9.78/3.21 YES 9.78/3.24 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 9.78/3.24 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.78/3.24 9.78/3.24 9.78/3.24 Termination of the given CSR could be proven: 9.78/3.24 9.78/3.24 (0) CSR 9.78/3.24 (1) CSRRRRProof [EQUIVALENT, 158 ms] 9.78/3.24 (2) CSR 9.78/3.24 (3) CSRRRRProof [EQUIVALENT, 47 ms] 9.78/3.24 (4) CSR 9.78/3.24 (5) CSRRRRProof [EQUIVALENT, 0 ms] 9.78/3.24 (6) CSR 9.78/3.24 (7) CSDependencyPairsProof [EQUIVALENT, 117 ms] 9.78/3.24 (8) QCSDP 9.78/3.24 (9) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 9.78/3.24 (10) AND 9.78/3.24 (11) QCSDP 9.78/3.24 (12) QCSUsableRulesProof [EQUIVALENT, 0 ms] 9.78/3.24 (13) QCSDP 9.78/3.24 (14) QCSDPMuMonotonicPoloProof [EQUIVALENT, 0 ms] 9.78/3.24 (15) QCSDP 9.78/3.24 (16) PIsEmptyProof [EQUIVALENT, 0 ms] 9.78/3.24 (17) YES 9.78/3.24 (18) QCSDP 9.78/3.24 (19) QCSUsableRulesProof [EQUIVALENT, 10 ms] 9.78/3.24 (20) QCSDP 9.78/3.24 (21) QCSDPMuMonotonicPoloProof [EQUIVALENT, 59 ms] 9.78/3.24 (22) QCSDP 9.78/3.24 (23) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 9.78/3.24 (24) QCSDP 9.78/3.24 (25) QCSDPSubtermProof [EQUIVALENT, 0 ms] 9.78/3.24 (26) QCSDP 9.78/3.24 (27) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 9.78/3.24 (28) TRUE 9.78/3.24 (29) QCSDP 9.78/3.24 (30) QCSDPSubtermProof [EQUIVALENT, 0 ms] 9.78/3.24 (31) QCSDP 9.78/3.24 (32) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 9.78/3.24 (33) TRUE 9.78/3.24 (34) QCSDP 9.78/3.24 (35) QCSDPReductionPairProof [EQUIVALENT, 82 ms] 9.78/3.24 (36) QCSDP 9.78/3.24 (37) QCSDependencyGraphProof [EQUIVALENT, 0 ms] 9.78/3.24 (38) TRUE 9.78/3.24 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (0) 9.78/3.24 Obligation: 9.78/3.24 Context-sensitive rewrite system: 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U121(tt, IL) -> U122(isNatIListKind(IL)) 9.78/3.24 U122(tt) -> nil 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(nil) -> 0 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(0, IL) -> U121(isNatIList(IL), IL) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 The replacement map contains the following entries: 9.78/3.24 9.78/3.24 zeros: empty set 9.78/3.24 cons: {1} 9.78/3.24 0: empty set 9.78/3.24 U101: {1} 9.78/3.24 tt: empty set 9.78/3.24 U102: {1} 9.78/3.24 isNatKind: empty set 9.78/3.24 U103: {1} 9.78/3.24 isNatIListKind: empty set 9.78/3.24 U104: {1} 9.78/3.24 U105: {1} 9.78/3.24 isNat: empty set 9.78/3.24 U106: {1} 9.78/3.24 isNatIList: empty set 9.78/3.24 U11: {1} 9.78/3.24 U12: {1} 9.78/3.24 U111: {1} 9.78/3.24 U112: {1} 9.78/3.24 U113: {1} 9.78/3.24 U114: {1} 9.78/3.24 s: {1} 9.78/3.24 length: {1} 9.78/3.24 U13: {1} 9.78/3.24 isNatList: empty set 9.78/3.24 U121: {1} 9.78/3.24 U122: {1} 9.78/3.24 nil: empty set 9.78/3.24 U131: {1} 9.78/3.24 U132: {1} 9.78/3.24 U133: {1} 9.78/3.24 U134: {1} 9.78/3.24 U135: {1} 9.78/3.24 U136: {1} 9.78/3.24 take: {1, 2} 9.78/3.24 U21: {1} 9.78/3.24 U22: {1} 9.78/3.24 U23: {1} 9.78/3.24 U31: {1} 9.78/3.24 U32: {1} 9.78/3.24 U33: {1} 9.78/3.24 U41: {1} 9.78/3.24 U42: {1} 9.78/3.24 U43: {1} 9.78/3.24 U44: {1} 9.78/3.24 U45: {1} 9.78/3.24 U46: {1} 9.78/3.24 U51: {1} 9.78/3.24 U52: {1} 9.78/3.24 U61: {1} 9.78/3.24 U62: {1} 9.78/3.24 U71: {1} 9.78/3.24 U81: {1} 9.78/3.24 U91: {1} 9.78/3.24 U92: {1} 9.78/3.24 U93: {1} 9.78/3.24 U94: {1} 9.78/3.24 U95: {1} 9.78/3.24 U96: {1} 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (1) CSRRRRProof (EQUIVALENT) 9.78/3.24 The following CSR is given: Context-sensitive rewrite system: 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U121(tt, IL) -> U122(isNatIListKind(IL)) 9.78/3.24 U122(tt) -> nil 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(nil) -> 0 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(0, IL) -> U121(isNatIList(IL), IL) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 The replacement map contains the following entries: 9.78/3.24 9.78/3.24 zeros: empty set 9.78/3.24 cons: {1} 9.78/3.24 0: empty set 9.78/3.24 U101: {1} 9.78/3.24 tt: empty set 9.78/3.24 U102: {1} 9.78/3.24 isNatKind: empty set 9.78/3.24 U103: {1} 9.78/3.24 isNatIListKind: empty set 9.78/3.24 U104: {1} 9.78/3.24 U105: {1} 9.78/3.24 isNat: empty set 9.78/3.24 U106: {1} 9.78/3.24 isNatIList: empty set 9.78/3.24 U11: {1} 9.78/3.24 U12: {1} 9.78/3.24 U111: {1} 9.78/3.24 U112: {1} 9.78/3.24 U113: {1} 9.78/3.24 U114: {1} 9.78/3.24 s: {1} 9.78/3.24 length: {1} 9.78/3.24 U13: {1} 9.78/3.24 isNatList: empty set 9.78/3.24 U121: {1} 9.78/3.24 U122: {1} 9.78/3.24 nil: empty set 9.78/3.24 U131: {1} 9.78/3.24 U132: {1} 9.78/3.24 U133: {1} 9.78/3.24 U134: {1} 9.78/3.24 U135: {1} 9.78/3.24 U136: {1} 9.78/3.24 take: {1, 2} 9.78/3.24 U21: {1} 9.78/3.24 U22: {1} 9.78/3.24 U23: {1} 9.78/3.24 U31: {1} 9.78/3.24 U32: {1} 9.78/3.24 U33: {1} 9.78/3.24 U41: {1} 9.78/3.24 U42: {1} 9.78/3.24 U43: {1} 9.78/3.24 U44: {1} 9.78/3.24 U45: {1} 9.78/3.24 U46: {1} 9.78/3.24 U51: {1} 9.78/3.24 U52: {1} 9.78/3.24 U61: {1} 9.78/3.24 U62: {1} 9.78/3.24 U71: {1} 9.78/3.24 U81: {1} 9.78/3.24 U91: {1} 9.78/3.24 U92: {1} 9.78/3.24 U93: {1} 9.78/3.24 U94: {1} 9.78/3.24 U95: {1} 9.78/3.24 U96: {1} 9.78/3.24 Used ordering: 9.78/3.24 Polynomial interpretation [POLO]: 9.78/3.24 9.78/3.24 POL(0) = 0 9.78/3.24 POL(U101(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U102(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U103(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U104(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U105(x_1, x_2)) = x_1 9.78/3.24 POL(U106(x_1)) = x_1 9.78/3.24 POL(U11(x_1, x_2)) = x_1 9.78/3.24 POL(U111(x_1, x_2, x_3)) = x_1 + x_2 + x_3 9.78/3.24 POL(U112(x_1, x_2, x_3)) = x_1 + x_2 + x_3 9.78/3.24 POL(U113(x_1, x_2, x_3)) = x_1 + x_2 + x_3 9.78/3.24 POL(U114(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(U12(x_1, x_2)) = x_1 9.78/3.24 POL(U121(x_1, x_2)) = 1 + x_1 + x_2 9.78/3.24 POL(U122(x_1)) = x_1 9.78/3.24 POL(U13(x_1)) = x_1 9.78/3.24 POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U21(x_1, x_2)) = x_1 9.78/3.24 POL(U22(x_1, x_2)) = x_1 9.78/3.24 POL(U23(x_1)) = x_1 9.78/3.24 POL(U31(x_1, x_2)) = x_1 9.78/3.24 POL(U32(x_1, x_2)) = x_1 9.78/3.24 POL(U33(x_1)) = x_1 9.78/3.24 POL(U41(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U42(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U43(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U44(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U45(x_1, x_2)) = x_1 9.78/3.24 POL(U46(x_1)) = x_1 9.78/3.24 POL(U51(x_1, x_2)) = x_1 9.78/3.24 POL(U52(x_1)) = x_1 9.78/3.24 POL(U61(x_1, x_2)) = x_1 9.78/3.24 POL(U62(x_1)) = x_1 9.78/3.24 POL(U71(x_1)) = x_1 9.78/3.24 POL(U81(x_1)) = x_1 9.78/3.24 POL(U91(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U92(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U93(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U94(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U95(x_1, x_2)) = x_1 9.78/3.24 POL(U96(x_1)) = x_1 9.78/3.24 POL(cons(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(isNat(x_1)) = 0 9.78/3.24 POL(isNatIList(x_1)) = 0 9.78/3.24 POL(isNatIListKind(x_1)) = 0 9.78/3.24 POL(isNatKind(x_1)) = 0 9.78/3.24 POL(isNatList(x_1)) = 0 9.78/3.24 POL(length(x_1)) = x_1 9.78/3.24 POL(nil) = 0 9.78/3.24 POL(s(x_1)) = x_1 9.78/3.24 POL(take(x_1, x_2)) = 1 + x_1 + x_2 9.78/3.24 POL(tt) = 0 9.78/3.24 POL(zeros) = 1 9.78/3.24 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 9.78/3.24 9.78/3.24 U121(tt, IL) -> U122(isNatIListKind(IL)) 9.78/3.24 9.78/3.24 9.78/3.24 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (2) 9.78/3.24 Obligation: 9.78/3.24 Context-sensitive rewrite system: 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U122(tt) -> nil 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(nil) -> 0 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(0, IL) -> U121(isNatIList(IL), IL) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 The replacement map contains the following entries: 9.78/3.24 9.78/3.24 zeros: empty set 9.78/3.24 cons: {1} 9.78/3.24 0: empty set 9.78/3.24 U101: {1} 9.78/3.24 tt: empty set 9.78/3.24 U102: {1} 9.78/3.24 isNatKind: empty set 9.78/3.24 U103: {1} 9.78/3.24 isNatIListKind: empty set 9.78/3.24 U104: {1} 9.78/3.24 U105: {1} 9.78/3.24 isNat: empty set 9.78/3.24 U106: {1} 9.78/3.24 isNatIList: empty set 9.78/3.24 U11: {1} 9.78/3.24 U12: {1} 9.78/3.24 U111: {1} 9.78/3.24 U112: {1} 9.78/3.24 U113: {1} 9.78/3.24 U114: {1} 9.78/3.24 s: {1} 9.78/3.24 length: {1} 9.78/3.24 U13: {1} 9.78/3.24 isNatList: empty set 9.78/3.24 U121: {1} 9.78/3.24 U122: {1} 9.78/3.24 nil: empty set 9.78/3.24 U131: {1} 9.78/3.24 U132: {1} 9.78/3.24 U133: {1} 9.78/3.24 U134: {1} 9.78/3.24 U135: {1} 9.78/3.24 U136: {1} 9.78/3.24 take: {1, 2} 9.78/3.24 U21: {1} 9.78/3.24 U22: {1} 9.78/3.24 U23: {1} 9.78/3.24 U31: {1} 9.78/3.24 U32: {1} 9.78/3.24 U33: {1} 9.78/3.24 U41: {1} 9.78/3.24 U42: {1} 9.78/3.24 U43: {1} 9.78/3.24 U44: {1} 9.78/3.24 U45: {1} 9.78/3.24 U46: {1} 9.78/3.24 U51: {1} 9.78/3.24 U52: {1} 9.78/3.24 U61: {1} 9.78/3.24 U62: {1} 9.78/3.24 U71: {1} 9.78/3.24 U81: {1} 9.78/3.24 U91: {1} 9.78/3.24 U92: {1} 9.78/3.24 U93: {1} 9.78/3.24 U94: {1} 9.78/3.24 U95: {1} 9.78/3.24 U96: {1} 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (3) CSRRRRProof (EQUIVALENT) 9.78/3.24 The following CSR is given: Context-sensitive rewrite system: 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U122(tt) -> nil 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(nil) -> 0 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(0, IL) -> U121(isNatIList(IL), IL) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 The replacement map contains the following entries: 9.78/3.24 9.78/3.24 zeros: empty set 9.78/3.24 cons: {1} 9.78/3.24 0: empty set 9.78/3.24 U101: {1} 9.78/3.24 tt: empty set 9.78/3.24 U102: {1} 9.78/3.24 isNatKind: empty set 9.78/3.24 U103: {1} 9.78/3.24 isNatIListKind: empty set 9.78/3.24 U104: {1} 9.78/3.24 U105: {1} 9.78/3.24 isNat: empty set 9.78/3.24 U106: {1} 9.78/3.24 isNatIList: empty set 9.78/3.24 U11: {1} 9.78/3.24 U12: {1} 9.78/3.24 U111: {1} 9.78/3.24 U112: {1} 9.78/3.24 U113: {1} 9.78/3.24 U114: {1} 9.78/3.24 s: {1} 9.78/3.24 length: {1} 9.78/3.24 U13: {1} 9.78/3.24 isNatList: empty set 9.78/3.24 U121: {1} 9.78/3.24 U122: {1} 9.78/3.24 nil: empty set 9.78/3.24 U131: {1} 9.78/3.24 U132: {1} 9.78/3.24 U133: {1} 9.78/3.24 U134: {1} 9.78/3.24 U135: {1} 9.78/3.24 U136: {1} 9.78/3.24 take: {1, 2} 9.78/3.24 U21: {1} 9.78/3.24 U22: {1} 9.78/3.24 U23: {1} 9.78/3.24 U31: {1} 9.78/3.24 U32: {1} 9.78/3.24 U33: {1} 9.78/3.24 U41: {1} 9.78/3.24 U42: {1} 9.78/3.24 U43: {1} 9.78/3.24 U44: {1} 9.78/3.24 U45: {1} 9.78/3.24 U46: {1} 9.78/3.24 U51: {1} 9.78/3.24 U52: {1} 9.78/3.24 U61: {1} 9.78/3.24 U62: {1} 9.78/3.24 U71: {1} 9.78/3.24 U81: {1} 9.78/3.24 U91: {1} 9.78/3.24 U92: {1} 9.78/3.24 U93: {1} 9.78/3.24 U94: {1} 9.78/3.24 U95: {1} 9.78/3.24 U96: {1} 9.78/3.24 Used ordering: 9.78/3.24 Polynomial interpretation [POLO]: 9.78/3.24 9.78/3.24 POL(0) = 0 9.78/3.24 POL(U101(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U102(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U103(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U104(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U105(x_1, x_2)) = x_1 9.78/3.24 POL(U106(x_1)) = x_1 9.78/3.24 POL(U11(x_1, x_2)) = x_1 9.78/3.24 POL(U111(x_1, x_2, x_3)) = x_1 + x_2 + x_3 9.78/3.24 POL(U112(x_1, x_2, x_3)) = x_1 + x_2 + x_3 9.78/3.24 POL(U113(x_1, x_2, x_3)) = x_1 + x_2 + x_3 9.78/3.24 POL(U114(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(U12(x_1, x_2)) = x_1 9.78/3.24 POL(U121(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(U122(x_1)) = 1 + x_1 9.78/3.24 POL(U13(x_1)) = x_1 9.78/3.24 POL(U131(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U132(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U133(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U134(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U135(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U136(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U21(x_1, x_2)) = x_1 9.78/3.24 POL(U22(x_1, x_2)) = x_1 9.78/3.24 POL(U23(x_1)) = x_1 9.78/3.24 POL(U31(x_1, x_2)) = x_1 9.78/3.24 POL(U32(x_1, x_2)) = x_1 9.78/3.24 POL(U33(x_1)) = x_1 9.78/3.24 POL(U41(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U42(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U43(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U44(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U45(x_1, x_2)) = x_1 9.78/3.24 POL(U46(x_1)) = x_1 9.78/3.24 POL(U51(x_1, x_2)) = x_1 9.78/3.24 POL(U52(x_1)) = x_1 9.78/3.24 POL(U61(x_1, x_2)) = x_1 9.78/3.24 POL(U62(x_1)) = x_1 9.78/3.24 POL(U71(x_1)) = x_1 9.78/3.24 POL(U81(x_1)) = x_1 9.78/3.24 POL(U91(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U92(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U93(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U94(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U95(x_1, x_2)) = x_1 9.78/3.24 POL(U96(x_1)) = x_1 9.78/3.24 POL(cons(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(isNat(x_1)) = 1 9.78/3.24 POL(isNatIList(x_1)) = 1 9.78/3.24 POL(isNatIListKind(x_1)) = 1 9.78/3.24 POL(isNatKind(x_1)) = 1 9.78/3.24 POL(isNatList(x_1)) = 1 9.78/3.24 POL(length(x_1)) = 1 + x_1 9.78/3.24 POL(nil) = 0 9.78/3.24 POL(s(x_1)) = x_1 9.78/3.24 POL(take(x_1, x_2)) = 1 + x_1 + x_2 9.78/3.24 POL(tt) = 1 9.78/3.24 POL(zeros) = 1 9.78/3.24 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 9.78/3.24 9.78/3.24 U122(tt) -> nil 9.78/3.24 length(nil) -> 0 9.78/3.24 9.78/3.24 9.78/3.24 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (4) 9.78/3.24 Obligation: 9.78/3.24 Context-sensitive rewrite system: 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(0, IL) -> U121(isNatIList(IL), IL) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 The replacement map contains the following entries: 9.78/3.24 9.78/3.24 zeros: empty set 9.78/3.24 cons: {1} 9.78/3.24 0: empty set 9.78/3.24 U101: {1} 9.78/3.24 tt: empty set 9.78/3.24 U102: {1} 9.78/3.24 isNatKind: empty set 9.78/3.24 U103: {1} 9.78/3.24 isNatIListKind: empty set 9.78/3.24 U104: {1} 9.78/3.24 U105: {1} 9.78/3.24 isNat: empty set 9.78/3.24 U106: {1} 9.78/3.24 isNatIList: empty set 9.78/3.24 U11: {1} 9.78/3.24 U12: {1} 9.78/3.24 U111: {1} 9.78/3.24 U112: {1} 9.78/3.24 U113: {1} 9.78/3.24 U114: {1} 9.78/3.24 s: {1} 9.78/3.24 length: {1} 9.78/3.24 U13: {1} 9.78/3.24 isNatList: empty set 9.78/3.24 U121: {1} 9.78/3.24 nil: empty set 9.78/3.24 U131: {1} 9.78/3.24 U132: {1} 9.78/3.24 U133: {1} 9.78/3.24 U134: {1} 9.78/3.24 U135: {1} 9.78/3.24 U136: {1} 9.78/3.24 take: {1, 2} 9.78/3.24 U21: {1} 9.78/3.24 U22: {1} 9.78/3.24 U23: {1} 9.78/3.24 U31: {1} 9.78/3.24 U32: {1} 9.78/3.24 U33: {1} 9.78/3.24 U41: {1} 9.78/3.24 U42: {1} 9.78/3.24 U43: {1} 9.78/3.24 U44: {1} 9.78/3.24 U45: {1} 9.78/3.24 U46: {1} 9.78/3.24 U51: {1} 9.78/3.24 U52: {1} 9.78/3.24 U61: {1} 9.78/3.24 U62: {1} 9.78/3.24 U71: {1} 9.78/3.24 U81: {1} 9.78/3.24 U91: {1} 9.78/3.24 U92: {1} 9.78/3.24 U93: {1} 9.78/3.24 U94: {1} 9.78/3.24 U95: {1} 9.78/3.24 U96: {1} 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (5) CSRRRRProof (EQUIVALENT) 9.78/3.24 The following CSR is given: Context-sensitive rewrite system: 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(0, IL) -> U121(isNatIList(IL), IL) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 The replacement map contains the following entries: 9.78/3.24 9.78/3.24 zeros: empty set 9.78/3.24 cons: {1} 9.78/3.24 0: empty set 9.78/3.24 U101: {1} 9.78/3.24 tt: empty set 9.78/3.24 U102: {1} 9.78/3.24 isNatKind: empty set 9.78/3.24 U103: {1} 9.78/3.24 isNatIListKind: empty set 9.78/3.24 U104: {1} 9.78/3.24 U105: {1} 9.78/3.24 isNat: empty set 9.78/3.24 U106: {1} 9.78/3.24 isNatIList: empty set 9.78/3.24 U11: {1} 9.78/3.24 U12: {1} 9.78/3.24 U111: {1} 9.78/3.24 U112: {1} 9.78/3.24 U113: {1} 9.78/3.24 U114: {1} 9.78/3.24 s: {1} 9.78/3.24 length: {1} 9.78/3.24 U13: {1} 9.78/3.24 isNatList: empty set 9.78/3.24 U121: {1} 9.78/3.24 nil: empty set 9.78/3.24 U131: {1} 9.78/3.24 U132: {1} 9.78/3.24 U133: {1} 9.78/3.24 U134: {1} 9.78/3.24 U135: {1} 9.78/3.24 U136: {1} 9.78/3.24 take: {1, 2} 9.78/3.24 U21: {1} 9.78/3.24 U22: {1} 9.78/3.24 U23: {1} 9.78/3.24 U31: {1} 9.78/3.24 U32: {1} 9.78/3.24 U33: {1} 9.78/3.24 U41: {1} 9.78/3.24 U42: {1} 9.78/3.24 U43: {1} 9.78/3.24 U44: {1} 9.78/3.24 U45: {1} 9.78/3.24 U46: {1} 9.78/3.24 U51: {1} 9.78/3.24 U52: {1} 9.78/3.24 U61: {1} 9.78/3.24 U62: {1} 9.78/3.24 U71: {1} 9.78/3.24 U81: {1} 9.78/3.24 U91: {1} 9.78/3.24 U92: {1} 9.78/3.24 U93: {1} 9.78/3.24 U94: {1} 9.78/3.24 U95: {1} 9.78/3.24 U96: {1} 9.78/3.24 Used ordering: 9.78/3.24 Polynomial interpretation [POLO]: 9.78/3.24 9.78/3.24 POL(0) = 0 9.78/3.24 POL(U101(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U102(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U103(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U104(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U105(x_1, x_2)) = x_1 9.78/3.24 POL(U106(x_1)) = x_1 9.78/3.24 POL(U11(x_1, x_2)) = x_1 9.78/3.24 POL(U111(x_1, x_2, x_3)) = x_1 + x_2 + x_3 9.78/3.24 POL(U112(x_1, x_2, x_3)) = x_1 + x_2 + x_3 9.78/3.24 POL(U113(x_1, x_2, x_3)) = x_1 + x_2 + x_3 9.78/3.24 POL(U114(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(U12(x_1, x_2)) = x_1 9.78/3.24 POL(U121(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(U13(x_1)) = x_1 9.78/3.24 POL(U131(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U132(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U133(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U134(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U135(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U136(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 9.78/3.24 POL(U21(x_1, x_2)) = x_1 9.78/3.24 POL(U22(x_1, x_2)) = x_1 9.78/3.24 POL(U23(x_1)) = x_1 9.78/3.24 POL(U31(x_1, x_2)) = x_1 9.78/3.24 POL(U32(x_1, x_2)) = x_1 9.78/3.24 POL(U33(x_1)) = x_1 9.78/3.24 POL(U41(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U42(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U43(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U44(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U45(x_1, x_2)) = x_1 9.78/3.24 POL(U46(x_1)) = x_1 9.78/3.24 POL(U51(x_1, x_2)) = x_1 9.78/3.24 POL(U52(x_1)) = x_1 9.78/3.24 POL(U61(x_1, x_2)) = x_1 9.78/3.24 POL(U62(x_1)) = x_1 9.78/3.24 POL(U71(x_1)) = x_1 9.78/3.24 POL(U81(x_1)) = x_1 9.78/3.24 POL(U91(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U92(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U93(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U94(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U95(x_1, x_2)) = x_1 9.78/3.24 POL(U96(x_1)) = x_1 9.78/3.24 POL(cons(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(isNat(x_1)) = 0 9.78/3.24 POL(isNatIList(x_1)) = 0 9.78/3.24 POL(isNatIListKind(x_1)) = 0 9.78/3.24 POL(isNatKind(x_1)) = 0 9.78/3.24 POL(isNatList(x_1)) = 0 9.78/3.24 POL(length(x_1)) = x_1 9.78/3.24 POL(nil) = 1 9.78/3.24 POL(s(x_1)) = x_1 9.78/3.24 POL(take(x_1, x_2)) = 1 + x_1 + x_2 9.78/3.24 POL(tt) = 0 9.78/3.24 POL(zeros) = 1 9.78/3.24 With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly: 9.78/3.24 9.78/3.24 take(0, IL) -> U121(isNatIList(IL), IL) 9.78/3.24 9.78/3.24 9.78/3.24 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (6) 9.78/3.24 Obligation: 9.78/3.24 Context-sensitive rewrite system: 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 The replacement map contains the following entries: 9.78/3.24 9.78/3.24 zeros: empty set 9.78/3.24 cons: {1} 9.78/3.24 0: empty set 9.78/3.24 U101: {1} 9.78/3.24 tt: empty set 9.78/3.24 U102: {1} 9.78/3.24 isNatKind: empty set 9.78/3.24 U103: {1} 9.78/3.24 isNatIListKind: empty set 9.78/3.24 U104: {1} 9.78/3.24 U105: {1} 9.78/3.24 isNat: empty set 9.78/3.24 U106: {1} 9.78/3.24 isNatIList: empty set 9.78/3.24 U11: {1} 9.78/3.24 U12: {1} 9.78/3.24 U111: {1} 9.78/3.24 U112: {1} 9.78/3.24 U113: {1} 9.78/3.24 U114: {1} 9.78/3.24 s: {1} 9.78/3.24 length: {1} 9.78/3.24 U13: {1} 9.78/3.24 isNatList: empty set 9.78/3.24 nil: empty set 9.78/3.24 U131: {1} 9.78/3.24 U132: {1} 9.78/3.24 U133: {1} 9.78/3.24 U134: {1} 9.78/3.24 U135: {1} 9.78/3.24 U136: {1} 9.78/3.24 take: {1, 2} 9.78/3.24 U21: {1} 9.78/3.24 U22: {1} 9.78/3.24 U23: {1} 9.78/3.24 U31: {1} 9.78/3.24 U32: {1} 9.78/3.24 U33: {1} 9.78/3.24 U41: {1} 9.78/3.24 U42: {1} 9.78/3.24 U43: {1} 9.78/3.24 U44: {1} 9.78/3.24 U45: {1} 9.78/3.24 U46: {1} 9.78/3.24 U51: {1} 9.78/3.24 U52: {1} 9.78/3.24 U61: {1} 9.78/3.24 U62: {1} 9.78/3.24 U71: {1} 9.78/3.24 U81: {1} 9.78/3.24 U91: {1} 9.78/3.24 U92: {1} 9.78/3.24 U93: {1} 9.78/3.24 U94: {1} 9.78/3.24 U95: {1} 9.78/3.24 U96: {1} 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (7) CSDependencyPairsProof (EQUIVALENT) 9.78/3.24 Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem. 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (8) 9.78/3.24 Obligation: 9.78/3.24 Q-restricted context-sensitive dependency pair problem: 9.78/3.24 The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, U106'_1, LENGTH_1, U13'_1, U23'_1, U33'_1, U46'_1, U52'_1, U62'_1, U96'_1, U71'_1, U81'_1, TAKE_2} are replacing on all positions. 9.78/3.24 For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U102'_3, U101'_3, U103'_3, U104'_3, U105'_2, U12'_2, U11'_2, U112'_3, U111'_3, U113'_3, U114'_2, U132'_4, U131'_4, U133'_4, U134'_4, U135'_4, U136'_4, U22'_2, U21'_2, U32'_2, U31'_2, U42'_3, U41'_3, U43'_3, U44'_3, U45'_2, U51'_2, U61'_2, U92'_3, U91'_3, U93'_3, U94'_3, U95'_2} we have mu(f) = {1}. 9.78/3.24 The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, ISNATKIND_1, ISNATILISTKIND_1, ISNAT_1, ISNATILIST_1, ISNATLIST_1, U_1} are not replacing on any position. 9.78/3.24 9.78/3.24 The ordinary context-sensitive dependency pairs DP_o are: 9.78/3.24 U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2) 9.78/3.24 U101'(tt, V1, V2) -> ISNATKIND(V1) 9.78/3.24 U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2) 9.78/3.24 U102'(tt, V1, V2) -> ISNATILISTKIND(V2) 9.78/3.24 U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2) 9.78/3.24 U103'(tt, V1, V2) -> ISNATILISTKIND(V2) 9.78/3.24 U104'(tt, V1, V2) -> U105'(isNat(V1), V2) 9.78/3.24 U104'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 U105'(tt, V2) -> U106'(isNatIList(V2)) 9.78/3.24 U105'(tt, V2) -> ISNATILIST(V2) 9.78/3.24 U11'(tt, V1) -> U12'(isNatIListKind(V1), V1) 9.78/3.24 U11'(tt, V1) -> ISNATILISTKIND(V1) 9.78/3.24 U111'(tt, L, N) -> U112'(isNatIListKind(L), L, N) 9.78/3.24 U111'(tt, L, N) -> ISNATILISTKIND(L) 9.78/3.24 U112'(tt, L, N) -> U113'(isNat(N), L, N) 9.78/3.24 U112'(tt, L, N) -> ISNAT(N) 9.78/3.24 U113'(tt, L, N) -> U114'(isNatKind(N), L) 9.78/3.24 U113'(tt, L, N) -> ISNATKIND(N) 9.78/3.24 U114'(tt, L) -> LENGTH(L) 9.78/3.24 U12'(tt, V1) -> U13'(isNatList(V1)) 9.78/3.24 U12'(tt, V1) -> ISNATLIST(V1) 9.78/3.24 U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N) 9.78/3.24 U131'(tt, IL, M, N) -> ISNATILISTKIND(IL) 9.78/3.24 U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N) 9.78/3.24 U132'(tt, IL, M, N) -> ISNAT(M) 9.78/3.24 U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N) 9.78/3.24 U133'(tt, IL, M, N) -> ISNATKIND(M) 9.78/3.24 U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N) 9.78/3.24 U134'(tt, IL, M, N) -> ISNAT(N) 9.78/3.24 U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N) 9.78/3.24 U135'(tt, IL, M, N) -> ISNATKIND(N) 9.78/3.24 U21'(tt, V1) -> U22'(isNatKind(V1), V1) 9.78/3.24 U21'(tt, V1) -> ISNATKIND(V1) 9.78/3.24 U22'(tt, V1) -> U23'(isNat(V1)) 9.78/3.24 U22'(tt, V1) -> ISNAT(V1) 9.78/3.24 U31'(tt, V) -> U32'(isNatIListKind(V), V) 9.78/3.24 U31'(tt, V) -> ISNATILISTKIND(V) 9.78/3.24 U32'(tt, V) -> U33'(isNatList(V)) 9.78/3.24 U32'(tt, V) -> ISNATLIST(V) 9.78/3.24 U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2) 9.78/3.24 U41'(tt, V1, V2) -> ISNATKIND(V1) 9.78/3.24 U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2) 9.78/3.24 U42'(tt, V1, V2) -> ISNATILISTKIND(V2) 9.78/3.24 U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2) 9.78/3.24 U43'(tt, V1, V2) -> ISNATILISTKIND(V2) 9.78/3.24 U44'(tt, V1, V2) -> U45'(isNat(V1), V2) 9.78/3.24 U44'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 U45'(tt, V2) -> U46'(isNatIList(V2)) 9.78/3.24 U45'(tt, V2) -> ISNATILIST(V2) 9.78/3.24 U51'(tt, V2) -> U52'(isNatIListKind(V2)) 9.78/3.24 U51'(tt, V2) -> ISNATILISTKIND(V2) 9.78/3.24 U61'(tt, V2) -> U62'(isNatIListKind(V2)) 9.78/3.24 U61'(tt, V2) -> ISNATILISTKIND(V2) 9.78/3.24 U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2) 9.78/3.24 U91'(tt, V1, V2) -> ISNATKIND(V1) 9.78/3.24 U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2) 9.78/3.24 U92'(tt, V1, V2) -> ISNATILISTKIND(V2) 9.78/3.24 U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2) 9.78/3.24 U93'(tt, V1, V2) -> ISNATILISTKIND(V2) 9.78/3.24 U94'(tt, V1, V2) -> U95'(isNat(V1), V2) 9.78/3.24 U94'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 U95'(tt, V2) -> U96'(isNatList(V2)) 9.78/3.24 U95'(tt, V2) -> ISNATLIST(V2) 9.78/3.24 ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1) 9.78/3.24 ISNAT(length(V1)) -> ISNATILISTKIND(V1) 9.78/3.24 ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) 9.78/3.24 ISNAT(s(V1)) -> ISNATKIND(V1) 9.78/3.24 ISNATILIST(V) -> U31'(isNatIListKind(V), V) 9.78/3.24 ISNATILIST(V) -> ISNATILISTKIND(V) 9.78/3.24 ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2) 9.78/3.24 ISNATILIST(cons(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2) 9.78/3.24 ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2) 9.78/3.24 ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 ISNATKIND(length(V1)) -> U71'(isNatIListKind(V1)) 9.78/3.24 ISNATKIND(length(V1)) -> ISNATILISTKIND(V1) 9.78/3.24 ISNATKIND(s(V1)) -> U81'(isNatKind(V1)) 9.78/3.24 ISNATKIND(s(V1)) -> ISNATKIND(V1) 9.78/3.24 ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2) 9.78/3.24 ISNATLIST(cons(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2) 9.78/3.24 ISNATLIST(take(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 LENGTH(cons(N, L)) -> U111'(isNatList(L), L, N) 9.78/3.24 LENGTH(cons(N, L)) -> ISNATLIST(L) 9.78/3.24 TAKE(s(M), cons(N, IL)) -> U131'(isNatIList(IL), IL, M, N) 9.78/3.24 TAKE(s(M), cons(N, IL)) -> ISNATILIST(IL) 9.78/3.24 9.78/3.24 The collapsing dependency pairs are DP_c: 9.78/3.24 U114'(tt, L) -> L 9.78/3.24 U136'(tt, IL, M, N) -> N 9.78/3.24 9.78/3.24 9.78/3.24 The hidden terms of R are: 9.78/3.24 9.78/3.24 zeros 9.78/3.24 take(x0, x1) 9.78/3.24 9.78/3.24 Every hiding context is built from: 9.78/3.24 aprove.DPFramework.CSDPProblem.QCSDPProblem$1@1cd855b6 9.78/3.24 9.78/3.24 Hence, the new unhiding pairs DP_u are : 9.78/3.24 U114'(tt, L) -> U(L) 9.78/3.24 U136'(tt, IL, M, N) -> U(N) 9.78/3.24 U(take(x_0, x_1)) -> U(x_0) 9.78/3.24 U(take(x_0, x_1)) -> U(x_1) 9.78/3.24 U(zeros) -> ZEROS 9.78/3.24 U(take(x0, x1)) -> TAKE(x0, x1) 9.78/3.24 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 Q is empty. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (9) QCSDependencyGraphProof (EQUIVALENT) 9.78/3.24 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 4 SCCs with 40 less nodes. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (10) 9.78/3.24 Complex Obligation (AND) 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (11) 9.78/3.24 Obligation: 9.78/3.24 Q-restricted context-sensitive dependency pair problem: 9.78/3.24 The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1} are replacing on all positions. 9.78/3.24 For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U51'_2, U61'_2} we have mu(f) = {1}. 9.78/3.24 The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, ISNATILISTKIND_1, ISNATKIND_1} are not replacing on any position. 9.78/3.24 9.78/3.24 The TRS P consists of the following rules: 9.78/3.24 9.78/3.24 ISNATKIND(length(V1)) -> ISNATILISTKIND(V1) 9.78/3.24 ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2) 9.78/3.24 U51'(tt, V2) -> ISNATILISTKIND(V2) 9.78/3.24 ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 ISNATKIND(s(V1)) -> ISNATKIND(V1) 9.78/3.24 ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2) 9.78/3.24 U61'(tt, V2) -> ISNATILISTKIND(V2) 9.78/3.24 ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 Q is empty. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (12) QCSUsableRulesProof (EQUIVALENT) 9.78/3.24 The following rules are not useable [DA_EMMES] and can be deleted: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, x0, x1) -> U102(isNatKind(x0), x0, x1) 9.78/3.24 U102(tt, x0, x1) -> U103(isNatIListKind(x1), x0, x1) 9.78/3.24 U103(tt, x0, x1) -> U104(isNatIListKind(x1), x0, x1) 9.78/3.24 U104(tt, x0, x1) -> U105(isNat(x0), x1) 9.78/3.24 U105(tt, x0) -> U106(isNatIList(x0)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, x0) -> U12(isNatIListKind(x0), x0) 9.78/3.24 U111(tt, x0, x1) -> U112(isNatIListKind(x0), x0, x1) 9.78/3.24 U112(tt, x0, x1) -> U113(isNat(x1), x0, x1) 9.78/3.24 U113(tt, x0, x1) -> U114(isNatKind(x1), x0) 9.78/3.24 U114(tt, x0) -> s(length(x0)) 9.78/3.24 U12(tt, x0) -> U13(isNatList(x0)) 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, x0, x1, x2) -> U132(isNatIListKind(x0), x0, x1, x2) 9.78/3.24 U132(tt, x0, x1, x2) -> U133(isNat(x1), x0, x1, x2) 9.78/3.24 U133(tt, x0, x1, x2) -> U134(isNatKind(x1), x0, x1, x2) 9.78/3.24 U134(tt, x0, x1, x2) -> U135(isNat(x2), x0, x1, x2) 9.78/3.24 U135(tt, x0, x1, x2) -> U136(isNatKind(x2), x0, x1, x2) 9.78/3.24 U136(tt, x0, x1, x2) -> cons(x2, take(x1, x0)) 9.78/3.24 U21(tt, x0) -> U22(isNatKind(x0), x0) 9.78/3.24 U22(tt, x0) -> U23(isNat(x0)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, x0) -> U32(isNatIListKind(x0), x0) 9.78/3.24 U32(tt, x0) -> U33(isNatList(x0)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, x0, x1) -> U42(isNatKind(x0), x0, x1) 9.78/3.24 U42(tt, x0, x1) -> U43(isNatIListKind(x1), x0, x1) 9.78/3.24 U43(tt, x0, x1) -> U44(isNatIListKind(x1), x0, x1) 9.78/3.24 U44(tt, x0, x1) -> U45(isNat(x0), x1) 9.78/3.24 U45(tt, x0) -> U46(isNatIList(x0)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U91(tt, x0, x1) -> U92(isNatKind(x0), x0, x1) 9.78/3.24 U92(tt, x0, x1) -> U93(isNatIListKind(x1), x0, x1) 9.78/3.24 U93(tt, x0, x1) -> U94(isNatIListKind(x1), x0, x1) 9.78/3.24 U94(tt, x0, x1) -> U95(isNat(x0), x1) 9.78/3.24 U95(tt, x0) -> U96(isNatList(x0)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(x0)) -> U11(isNatIListKind(x0), x0) 9.78/3.24 isNat(s(x0)) -> U21(isNatKind(x0), x0) 9.78/3.24 isNatIList(x0) -> U31(isNatIListKind(x0), x0) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(x0, x1)) -> U41(isNatKind(x0), x0, x1) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(x0, x1)) -> U91(isNatKind(x0), x0, x1) 9.78/3.24 isNatList(take(x0, x1)) -> U101(isNatKind(x0), x0, x1) 9.78/3.24 length(cons(x0, x1)) -> U111(isNatList(x1), x1, x0) 9.78/3.24 take(s(x0), cons(x1, x2)) -> U131(isNatIList(x2), x2, x0, x1) 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (13) 9.78/3.24 Obligation: 9.78/3.24 Q-restricted context-sensitive dependency pair problem: 9.78/3.24 The symbols in {length_1, U71_1, s_1, U81_1, U52_1, take_2, U62_1} are replacing on all positions. 9.78/3.24 For all symbols f in {cons_2, U51_2, U61_2, U51'_2, U61'_2} we have mu(f) = {1}. 9.78/3.24 The symbols in {isNatKind_1, isNatIListKind_1, ISNATILISTKIND_1, ISNATKIND_1} are not replacing on any position. 9.78/3.24 9.78/3.24 The TRS P consists of the following rules: 9.78/3.24 9.78/3.24 ISNATKIND(length(V1)) -> ISNATILISTKIND(V1) 9.78/3.24 ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2) 9.78/3.24 U51'(tt, V2) -> ISNATILISTKIND(V2) 9.78/3.24 ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 ISNATKIND(s(V1)) -> ISNATKIND(V1) 9.78/3.24 ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2) 9.78/3.24 U61'(tt, V2) -> ISNATILISTKIND(V2) 9.78/3.24 ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 U81(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U52(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 9.78/3.24 Q is empty. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (14) QCSDPMuMonotonicPoloProof (EQUIVALENT) 9.78/3.24 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. 9.78/3.24 9.78/3.24 Strictly oriented dependency pairs: 9.78/3.24 9.78/3.24 ISNATKIND(length(V1)) -> ISNATILISTKIND(V1) 9.78/3.24 ISNATILISTKIND(cons(V1, V2)) -> U51'(isNatKind(V1), V2) 9.78/3.24 U51'(tt, V2) -> ISNATILISTKIND(V2) 9.78/3.24 ISNATILISTKIND(cons(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 ISNATKIND(s(V1)) -> ISNATKIND(V1) 9.78/3.24 ISNATILISTKIND(take(V1, V2)) -> U61'(isNatKind(V1), V2) 9.78/3.24 U61'(tt, V2) -> ISNATILISTKIND(V2) 9.78/3.24 ISNATILISTKIND(take(V1, V2)) -> ISNATKIND(V1) 9.78/3.24 9.78/3.24 Strictly oriented rules of the TRS R: 9.78/3.24 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 U81(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U52(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 9.78/3.24 Used ordering: POLO with Polynomial interpretation [POLO]: 9.78/3.24 9.78/3.24 POL(0) = 2 9.78/3.24 POL(ISNATILISTKIND(x_1)) = 1 + 2*x_1 9.78/3.24 POL(ISNATKIND(x_1)) = 2 + 2*x_1 9.78/3.24 POL(U51(x_1, x_2)) = 2 + x_1 + 2*x_2 9.78/3.24 POL(U51'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 9.78/3.24 POL(U52(x_1)) = 1 + 2*x_1 9.78/3.24 POL(U61(x_1, x_2)) = x_1 + 2*x_2 9.78/3.24 POL(U61'(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 9.78/3.24 POL(U62(x_1)) = 1 + 2*x_1 9.78/3.24 POL(U71(x_1)) = 1 + 2*x_1 9.78/3.24 POL(U81(x_1)) = 1 + 2*x_1 9.78/3.24 POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 9.78/3.24 POL(isNatIListKind(x_1)) = x_1 9.78/3.24 POL(isNatKind(x_1)) = 2*x_1 9.78/3.24 POL(length(x_1)) = 2 + 2*x_1 9.78/3.24 POL(nil) = 2 9.78/3.24 POL(s(x_1)) = 2 + 2*x_1 9.78/3.24 POL(take(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 9.78/3.24 POL(tt) = 2 9.78/3.24 POL(zeros) = 2 9.78/3.24 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (15) 9.78/3.24 Obligation: 9.78/3.24 Q-restricted context-sensitive dependency pair problem: 9.78/3.24 For all symbols f in {cons_2, U51_2} we have mu(f) = {1}. 9.78/3.24 The symbols in {isNatIListKind_1, isNatKind_1} are not replacing on any position. 9.78/3.24 9.78/3.24 The TRS P consists of the following rules: 9.78/3.24 none 9.78/3.24 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 9.78/3.24 Q is empty. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (16) PIsEmptyProof (EQUIVALENT) 9.78/3.24 The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain. 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (17) 9.78/3.24 YES 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (18) 9.78/3.24 Obligation: 9.78/3.24 Q-restricted context-sensitive dependency pair problem: 9.78/3.24 The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1} are replacing on all positions. 9.78/3.24 For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U103'_3, U102'_3, U104'_3, U105'_2, U31'_2, U32'_2, U91'_3, U92'_3, U93'_3, U94'_3, U95'_2, U101'_3, U11'_2, U12'_2, U21'_2, U22'_2, U41'_3, U42'_3, U43'_3, U44'_3, U45'_2} we have mu(f) = {1}. 9.78/3.24 The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1} are not replacing on any position. 9.78/3.24 9.78/3.24 The TRS P consists of the following rules: 9.78/3.24 9.78/3.24 U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2) 9.78/3.24 U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2) 9.78/3.24 U104'(tt, V1, V2) -> U105'(isNat(V1), V2) 9.78/3.24 U105'(tt, V2) -> ISNATILIST(V2) 9.78/3.24 ISNATILIST(V) -> U31'(isNatIListKind(V), V) 9.78/3.24 U31'(tt, V) -> U32'(isNatIListKind(V), V) 9.78/3.24 U32'(tt, V) -> ISNATLIST(V) 9.78/3.24 ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2) 9.78/3.24 U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2) 9.78/3.24 U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2) 9.78/3.24 U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2) 9.78/3.24 U94'(tt, V1, V2) -> U95'(isNat(V1), V2) 9.78/3.24 U95'(tt, V2) -> ISNATLIST(V2) 9.78/3.24 ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2) 9.78/3.24 U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2) 9.78/3.24 U94'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1) 9.78/3.24 U11'(tt, V1) -> U12'(isNatIListKind(V1), V1) 9.78/3.24 U12'(tt, V1) -> ISNATLIST(V1) 9.78/3.24 ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) 9.78/3.24 U21'(tt, V1) -> U22'(isNatKind(V1), V1) 9.78/3.24 U22'(tt, V1) -> ISNAT(V1) 9.78/3.24 ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2) 9.78/3.24 U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2) 9.78/3.24 U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2) 9.78/3.24 U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2) 9.78/3.24 U44'(tt, V1, V2) -> U45'(isNat(V1), V2) 9.78/3.24 U45'(tt, V2) -> ISNATILIST(V2) 9.78/3.24 U44'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 U104'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 U106(tt) -> tt 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.24 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.24 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.24 U114(tt, L) -> s(length(L)) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 U13(tt) -> tt 9.78/3.24 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.24 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.24 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.24 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.24 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.24 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U81(tt) -> tt 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 U96(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.24 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.24 9.78/3.24 Q is empty. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (19) QCSUsableRulesProof (EQUIVALENT) 9.78/3.24 The following rules are not useable [DA_EMMES] and can be deleted: 9.78/3.24 9.78/3.24 zeros -> cons(0, zeros) 9.78/3.24 U111(tt, x0, x1) -> U112(isNatIListKind(x0), x0, x1) 9.78/3.24 U112(tt, x0, x1) -> U113(isNat(x1), x0, x1) 9.78/3.24 U113(tt, x0, x1) -> U114(isNatKind(x1), x0) 9.78/3.24 U114(tt, x0) -> s(length(x0)) 9.78/3.24 U131(tt, x0, x1, x2) -> U132(isNatIListKind(x0), x0, x1, x2) 9.78/3.24 U132(tt, x0, x1, x2) -> U133(isNat(x1), x0, x1, x2) 9.78/3.24 U133(tt, x0, x1, x2) -> U134(isNatKind(x1), x0, x1, x2) 9.78/3.24 U134(tt, x0, x1, x2) -> U135(isNat(x2), x0, x1, x2) 9.78/3.24 U135(tt, x0, x1, x2) -> U136(isNatKind(x2), x0, x1, x2) 9.78/3.24 U136(tt, x0, x1, x2) -> cons(x2, take(x1, x0)) 9.78/3.24 length(cons(x0, x1)) -> U111(isNatList(x1), x1, x0) 9.78/3.24 take(s(x0), cons(x1, x2)) -> U131(isNatIList(x2), x2, x0, x1) 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (20) 9.78/3.24 Obligation: 9.78/3.24 Q-restricted context-sensitive dependency pair problem: 9.78/3.24 The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions. 9.78/3.24 For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U103'_3, U102'_3, U104'_3, U105'_2, U31'_2, U32'_2, U91'_3, U92'_3, U93'_3, U94'_3, U95'_2, U101'_3, U11'_2, U12'_2, U21'_2, U22'_2, U41'_3, U42'_3, U43'_3, U44'_3, U45'_2} we have mu(f) = {1}. 9.78/3.24 The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1} are not replacing on any position. 9.78/3.24 9.78/3.24 The TRS P consists of the following rules: 9.78/3.24 9.78/3.24 U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2) 9.78/3.24 U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2) 9.78/3.24 U104'(tt, V1, V2) -> U105'(isNat(V1), V2) 9.78/3.24 U105'(tt, V2) -> ISNATILIST(V2) 9.78/3.24 ISNATILIST(V) -> U31'(isNatIListKind(V), V) 9.78/3.24 U31'(tt, V) -> U32'(isNatIListKind(V), V) 9.78/3.24 U32'(tt, V) -> ISNATLIST(V) 9.78/3.24 ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2) 9.78/3.24 U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2) 9.78/3.24 U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2) 9.78/3.24 U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2) 9.78/3.24 U94'(tt, V1, V2) -> U95'(isNat(V1), V2) 9.78/3.24 U95'(tt, V2) -> ISNATLIST(V2) 9.78/3.24 ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2) 9.78/3.24 U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2) 9.78/3.24 U94'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1) 9.78/3.24 U11'(tt, V1) -> U12'(isNatIListKind(V1), V1) 9.78/3.24 U12'(tt, V1) -> ISNATLIST(V1) 9.78/3.24 ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) 9.78/3.24 U21'(tt, V1) -> U22'(isNatKind(V1), V1) 9.78/3.24 U22'(tt, V1) -> ISNAT(V1) 9.78/3.24 ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2) 9.78/3.24 U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2) 9.78/3.24 U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2) 9.78/3.24 U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2) 9.78/3.24 U44'(tt, V1, V2) -> U45'(isNat(V1), V2) 9.78/3.24 U45'(tt, V2) -> ISNATILIST(V2) 9.78/3.24 U44'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 U104'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 U81(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U106(tt) -> tt 9.78/3.24 U96(tt) -> tt 9.78/3.24 U13(tt) -> tt 9.78/3.24 9.78/3.24 Q is empty. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (21) QCSDPMuMonotonicPoloProof (EQUIVALENT) 9.78/3.24 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. 9.78/3.24 9.78/3.24 Strictly oriented dependency pairs: 9.78/3.24 9.78/3.24 U102'(tt, V1, V2) -> U103'(isNatIListKind(V2), V1, V2) 9.78/3.24 ISNATLIST(cons(V1, V2)) -> U91'(isNatKind(V1), V1, V2) 9.78/3.24 U92'(tt, V1, V2) -> U93'(isNatIListKind(V2), V1, V2) 9.78/3.24 U95'(tt, V2) -> ISNATLIST(V2) 9.78/3.24 ISNATLIST(take(V1, V2)) -> U101'(isNatKind(V1), V1, V2) 9.78/3.24 U101'(tt, V1, V2) -> U102'(isNatKind(V1), V1, V2) 9.78/3.24 U94'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 ISNAT(length(V1)) -> U11'(isNatIListKind(V1), V1) 9.78/3.24 U12'(tt, V1) -> ISNATLIST(V1) 9.78/3.24 ISNATILIST(cons(V1, V2)) -> U41'(isNatKind(V1), V1, V2) 9.78/3.24 U42'(tt, V1, V2) -> U43'(isNatIListKind(V2), V1, V2) 9.78/3.24 U45'(tt, V2) -> ISNATILIST(V2) 9.78/3.24 U44'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 9.78/3.24 9.78/3.24 Used ordering: POLO with Polynomial interpretation [POLO]: 9.78/3.24 9.78/3.24 POL(0) = 0 9.78/3.24 POL(ISNAT(x_1)) = x_1 9.78/3.24 POL(ISNATILIST(x_1)) = 2*x_1 9.78/3.24 POL(ISNATLIST(x_1)) = 2*x_1 9.78/3.24 POL(U101(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U101'(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + 2*x_3 9.78/3.24 POL(U102(x_1, x_2, x_3)) = 2*x_1 9.78/3.24 POL(U102'(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + 2*x_3 9.78/3.24 POL(U103(x_1, x_2, x_3)) = 2*x_1 9.78/3.24 POL(U103'(x_1, x_2, x_3)) = 2*x_1 + x_2 + 2*x_3 9.78/3.24 POL(U104(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U104'(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3 9.78/3.24 POL(U105(x_1, x_2)) = 2*x_1 9.78/3.24 POL(U105'(x_1, x_2)) = 2*x_1 + 2*x_2 9.78/3.24 POL(U106(x_1)) = x_1 9.78/3.24 POL(U11(x_1, x_2)) = x_1 9.78/3.24 POL(U11'(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 9.78/3.24 POL(U12(x_1, x_2)) = 2*x_1 9.78/3.24 POL(U12'(x_1, x_2)) = 1 + x_1 + 2*x_2 9.78/3.24 POL(U13(x_1)) = 2*x_1 9.78/3.24 POL(U21(x_1, x_2)) = x_1 9.78/3.24 POL(U21'(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(U22(x_1, x_2)) = x_1 9.78/3.24 POL(U22'(x_1, x_2)) = x_1 + x_2 9.78/3.24 POL(U23(x_1)) = 2*x_1 9.78/3.24 POL(U31(x_1, x_2)) = 2*x_1 9.78/3.24 POL(U31'(x_1, x_2)) = 2*x_1 + 2*x_2 9.78/3.24 POL(U32(x_1, x_2)) = x_1 9.78/3.24 POL(U32'(x_1, x_2)) = 2*x_1 + 2*x_2 9.78/3.24 POL(U33(x_1)) = 2*x_1 9.78/3.24 POL(U41(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U41'(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + 2*x_3 9.78/3.24 POL(U42(x_1, x_2, x_3)) = 2*x_1 9.78/3.24 POL(U42'(x_1, x_2, x_3)) = 2 + 2*x_1 + x_2 + 2*x_3 9.78/3.24 POL(U43(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U43'(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + 2*x_3 9.78/3.24 POL(U44(x_1, x_2, x_3)) = 2*x_1 9.78/3.24 POL(U44'(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3 9.78/3.24 POL(U45(x_1, x_2)) = x_1 9.78/3.24 POL(U45'(x_1, x_2)) = 1 + x_1 + 2*x_2 9.78/3.24 POL(U46(x_1)) = 2*x_1 9.78/3.24 POL(U51(x_1, x_2)) = 2*x_1 9.78/3.24 POL(U52(x_1)) = x_1 9.78/3.24 POL(U61(x_1, x_2)) = 2*x_1 9.78/3.24 POL(U62(x_1)) = x_1 9.78/3.24 POL(U71(x_1)) = 2*x_1 9.78/3.24 POL(U81(x_1)) = 2*x_1 9.78/3.24 POL(U91(x_1, x_2, x_3)) = 2*x_1 9.78/3.24 POL(U91'(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3 9.78/3.24 POL(U92(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U92'(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3 9.78/3.24 POL(U93(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U93'(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3 9.78/3.24 POL(U94(x_1, x_2, x_3)) = x_1 9.78/3.24 POL(U94'(x_1, x_2, x_3)) = 1 + x_1 + x_2 + 2*x_3 9.78/3.24 POL(U95(x_1, x_2)) = 2*x_1 9.78/3.24 POL(U95'(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 9.78/3.24 POL(U96(x_1)) = x_1 9.78/3.24 POL(cons(x_1, x_2)) = 2 + x_1 + 2*x_2 9.78/3.24 POL(isNat(x_1)) = 0 9.78/3.24 POL(isNatIList(x_1)) = 0 9.78/3.24 POL(isNatIListKind(x_1)) = 0 9.78/3.24 POL(isNatKind(x_1)) = 0 9.78/3.24 POL(isNatList(x_1)) = 0 9.78/3.24 POL(length(x_1)) = 2 + 2*x_1 9.78/3.24 POL(nil) = 0 9.78/3.24 POL(s(x_1)) = 2*x_1 9.78/3.24 POL(take(x_1, x_2)) = 2 + 2*x_1 + x_2 9.78/3.24 POL(tt) = 0 9.78/3.24 POL(zeros) = 0 9.78/3.24 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (22) 9.78/3.24 Obligation: 9.78/3.24 Q-restricted context-sensitive dependency pair problem: 9.78/3.24 The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions. 9.78/3.24 For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U104'_3, U103'_3, U105'_2, U31'_2, U32'_2, U92'_3, U91'_3, U94'_3, U93'_3, U95'_2, U12'_2, U11'_2, U21'_2, U22'_2, U42'_3, U41'_3, U44'_3, U43'_3, U45'_2} we have mu(f) = {1}. 9.78/3.24 The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNATILIST_1, ISNATLIST_1, ISNAT_1} are not replacing on any position. 9.78/3.24 9.78/3.24 The TRS P consists of the following rules: 9.78/3.24 9.78/3.24 U103'(tt, V1, V2) -> U104'(isNatIListKind(V2), V1, V2) 9.78/3.24 U104'(tt, V1, V2) -> U105'(isNat(V1), V2) 9.78/3.24 U105'(tt, V2) -> ISNATILIST(V2) 9.78/3.24 ISNATILIST(V) -> U31'(isNatIListKind(V), V) 9.78/3.24 U31'(tt, V) -> U32'(isNatIListKind(V), V) 9.78/3.24 U32'(tt, V) -> ISNATLIST(V) 9.78/3.24 U91'(tt, V1, V2) -> U92'(isNatKind(V1), V1, V2) 9.78/3.24 U93'(tt, V1, V2) -> U94'(isNatIListKind(V2), V1, V2) 9.78/3.24 U94'(tt, V1, V2) -> U95'(isNat(V1), V2) 9.78/3.24 U11'(tt, V1) -> U12'(isNatIListKind(V1), V1) 9.78/3.24 ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) 9.78/3.24 U21'(tt, V1) -> U22'(isNatKind(V1), V1) 9.78/3.24 U22'(tt, V1) -> ISNAT(V1) 9.78/3.24 U41'(tt, V1, V2) -> U42'(isNatKind(V1), V1, V2) 9.78/3.24 U43'(tt, V1, V2) -> U44'(isNatIListKind(V2), V1, V2) 9.78/3.24 U44'(tt, V1, V2) -> U45'(isNat(V1), V2) 9.78/3.24 U104'(tt, V1, V2) -> ISNAT(V1) 9.78/3.24 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 U81(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U106(tt) -> tt 9.78/3.24 U96(tt) -> tt 9.78/3.24 U13(tt) -> tt 9.78/3.24 9.78/3.24 Q is empty. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (23) QCSDependencyGraphProof (EQUIVALENT) 9.78/3.24 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 1 SCC with 14 less nodes. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (24) 9.78/3.24 Obligation: 9.78/3.24 Q-restricted context-sensitive dependency pair problem: 9.78/3.24 The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions. 9.78/3.24 For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U22'_2, U21'_2} we have mu(f) = {1}. 9.78/3.24 The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNAT_1} are not replacing on any position. 9.78/3.24 9.78/3.24 The TRS P consists of the following rules: 9.78/3.24 9.78/3.24 U21'(tt, V1) -> U22'(isNatKind(V1), V1) 9.78/3.24 U22'(tt, V1) -> ISNAT(V1) 9.78/3.24 ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) 9.78/3.24 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 U81(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U106(tt) -> tt 9.78/3.24 U96(tt) -> tt 9.78/3.24 U13(tt) -> tt 9.78/3.24 9.78/3.24 Q is empty. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (25) QCSDPSubtermProof (EQUIVALENT) 9.78/3.24 We use the subterm processor [DA_EMMES]. 9.78/3.24 9.78/3.24 9.78/3.24 The following pairs can be oriented strictly and are deleted. 9.78/3.24 9.78/3.24 ISNAT(s(V1)) -> U21'(isNatKind(V1), V1) 9.78/3.24 The remaining pairs can at least be oriented weakly. 9.78/3.24 9.78/3.24 U21'(tt, V1) -> U22'(isNatKind(V1), V1) 9.78/3.24 U22'(tt, V1) -> ISNAT(V1) 9.78/3.24 Used ordering: Combined order from the following AFS and order. 9.78/3.24 U22'(x1, x2) = x2 9.78/3.24 9.78/3.24 U21'(x1, x2) = x2 9.78/3.24 9.78/3.24 ISNAT(x1) = x1 9.78/3.24 9.78/3.24 9.78/3.24 Subterm Order 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (26) 9.78/3.24 Obligation: 9.78/3.24 Q-restricted context-sensitive dependency pair problem: 9.78/3.24 The symbols in {length_1, U71_1, take_2, s_1, U81_1, U62_1, U52_1, U13_1, U23_1, U96_1, U106_1, U33_1, U46_1} are replacing on all positions. 9.78/3.24 For all symbols f in {cons_2, U51_2, U61_2, U11_2, U12_2, U91_3, U92_3, U93_3, U94_3, U95_2, U21_2, U22_2, U101_3, U102_3, U103_3, U104_3, U105_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U22'_2, U21'_2} we have mu(f) = {1}. 9.78/3.24 The symbols in {isNatIListKind_1, isNatKind_1, isNat_1, isNatList_1, isNatIList_1, ISNAT_1} are not replacing on any position. 9.78/3.24 9.78/3.24 The TRS P consists of the following rules: 9.78/3.24 9.78/3.24 U21'(tt, V1) -> U22'(isNatKind(V1), V1) 9.78/3.24 U22'(tt, V1) -> ISNAT(V1) 9.78/3.24 9.78/3.24 The TRS R consists of the following rules: 9.78/3.24 9.78/3.24 isNatIListKind(nil) -> tt 9.78/3.24 isNatIListKind(zeros) -> tt 9.78/3.24 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.24 isNatKind(0) -> tt 9.78/3.24 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.24 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.24 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.24 U81(tt) -> tt 9.78/3.24 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.24 U62(tt) -> tt 9.78/3.24 U71(tt) -> tt 9.78/3.24 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.24 U52(tt) -> tt 9.78/3.24 isNat(0) -> tt 9.78/3.24 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.24 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.24 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.24 isNatList(nil) -> tt 9.78/3.24 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.24 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.24 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.24 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.24 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.24 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.24 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.24 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.24 U23(tt) -> tt 9.78/3.24 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.24 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.24 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.24 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.24 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.24 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.24 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.24 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.24 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.24 U32(tt, V) -> U33(isNatList(V)) 9.78/3.24 U33(tt) -> tt 9.78/3.24 isNatIList(zeros) -> tt 9.78/3.24 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.24 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.24 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.24 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.24 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.24 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.24 U46(tt) -> tt 9.78/3.24 U106(tt) -> tt 9.78/3.24 U96(tt) -> tt 9.78/3.24 U13(tt) -> tt 9.78/3.24 9.78/3.24 Q is empty. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (27) QCSDependencyGraphProof (EQUIVALENT) 9.78/3.24 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes. 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.24 9.78/3.24 (28) 9.78/3.24 TRUE 9.78/3.24 9.78/3.24 ---------------------------------------- 9.78/3.25 9.78/3.25 (29) 9.78/3.25 Obligation: 9.78/3.25 Q-restricted context-sensitive dependency pair problem: 9.78/3.25 The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, TAKE_2} are replacing on all positions. 9.78/3.25 For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U133'_4, U132'_4, U134'_4, U135'_4, U136'_4, U131'_4} we have mu(f) = {1}. 9.78/3.25 The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, U_1} are not replacing on any position. 9.78/3.25 9.78/3.25 The TRS P consists of the following rules: 9.78/3.25 9.78/3.25 U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N) 9.78/3.25 U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N) 9.78/3.25 U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N) 9.78/3.25 U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N) 9.78/3.25 U136'(tt, IL, M, N) -> U(N) 9.78/3.25 U(take(x_0, x_1)) -> U(x_0) 9.78/3.25 U(take(x_0, x_1)) -> U(x_1) 9.78/3.25 U(take(x0, x1)) -> TAKE(x0, x1) 9.78/3.25 TAKE(s(M), cons(N, IL)) -> U131'(isNatIList(IL), IL, M, N) 9.78/3.25 U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N) 9.78/3.25 9.78/3.25 The TRS R consists of the following rules: 9.78/3.25 9.78/3.25 zeros -> cons(0, zeros) 9.78/3.25 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.25 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.25 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.25 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.25 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.25 U106(tt) -> tt 9.78/3.25 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.25 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.25 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.25 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.25 U114(tt, L) -> s(length(L)) 9.78/3.25 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.25 U13(tt) -> tt 9.78/3.25 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.25 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.25 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.25 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.25 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.25 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.25 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.25 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.25 U23(tt) -> tt 9.78/3.25 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.25 U32(tt, V) -> U33(isNatList(V)) 9.78/3.25 U33(tt) -> tt 9.78/3.25 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.25 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.25 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.25 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.25 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.25 U46(tt) -> tt 9.78/3.25 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.25 U52(tt) -> tt 9.78/3.25 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.25 U62(tt) -> tt 9.78/3.25 U71(tt) -> tt 9.78/3.25 U81(tt) -> tt 9.78/3.25 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.25 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.25 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.25 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.25 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.25 U96(tt) -> tt 9.78/3.25 isNat(0) -> tt 9.78/3.25 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.25 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.25 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.25 isNatIList(zeros) -> tt 9.78/3.25 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.25 isNatIListKind(nil) -> tt 9.78/3.25 isNatIListKind(zeros) -> tt 9.78/3.25 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.25 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.25 isNatKind(0) -> tt 9.78/3.25 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.25 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.25 isNatList(nil) -> tt 9.78/3.25 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.25 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.25 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.25 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.25 9.78/3.25 Q is empty. 9.78/3.25 9.78/3.25 ---------------------------------------- 9.78/3.25 9.78/3.25 (30) QCSDPSubtermProof (EQUIVALENT) 9.78/3.25 We use the subterm processor [DA_EMMES]. 9.78/3.25 9.78/3.25 9.78/3.25 The following pairs can be oriented strictly and are deleted. 9.78/3.25 9.78/3.25 U(take(x_0, x_1)) -> U(x_0) 9.78/3.25 U(take(x_0, x_1)) -> U(x_1) 9.78/3.25 U(take(x0, x1)) -> TAKE(x0, x1) 9.78/3.25 TAKE(s(M), cons(N, IL)) -> U131'(isNatIList(IL), IL, M, N) 9.78/3.25 The remaining pairs can at least be oriented weakly. 9.78/3.25 9.78/3.25 U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N) 9.78/3.25 U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N) 9.78/3.25 U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N) 9.78/3.25 U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N) 9.78/3.25 U136'(tt, IL, M, N) -> U(N) 9.78/3.25 U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N) 9.78/3.25 Used ordering: Combined order from the following AFS and order. 9.78/3.25 U133'(x1, x2, x3, x4) = x4 9.78/3.25 9.78/3.25 U132'(x1, x2, x3, x4) = x4 9.78/3.25 9.78/3.25 U134'(x1, x2, x3, x4) = x4 9.78/3.25 9.78/3.25 U135'(x1, x2, x3, x4) = x4 9.78/3.25 9.78/3.25 U136'(x1, x2, x3, x4) = x4 9.78/3.25 9.78/3.25 U(x1) = x1 9.78/3.25 9.78/3.25 TAKE(x1, x2) = x2 9.78/3.25 9.78/3.25 U131'(x1, x2, x3, x4) = x4 9.78/3.25 9.78/3.25 9.78/3.25 Subterm Order 9.78/3.25 9.78/3.25 ---------------------------------------- 9.78/3.25 9.78/3.25 (31) 9.78/3.25 Obligation: 9.78/3.25 Q-restricted context-sensitive dependency pair problem: 9.78/3.25 The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1} are replacing on all positions. 9.78/3.25 For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U133'_4, U132'_4, U134'_4, U135'_4, U136'_4, U131'_4} we have mu(f) = {1}. 9.78/3.25 The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1, U_1} are not replacing on any position. 9.78/3.25 9.78/3.25 The TRS P consists of the following rules: 9.78/3.25 9.78/3.25 U132'(tt, IL, M, N) -> U133'(isNat(M), IL, M, N) 9.78/3.25 U133'(tt, IL, M, N) -> U134'(isNatKind(M), IL, M, N) 9.78/3.25 U134'(tt, IL, M, N) -> U135'(isNat(N), IL, M, N) 9.78/3.25 U135'(tt, IL, M, N) -> U136'(isNatKind(N), IL, M, N) 9.78/3.25 U136'(tt, IL, M, N) -> U(N) 9.78/3.25 U131'(tt, IL, M, N) -> U132'(isNatIListKind(IL), IL, M, N) 9.78/3.25 9.78/3.25 The TRS R consists of the following rules: 9.78/3.25 9.78/3.25 zeros -> cons(0, zeros) 9.78/3.25 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.25 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.25 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.25 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.25 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.25 U106(tt) -> tt 9.78/3.25 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.25 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.25 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.25 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.25 U114(tt, L) -> s(length(L)) 9.78/3.25 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.25 U13(tt) -> tt 9.78/3.25 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.25 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.25 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.25 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.25 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.25 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.25 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.25 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.25 U23(tt) -> tt 9.78/3.25 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.25 U32(tt, V) -> U33(isNatList(V)) 9.78/3.25 U33(tt) -> tt 9.78/3.25 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.25 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.25 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.25 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.25 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.25 U46(tt) -> tt 9.78/3.25 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.25 U52(tt) -> tt 9.78/3.25 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.25 U62(tt) -> tt 9.78/3.25 U71(tt) -> tt 9.78/3.25 U81(tt) -> tt 9.78/3.25 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.25 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.25 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.25 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.25 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.25 U96(tt) -> tt 9.78/3.25 isNat(0) -> tt 9.78/3.25 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.25 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.25 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.25 isNatIList(zeros) -> tt 9.78/3.25 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.25 isNatIListKind(nil) -> tt 9.78/3.25 isNatIListKind(zeros) -> tt 9.78/3.25 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.25 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.25 isNatKind(0) -> tt 9.78/3.25 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.25 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.25 isNatList(nil) -> tt 9.78/3.25 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.25 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.25 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.25 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.25 9.78/3.25 Q is empty. 9.78/3.25 9.78/3.25 ---------------------------------------- 9.78/3.25 9.78/3.25 (32) QCSDependencyGraphProof (EQUIVALENT) 9.78/3.25 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 6 less nodes. 9.78/3.25 9.78/3.25 ---------------------------------------- 9.78/3.25 9.78/3.25 (33) 9.78/3.25 TRUE 9.78/3.25 9.78/3.25 ---------------------------------------- 9.78/3.25 9.78/3.25 (34) 9.78/3.25 Obligation: 9.78/3.25 Q-restricted context-sensitive dependency pair problem: 9.78/3.25 The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, LENGTH_1} are replacing on all positions. 9.78/3.25 For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U113'_3, U112'_3, U114'_2, U111'_3} we have mu(f) = {1}. 9.78/3.25 The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1} are not replacing on any position. 9.78/3.25 9.78/3.25 The TRS P consists of the following rules: 9.78/3.25 9.78/3.25 U112'(tt, L, N) -> U113'(isNat(N), L, N) 9.78/3.25 U113'(tt, L, N) -> U114'(isNatKind(N), L) 9.78/3.25 U114'(tt, L) -> LENGTH(L) 9.78/3.25 LENGTH(cons(N, L)) -> U111'(isNatList(L), L, N) 9.78/3.25 U111'(tt, L, N) -> U112'(isNatIListKind(L), L, N) 9.78/3.25 9.78/3.25 The TRS R consists of the following rules: 9.78/3.25 9.78/3.25 zeros -> cons(0, zeros) 9.78/3.25 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.25 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.25 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.25 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.25 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.25 U106(tt) -> tt 9.78/3.25 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.25 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.25 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.25 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.25 U114(tt, L) -> s(length(L)) 9.78/3.25 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.25 U13(tt) -> tt 9.78/3.25 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.25 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.25 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.25 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.25 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.25 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.25 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.25 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.25 U23(tt) -> tt 9.78/3.25 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.25 U32(tt, V) -> U33(isNatList(V)) 9.78/3.25 U33(tt) -> tt 9.78/3.25 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.25 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.25 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.25 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.25 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.25 U46(tt) -> tt 9.78/3.25 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.25 U52(tt) -> tt 9.78/3.25 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.25 U62(tt) -> tt 9.78/3.25 U71(tt) -> tt 9.78/3.25 U81(tt) -> tt 9.78/3.25 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.25 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.25 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.25 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.25 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.25 U96(tt) -> tt 9.78/3.25 isNat(0) -> tt 9.78/3.25 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.25 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.25 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.25 isNatIList(zeros) -> tt 9.78/3.25 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.25 isNatIListKind(nil) -> tt 9.78/3.25 isNatIListKind(zeros) -> tt 9.78/3.25 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.25 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.25 isNatKind(0) -> tt 9.78/3.25 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.25 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.25 isNatList(nil) -> tt 9.78/3.25 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.25 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.25 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.25 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.25 9.78/3.25 Q is empty. 9.78/3.25 9.78/3.25 ---------------------------------------- 9.78/3.25 9.78/3.25 (35) QCSDPReductionPairProof (EQUIVALENT) 9.78/3.25 Using the order 9.78/3.25 9.78/3.25 Polynomial interpretation [POLO]: 9.78/3.25 9.78/3.25 POL(0) = 2 9.78/3.25 POL(LENGTH(x_1)) = 2 + 2*x_1 9.78/3.25 POL(U101(x_1, x_2, x_3)) = x_2 9.78/3.25 POL(U102(x_1, x_2, x_3)) = x_2 9.78/3.25 POL(U103(x_1, x_2, x_3)) = x_2 9.78/3.25 POL(U104(x_1, x_2, x_3)) = x_2 9.78/3.25 POL(U105(x_1, x_2)) = x_1 9.78/3.25 POL(U106(x_1)) = 2 9.78/3.25 POL(U11(x_1, x_2)) = x_2 9.78/3.25 POL(U111(x_1, x_2, x_3)) = 2*x_2 9.78/3.25 POL(U111'(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 9.78/3.25 POL(U112(x_1, x_2, x_3)) = 2*x_2 9.78/3.25 POL(U112'(x_1, x_2, x_3)) = 2 + 2*x_2 9.78/3.25 POL(U113(x_1, x_2, x_3)) = 2*x_2 9.78/3.25 POL(U113'(x_1, x_2, x_3)) = 2 + 2*x_2 9.78/3.25 POL(U114(x_1, x_2)) = 2*x_2 9.78/3.25 POL(U114'(x_1, x_2)) = 2 + 2*x_2 9.78/3.25 POL(U12(x_1, x_2)) = x_2 9.78/3.25 POL(U13(x_1)) = x_1 9.78/3.25 POL(U131(x_1, x_2, x_3, x_4)) = 2*x_3 9.78/3.25 POL(U132(x_1, x_2, x_3, x_4)) = 2*x_3 9.78/3.25 POL(U133(x_1, x_2, x_3, x_4)) = 2*x_3 9.78/3.25 POL(U134(x_1, x_2, x_3, x_4)) = 2*x_3 9.78/3.25 POL(U135(x_1, x_2, x_3, x_4)) = 2*x_3 9.78/3.25 POL(U136(x_1, x_2, x_3, x_4)) = 2*x_3 9.78/3.25 POL(U21(x_1, x_2)) = 2*x_2 9.78/3.25 POL(U22(x_1, x_2)) = 2*x_2 9.78/3.25 POL(U23(x_1)) = 2*x_1 9.78/3.25 POL(U31(x_1, x_2)) = 2*x_2 9.78/3.25 POL(U32(x_1, x_2)) = 2*x_2 9.78/3.25 POL(U33(x_1)) = 2*x_1 9.78/3.25 POL(U41(x_1, x_2, x_3)) = 2 + 2*x_3 9.78/3.25 POL(U42(x_1, x_2, x_3)) = 2 9.78/3.25 POL(U43(x_1, x_2, x_3)) = x_1 9.78/3.25 POL(U44(x_1, x_2, x_3)) = x_1 9.78/3.25 POL(U45(x_1, x_2)) = 2 9.78/3.25 POL(U46(x_1)) = 2 9.78/3.25 POL(U51(x_1, x_2)) = 2 9.78/3.25 POL(U52(x_1)) = x_1 9.78/3.25 POL(U61(x_1, x_2)) = 2 9.78/3.25 POL(U62(x_1)) = 2 9.78/3.25 POL(U71(x_1)) = 2 9.78/3.25 POL(U81(x_1)) = 2 9.78/3.25 POL(U91(x_1, x_2, x_3)) = 2*x_3 9.78/3.25 POL(U92(x_1, x_2, x_3)) = x_3 9.78/3.25 POL(U93(x_1, x_2, x_3)) = x_3 9.78/3.25 POL(U94(x_1, x_2, x_3)) = x_3 9.78/3.25 POL(U95(x_1, x_2)) = x_2 9.78/3.25 POL(U96(x_1)) = x_1 9.78/3.25 POL(cons(x_1, x_2)) = 2*x_2 9.78/3.25 POL(isNat(x_1)) = x_1 9.78/3.25 POL(isNatIList(x_1)) = 2 + 2*x_1 9.78/3.25 POL(isNatIListKind(x_1)) = 2 9.78/3.25 POL(isNatKind(x_1)) = 2 9.78/3.25 POL(isNatList(x_1)) = x_1 9.78/3.25 POL(length(x_1)) = x_1 9.78/3.25 POL(nil) = 2 9.78/3.25 POL(s(x_1)) = 2*x_1 9.78/3.25 POL(take(x_1, x_2)) = x_1 9.78/3.25 POL(tt) = 2 9.78/3.25 POL(zeros) = 0 9.78/3.25 9.78/3.25 9.78/3.25 the following usable rules 9.78/3.25 9.78/3.25 9.78/3.25 isNat(0) -> tt 9.78/3.25 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.25 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.25 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.25 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.25 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.25 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.25 U114(tt, L) -> s(length(L)) 9.78/3.25 isNatKind(0) -> tt 9.78/3.25 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.25 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.25 U71(tt) -> tt 9.78/3.25 isNatIListKind(nil) -> tt 9.78/3.25 isNatIListKind(zeros) -> tt 9.78/3.25 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.25 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.25 zeros -> cons(0, zeros) 9.78/3.25 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.25 U52(tt) -> tt 9.78/3.25 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.25 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.25 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.25 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.25 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.25 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.25 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.25 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.25 isNatIList(zeros) -> tt 9.78/3.25 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.25 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.25 U32(tt, V) -> U33(isNatList(V)) 9.78/3.25 U33(tt) -> tt 9.78/3.25 isNatList(nil) -> tt 9.78/3.25 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.25 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.25 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.25 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.25 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.25 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.25 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.25 U96(tt) -> tt 9.78/3.25 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.25 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.25 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.25 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.25 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.25 U106(tt) -> tt 9.78/3.25 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.25 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.25 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.25 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.25 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.25 U46(tt) -> tt 9.78/3.25 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.25 U62(tt) -> tt 9.78/3.25 U81(tt) -> tt 9.78/3.25 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.25 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.25 U13(tt) -> tt 9.78/3.25 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.25 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.25 U23(tt) -> tt 9.78/3.25 9.78/3.25 9.78/3.25 could all be oriented weakly. 9.78/3.25 9.78/3.25 Furthermore, the pairs 9.78/3.25 9.78/3.25 9.78/3.25 LENGTH(cons(N, L)) -> U111'(isNatList(L), L, N) 9.78/3.25 U111'(tt, L, N) -> U112'(isNatIListKind(L), L, N) 9.78/3.25 9.78/3.25 9.78/3.25 could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES]. 9.78/3.25 9.78/3.25 9.78/3.25 ---------------------------------------- 9.78/3.25 9.78/3.25 (36) 9.78/3.25 Obligation: 9.78/3.25 Q-restricted context-sensitive dependency pair problem: 9.78/3.25 The symbols in {U106_1, s_1, length_1, U13_1, take_2, U23_1, U33_1, U46_1, U52_1, U62_1, U71_1, U81_1, U96_1, LENGTH_1} are replacing on all positions. 9.78/3.25 For all symbols f in {cons_2, U101_3, U102_3, U103_3, U104_3, U105_2, U11_2, U12_2, U111_3, U112_3, U113_3, U114_2, U131_4, U132_4, U133_4, U134_4, U135_4, U136_4, U21_2, U22_2, U31_2, U32_2, U41_3, U42_3, U43_3, U44_3, U45_2, U51_2, U61_2, U91_3, U92_3, U93_3, U94_3, U95_2, U113'_3, U112'_3, U114'_2} we have mu(f) = {1}. 9.78/3.25 The symbols in {isNatKind_1, isNatIListKind_1, isNat_1, isNatIList_1, isNatList_1} are not replacing on any position. 9.78/3.25 9.78/3.25 The TRS P consists of the following rules: 9.78/3.25 9.78/3.25 U112'(tt, L, N) -> U113'(isNat(N), L, N) 9.78/3.25 U113'(tt, L, N) -> U114'(isNatKind(N), L) 9.78/3.25 U114'(tt, L) -> LENGTH(L) 9.78/3.25 9.78/3.25 The TRS R consists of the following rules: 9.78/3.25 9.78/3.25 zeros -> cons(0, zeros) 9.78/3.25 U101(tt, V1, V2) -> U102(isNatKind(V1), V1, V2) 9.78/3.25 U102(tt, V1, V2) -> U103(isNatIListKind(V2), V1, V2) 9.78/3.25 U103(tt, V1, V2) -> U104(isNatIListKind(V2), V1, V2) 9.78/3.25 U104(tt, V1, V2) -> U105(isNat(V1), V2) 9.78/3.25 U105(tt, V2) -> U106(isNatIList(V2)) 9.78/3.25 U106(tt) -> tt 9.78/3.25 U11(tt, V1) -> U12(isNatIListKind(V1), V1) 9.78/3.25 U111(tt, L, N) -> U112(isNatIListKind(L), L, N) 9.78/3.25 U112(tt, L, N) -> U113(isNat(N), L, N) 9.78/3.25 U113(tt, L, N) -> U114(isNatKind(N), L) 9.78/3.25 U114(tt, L) -> s(length(L)) 9.78/3.25 U12(tt, V1) -> U13(isNatList(V1)) 9.78/3.25 U13(tt) -> tt 9.78/3.25 U131(tt, IL, M, N) -> U132(isNatIListKind(IL), IL, M, N) 9.78/3.25 U132(tt, IL, M, N) -> U133(isNat(M), IL, M, N) 9.78/3.25 U133(tt, IL, M, N) -> U134(isNatKind(M), IL, M, N) 9.78/3.25 U134(tt, IL, M, N) -> U135(isNat(N), IL, M, N) 9.78/3.25 U135(tt, IL, M, N) -> U136(isNatKind(N), IL, M, N) 9.78/3.25 U136(tt, IL, M, N) -> cons(N, take(M, IL)) 9.78/3.25 U21(tt, V1) -> U22(isNatKind(V1), V1) 9.78/3.25 U22(tt, V1) -> U23(isNat(V1)) 9.78/3.25 U23(tt) -> tt 9.78/3.25 U31(tt, V) -> U32(isNatIListKind(V), V) 9.78/3.25 U32(tt, V) -> U33(isNatList(V)) 9.78/3.25 U33(tt) -> tt 9.78/3.25 U41(tt, V1, V2) -> U42(isNatKind(V1), V1, V2) 9.78/3.25 U42(tt, V1, V2) -> U43(isNatIListKind(V2), V1, V2) 9.78/3.25 U43(tt, V1, V2) -> U44(isNatIListKind(V2), V1, V2) 9.78/3.25 U44(tt, V1, V2) -> U45(isNat(V1), V2) 9.78/3.25 U45(tt, V2) -> U46(isNatIList(V2)) 9.78/3.25 U46(tt) -> tt 9.78/3.25 U51(tt, V2) -> U52(isNatIListKind(V2)) 9.78/3.25 U52(tt) -> tt 9.78/3.25 U61(tt, V2) -> U62(isNatIListKind(V2)) 9.78/3.25 U62(tt) -> tt 9.78/3.25 U71(tt) -> tt 9.78/3.25 U81(tt) -> tt 9.78/3.25 U91(tt, V1, V2) -> U92(isNatKind(V1), V1, V2) 9.78/3.25 U92(tt, V1, V2) -> U93(isNatIListKind(V2), V1, V2) 9.78/3.25 U93(tt, V1, V2) -> U94(isNatIListKind(V2), V1, V2) 9.78/3.25 U94(tt, V1, V2) -> U95(isNat(V1), V2) 9.78/3.25 U95(tt, V2) -> U96(isNatList(V2)) 9.78/3.25 U96(tt) -> tt 9.78/3.25 isNat(0) -> tt 9.78/3.25 isNat(length(V1)) -> U11(isNatIListKind(V1), V1) 9.78/3.25 isNat(s(V1)) -> U21(isNatKind(V1), V1) 9.78/3.25 isNatIList(V) -> U31(isNatIListKind(V), V) 9.78/3.25 isNatIList(zeros) -> tt 9.78/3.25 isNatIList(cons(V1, V2)) -> U41(isNatKind(V1), V1, V2) 9.78/3.25 isNatIListKind(nil) -> tt 9.78/3.25 isNatIListKind(zeros) -> tt 9.78/3.25 isNatIListKind(cons(V1, V2)) -> U51(isNatKind(V1), V2) 9.78/3.25 isNatIListKind(take(V1, V2)) -> U61(isNatKind(V1), V2) 9.78/3.25 isNatKind(0) -> tt 9.78/3.25 isNatKind(length(V1)) -> U71(isNatIListKind(V1)) 9.78/3.25 isNatKind(s(V1)) -> U81(isNatKind(V1)) 9.78/3.25 isNatList(nil) -> tt 9.78/3.25 isNatList(cons(V1, V2)) -> U91(isNatKind(V1), V1, V2) 9.78/3.25 isNatList(take(V1, V2)) -> U101(isNatKind(V1), V1, V2) 9.78/3.25 length(cons(N, L)) -> U111(isNatList(L), L, N) 9.78/3.25 take(s(M), cons(N, IL)) -> U131(isNatIList(IL), IL, M, N) 9.78/3.25 9.78/3.25 Q is empty. 9.78/3.25 9.78/3.25 ---------------------------------------- 9.78/3.25 9.78/3.25 (37) QCSDependencyGraphProof (EQUIVALENT) 9.78/3.25 The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 3 less nodes. 9.78/3.25 9.78/3.25 ---------------------------------------- 9.78/3.25 9.78/3.25 (38) 9.78/3.25 TRUE 10.16/3.34 EOF