -> tt 12.55/3.87 U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) 12.55/3.87 U111(tt, V2) -> U112(isLNatKind(V2)) 12.55/3.87 U112(tt) -> tt 12.55/3.87 U12(tt, N, XS) -> U13(isLNat(XS), N, XS) 12.55/3.87 U121(tt, V2) -> U122(isLNatKind(V2)) 12.55/3.87 U122(tt) -> tt 12.55/3.87 U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) 12.55/3.87 U131(tt) -> tt 12.55/3.87 U14(tt, N, XS) -> snd(splitAt(N, XS)) 12.55/3.87 U141(tt) -> tt 12.55/3.87 U151(tt) -> tt 12.55/3.87 U161(tt) -> tt 12.55/3.87 U171(tt, V2) -> U172(isLNatKind(V2)) 12.55/3.87 U172(tt) -> tt 12.55/3.87 U181(tt, V1) -> U182(isLNatKind(V1), V1) 12.55/3.87 U182(tt, V1) -> U183(isLNat(V1)) 12.55/3.87 U183(tt) -> tt 12.55/3.87 U191(tt, V1) -> U192(isNaturalKind(V1), V1) 12.55/3.87 U192(tt, V1) -> U193(isNatural(V1)) 12.55/3.87 U193(tt) -> tt 12.55/3.87 U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) 12.55/3.87 U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) 12.55/3.87 U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) 12.55/3.87 U204(tt, V1, V2) -> U205(isNatural(V1), V2) 12.55/3.87 U205(tt, V2) -> U206(isLNat(V2)) 12.55/3.87 U206(tt) -> tt 12.55/3.87 U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) 12.55/3.87 U211(tt) -> tt 12.55/3.87 U22(tt, X, Y) -> U23(isLNat(Y), X, Y) 12.55/3.87 U221(tt) -> tt 12.55/3.87 U23(tt, X, Y) -> U24(isLNatKind(Y), X) 12.55/3.87 U231(tt, V2) -> U232(isLNatKind(V2)) 12.55/3.87 U232(tt) -> tt 12.55/3.87 U24(tt, X) -> X 12.55/3.87 U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) 12.55/3.87 U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) 12.55/3.87 U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) 12.55/3.87 U244(tt, V1, V2) -> U245(isLNat(V1), V2) 12.55/3.87 U245(tt, V2) -> U246(isLNat(V2)) 12.55/3.87 U246(tt) -> tt 12.55/3.87 U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) 12.55/3.87 U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) 12.55/3.87 U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) 12.55/3.87 U254(tt, V1, V2) -> U255(isNatural(V1), V2) 12.55/3.87 U255(tt, V2) -> U256(isLNat(V2)) 12.55/3.87 U256(tt) -> tt 12.55/3.87 U261(tt, V2) -> U262(isLNatKind(V2)) 12.55/3.87 U262(tt) -> tt 12.55/3.87 U271(tt, V2) -> U272(isLNatKind(V2)) 12.55/3.87 U272(tt) -> tt 12.55/3.87 U281(tt, N) -> U282(isNaturalKind(N), N) 12.55/3.87 U282(tt, N) -> cons(N, natsFrom(s(N))) 12.55/3.87 U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) 12.55/3.87 U292(tt, N, XS) -> U293(isLNat(XS), N, XS) 12.55/3.87 U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) 12.55/3.87 U294(tt, N, XS) -> head(afterNth(N, XS)) 12.55/3.87 U301(tt, X, Y) -> U302(isLNatKind(X), Y) 12.55/3.87 U302(tt, Y) -> U303(isLNat(Y), Y) 12.55/3.87 U303(tt, Y) -> U304(isLNatKind(Y), Y) 12.55/3.87 U304(tt, Y) -> Y 12.55/3.87 U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) 12.55/3.87 U311(tt, XS) -> U312(isLNatKind(XS), XS) 12.55/3.87 U312(tt, XS) -> pair(nil, XS) 12.55/3.87 U32(tt, N, XS) -> U33(isLNat(XS), N, XS) 12.55/3.87 U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) 12.55/3.87 U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) 12.55/3.87 U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) 12.55/3.87 U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) 12.55/3.87 U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) 12.55/3.87 U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) 12.55/3.87 U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) 12.55/3.87 U33(tt, N, XS) -> U34(isLNatKind(XS), N) 12.55/3.87 U331(tt, N, XS) -> U332(isNaturalKind(N), XS) 12.55/3.87 U332(tt, XS) -> U333(isLNat(XS), XS) 12.55/3.87 U333(tt, XS) -> U334(isLNatKind(XS), XS) 12.55/3.87 U334(tt, XS) -> XS 12.55/3.87 U34(tt, N) -> N 12.55/3.87 U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) 12.55/3.87 U342(tt, N, XS) -> U343(isLNat(XS), N, XS) 12.55/3.87 U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) 12.55/3.87 U344(tt, N, XS) -> fst(splitAt(N, XS)) 12.55/3.87 U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) 12.55/3.87 U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) 12.55/3.87 U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) 12.55/3.87 U44(tt, V1, V2) -> U45(isNatural(V1), V2) 12.55/3.87 U45(tt, V2) -> U46(isLNat(V2)) 12.55/3.87 U46(tt) -> tt 12.55/3.87 U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) 12.55/3.87 U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) 12.55/3.87 U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) 12.55/3.87 U54(tt, V1, V2) -> U55(isNatural(V1), V2) 12.55/3.87 U55(tt, V2) -> U56(isLNat(V2)) 12.55/3.87 U56(tt) -> tt 12.55/3.87 U61(tt, V1) -> U62(isPLNatKind(V1), V1) 12.55/3.87 U62(tt, V1) -> U63(isPLNat(V1)) 12.55/3.87 U63(tt) -> tt 12.55/3.87 U71(tt, V1) -> U72(isNaturalKind(V1), V1) 12.55/3.87 U72(tt, V1) -> U73(isNatural(V1)) 12.55/3.87 U73(tt) -> tt 12.55/3.87 U81(tt, V1) -> U82(isPLNatKind(V1), V1) 12.55/3.87 U82(tt, V1) -> U83(isPLNat(V1)) 12.55/3.87 U83(tt) -> tt 12.55/3.87 U91(tt, V1) -> U92(isLNatKind(V1), V1) 12.55/3.87 U92(tt, V1) -> U93(isLNat(V1)) 12.55/3.87 U93(tt) -> tt 12.55/3.87 afterNth(N, XS) -> U11(isNatural(N), N, XS) 12.55/3.87 fst(pair(X, Y)) -> U21(isLNat(X), X, Y) 12.55/3.87 head(cons(N, XS)) -> U31(isNatural(N), N, XS) 12.55/3.87 isLNat(nil) -> tt 12.55/3.87 isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) 12.55/3.87 isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) 12.55/3.87 isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) 12.55/3.87 isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) 12.55/3.87 isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) 12.55/3.87 isLNatKind(nil) -> tt 12.55/3.87 isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) 12.55/3.87 isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) 12.55/3.87 isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) 12.55/3.87 isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) 12.55/3.87 isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) 12.55/3.87 isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) 12.55/3.87 isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) 12.55/3.87 isNatural(0) -> tt 12.55/3.87 isNatural(head(V1)) -> U181(isLNatKind(V1), V1) 12.55/3.87 isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) 12.55/3.87 isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) 12.55/3.87 isNaturalKind(0) -> tt 12.55/3.87 isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) 12.55/3.87 isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) 12.55/3.87 isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) 12.55/3.87 isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) 12.55/3.87 isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) 12.55/3.87 isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) 12.55/3.87 isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) 12.55/3.87 natsFrom(N) -> U281(isNatural(N), N) 12.55/3.87 sel(N, XS) -> U291(isNatural(N), N, XS) 12.55/3.87 snd(pair(X, Y)) -> U301(isLNat(X), X, Y) 12.55/3.87 splitAt(0, XS) -> U311(isLNat(XS), XS) 12.55/3.87 splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) 12.55/3.87 tail(cons(N, XS)) -> U331(isNatural(N), N, XS) 12.55/3.87 take(N, XS) -> U341(isNatural(N), N, XS) 12.55/3.87 12.55/3.87 Q is empty. 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (29) QCSDependencyGraphProof (EQUIVALENT) 12.55/3.87 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 6 less nodes. 12.55/3.87 12.55/3.87 ---------------------------------------- 12.55/3.87 12.55/3.87 (30) 12.55/3.87 TRUE 12.55/3.93 EOF