/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination of the given CSR could be proven: (0) CSR (1) CSDependencyPairsProof [EQUIVALENT, 552 ms] (2) QCSDP (3) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) QCSDP (6) QCSUsableRulesProof [EQUIVALENT, 0 ms] (7) QCSDP (8) QCSDPMuMonotonicPoloProof [EQUIVALENT, 64 ms] (9) QCSDP (10) QCSDPSubtermProof [EQUIVALENT, 0 ms] (11) QCSDP (12) PIsEmptyProof [EQUIVALENT, 0 ms] (13) YES (14) QCSDP (15) QCSUsableRulesProof [EQUIVALENT, 90 ms] (16) QCSDP (17) QCSDPMuMonotonicPoloProof [EQUIVALENT, 296 ms] (18) QCSDP (19) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (20) TRUE (21) QCSDP (22) QCSDPSubtermProof [EQUIVALENT, 0 ms] (23) QCSDP (24) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (25) TRUE (26) QCSDP (27) QCSDPSubtermProof [EQUIVALENT, 12 ms] (28) QCSDP (29) QCSDependencyGraphProof [EQUIVALENT, 0 ms] (30) TRUE ---------------------------------------- (0) Obligation: Context-sensitive rewrite system: The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(XS), N, XS) U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(N, XS)) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(Y), X, Y) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(Y), X) U231(tt, V2) -> U232(isLNatKind(V2)) U232(tt) -> tt U24(tt, X) -> X U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) U244(tt, V1, V2) -> U245(isLNat(V1), V2) U245(tt, V2) -> U246(isLNat(V2)) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U254(tt, V1, V2) -> U255(isNatural(V1), V2) U255(tt, V2) -> U256(isLNat(V2)) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(V2)) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(N), N) U282(tt, N) -> cons(N, natsFrom(s(N))) U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) U292(tt, N, XS) -> U293(isLNat(XS), N, XS) U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) U294(tt, N, XS) -> head(afterNth(N, XS)) U301(tt, X, Y) -> U302(isLNatKind(X), Y) U302(tt, Y) -> U303(isLNat(Y), Y) U303(tt, Y) -> U304(isLNatKind(Y), Y) U304(tt, Y) -> Y U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) U311(tt, XS) -> U312(isLNatKind(XS), XS) U312(tt, XS) -> pair(nil, XS) U32(tt, N, XS) -> U33(isLNat(XS), N, XS) U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(XS), N) U331(tt, N, XS) -> U332(isNaturalKind(N), XS) U332(tt, XS) -> U333(isLNat(XS), XS) U333(tt, XS) -> U334(isLNatKind(XS), XS) U334(tt, XS) -> XS U34(tt, N) -> N U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) U342(tt, N, XS) -> U343(isLNat(XS), N, XS) U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) U344(tt, N, XS) -> fst(splitAt(N, XS)) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(V1), V1) U82(tt, V1) -> U83(isPLNat(V1)) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, XS) isLNat(nil) -> tt isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) tail(cons(N, XS)) -> U331(isNatural(N), N, XS) take(N, XS) -> U341(isNatural(N), N, XS) The replacement map contains the following entries: U101: {1} tt: empty set U102: {1} isNaturalKind: empty set U103: {1} isLNatKind: empty set U104: {1} U105: {1} isNatural: empty set U106: {1} isLNat: empty set U11: {1} U12: {1} U111: {1} U112: {1} U13: {1} U121: {1} U122: {1} U14: {1} U131: {1} snd: {1} splitAt: {1, 2} U141: {1} U151: {1} U161: {1} U171: {1} U172: {1} U181: {1} U182: {1} U183: {1} U191: {1} U192: {1} U193: {1} U201: {1} U202: {1} U203: {1} U204: {1} U205: {1} U206: {1} U21: {1} U22: {1} U211: {1} U23: {1} U221: {1} U24: {1} U231: {1} U232: {1} U241: {1} U242: {1} U243: {1} U244: {1} U245: {1} U246: {1} U251: {1} U252: {1} U253: {1} U254: {1} U255: {1} U256: {1} U261: {1} U262: {1} U271: {1} U272: {1} U281: {1} U282: {1} cons: {1} natsFrom: {1} s: {1} U291: {1} U292: {1} U293: {1} U294: {1} head: {1} afterNth: {1, 2} U301: {1} U302: {1} U303: {1} U304: {1} U31: {1} U32: {1} U311: {1} U312: {1} pair: {1, 2} nil: empty set U33: {1} U321: {1} U322: {1} U323: {1} U324: {1} U325: {1} U326: {1} U327: {1} U34: {1} U331: {1} U332: {1} U333: {1} U334: {1} U341: {1} U342: {1} U343: {1} U344: {1} fst: {1} U41: {1} U42: {1} U43: {1} U44: {1} U45: {1} U46: {1} U51: {1} U52: {1} U53: {1} U54: {1} U55: {1} U56: {1} U61: {1} U62: {1} isPLNatKind: empty set U63: {1} isPLNat: empty set U71: {1} U72: {1} U73: {1} U81: {1} U82: {1} U83: {1} U91: {1} U92: {1} U93: {1} tail: {1} take: {1, 2} 0: empty set sel: {1, 2} ---------------------------------------- (1) CSDependencyPairsProof (EQUIVALENT) Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem. ---------------------------------------- (2) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. 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. The ordinary context-sensitive dependency pairs DP_o are: U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) U101'(tt, V1, V2) -> ISNATURALKIND(V1) U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) U102'(tt, V1, V2) -> ISLNATKIND(V2) U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) U103'(tt, V1, V2) -> ISLNATKIND(V2) U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) U104'(tt, V1, V2) -> ISNATURAL(V1) U105'(tt, V2) -> U106'(isLNat(V2)) U105'(tt, V2) -> ISLNAT(V2) U11'(tt, N, XS) -> U12'(isNaturalKind(N), N, XS) U11'(tt, N, XS) -> ISNATURALKIND(N) U111'(tt, V2) -> U112'(isLNatKind(V2)) U111'(tt, V2) -> ISLNATKIND(V2) U12'(tt, N, XS) -> U13'(isLNat(XS), N, XS) U12'(tt, N, XS) -> ISLNAT(XS) U121'(tt, V2) -> U122'(isLNatKind(V2)) U121'(tt, V2) -> ISLNATKIND(V2) U13'(tt, N, XS) -> U14'(isLNatKind(XS), N, XS) U13'(tt, N, XS) -> ISLNATKIND(XS) U14'(tt, N, XS) -> SND(splitAt(N, XS)) U14'(tt, N, XS) -> SPLITAT(N, XS) U171'(tt, V2) -> U172'(isLNatKind(V2)) U171'(tt, V2) -> ISLNATKIND(V2) U181'(tt, V1) -> U182'(isLNatKind(V1), V1) U181'(tt, V1) -> ISLNATKIND(V1) U182'(tt, V1) -> U183'(isLNat(V1)) U182'(tt, V1) -> ISLNAT(V1) U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) U191'(tt, V1) -> ISNATURALKIND(V1) U192'(tt, V1) -> U193'(isNatural(V1)) U192'(tt, V1) -> ISNATURAL(V1) U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) U201'(tt, V1, V2) -> ISNATURALKIND(V1) U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) U202'(tt, V1, V2) -> ISLNATKIND(V2) U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) U203'(tt, V1, V2) -> ISLNATKIND(V2) U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) U204'(tt, V1, V2) -> ISNATURAL(V1) U205'(tt, V2) -> U206'(isLNat(V2)) U205'(tt, V2) -> ISLNAT(V2) U21'(tt, X, Y) -> U22'(isLNatKind(X), X, Y) U21'(tt, X, Y) -> ISLNATKIND(X) U22'(tt, X, Y) -> U23'(isLNat(Y), X, Y) U22'(tt, X, Y) -> ISLNAT(Y) U23'(tt, X, Y) -> U24'(isLNatKind(Y), X) U23'(tt, X, Y) -> ISLNATKIND(Y) U231'(tt, V2) -> U232'(isLNatKind(V2)) U231'(tt, V2) -> ISLNATKIND(V2) U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) U241'(tt, V1, V2) -> ISLNATKIND(V1) U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) U242'(tt, V1, V2) -> ISLNATKIND(V2) U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) U243'(tt, V1, V2) -> ISLNATKIND(V2) U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) U244'(tt, V1, V2) -> ISLNAT(V1) U245'(tt, V2) -> U246'(isLNat(V2)) U245'(tt, V2) -> ISLNAT(V2) U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) U251'(tt, V1, V2) -> ISNATURALKIND(V1) U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) U252'(tt, V1, V2) -> ISLNATKIND(V2) U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) U253'(tt, V1, V2) -> ISLNATKIND(V2) U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) U254'(tt, V1, V2) -> ISNATURAL(V1) U255'(tt, V2) -> U256'(isLNat(V2)) U255'(tt, V2) -> ISLNAT(V2) U261'(tt, V2) -> U262'(isLNatKind(V2)) U261'(tt, V2) -> ISLNATKIND(V2) U271'(tt, V2) -> U272'(isLNatKind(V2)) U271'(tt, V2) -> ISLNATKIND(V2) U281'(tt, N) -> U282'(isNaturalKind(N), N) U281'(tt, N) -> ISNATURALKIND(N) U291'(tt, N, XS) -> U292'(isNaturalKind(N), N, XS) U291'(tt, N, XS) -> ISNATURALKIND(N) U292'(tt, N, XS) -> U293'(isLNat(XS), N, XS) U292'(tt, N, XS) -> ISLNAT(XS) U293'(tt, N, XS) -> U294'(isLNatKind(XS), N, XS) U293'(tt, N, XS) -> ISLNATKIND(XS) U294'(tt, N, XS) -> HEAD(afterNth(N, XS)) U294'(tt, N, XS) -> AFTERNTH(N, XS) U301'(tt, X, Y) -> U302'(isLNatKind(X), Y) U301'(tt, X, Y) -> ISLNATKIND(X) U302'(tt, Y) -> U303'(isLNat(Y), Y) U302'(tt, Y) -> ISLNAT(Y) U303'(tt, Y) -> U304'(isLNatKind(Y), Y) U303'(tt, Y) -> ISLNATKIND(Y) U31'(tt, N, XS) -> U32'(isNaturalKind(N), N, XS) U31'(tt, N, XS) -> ISNATURALKIND(N) U311'(tt, XS) -> U312'(isLNatKind(XS), XS) U311'(tt, XS) -> ISLNATKIND(XS) U32'(tt, N, XS) -> U33'(isLNat(XS), N, XS) U32'(tt, N, XS) -> ISLNAT(XS) U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) U321'(tt, N, X, XS) -> ISNATURALKIND(N) U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) U322'(tt, N, X, XS) -> ISNATURAL(X) U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) U323'(tt, N, X, XS) -> ISNATURALKIND(X) U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) U324'(tt, N, X, XS) -> ISLNAT(XS) U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) U325'(tt, N, X, XS) -> ISLNATKIND(XS) U326'(tt, N, X, XS) -> U327'(splitAt(N, XS), X) U326'(tt, N, X, XS) -> SPLITAT(N, XS) U33'(tt, N, XS) -> U34'(isLNatKind(XS), N) U33'(tt, N, XS) -> ISLNATKIND(XS) U331'(tt, N, XS) -> U332'(isNaturalKind(N), XS) U331'(tt, N, XS) -> ISNATURALKIND(N) U332'(tt, XS) -> U333'(isLNat(XS), XS) U332'(tt, XS) -> ISLNAT(XS) U333'(tt, XS) -> U334'(isLNatKind(XS), XS) U333'(tt, XS) -> ISLNATKIND(XS) U341'(tt, N, XS) -> U342'(isNaturalKind(N), N, XS) U341'(tt, N, XS) -> ISNATURALKIND(N) U342'(tt, N, XS) -> U343'(isLNat(XS), N, XS) U342'(tt, N, XS) -> ISLNAT(XS) U343'(tt, N, XS) -> U344'(isLNatKind(XS), N, XS) U343'(tt, N, XS) -> ISLNATKIND(XS) U344'(tt, N, XS) -> FST(splitAt(N, XS)) U344'(tt, N, XS) -> SPLITAT(N, XS) U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) U41'(tt, V1, V2) -> ISNATURALKIND(V1) U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) U42'(tt, V1, V2) -> ISLNATKIND(V2) U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) U43'(tt, V1, V2) -> ISLNATKIND(V2) U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) U44'(tt, V1, V2) -> ISNATURAL(V1) U45'(tt, V2) -> U46'(isLNat(V2)) U45'(tt, V2) -> ISLNAT(V2) U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) U51'(tt, V1, V2) -> ISNATURALKIND(V1) U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) U52'(tt, V1, V2) -> ISLNATKIND(V2) U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) U53'(tt, V1, V2) -> ISLNATKIND(V2) U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) U54'(tt, V1, V2) -> ISNATURAL(V1) U55'(tt, V2) -> U56'(isLNat(V2)) U55'(tt, V2) -> ISLNAT(V2) U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) U61'(tt, V1) -> ISPLNATKIND(V1) U62'(tt, V1) -> U63'(isPLNat(V1)) U62'(tt, V1) -> ISPLNAT(V1) U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) U71'(tt, V1) -> ISNATURALKIND(V1) U72'(tt, V1) -> U73'(isNatural(V1)) U72'(tt, V1) -> ISNATURAL(V1) U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) U81'(tt, V1) -> ISPLNATKIND(V1) U82'(tt, V1) -> U83'(isPLNat(V1)) U82'(tt, V1) -> ISPLNAT(V1) U91'(tt, V1) -> U92'(isLNatKind(V1), V1) U91'(tt, V1) -> ISLNATKIND(V1) U92'(tt, V1) -> U93'(isLNat(V1)) U92'(tt, V1) -> ISLNAT(V1) AFTERNTH(N, XS) -> U11'(isNatural(N), N, XS) AFTERNTH(N, XS) -> ISNATURAL(N) FST(pair(X, Y)) -> U21'(isLNat(X), X, Y) FST(pair(X, Y)) -> ISLNAT(X) HEAD(cons(N, XS)) -> U31'(isNatural(N), N, XS) HEAD(cons(N, XS)) -> ISNATURAL(N) ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) ISLNAT(afterNth(V1, V2)) -> ISNATURALKIND(V1) ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) ISLNAT(cons(V1, V2)) -> ISNATURALKIND(V1) ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) ISLNAT(fst(V1)) -> ISPLNATKIND(V1) ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) ISLNAT(natsFrom(V1)) -> ISNATURALKIND(V1) ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) ISLNAT(snd(V1)) -> ISPLNATKIND(V1) ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) ISLNAT(tail(V1)) -> ISLNATKIND(V1) ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) ISLNAT(take(V1, V2)) -> ISNATURALKIND(V1) ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) ISLNATKIND(fst(V1)) -> U131'(isPLNatKind(V1)) ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) ISLNATKIND(natsFrom(V1)) -> U141'(isNaturalKind(V1)) ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) ISLNATKIND(snd(V1)) -> U151'(isPLNatKind(V1)) ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) ISLNATKIND(tail(V1)) -> U161'(isLNatKind(V1)) ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) ISNATURAL(head(V1)) -> ISLNATKIND(V1) ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) ISNATURAL(s(V1)) -> ISNATURALKIND(V1) ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) ISNATURAL(sel(V1, V2)) -> ISNATURALKIND(V1) ISNATURALKIND(head(V1)) -> U211'(isLNatKind(V1)) ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) ISNATURALKIND(s(V1)) -> U221'(isNaturalKind(V1)) ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) ISPLNAT(pair(V1, V2)) -> ISLNATKIND(V1) ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) ISPLNAT(splitAt(V1, V2)) -> ISNATURALKIND(V1) ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) NATSFROM(N) -> U281'(isNatural(N), N) NATSFROM(N) -> ISNATURAL(N) SEL(N, XS) -> U291'(isNatural(N), N, XS) SEL(N, XS) -> ISNATURAL(N) SND(pair(X, Y)) -> U301'(isLNat(X), X, Y) SND(pair(X, Y)) -> ISLNAT(X) SPLITAT(0, XS) -> U311'(isLNat(XS), XS) SPLITAT(0, XS) -> ISLNAT(XS) SPLITAT(s(N), cons(X, XS)) -> U321'(isNatural(N), N, X, XS) SPLITAT(s(N), cons(X, XS)) -> ISNATURAL(N) TAIL(cons(N, XS)) -> U331'(isNatural(N), N, XS) TAIL(cons(N, XS)) -> ISNATURAL(N) TAKE(N, XS) -> U341'(isNatural(N), N, XS) TAKE(N, XS) -> ISNATURAL(N) The collapsing dependency pairs are DP_c: U14'(tt, N, XS) -> N U14'(tt, N, XS) -> XS U24'(tt, X) -> X U282'(tt, N) -> N U294'(tt, N, XS) -> N U294'(tt, N, XS) -> XS U304'(tt, Y) -> Y U312'(tt, XS) -> XS U326'(tt, N, X, XS) -> N U326'(tt, N, X, XS) -> XS U327'(pair(YS, ZS), X) -> X U334'(tt, XS) -> XS U34'(tt, N) -> N U344'(tt, N, XS) -> N U344'(tt, N, XS) -> XS The hidden terms of R are: natsFrom(s(x0)) Every hiding context is built from: aprove.DPFramework.CSDPProblem.QCSDPProblem$1@25e7e031 aprove.DPFramework.CSDPProblem.QCSDPProblem$1@6f1f562a Hence, the new unhiding pairs DP_u are : U14'(tt, N, XS) -> U(N) U14'(tt, N, XS) -> U(XS) U24'(tt, X) -> U(X) U282'(tt, N) -> U(N) U294'(tt, N, XS) -> U(N) U294'(tt, N, XS) -> U(XS) U304'(tt, Y) -> U(Y) U312'(tt, XS) -> U(XS) U326'(tt, N, X, XS) -> U(N) U326'(tt, N, X, XS) -> U(XS) U327'(pair(YS, ZS), X) -> U(X) U334'(tt, XS) -> U(XS) U34'(tt, N) -> U(N) U344'(tt, N, XS) -> U(N) U344'(tt, N, XS) -> U(XS) U(s(x_0)) -> U(x_0) U(natsFrom(x_0)) -> U(x_0) U(natsFrom(s(x0))) -> NATSFROM(s(x0)) The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(XS), N, XS) U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(N, XS)) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(Y), X, Y) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(Y), X) U231(tt, V2) -> U232(isLNatKind(V2)) U232(tt) -> tt U24(tt, X) -> X U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) U244(tt, V1, V2) -> U245(isLNat(V1), V2) U245(tt, V2) -> U246(isLNat(V2)) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U254(tt, V1, V2) -> U255(isNatural(V1), V2) U255(tt, V2) -> U256(isLNat(V2)) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(V2)) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(N), N) U282(tt, N) -> cons(N, natsFrom(s(N))) U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) U292(tt, N, XS) -> U293(isLNat(XS), N, XS) U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) U294(tt, N, XS) -> head(afterNth(N, XS)) U301(tt, X, Y) -> U302(isLNatKind(X), Y) U302(tt, Y) -> U303(isLNat(Y), Y) U303(tt, Y) -> U304(isLNatKind(Y), Y) U304(tt, Y) -> Y U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) U311(tt, XS) -> U312(isLNatKind(XS), XS) U312(tt, XS) -> pair(nil, XS) U32(tt, N, XS) -> U33(isLNat(XS), N, XS) U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(XS), N) U331(tt, N, XS) -> U332(isNaturalKind(N), XS) U332(tt, XS) -> U333(isLNat(XS), XS) U333(tt, XS) -> U334(isLNatKind(XS), XS) U334(tt, XS) -> XS U34(tt, N) -> N U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) U342(tt, N, XS) -> U343(isLNat(XS), N, XS) U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) U344(tt, N, XS) -> fst(splitAt(N, XS)) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(V1), V1) U82(tt, V1) -> U83(isPLNat(V1)) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, XS) isLNat(nil) -> tt isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) tail(cons(N, XS)) -> U331(isNatural(N), N, XS) take(N, XS) -> U341(isNatural(N), N, XS) Q is empty. ---------------------------------------- (3) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 149 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. 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. The TRS P consists of the following rules: ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) U111'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) U231'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) U121'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) U261'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) U171'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) U271'(tt, V2) -> ISLNATKIND(V2) ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(XS), N, XS) U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(N, XS)) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(Y), X, Y) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(Y), X) U231(tt, V2) -> U232(isLNatKind(V2)) U232(tt) -> tt U24(tt, X) -> X U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) U244(tt, V1, V2) -> U245(isLNat(V1), V2) U245(tt, V2) -> U246(isLNat(V2)) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U254(tt, V1, V2) -> U255(isNatural(V1), V2) U255(tt, V2) -> U256(isLNat(V2)) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(V2)) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(N), N) U282(tt, N) -> cons(N, natsFrom(s(N))) U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) U292(tt, N, XS) -> U293(isLNat(XS), N, XS) U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) U294(tt, N, XS) -> head(afterNth(N, XS)) U301(tt, X, Y) -> U302(isLNatKind(X), Y) U302(tt, Y) -> U303(isLNat(Y), Y) U303(tt, Y) -> U304(isLNatKind(Y), Y) U304(tt, Y) -> Y U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) U311(tt, XS) -> U312(isLNatKind(XS), XS) U312(tt, XS) -> pair(nil, XS) U32(tt, N, XS) -> U33(isLNat(XS), N, XS) U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(XS), N) U331(tt, N, XS) -> U332(isNaturalKind(N), XS) U332(tt, XS) -> U333(isLNat(XS), XS) U333(tt, XS) -> U334(isLNatKind(XS), XS) U334(tt, XS) -> XS U34(tt, N) -> N U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) U342(tt, N, XS) -> U343(isLNat(XS), N, XS) U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) U344(tt, N, XS) -> fst(splitAt(N, XS)) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(V1), V1) U82(tt, V1) -> U83(isPLNat(V1)) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, XS) isLNat(nil) -> tt isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) tail(cons(N, XS)) -> U331(isNatural(N), N, XS) take(N, XS) -> U341(isNatural(N), N, XS) Q is empty. ---------------------------------------- (6) QCSUsableRulesProof (EQUIVALENT) The following rules are not useable [DA_EMMES] and can be deleted: U101(tt, x0, x1) -> U102(isNaturalKind(x0), x0, x1) U102(tt, x0, x1) -> U103(isLNatKind(x1), x0, x1) U103(tt, x0, x1) -> U104(isLNatKind(x1), x0, x1) U104(tt, x0, x1) -> U105(isNatural(x0), x1) U105(tt, x0) -> U106(isLNat(x0)) U106(tt) -> tt U11(tt, x0, x1) -> U12(isNaturalKind(x0), x0, x1) U12(tt, x0, x1) -> U13(isLNat(x1), x0, x1) U13(tt, x0, x1) -> U14(isLNatKind(x1), x0, x1) U14(tt, x0, x1) -> snd(splitAt(x0, x1)) U181(tt, x0) -> U182(isLNatKind(x0), x0) U182(tt, x0) -> U183(isLNat(x0)) U183(tt) -> tt U191(tt, x0) -> U192(isNaturalKind(x0), x0) U192(tt, x0) -> U193(isNatural(x0)) U193(tt) -> tt U201(tt, x0, x1) -> U202(isNaturalKind(x0), x0, x1) U202(tt, x0, x1) -> U203(isLNatKind(x1), x0, x1) U203(tt, x0, x1) -> U204(isLNatKind(x1), x0, x1) U204(tt, x0, x1) -> U205(isNatural(x0), x1) U205(tt, x0) -> U206(isLNat(x0)) U206(tt) -> tt U21(tt, x0, x1) -> U22(isLNatKind(x0), x0, x1) U22(tt, x0, x1) -> U23(isLNat(x1), x0, x1) U23(tt, x0, x1) -> U24(isLNatKind(x1), x0) U24(tt, x0) -> x0 U241(tt, x0, x1) -> U242(isLNatKind(x0), x0, x1) U242(tt, x0, x1) -> U243(isLNatKind(x1), x0, x1) U243(tt, x0, x1) -> U244(isLNatKind(x1), x0, x1) U244(tt, x0, x1) -> U245(isLNat(x0), x1) U245(tt, x0) -> U246(isLNat(x0)) U246(tt) -> tt U251(tt, x0, x1) -> U252(isNaturalKind(x0), x0, x1) U252(tt, x0, x1) -> U253(isLNatKind(x1), x0, x1) U253(tt, x0, x1) -> U254(isLNatKind(x1), x0, x1) U254(tt, x0, x1) -> U255(isNatural(x0), x1) U255(tt, x0) -> U256(isLNat(x0)) U256(tt) -> tt U281(tt, x0) -> U282(isNaturalKind(x0), x0) U282(tt, x0) -> cons(x0, natsFrom(s(x0))) U291(tt, x0, x1) -> U292(isNaturalKind(x0), x0, x1) U292(tt, x0, x1) -> U293(isLNat(x1), x0, x1) U293(tt, x0, x1) -> U294(isLNatKind(x1), x0, x1) U294(tt, x0, x1) -> head(afterNth(x0, x1)) U301(tt, x0, x1) -> U302(isLNatKind(x0), x1) U302(tt, x0) -> U303(isLNat(x0), x0) U303(tt, x0) -> U304(isLNatKind(x0), x0) U304(tt, x0) -> x0 U31(tt, x0, x1) -> U32(isNaturalKind(x0), x0, x1) U311(tt, x0) -> U312(isLNatKind(x0), x0) U312(tt, x0) -> pair(nil, x0) U32(tt, x0, x1) -> U33(isLNat(x1), x0, x1) U321(tt, x0, x1, x2) -> U322(isNaturalKind(x0), x0, x1, x2) U322(tt, x0, x1, x2) -> U323(isNatural(x1), x0, x1, x2) U323(tt, x0, x1, x2) -> U324(isNaturalKind(x1), x0, x1, x2) U324(tt, x0, x1, x2) -> U325(isLNat(x2), x0, x1, x2) U325(tt, x0, x1, x2) -> U326(isLNatKind(x2), x0, x1, x2) U326(tt, x0, x1, x2) -> U327(splitAt(x0, x2), x1) U327(pair(x0, x1), x2) -> pair(cons(x2, x0), x1) U33(tt, x0, x1) -> U34(isLNatKind(x1), x0) U331(tt, x0, x1) -> U332(isNaturalKind(x0), x1) U332(tt, x0) -> U333(isLNat(x0), x0) U333(tt, x0) -> U334(isLNatKind(x0), x0) U334(tt, x0) -> x0 U34(tt, x0) -> x0 U341(tt, x0, x1) -> U342(isNaturalKind(x0), x0, x1) U342(tt, x0, x1) -> U343(isLNat(x1), x0, x1) U343(tt, x0, x1) -> U344(isLNatKind(x1), x0, x1) U344(tt, x0, x1) -> fst(splitAt(x0, x1)) U41(tt, x0, x1) -> U42(isNaturalKind(x0), x0, x1) U42(tt, x0, x1) -> U43(isLNatKind(x1), x0, x1) U43(tt, x0, x1) -> U44(isLNatKind(x1), x0, x1) U44(tt, x0, x1) -> U45(isNatural(x0), x1) U45(tt, x0) -> U46(isLNat(x0)) U46(tt) -> tt U51(tt, x0, x1) -> U52(isNaturalKind(x0), x0, x1) U52(tt, x0, x1) -> U53(isLNatKind(x1), x0, x1) U53(tt, x0, x1) -> U54(isLNatKind(x1), x0, x1) U54(tt, x0, x1) -> U55(isNatural(x0), x1) U55(tt, x0) -> U56(isLNat(x0)) U56(tt) -> tt U61(tt, x0) -> U62(isPLNatKind(x0), x0) U62(tt, x0) -> U63(isPLNat(x0)) U63(tt) -> tt U71(tt, x0) -> U72(isNaturalKind(x0), x0) U72(tt, x0) -> U73(isNatural(x0)) U73(tt) -> tt U81(tt, x0) -> U82(isPLNatKind(x0), x0) U82(tt, x0) -> U83(isPLNat(x0)) U83(tt) -> tt U91(tt, x0) -> U92(isLNatKind(x0), x0) U92(tt, x0) -> U93(isLNat(x0)) U93(tt) -> tt afterNth(x0, x1) -> U11(isNatural(x0), x0, x1) fst(pair(x0, x1)) -> U21(isLNat(x0), x0, x1) head(cons(x0, x1)) -> U31(isNatural(x0), x0, x1) isLNat(nil) -> tt isLNat(afterNth(x0, x1)) -> U41(isNaturalKind(x0), x0, x1) isLNat(cons(x0, x1)) -> U51(isNaturalKind(x0), x0, x1) isLNat(fst(x0)) -> U61(isPLNatKind(x0), x0) isLNat(natsFrom(x0)) -> U71(isNaturalKind(x0), x0) isLNat(snd(x0)) -> U81(isPLNatKind(x0), x0) isLNat(tail(x0)) -> U91(isLNatKind(x0), x0) isLNat(take(x0, x1)) -> U101(isNaturalKind(x0), x0, x1) isNatural(0) -> tt isNatural(head(x0)) -> U181(isLNatKind(x0), x0) isNatural(s(x0)) -> U191(isNaturalKind(x0), x0) isNatural(sel(x0, x1)) -> U201(isNaturalKind(x0), x0, x1) isPLNat(pair(x0, x1)) -> U241(isLNatKind(x0), x0, x1) isPLNat(splitAt(x0, x1)) -> U251(isNaturalKind(x0), x0, x1) natsFrom(x0) -> U281(isNatural(x0), x0) sel(x0, x1) -> U291(isNatural(x0), x0, x1) snd(pair(x0, x1)) -> U301(isLNat(x0), x0, x1) splitAt(0, x0) -> U311(isLNat(x0), x0) splitAt(s(x0), cons(x1, x2)) -> U321(isNatural(x0), x0, x1, x2) tail(cons(x0, x1)) -> U331(isNatural(x0), x0, x1) take(x0, x1) -> U341(isNatural(x0), x0, x1) ---------------------------------------- (7) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. The symbols in {isNaturalKind_1, isLNatKind_1, isPLNatKind_1, ISLNATKIND_1, ISNATURALKIND_1, ISPLNATKIND_1} are not replacing on any position. The TRS P consists of the following rules: ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) U111'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) U231'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) U121'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) U261'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) U171'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) U271'(tt, V2) -> ISLNATKIND(V2) ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) The TRS R consists of the following rules: isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) U231(tt, V2) -> U232(isLNatKind(V2)) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) U121(tt, V2) -> U122(isLNatKind(V2)) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) U141(tt) -> tt isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) U271(tt, V2) -> U272(isLNatKind(V2)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U161(tt) -> tt U272(tt) -> tt U151(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U131(tt) -> tt U122(tt) -> tt U232(tt) -> tt U221(tt) -> tt U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U211(tt) -> tt Q is empty. ---------------------------------------- (8) QCSDPMuMonotonicPoloProof (EQUIVALENT) 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. Strictly oriented dependency pairs: ISNATURALKIND(head(V1)) -> ISLNATKIND(V1) ISLNATKIND(afterNth(V1, V2)) -> U111'(isNaturalKind(V1), V2) U111'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(afterNth(V1, V2)) -> ISNATURALKIND(V1) ISNATURALKIND(s(V1)) -> ISNATURALKIND(V1) ISNATURALKIND(sel(V1, V2)) -> U231'(isNaturalKind(V1), V2) U231'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(cons(V1, V2)) -> U121'(isNaturalKind(V1), V2) U121'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(cons(V1, V2)) -> ISNATURALKIND(V1) ISNATURALKIND(sel(V1, V2)) -> ISNATURALKIND(V1) ISLNATKIND(fst(V1)) -> ISPLNATKIND(V1) ISPLNATKIND(pair(V1, V2)) -> U261'(isLNatKind(V1), V2) U261'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(natsFrom(V1)) -> ISNATURALKIND(V1) ISLNATKIND(snd(V1)) -> ISPLNATKIND(V1) ISPLNATKIND(pair(V1, V2)) -> ISLNATKIND(V1) ISLNATKIND(take(V1, V2)) -> U171'(isNaturalKind(V1), V2) U171'(tt, V2) -> ISLNATKIND(V2) ISLNATKIND(take(V1, V2)) -> ISNATURALKIND(V1) ISPLNATKIND(splitAt(V1, V2)) -> U271'(isNaturalKind(V1), V2) U271'(tt, V2) -> ISLNATKIND(V2) ISPLNATKIND(splitAt(V1, V2)) -> ISNATURALKIND(V1) Strictly oriented rules of the TRS R: isNaturalKind(0) -> tt isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) U231(tt, V2) -> U232(isLNatKind(V2)) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) U141(tt) -> tt isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) U271(tt, V2) -> U272(isLNatKind(V2)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U272(tt) -> tt U151(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U131(tt) -> tt U122(tt) -> tt U232(tt) -> tt U221(tt) -> tt U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt Used ordering: POLO with Polynomial interpretation [POLO]: POL(0) = 2 POL(ISLNATKIND(x_1)) = 1 + 2*x_1 POL(ISNATURALKIND(x_1)) = 2 + x_1 POL(ISPLNATKIND(x_1)) = 1 + 2*x_1 POL(U111(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U111'(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U112(x_1)) = 1 + x_1 POL(U121(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U121'(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U122(x_1)) = 1 + x_1 POL(U131(x_1)) = 1 + x_1 POL(U141(x_1)) = 1 + x_1 POL(U151(x_1)) = 1 + 2*x_1 POL(U161(x_1)) = x_1 POL(U171(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U171'(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U172(x_1)) = 1 + x_1 POL(U211(x_1)) = x_1 POL(U221(x_1)) = 1 + 2*x_1 POL(U231(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U231'(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U232(x_1)) = 1 + x_1 POL(U261(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U261'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(U262(x_1)) = 1 + x_1 POL(U271(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U271'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(U272(x_1)) = 1 + x_1 POL(afterNth(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(cons(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(fst(x_1)) = 1 + 2*x_1 POL(head(x_1)) = 2*x_1 POL(isLNatKind(x_1)) = 2*x_1 POL(isNaturalKind(x_1)) = 2*x_1 POL(isPLNatKind(x_1)) = 1 + 2*x_1 POL(natsFrom(x_1)) = 2 + 2*x_1 POL(nil) = 1 POL(pair(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(s(x_1)) = 1 + 2*x_1 POL(sel(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(snd(x_1)) = 2 + 2*x_1 POL(splitAt(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(tail(x_1)) = x_1 POL(take(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(tt) = 0 ---------------------------------------- (9) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {head_1, U211_1, sel_2, U122_1, fst_1, U131_1, tail_1, U161_1} are replacing on all positions. For all symbols f in {U231_2, U121_2} we have mu(f) = {1}. The symbols in {isNaturalKind_1, isLNatKind_1, isPLNatKind_1, ISLNATKIND_1} are not replacing on any position. The TRS P consists of the following rules: ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) The TRS R consists of the following rules: isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) U121(tt, V2) -> U122(isLNatKind(V2)) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) U161(tt) -> tt U211(tt) -> tt Q is empty. ---------------------------------------- (10) QCSDPSubtermProof (EQUIVALENT) We use the subterm processor [DA_EMMES]. The following pairs can be oriented strictly and are deleted. ISLNATKIND(tail(V1)) -> ISLNATKIND(V1) The remaining pairs can at least be oriented weakly. none Used ordering: Combined order from the following AFS and order. ISLNATKIND(x1) = x1 Subterm Order ---------------------------------------- (11) Obligation: Q-restricted context-sensitive dependency pair problem: The symbols in {head_1, U211_1, sel_2, U122_1, fst_1, U131_1, tail_1, U161_1} are replacing on all positions. For all symbols f in {U231_2, U121_2} we have mu(f) = {1}. The symbols in {isNaturalKind_1, isLNatKind_1, isPLNatKind_1} are not replacing on any position. The TRS P consists of the following rules: none The TRS R consists of the following rules: isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) U121(tt, V2) -> U122(isLNatKind(V2)) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) U161(tt) -> tt U211(tt) -> tt Q is empty. ---------------------------------------- (12) PIsEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain. ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. 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. The TRS P consists of the following rules: U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) U105'(tt, V2) -> ISLNAT(V2) ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) U45'(tt, V2) -> ISLNAT(V2) ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) U55'(tt, V2) -> ISLNAT(V2) ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) U62'(tt, V1) -> ISPLNAT(V1) ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) U245'(tt, V2) -> ISLNAT(V2) ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) U72'(tt, V1) -> ISNATURAL(V1) ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) U181'(tt, V1) -> U182'(isLNatKind(V1), V1) U182'(tt, V1) -> ISLNAT(V1) ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) U82'(tt, V1) -> ISPLNAT(V1) ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) U255'(tt, V2) -> ISLNAT(V2) ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) U91'(tt, V1) -> U92'(isLNatKind(V1), V1) U92'(tt, V1) -> ISLNAT(V1) ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) U254'(tt, V1, V2) -> ISNATURAL(V1) ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) U192'(tt, V1) -> ISNATURAL(V1) ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) U205'(tt, V2) -> ISLNAT(V2) U204'(tt, V1, V2) -> ISNATURAL(V1) U244'(tt, V1, V2) -> ISLNAT(V1) U54'(tt, V1, V2) -> ISNATURAL(V1) U44'(tt, V1, V2) -> ISNATURAL(V1) U104'(tt, V1, V2) -> ISNATURAL(V1) The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(XS), N, XS) U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(N, XS)) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(Y), X, Y) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(Y), X) U231(tt, V2) -> U232(isLNatKind(V2)) U232(tt) -> tt U24(tt, X) -> X U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) U244(tt, V1, V2) -> U245(isLNat(V1), V2) U245(tt, V2) -> U246(isLNat(V2)) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U254(tt, V1, V2) -> U255(isNatural(V1), V2) U255(tt, V2) -> U256(isLNat(V2)) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(V2)) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(N), N) U282(tt, N) -> cons(N, natsFrom(s(N))) U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) U292(tt, N, XS) -> U293(isLNat(XS), N, XS) U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) U294(tt, N, XS) -> head(afterNth(N, XS)) U301(tt, X, Y) -> U302(isLNatKind(X), Y) U302(tt, Y) -> U303(isLNat(Y), Y) U303(tt, Y) -> U304(isLNatKind(Y), Y) U304(tt, Y) -> Y U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) U311(tt, XS) -> U312(isLNatKind(XS), XS) U312(tt, XS) -> pair(nil, XS) U32(tt, N, XS) -> U33(isLNat(XS), N, XS) U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(XS), N) U331(tt, N, XS) -> U332(isNaturalKind(N), XS) U332(tt, XS) -> U333(isLNat(XS), XS) U333(tt, XS) -> U334(isLNatKind(XS), XS) U334(tt, XS) -> XS U34(tt, N) -> N U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) U342(tt, N, XS) -> U343(isLNat(XS), N, XS) U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) U344(tt, N, XS) -> fst(splitAt(N, XS)) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(V1), V1) U82(tt, V1) -> U83(isPLNat(V1)) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, XS) isLNat(nil) -> tt isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) tail(cons(N, XS)) -> U331(isNatural(N), N, XS) take(N, XS) -> U341(isNatural(N), N, XS) Q is empty. ---------------------------------------- (15) QCSUsableRulesProof (EQUIVALENT) The following rules are not useable [DA_EMMES] and can be deleted: U11(tt, x0, x1) -> U12(isNaturalKind(x0), x0, x1) U12(tt, x0, x1) -> U13(isLNat(x1), x0, x1) U13(tt, x0, x1) -> U14(isLNatKind(x1), x0, x1) U14(tt, x0, x1) -> snd(splitAt(x0, x1)) U21(tt, x0, x1) -> U22(isLNatKind(x0), x0, x1) U22(tt, x0, x1) -> U23(isLNat(x1), x0, x1) U23(tt, x0, x1) -> U24(isLNatKind(x1), x0) U24(tt, x0) -> x0 U281(tt, x0) -> U282(isNaturalKind(x0), x0) U282(tt, x0) -> cons(x0, natsFrom(s(x0))) U291(tt, x0, x1) -> U292(isNaturalKind(x0), x0, x1) U292(tt, x0, x1) -> U293(isLNat(x1), x0, x1) U293(tt, x0, x1) -> U294(isLNatKind(x1), x0, x1) U294(tt, x0, x1) -> head(afterNth(x0, x1)) U301(tt, x0, x1) -> U302(isLNatKind(x0), x1) U302(tt, x0) -> U303(isLNat(x0), x0) U303(tt, x0) -> U304(isLNatKind(x0), x0) U304(tt, x0) -> x0 U31(tt, x0, x1) -> U32(isNaturalKind(x0), x0, x1) U311(tt, x0) -> U312(isLNatKind(x0), x0) U312(tt, x0) -> pair(nil, x0) U32(tt, x0, x1) -> U33(isLNat(x1), x0, x1) U321(tt, x0, x1, x2) -> U322(isNaturalKind(x0), x0, x1, x2) U322(tt, x0, x1, x2) -> U323(isNatural(x1), x0, x1, x2) U323(tt, x0, x1, x2) -> U324(isNaturalKind(x1), x0, x1, x2) U324(tt, x0, x1, x2) -> U325(isLNat(x2), x0, x1, x2) U325(tt, x0, x1, x2) -> U326(isLNatKind(x2), x0, x1, x2) U326(tt, x0, x1, x2) -> U327(splitAt(x0, x2), x1) U327(pair(x0, x1), x2) -> pair(cons(x2, x0), x1) U33(tt, x0, x1) -> U34(isLNatKind(x1), x0) U331(tt, x0, x1) -> U332(isNaturalKind(x0), x1) U332(tt, x0) -> U333(isLNat(x0), x0) U333(tt, x0) -> U334(isLNatKind(x0), x0) U334(tt, x0) -> x0 U34(tt, x0) -> x0 U341(tt, x0, x1) -> U342(isNaturalKind(x0), x0, x1) U342(tt, x0, x1) -> U343(isLNat(x1), x0, x1) U343(tt, x0, x1) -> U344(isLNatKind(x1), x0, x1) U344(tt, x0, x1) -> fst(splitAt(x0, x1)) afterNth(x0, x1) -> U11(isNatural(x0), x0, x1) fst(pair(x0, x1)) -> U21(isLNat(x0), x0, x1) head(cons(x0, x1)) -> U31(isNatural(x0), x0, x1) natsFrom(x0) -> U281(isNatural(x0), x0) sel(x0, x1) -> U291(isNatural(x0), x0, x1) snd(pair(x0, x1)) -> U301(isLNat(x0), x0, x1) splitAt(0, x0) -> U311(isLNat(x0), x0) splitAt(s(x0), cons(x1, x2)) -> U321(isNatural(x0), x0, x1, x2) tail(cons(x0, x1)) -> U331(isNatural(x0), x0, x1) take(x0, x1) -> U341(isNatural(x0), x0, x1) ---------------------------------------- (16) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. 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. The TRS P consists of the following rules: U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) U105'(tt, V2) -> ISLNAT(V2) ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) U45'(tt, V2) -> ISLNAT(V2) ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) U55'(tt, V2) -> ISLNAT(V2) ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) U62'(tt, V1) -> ISPLNAT(V1) ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) U245'(tt, V2) -> ISLNAT(V2) ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) U72'(tt, V1) -> ISNATURAL(V1) ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) U181'(tt, V1) -> U182'(isLNatKind(V1), V1) U182'(tt, V1) -> ISLNAT(V1) ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) U82'(tt, V1) -> ISPLNAT(V1) ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) U255'(tt, V2) -> ISLNAT(V2) ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) U91'(tt, V1) -> U92'(isLNatKind(V1), V1) U92'(tt, V1) -> ISLNAT(V1) ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) U254'(tt, V1, V2) -> ISNATURAL(V1) ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) U192'(tt, V1) -> ISNATURAL(V1) ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) U205'(tt, V2) -> ISLNAT(V2) U204'(tt, V1, V2) -> ISNATURAL(V1) U244'(tt, V1, V2) -> ISLNAT(V1) U54'(tt, V1, V2) -> ISNATURAL(V1) U44'(tt, V1, V2) -> ISNATURAL(V1) U104'(tt, V1, V2) -> ISNATURAL(V1) The TRS R consists of the following rules: isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) U231(tt, V2) -> U232(isLNatKind(V2)) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) U141(tt) -> tt isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) U271(tt, V2) -> U272(isLNatKind(V2)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U161(tt) -> tt U272(tt) -> tt U151(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U131(tt) -> tt U232(tt) -> tt U221(tt) -> tt U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U211(tt) -> tt U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) U181(tt, V1) -> U182(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) isLNat(nil) -> tt isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) U244(tt, V1, V2) -> U245(isLNat(V1), V2) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) U71(tt, V1) -> U72(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) U73(tt) -> tt isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) U81(tt, V1) -> U82(isPLNatKind(V1), V1) U82(tt, V1) -> U83(isPLNat(V1)) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U254(tt, V1, V2) -> U255(isNatural(V1), V2) U255(tt, V2) -> U256(isLNat(V2)) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U93(tt) -> tt U256(tt) -> tt U83(tt) -> tt U245(tt, V2) -> U246(isLNat(V2)) U246(tt) -> tt U63(tt) -> tt U56(tt) -> tt U206(tt) -> tt U193(tt) -> tt U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt U183(tt) -> tt Q is empty. ---------------------------------------- (17) QCSDPMuMonotonicPoloProof (EQUIVALENT) 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. Strictly oriented dependency pairs: U104'(tt, V1, V2) -> U105'(isNatural(V1), V2) U105'(tt, V2) -> ISLNAT(V2) U43'(tt, V1, V2) -> U44'(isLNatKind(V2), V1, V2) U45'(tt, V2) -> ISLNAT(V2) U51'(tt, V1, V2) -> U52'(isNaturalKind(V1), V1, V2) U52'(tt, V1, V2) -> U53'(isLNatKind(V2), V1, V2) ISLNAT(fst(V1)) -> U61'(isPLNatKind(V1), V1) U61'(tt, V1) -> U62'(isPLNatKind(V1), V1) ISPLNAT(pair(V1, V2)) -> U241'(isLNatKind(V1), V1, V2) U241'(tt, V1, V2) -> U242'(isLNatKind(V1), V1, V2) U242'(tt, V1, V2) -> U243'(isLNatKind(V2), V1, V2) U71'(tt, V1) -> U72'(isNaturalKind(V1), V1) U72'(tt, V1) -> ISNATURAL(V1) U181'(tt, V1) -> U182'(isLNatKind(V1), V1) U182'(tt, V1) -> ISLNAT(V1) U81'(tt, V1) -> U82'(isPLNatKind(V1), V1) U82'(tt, V1) -> ISPLNAT(V1) ISPLNAT(splitAt(V1, V2)) -> U251'(isNaturalKind(V1), V1, V2) U253'(tt, V1, V2) -> U254'(isLNatKind(V2), V1, V2) U254'(tt, V1, V2) -> U255'(isNatural(V1), V2) ISLNAT(tail(V1)) -> U91'(isLNatKind(V1), V1) U91'(tt, V1) -> U92'(isLNatKind(V1), V1) U254'(tt, V1, V2) -> ISNATURAL(V1) U191'(tt, V1) -> U192'(isNaturalKind(V1), V1) U192'(tt, V1) -> ISNATURAL(V1) ISNATURAL(sel(V1, V2)) -> U201'(isNaturalKind(V1), V1, V2) U44'(tt, V1, V2) -> ISNATURAL(V1) U104'(tt, V1, V2) -> ISNATURAL(V1) Strictly oriented rules of the TRS R: isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) isLNat(nil) -> tt U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) U244(tt, V1, V2) -> U245(isLNat(V1), V2) U71(tt, V1) -> U72(isNaturalKind(V1), V1) U73(tt) -> tt U82(tt, V1) -> U83(isPLNat(V1)) U254(tt, V1, V2) -> U255(isNatural(V1), V2) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U93(tt) -> tt U83(tt) -> tt U245(tt, V2) -> U246(isLNat(V2)) U63(tt) -> tt U183(tt) -> tt Used ordering: POLO with Polynomial interpretation [POLO]: POL(0) = 2 POL(ISLNAT(x_1)) = x_1 POL(ISNATURAL(x_1)) = x_1 POL(ISPLNAT(x_1)) = 2*x_1 POL(U101(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(U101'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 POL(U102(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 POL(U102'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(U103(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 POL(U103'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 POL(U104(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 POL(U104'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(U105(x_1, x_2)) = x_1 + x_2 POL(U105'(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U106(x_1)) = x_1 POL(U111(x_1, x_2)) = 2*x_1 POL(U112(x_1)) = 2*x_1 POL(U121(x_1, x_2)) = 2*x_1 POL(U122(x_1)) = x_1 POL(U131(x_1)) = 2*x_1 POL(U141(x_1)) = 2*x_1 POL(U151(x_1)) = 2*x_1 POL(U161(x_1)) = 2*x_1 POL(U171(x_1, x_2)) = x_1 POL(U172(x_1)) = 2*x_1 POL(U181(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U181'(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(U182(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(U182'(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(U183(x_1)) = 1 + x_1 POL(U191(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U191'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(U192(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U192'(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U193(x_1)) = x_1 POL(U201(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U201'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U202(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U202'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U203(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U203'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U204(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 POL(U204'(x_1, x_2, x_3)) = x_1 + 2*x_2 + x_3 POL(U205(x_1, x_2)) = x_1 + x_2 POL(U205'(x_1, x_2)) = x_1 + x_2 POL(U206(x_1)) = x_1 POL(U211(x_1)) = 2*x_1 POL(U221(x_1)) = 2*x_1 POL(U231(x_1, x_2)) = x_1 POL(U232(x_1)) = 2*x_1 POL(U241(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3 POL(U241'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 POL(U242(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3 POL(U242'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + x_3 POL(U243(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + x_3 POL(U243'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U244(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3 POL(U244'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U245(x_1, x_2)) = 1 + x_1 + x_2 POL(U245'(x_1, x_2)) = 2*x_1 + x_2 POL(U246(x_1)) = x_1 POL(U251(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 POL(U251'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(U252(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 POL(U252'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 POL(U253(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3 POL(U253'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 POL(U254(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 POL(U254'(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 POL(U255(x_1, x_2)) = x_1 + x_2 POL(U255'(x_1, x_2)) = x_1 + x_2 POL(U256(x_1)) = x_1 POL(U261(x_1, x_2)) = x_1 POL(U262(x_1)) = x_1 POL(U271(x_1, x_2)) = 2*x_1 POL(U272(x_1)) = x_1 POL(U41(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 POL(U41'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(U42(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 POL(U42'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 POL(U43(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3 POL(U43'(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + x_3 POL(U44(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U44'(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + x_3 POL(U45(x_1, x_2)) = x_1 + 2*x_2 POL(U45'(x_1, x_2)) = 1 + x_1 + x_2 POL(U46(x_1)) = 2*x_1 POL(U51(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 POL(U51'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 POL(U52(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U52'(x_1, x_2, x_3)) = 1 + 2*x_1 + 2*x_2 + 2*x_3 POL(U53(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3 POL(U53'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U54(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3 POL(U54'(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + x_3 POL(U55(x_1, x_2)) = x_1 + 2*x_2 POL(U55'(x_1, x_2)) = x_1 + x_2 POL(U56(x_1)) = 2*x_1 POL(U61(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(U61'(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U62(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(U62'(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U63(x_1)) = 1 + x_1 POL(U71(x_1, x_2)) = 2 + x_1 + 2*x_2 POL(U71'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(U72(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U72'(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U73(x_1)) = 1 + x_1 POL(U81(x_1, x_2)) = 2 + x_1 + x_2 POL(U81'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(U82(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(U82'(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 POL(U83(x_1)) = 1 + x_1 POL(U91(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(U91'(x_1, x_2)) = 1 + x_1 + 2*x_2 POL(U92(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(U92'(x_1, x_2)) = x_1 + 2*x_2 POL(U93(x_1)) = 1 + x_1 POL(afterNth(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(fst(x_1)) = 2 + 2*x_1 POL(head(x_1)) = 2 + 2*x_1 POL(isLNat(x_1)) = x_1 POL(isLNatKind(x_1)) = 0 POL(isNatural(x_1)) = 2*x_1 POL(isNaturalKind(x_1)) = 0 POL(isPLNat(x_1)) = x_1 POL(isPLNatKind(x_1)) = 0 POL(natsFrom(x_1)) = 2 + 2*x_1 POL(nil) = 2 POL(pair(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(s(x_1)) = 2 + 2*x_1 POL(sel(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(snd(x_1)) = 2 + 2*x_1 POL(splitAt(x_1, x_2)) = 2 + 2*x_1 + x_2 POL(tail(x_1)) = 2 + 2*x_1 POL(take(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 POL(tt) = 0 ---------------------------------------- (18) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. 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. The TRS P consists of the following rules: U102'(tt, V1, V2) -> U103'(isLNatKind(V2), V1, V2) U103'(tt, V1, V2) -> U104'(isLNatKind(V2), V1, V2) ISLNAT(afterNth(V1, V2)) -> U41'(isNaturalKind(V1), V1, V2) U41'(tt, V1, V2) -> U42'(isNaturalKind(V1), V1, V2) U42'(tt, V1, V2) -> U43'(isLNatKind(V2), V1, V2) U44'(tt, V1, V2) -> U45'(isNatural(V1), V2) ISLNAT(cons(V1, V2)) -> U51'(isNaturalKind(V1), V1, V2) U53'(tt, V1, V2) -> U54'(isLNatKind(V2), V1, V2) U54'(tt, V1, V2) -> U55'(isNatural(V1), V2) U55'(tt, V2) -> ISLNAT(V2) U62'(tt, V1) -> ISPLNAT(V1) U243'(tt, V1, V2) -> U244'(isLNatKind(V2), V1, V2) U244'(tt, V1, V2) -> U245'(isLNat(V1), V2) U245'(tt, V2) -> ISLNAT(V2) ISLNAT(natsFrom(V1)) -> U71'(isNaturalKind(V1), V1) ISNATURAL(head(V1)) -> U181'(isLNatKind(V1), V1) ISLNAT(snd(V1)) -> U81'(isPLNatKind(V1), V1) U251'(tt, V1, V2) -> U252'(isNaturalKind(V1), V1, V2) U252'(tt, V1, V2) -> U253'(isLNatKind(V2), V1, V2) U255'(tt, V2) -> ISLNAT(V2) U92'(tt, V1) -> ISLNAT(V1) ISLNAT(take(V1, V2)) -> U101'(isNaturalKind(V1), V1, V2) U101'(tt, V1, V2) -> U102'(isNaturalKind(V1), V1, V2) ISNATURAL(s(V1)) -> U191'(isNaturalKind(V1), V1) U201'(tt, V1, V2) -> U202'(isNaturalKind(V1), V1, V2) U202'(tt, V1, V2) -> U203'(isLNatKind(V2), V1, V2) U203'(tt, V1, V2) -> U204'(isLNatKind(V2), V1, V2) U204'(tt, V1, V2) -> U205'(isNatural(V1), V2) U205'(tt, V2) -> ISLNAT(V2) U204'(tt, V1, V2) -> ISNATURAL(V1) U244'(tt, V1, V2) -> ISLNAT(V1) U54'(tt, V1, V2) -> ISNATURAL(V1) The TRS R consists of the following rules: isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) U231(tt, V2) -> U232(isLNatKind(V2)) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) U141(tt) -> tt isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) U271(tt, V2) -> U272(isLNatKind(V2)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U161(tt) -> tt U272(tt) -> tt U151(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U131(tt) -> tt U232(tt) -> tt U221(tt) -> tt U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U211(tt) -> tt U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U181(tt, V1) -> U182(isLNatKind(V1), V1) isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) U81(tt, V1) -> U82(isPLNatKind(V1), V1) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U255(tt, V2) -> U256(isLNat(V2)) U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U256(tt) -> tt U246(tt) -> tt U56(tt) -> tt U206(tt) -> tt U193(tt) -> tt U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt Q is empty. ---------------------------------------- (19) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 32 less nodes. ---------------------------------------- (20) TRUE ---------------------------------------- (21) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1, U_1} are not replacing on any position. The TRS P consists of the following rules: NATSFROM(N) -> U281'(isNatural(N), N) U281'(tt, N) -> U282'(isNaturalKind(N), N) U282'(tt, N) -> U(N) U(s(x_0)) -> U(x_0) U(natsFrom(x_0)) -> U(x_0) U(natsFrom(s(x0))) -> NATSFROM(s(x0)) The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(XS), N, XS) U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(N, XS)) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(Y), X, Y) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(Y), X) U231(tt, V2) -> U232(isLNatKind(V2)) U232(tt) -> tt U24(tt, X) -> X U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) U244(tt, V1, V2) -> U245(isLNat(V1), V2) U245(tt, V2) -> U246(isLNat(V2)) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U254(tt, V1, V2) -> U255(isNatural(V1), V2) U255(tt, V2) -> U256(isLNat(V2)) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(V2)) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(N), N) U282(tt, N) -> cons(N, natsFrom(s(N))) U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) U292(tt, N, XS) -> U293(isLNat(XS), N, XS) U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) U294(tt, N, XS) -> head(afterNth(N, XS)) U301(tt, X, Y) -> U302(isLNatKind(X), Y) U302(tt, Y) -> U303(isLNat(Y), Y) U303(tt, Y) -> U304(isLNatKind(Y), Y) U304(tt, Y) -> Y U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) U311(tt, XS) -> U312(isLNatKind(XS), XS) U312(tt, XS) -> pair(nil, XS) U32(tt, N, XS) -> U33(isLNat(XS), N, XS) U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(XS), N) U331(tt, N, XS) -> U332(isNaturalKind(N), XS) U332(tt, XS) -> U333(isLNat(XS), XS) U333(tt, XS) -> U334(isLNatKind(XS), XS) U334(tt, XS) -> XS U34(tt, N) -> N U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) U342(tt, N, XS) -> U343(isLNat(XS), N, XS) U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) U344(tt, N, XS) -> fst(splitAt(N, XS)) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(V1), V1) U82(tt, V1) -> U83(isPLNat(V1)) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, XS) isLNat(nil) -> tt isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) tail(cons(N, XS)) -> U331(isNatural(N), N, XS) take(N, XS) -> U341(isNatural(N), N, XS) Q is empty. ---------------------------------------- (22) QCSDPSubtermProof (EQUIVALENT) We use the subterm processor [DA_EMMES]. The following pairs can be oriented strictly and are deleted. U(s(x_0)) -> U(x_0) U(natsFrom(x_0)) -> U(x_0) U(natsFrom(s(x0))) -> NATSFROM(s(x0)) The remaining pairs can at least be oriented weakly. NATSFROM(N) -> U281'(isNatural(N), N) U281'(tt, N) -> U282'(isNaturalKind(N), N) U282'(tt, N) -> U(N) Used ordering: Combined order from the following AFS and order. U281'(x1, x2) = x2 NATSFROM(x1) = x1 U282'(x1, x2) = x2 U(x1) = x1 Subterm Order ---------------------------------------- (23) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1, U_1} are not replacing on any position. The TRS P consists of the following rules: NATSFROM(N) -> U281'(isNatural(N), N) U281'(tt, N) -> U282'(isNaturalKind(N), N) U282'(tt, N) -> U(N) The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(XS), N, XS) U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(N, XS)) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(Y), X, Y) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(Y), X) U231(tt, V2) -> U232(isLNatKind(V2)) U232(tt) -> tt U24(tt, X) -> X U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) U244(tt, V1, V2) -> U245(isLNat(V1), V2) U245(tt, V2) -> U246(isLNat(V2)) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U254(tt, V1, V2) -> U255(isNatural(V1), V2) U255(tt, V2) -> U256(isLNat(V2)) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(V2)) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(N), N) U282(tt, N) -> cons(N, natsFrom(s(N))) U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) U292(tt, N, XS) -> U293(isLNat(XS), N, XS) U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) U294(tt, N, XS) -> head(afterNth(N, XS)) U301(tt, X, Y) -> U302(isLNatKind(X), Y) U302(tt, Y) -> U303(isLNat(Y), Y) U303(tt, Y) -> U304(isLNatKind(Y), Y) U304(tt, Y) -> Y U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) U311(tt, XS) -> U312(isLNatKind(XS), XS) U312(tt, XS) -> pair(nil, XS) U32(tt, N, XS) -> U33(isLNat(XS), N, XS) U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(XS), N) U331(tt, N, XS) -> U332(isNaturalKind(N), XS) U332(tt, XS) -> U333(isLNat(XS), XS) U333(tt, XS) -> U334(isLNatKind(XS), XS) U334(tt, XS) -> XS U34(tt, N) -> N U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) U342(tt, N, XS) -> U343(isLNat(XS), N, XS) U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) U344(tt, N, XS) -> fst(splitAt(N, XS)) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(V1), V1) U82(tt, V1) -> U83(isPLNat(V1)) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, XS) isLNat(nil) -> tt isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) tail(cons(N, XS)) -> U331(isNatural(N), N, XS) take(N, XS) -> U341(isNatural(N), N, XS) Q is empty. ---------------------------------------- (24) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes. ---------------------------------------- (25) TRUE ---------------------------------------- (26) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1} are not replacing on any position. The TRS P consists of the following rules: U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) U326'(tt, N, X, XS) -> SPLITAT(N, XS) SPLITAT(s(N), cons(X, XS)) -> U321'(isNatural(N), N, X, XS) U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(XS), N, XS) U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(N, XS)) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(Y), X, Y) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(Y), X) U231(tt, V2) -> U232(isLNatKind(V2)) U232(tt) -> tt U24(tt, X) -> X U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) U244(tt, V1, V2) -> U245(isLNat(V1), V2) U245(tt, V2) -> U246(isLNat(V2)) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U254(tt, V1, V2) -> U255(isNatural(V1), V2) U255(tt, V2) -> U256(isLNat(V2)) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(V2)) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(N), N) U282(tt, N) -> cons(N, natsFrom(s(N))) U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) U292(tt, N, XS) -> U293(isLNat(XS), N, XS) U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) U294(tt, N, XS) -> head(afterNth(N, XS)) U301(tt, X, Y) -> U302(isLNatKind(X), Y) U302(tt, Y) -> U303(isLNat(Y), Y) U303(tt, Y) -> U304(isLNatKind(Y), Y) U304(tt, Y) -> Y U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) U311(tt, XS) -> U312(isLNatKind(XS), XS) U312(tt, XS) -> pair(nil, XS) U32(tt, N, XS) -> U33(isLNat(XS), N, XS) U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(XS), N) U331(tt, N, XS) -> U332(isNaturalKind(N), XS) U332(tt, XS) -> U333(isLNat(XS), XS) U333(tt, XS) -> U334(isLNatKind(XS), XS) U334(tt, XS) -> XS U34(tt, N) -> N U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) U342(tt, N, XS) -> U343(isLNat(XS), N, XS) U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) U344(tt, N, XS) -> fst(splitAt(N, XS)) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(V1), V1) U82(tt, V1) -> U83(isPLNat(V1)) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, XS) isLNat(nil) -> tt isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) tail(cons(N, XS)) -> U331(isNatural(N), N, XS) take(N, XS) -> U341(isNatural(N), N, XS) Q is empty. ---------------------------------------- (27) QCSDPSubtermProof (EQUIVALENT) We use the subterm processor [DA_EMMES]. The following pairs can be oriented strictly and are deleted. SPLITAT(s(N), cons(X, XS)) -> U321'(isNatural(N), N, X, XS) The remaining pairs can at least be oriented weakly. U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) U326'(tt, N, X, XS) -> SPLITAT(N, XS) U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) Used ordering: Combined order from the following AFS and order. U325'(x1, x2, x3, x4) = x2 U324'(x1, x2, x3, x4) = x2 U326'(x1, x2, x3, x4) = x2 SPLITAT(x1, x2) = x1 U321'(x1, x2, x3, x4) = x2 U322'(x1, x2, x3, x4) = x2 U323'(x1, x2, x3, x4) = x2 Subterm Order ---------------------------------------- (28) Obligation: Q-restricted context-sensitive dependency pair problem: 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. 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}. The symbols in {isNaturalKind_1, isLNatKind_1, isNatural_1, isLNat_1, isPLNatKind_1, isPLNat_1} are not replacing on any position. The TRS P consists of the following rules: U324'(tt, N, X, XS) -> U325'(isLNat(XS), N, X, XS) U325'(tt, N, X, XS) -> U326'(isLNatKind(XS), N, X, XS) U326'(tt, N, X, XS) -> SPLITAT(N, XS) U321'(tt, N, X, XS) -> U322'(isNaturalKind(N), N, X, XS) U322'(tt, N, X, XS) -> U323'(isNatural(X), N, X, XS) U323'(tt, N, X, XS) -> U324'(isNaturalKind(X), N, X, XS) The TRS R consists of the following rules: U101(tt, V1, V2) -> U102(isNaturalKind(V1), V1, V2) U102(tt, V1, V2) -> U103(isLNatKind(V2), V1, V2) U103(tt, V1, V2) -> U104(isLNatKind(V2), V1, V2) U104(tt, V1, V2) -> U105(isNatural(V1), V2) U105(tt, V2) -> U106(isLNat(V2)) U106(tt) -> tt U11(tt, N, XS) -> U12(isNaturalKind(N), N, XS) U111(tt, V2) -> U112(isLNatKind(V2)) U112(tt) -> tt U12(tt, N, XS) -> U13(isLNat(XS), N, XS) U121(tt, V2) -> U122(isLNatKind(V2)) U122(tt) -> tt U13(tt, N, XS) -> U14(isLNatKind(XS), N, XS) U131(tt) -> tt U14(tt, N, XS) -> snd(splitAt(N, XS)) U141(tt) -> tt U151(tt) -> tt U161(tt) -> tt U171(tt, V2) -> U172(isLNatKind(V2)) U172(tt) -> tt U181(tt, V1) -> U182(isLNatKind(V1), V1) U182(tt, V1) -> U183(isLNat(V1)) U183(tt) -> tt U191(tt, V1) -> U192(isNaturalKind(V1), V1) U192(tt, V1) -> U193(isNatural(V1)) U193(tt) -> tt U201(tt, V1, V2) -> U202(isNaturalKind(V1), V1, V2) U202(tt, V1, V2) -> U203(isLNatKind(V2), V1, V2) U203(tt, V1, V2) -> U204(isLNatKind(V2), V1, V2) U204(tt, V1, V2) -> U205(isNatural(V1), V2) U205(tt, V2) -> U206(isLNat(V2)) U206(tt) -> tt U21(tt, X, Y) -> U22(isLNatKind(X), X, Y) U211(tt) -> tt U22(tt, X, Y) -> U23(isLNat(Y), X, Y) U221(tt) -> tt U23(tt, X, Y) -> U24(isLNatKind(Y), X) U231(tt, V2) -> U232(isLNatKind(V2)) U232(tt) -> tt U24(tt, X) -> X U241(tt, V1, V2) -> U242(isLNatKind(V1), V1, V2) U242(tt, V1, V2) -> U243(isLNatKind(V2), V1, V2) U243(tt, V1, V2) -> U244(isLNatKind(V2), V1, V2) U244(tt, V1, V2) -> U245(isLNat(V1), V2) U245(tt, V2) -> U246(isLNat(V2)) U246(tt) -> tt U251(tt, V1, V2) -> U252(isNaturalKind(V1), V1, V2) U252(tt, V1, V2) -> U253(isLNatKind(V2), V1, V2) U253(tt, V1, V2) -> U254(isLNatKind(V2), V1, V2) U254(tt, V1, V2) -> U255(isNatural(V1), V2) U255(tt, V2) -> U256(isLNat(V2)) U256(tt) -> tt U261(tt, V2) -> U262(isLNatKind(V2)) U262(tt) -> tt U271(tt, V2) -> U272(isLNatKind(V2)) U272(tt) -> tt U281(tt, N) -> U282(isNaturalKind(N), N) U282(tt, N) -> cons(N, natsFrom(s(N))) U291(tt, N, XS) -> U292(isNaturalKind(N), N, XS) U292(tt, N, XS) -> U293(isLNat(XS), N, XS) U293(tt, N, XS) -> U294(isLNatKind(XS), N, XS) U294(tt, N, XS) -> head(afterNth(N, XS)) U301(tt, X, Y) -> U302(isLNatKind(X), Y) U302(tt, Y) -> U303(isLNat(Y), Y) U303(tt, Y) -> U304(isLNatKind(Y), Y) U304(tt, Y) -> Y U31(tt, N, XS) -> U32(isNaturalKind(N), N, XS) U311(tt, XS) -> U312(isLNatKind(XS), XS) U312(tt, XS) -> pair(nil, XS) U32(tt, N, XS) -> U33(isLNat(XS), N, XS) U321(tt, N, X, XS) -> U322(isNaturalKind(N), N, X, XS) U322(tt, N, X, XS) -> U323(isNatural(X), N, X, XS) U323(tt, N, X, XS) -> U324(isNaturalKind(X), N, X, XS) U324(tt, N, X, XS) -> U325(isLNat(XS), N, X, XS) U325(tt, N, X, XS) -> U326(isLNatKind(XS), N, X, XS) U326(tt, N, X, XS) -> U327(splitAt(N, XS), X) U327(pair(YS, ZS), X) -> pair(cons(X, YS), ZS) U33(tt, N, XS) -> U34(isLNatKind(XS), N) U331(tt, N, XS) -> U332(isNaturalKind(N), XS) U332(tt, XS) -> U333(isLNat(XS), XS) U333(tt, XS) -> U334(isLNatKind(XS), XS) U334(tt, XS) -> XS U34(tt, N) -> N U341(tt, N, XS) -> U342(isNaturalKind(N), N, XS) U342(tt, N, XS) -> U343(isLNat(XS), N, XS) U343(tt, N, XS) -> U344(isLNatKind(XS), N, XS) U344(tt, N, XS) -> fst(splitAt(N, XS)) U41(tt, V1, V2) -> U42(isNaturalKind(V1), V1, V2) U42(tt, V1, V2) -> U43(isLNatKind(V2), V1, V2) U43(tt, V1, V2) -> U44(isLNatKind(V2), V1, V2) U44(tt, V1, V2) -> U45(isNatural(V1), V2) U45(tt, V2) -> U46(isLNat(V2)) U46(tt) -> tt U51(tt, V1, V2) -> U52(isNaturalKind(V1), V1, V2) U52(tt, V1, V2) -> U53(isLNatKind(V2), V1, V2) U53(tt, V1, V2) -> U54(isLNatKind(V2), V1, V2) U54(tt, V1, V2) -> U55(isNatural(V1), V2) U55(tt, V2) -> U56(isLNat(V2)) U56(tt) -> tt U61(tt, V1) -> U62(isPLNatKind(V1), V1) U62(tt, V1) -> U63(isPLNat(V1)) U63(tt) -> tt U71(tt, V1) -> U72(isNaturalKind(V1), V1) U72(tt, V1) -> U73(isNatural(V1)) U73(tt) -> tt U81(tt, V1) -> U82(isPLNatKind(V1), V1) U82(tt, V1) -> U83(isPLNat(V1)) U83(tt) -> tt U91(tt, V1) -> U92(isLNatKind(V1), V1) U92(tt, V1) -> U93(isLNat(V1)) U93(tt) -> tt afterNth(N, XS) -> U11(isNatural(N), N, XS) fst(pair(X, Y)) -> U21(isLNat(X), X, Y) head(cons(N, XS)) -> U31(isNatural(N), N, XS) isLNat(nil) -> tt isLNat(afterNth(V1, V2)) -> U41(isNaturalKind(V1), V1, V2) isLNat(cons(V1, V2)) -> U51(isNaturalKind(V1), V1, V2) isLNat(fst(V1)) -> U61(isPLNatKind(V1), V1) isLNat(natsFrom(V1)) -> U71(isNaturalKind(V1), V1) isLNat(snd(V1)) -> U81(isPLNatKind(V1), V1) isLNat(tail(V1)) -> U91(isLNatKind(V1), V1) isLNat(take(V1, V2)) -> U101(isNaturalKind(V1), V1, V2) isLNatKind(nil) -> tt isLNatKind(afterNth(V1, V2)) -> U111(isNaturalKind(V1), V2) isLNatKind(cons(V1, V2)) -> U121(isNaturalKind(V1), V2) isLNatKind(fst(V1)) -> U131(isPLNatKind(V1)) isLNatKind(natsFrom(V1)) -> U141(isNaturalKind(V1)) isLNatKind(snd(V1)) -> U151(isPLNatKind(V1)) isLNatKind(tail(V1)) -> U161(isLNatKind(V1)) isLNatKind(take(V1, V2)) -> U171(isNaturalKind(V1), V2) isNatural(0) -> tt isNatural(head(V1)) -> U181(isLNatKind(V1), V1) isNatural(s(V1)) -> U191(isNaturalKind(V1), V1) isNatural(sel(V1, V2)) -> U201(isNaturalKind(V1), V1, V2) isNaturalKind(0) -> tt isNaturalKind(head(V1)) -> U211(isLNatKind(V1)) isNaturalKind(s(V1)) -> U221(isNaturalKind(V1)) isNaturalKind(sel(V1, V2)) -> U231(isNaturalKind(V1), V2) isPLNat(pair(V1, V2)) -> U241(isLNatKind(V1), V1, V2) isPLNat(splitAt(V1, V2)) -> U251(isNaturalKind(V1), V1, V2) isPLNatKind(pair(V1, V2)) -> U261(isLNatKind(V1), V2) isPLNatKind(splitAt(V1, V2)) -> U271(isNaturalKind(V1), V2) natsFrom(N) -> U281(isNatural(N), N) sel(N, XS) -> U291(isNatural(N), N, XS) snd(pair(X, Y)) -> U301(isLNat(X), X, Y) splitAt(0, XS) -> U311(isLNat(XS), XS) splitAt(s(N), cons(X, XS)) -> U321(isNatural(N), N, X, XS) tail(cons(N, XS)) -> U331(isNatural(N), N, XS) take(N, XS) -> U341(isNatural(N), N, XS) Q is empty. ---------------------------------------- (29) QCSDependencyGraphProof (EQUIVALENT) The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 6 less nodes. ---------------------------------------- (30) TRUE