41.90/12.86 YES 42.17/12.88 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 42.17/12.88 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 42.17/12.88 42.17/12.88 42.17/12.88 Termination w.r.t. Q of the given QTRS could be proven: 42.17/12.88 42.17/12.88 (0) QTRS 42.17/12.88 (1) DependencyPairsProof [EQUIVALENT, 322 ms] 42.17/12.88 (2) QDP 42.17/12.88 (3) DependencyGraphProof [EQUIVALENT, 0 ms] 42.17/12.88 (4) AND 42.17/12.88 (5) QDP 42.17/12.88 (6) UsableRulesProof [EQUIVALENT, 0 ms] 42.17/12.88 (7) QDP 42.17/12.88 (8) QReductionProof [EQUIVALENT, 0 ms] 42.17/12.88 (9) QDP 42.17/12.88 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 42.17/12.88 (11) YES 42.17/12.88 (12) QDP 42.17/12.88 (13) UsableRulesProof [EQUIVALENT, 0 ms] 42.17/12.88 (14) QDP 42.17/12.88 (15) QReductionProof [EQUIVALENT, 0 ms] 42.17/12.88 (16) QDP 42.17/12.88 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 42.17/12.88 (18) YES 42.17/12.88 (19) QDP 42.17/12.88 (20) QDPOrderProof [EQUIVALENT, 929 ms] 42.17/12.88 (21) QDP 42.17/12.88 (22) DependencyGraphProof [EQUIVALENT, 0 ms] 42.17/12.88 (23) QDP 42.17/12.88 (24) QDPOrderProof [EQUIVALENT, 606 ms] 42.17/12.88 (25) QDP 42.17/12.88 (26) DependencyGraphProof [EQUIVALENT, 0 ms] 42.17/12.88 (27) AND 42.17/12.88 (28) QDP 42.17/12.88 (29) UsableRulesProof [EQUIVALENT, 0 ms] 42.17/12.88 (30) QDP 42.17/12.88 (31) QReductionProof [EQUIVALENT, 0 ms] 42.17/12.88 (32) QDP 42.17/12.88 (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] 42.17/12.88 (34) YES 42.17/12.88 (35) QDP 42.17/12.88 (36) QDPOrderProof [EQUIVALENT, 804 ms] 42.17/12.88 (37) QDP 42.17/12.88 (38) DependencyGraphProof [EQUIVALENT, 0 ms] 42.17/12.88 (39) TRUE 42.17/12.88 42.17/12.88 42.17/12.88 ---------------------------------------- 42.17/12.88 42.17/12.88 (0) 42.17/12.88 Obligation: 42.17/12.88 Q restricted rewrite system: 42.17/12.88 The TRS R consists of the following rules: 42.17/12.88 42.17/12.88 a__zeros -> cons(0, zeros) 42.17/12.88 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.88 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.88 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.88 a__U106(tt) -> tt 42.17/12.88 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.88 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.17/12.88 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.17/12.88 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.17/12.88 a__U114(tt, L) -> s(a__length(mark(L))) 42.17/12.88 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.88 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.17/12.88 a__U122(tt) -> nil 42.17/12.88 a__U13(tt) -> tt 42.17/12.88 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.88 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.17/12.88 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.17/12.88 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.17/12.88 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.17/12.88 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.17/12.88 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.88 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.88 a__U23(tt) -> tt 42.17/12.88 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.88 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.88 a__U33(tt) -> tt 42.17/12.88 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.88 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.88 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.88 a__U46(tt) -> tt 42.17/12.88 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.88 a__U52(tt) -> tt 42.17/12.88 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.88 a__U62(tt) -> tt 42.17/12.88 a__U71(tt) -> tt 42.17/12.88 a__U81(tt) -> tt 42.17/12.88 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.88 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.88 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.88 a__U96(tt) -> tt 42.17/12.88 a__isNat(0) -> tt 42.17/12.88 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.88 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.88 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.88 a__isNatIList(zeros) -> tt 42.17/12.88 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.88 a__isNatIListKind(nil) -> tt 42.17/12.88 a__isNatIListKind(zeros) -> tt 42.17/12.88 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.88 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.88 a__isNatKind(0) -> tt 42.17/12.88 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.88 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.88 a__isNatList(nil) -> tt 42.17/12.88 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.88 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.88 a__length(nil) -> 0 42.17/12.88 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.17/12.88 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.17/12.88 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.17/12.88 mark(zeros) -> a__zeros 42.17/12.88 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.17/12.88 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.17/12.88 mark(isNatKind(X)) -> a__isNatKind(X) 42.17/12.88 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.17/12.88 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.17/12.88 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.17/12.88 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.17/12.88 mark(isNat(X)) -> a__isNat(X) 42.17/12.88 mark(U106(X)) -> a__U106(mark(X)) 42.17/12.88 mark(isNatIList(X)) -> a__isNatIList(X) 42.17/12.88 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.17/12.88 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.17/12.88 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.17/12.88 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.17/12.88 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.17/12.88 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.17/12.88 mark(length(X)) -> a__length(mark(X)) 42.17/12.88 mark(U13(X)) -> a__U13(mark(X)) 42.17/12.88 mark(isNatList(X)) -> a__isNatList(X) 42.17/12.88 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.17/12.88 mark(U122(X)) -> a__U122(mark(X)) 42.17/12.88 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.17/12.88 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.17/12.88 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.17/12.88 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.17/12.88 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.17/12.88 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.17/12.88 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.17/12.88 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.17/12.88 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.17/12.88 mark(U23(X)) -> a__U23(mark(X)) 42.17/12.88 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.17/12.88 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.17/12.88 mark(U33(X)) -> a__U33(mark(X)) 42.17/12.88 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.17/12.88 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.17/12.88 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.17/12.88 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.17/12.88 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.17/12.88 mark(U46(X)) -> a__U46(mark(X)) 42.17/12.88 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.17/12.88 mark(U52(X)) -> a__U52(mark(X)) 42.17/12.88 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.17/12.88 mark(U62(X)) -> a__U62(mark(X)) 42.17/12.88 mark(U71(X)) -> a__U71(mark(X)) 42.17/12.88 mark(U81(X)) -> a__U81(mark(X)) 42.17/12.88 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.17/12.88 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.17/12.88 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.17/12.88 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.17/12.88 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.17/12.88 mark(U96(X)) -> a__U96(mark(X)) 42.17/12.88 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.17/12.88 mark(0) -> 0 42.17/12.88 mark(tt) -> tt 42.17/12.88 mark(s(X)) -> s(mark(X)) 42.17/12.88 mark(nil) -> nil 42.17/12.88 a__zeros -> zeros 42.17/12.88 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.88 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.88 a__isNatKind(X) -> isNatKind(X) 42.17/12.88 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.88 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.88 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.88 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.88 a__isNat(X) -> isNat(X) 42.17/12.88 a__U106(X) -> U106(X) 42.17/12.88 a__isNatIList(X) -> isNatIList(X) 42.17/12.88 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.88 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.88 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.17/12.88 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.17/12.88 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.17/12.88 a__U114(X1, X2) -> U114(X1, X2) 42.17/12.88 a__length(X) -> length(X) 42.17/12.88 a__U13(X) -> U13(X) 42.17/12.88 a__isNatList(X) -> isNatList(X) 42.17/12.88 a__U121(X1, X2) -> U121(X1, X2) 42.17/12.88 a__U122(X) -> U122(X) 42.17/12.88 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.17/12.88 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.17/12.88 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.17/12.88 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.17/12.88 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.17/12.88 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.17/12.88 a__take(X1, X2) -> take(X1, X2) 42.17/12.88 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.88 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.88 a__U23(X) -> U23(X) 42.17/12.88 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.88 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.88 a__U33(X) -> U33(X) 42.17/12.88 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.88 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.88 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.88 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.88 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.88 a__U46(X) -> U46(X) 42.17/12.88 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.88 a__U52(X) -> U52(X) 42.17/12.88 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.88 a__U62(X) -> U62(X) 42.17/12.88 a__U71(X) -> U71(X) 42.17/12.88 a__U81(X) -> U81(X) 42.17/12.88 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.88 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.88 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.88 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.88 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.88 a__U96(X) -> U96(X) 42.17/12.88 42.17/12.88 The set Q consists of the following terms: 42.17/12.88 42.17/12.88 a__zeros 42.17/12.88 a__isNatIList(x0) 42.17/12.88 mark(zeros) 42.17/12.88 mark(U101(x0, x1, x2)) 42.17/12.88 mark(U102(x0, x1, x2)) 42.17/12.88 mark(isNatKind(x0)) 42.17/12.88 mark(U103(x0, x1, x2)) 42.17/12.88 mark(isNatIListKind(x0)) 42.17/12.88 mark(U104(x0, x1, x2)) 42.17/12.88 mark(U105(x0, x1)) 42.17/12.88 mark(isNat(x0)) 42.17/12.88 mark(U106(x0)) 42.17/12.88 mark(isNatIList(x0)) 42.17/12.88 mark(U11(x0, x1)) 42.17/12.88 mark(U12(x0, x1)) 42.17/12.88 mark(U111(x0, x1, x2)) 42.17/12.88 mark(U112(x0, x1, x2)) 42.17/12.88 mark(U113(x0, x1, x2)) 42.17/12.88 mark(U114(x0, x1)) 42.17/12.88 mark(length(x0)) 42.17/12.88 mark(U13(x0)) 42.17/12.88 mark(isNatList(x0)) 42.17/12.88 mark(U121(x0, x1)) 42.17/12.88 mark(U122(x0)) 42.17/12.88 mark(U131(x0, x1, x2, x3)) 42.17/12.88 mark(U132(x0, x1, x2, x3)) 42.17/12.88 mark(U133(x0, x1, x2, x3)) 42.17/12.88 mark(U134(x0, x1, x2, x3)) 42.17/12.88 mark(U135(x0, x1, x2, x3)) 42.17/12.88 mark(U136(x0, x1, x2, x3)) 42.17/12.88 mark(take(x0, x1)) 42.17/12.88 mark(U21(x0, x1)) 42.17/12.88 mark(U22(x0, x1)) 42.17/12.88 mark(U23(x0)) 42.17/12.88 mark(U31(x0, x1)) 42.17/12.88 mark(U32(x0, x1)) 42.17/12.88 mark(U33(x0)) 42.17/12.88 mark(U41(x0, x1, x2)) 42.17/12.88 mark(U42(x0, x1, x2)) 42.17/12.88 mark(U43(x0, x1, x2)) 42.17/12.88 mark(U44(x0, x1, x2)) 42.17/12.88 mark(U45(x0, x1)) 42.17/12.88 mark(U46(x0)) 42.17/12.88 mark(U51(x0, x1)) 42.17/12.88 mark(U52(x0)) 42.17/12.88 mark(U61(x0, x1)) 42.17/12.88 mark(U62(x0)) 42.17/12.88 mark(U71(x0)) 42.17/12.88 mark(U81(x0)) 42.17/12.88 mark(U91(x0, x1, x2)) 42.17/12.88 mark(U92(x0, x1, x2)) 42.17/12.88 mark(U93(x0, x1, x2)) 42.17/12.88 mark(U94(x0, x1, x2)) 42.17/12.88 mark(U95(x0, x1)) 42.17/12.88 mark(U96(x0)) 42.17/12.88 mark(cons(x0, x1)) 42.17/12.88 mark(0) 42.17/12.88 mark(tt) 42.17/12.88 mark(s(x0)) 42.17/12.88 mark(nil) 42.17/12.88 a__U101(x0, x1, x2) 42.17/12.88 a__U102(x0, x1, x2) 42.17/12.88 a__isNatKind(x0) 42.17/12.88 a__U103(x0, x1, x2) 42.17/12.88 a__isNatIListKind(x0) 42.17/12.88 a__U104(x0, x1, x2) 42.17/12.88 a__U105(x0, x1) 42.17/12.88 a__isNat(x0) 42.17/12.88 a__U106(x0) 42.17/12.88 a__U11(x0, x1) 42.17/12.88 a__U12(x0, x1) 42.17/12.88 a__U111(x0, x1, x2) 42.17/12.88 a__U112(x0, x1, x2) 42.17/12.88 a__U113(x0, x1, x2) 42.17/12.88 a__U114(x0, x1) 42.17/12.88 a__length(x0) 42.17/12.88 a__U13(x0) 42.17/12.88 a__isNatList(x0) 42.17/12.88 a__U121(x0, x1) 42.17/12.88 a__U122(x0) 42.17/12.88 a__U131(x0, x1, x2, x3) 42.17/12.88 a__U132(x0, x1, x2, x3) 42.17/12.88 a__U133(x0, x1, x2, x3) 42.17/12.88 a__U134(x0, x1, x2, x3) 42.17/12.88 a__U135(x0, x1, x2, x3) 42.17/12.88 a__U136(x0, x1, x2, x3) 42.17/12.88 a__take(x0, x1) 42.17/12.88 a__U21(x0, x1) 42.17/12.88 a__U22(x0, x1) 42.17/12.88 a__U23(x0) 42.17/12.88 a__U31(x0, x1) 42.17/12.88 a__U32(x0, x1) 42.17/12.88 a__U33(x0) 42.17/12.88 a__U41(x0, x1, x2) 42.17/12.88 a__U42(x0, x1, x2) 42.17/12.88 a__U43(x0, x1, x2) 42.17/12.88 a__U44(x0, x1, x2) 42.17/12.88 a__U45(x0, x1) 42.17/12.88 a__U46(x0) 42.17/12.88 a__U51(x0, x1) 42.17/12.88 a__U52(x0) 42.17/12.88 a__U61(x0, x1) 42.17/12.88 a__U62(x0) 42.17/12.88 a__U71(x0) 42.17/12.88 a__U81(x0) 42.17/12.88 a__U91(x0, x1, x2) 42.17/12.88 a__U92(x0, x1, x2) 42.17/12.88 a__U93(x0, x1, x2) 42.17/12.88 a__U94(x0, x1, x2) 42.17/12.88 a__U95(x0, x1) 42.17/12.88 a__U96(x0) 42.17/12.88 42.17/12.88 42.17/12.88 ---------------------------------------- 42.17/12.88 42.17/12.88 (1) DependencyPairsProof (EQUIVALENT) 42.17/12.88 Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. 42.17/12.88 ---------------------------------------- 42.17/12.88 42.17/12.88 (2) 42.17/12.88 Obligation: 42.17/12.88 Q DP problem: 42.17/12.88 The TRS P consists of the following rules: 42.17/12.88 42.17/12.88 A__U101(tt, V1, V2) -> A__U102(a__isNatKind(V1), V1, V2) 42.17/12.88 A__U101(tt, V1, V2) -> A__ISNATKIND(V1) 42.17/12.88 A__U102(tt, V1, V2) -> A__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.88 A__U102(tt, V1, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__U103(tt, V1, V2) -> A__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.88 A__U103(tt, V1, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__U104(tt, V1, V2) -> A__U105(a__isNat(V1), V2) 42.17/12.88 A__U104(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.88 A__U105(tt, V2) -> A__U106(a__isNatIList(V2)) 42.17/12.88 A__U105(tt, V2) -> A__ISNATILIST(V2) 42.17/12.88 A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 42.17/12.88 A__U11(tt, V1) -> A__ISNATILISTKIND(V1) 42.17/12.88 A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) 42.17/12.88 A__U111(tt, L, N) -> A__ISNATILISTKIND(L) 42.17/12.88 A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) 42.17/12.88 A__U112(tt, L, N) -> A__ISNAT(N) 42.17/12.88 A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) 42.17/12.88 A__U113(tt, L, N) -> A__ISNATKIND(N) 42.17/12.88 A__U114(tt, L) -> A__LENGTH(mark(L)) 42.17/12.88 A__U114(tt, L) -> MARK(L) 42.17/12.88 A__U12(tt, V1) -> A__U13(a__isNatList(V1)) 42.17/12.88 A__U12(tt, V1) -> A__ISNATLIST(V1) 42.17/12.88 A__U121(tt, IL) -> A__U122(a__isNatIListKind(IL)) 42.17/12.88 A__U121(tt, IL) -> A__ISNATILISTKIND(IL) 42.17/12.88 A__U131(tt, IL, M, N) -> A__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.88 A__U131(tt, IL, M, N) -> A__ISNATILISTKIND(IL) 42.17/12.88 A__U132(tt, IL, M, N) -> A__U133(a__isNat(M), IL, M, N) 42.17/12.88 A__U132(tt, IL, M, N) -> A__ISNAT(M) 42.17/12.88 A__U133(tt, IL, M, N) -> A__U134(a__isNatKind(M), IL, M, N) 42.17/12.88 A__U133(tt, IL, M, N) -> A__ISNATKIND(M) 42.17/12.88 A__U134(tt, IL, M, N) -> A__U135(a__isNat(N), IL, M, N) 42.17/12.88 A__U134(tt, IL, M, N) -> A__ISNAT(N) 42.17/12.88 A__U135(tt, IL, M, N) -> A__U136(a__isNatKind(N), IL, M, N) 42.17/12.88 A__U135(tt, IL, M, N) -> A__ISNATKIND(N) 42.17/12.88 A__U136(tt, IL, M, N) -> MARK(N) 42.17/12.88 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.17/12.88 A__U21(tt, V1) -> A__ISNATKIND(V1) 42.17/12.88 A__U22(tt, V1) -> A__U23(a__isNat(V1)) 42.17/12.88 A__U22(tt, V1) -> A__ISNAT(V1) 42.17/12.88 A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) 42.17/12.88 A__U31(tt, V) -> A__ISNATILISTKIND(V) 42.17/12.88 A__U32(tt, V) -> A__U33(a__isNatList(V)) 42.17/12.88 A__U32(tt, V) -> A__ISNATLIST(V) 42.17/12.88 A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 42.17/12.88 A__U41(tt, V1, V2) -> A__ISNATKIND(V1) 42.17/12.88 A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.88 A__U42(tt, V1, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.88 A__U43(tt, V1, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 42.17/12.88 A__U44(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.88 A__U45(tt, V2) -> A__U46(a__isNatIList(V2)) 42.17/12.88 A__U45(tt, V2) -> A__ISNATILIST(V2) 42.17/12.88 A__U51(tt, V2) -> A__U52(a__isNatIListKind(V2)) 42.17/12.88 A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__U61(tt, V2) -> A__U62(a__isNatIListKind(V2)) 42.17/12.88 A__U61(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__U91(tt, V1, V2) -> A__U92(a__isNatKind(V1), V1, V2) 42.17/12.88 A__U91(tt, V1, V2) -> A__ISNATKIND(V1) 42.17/12.88 A__U92(tt, V1, V2) -> A__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.88 A__U92(tt, V1, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__U93(tt, V1, V2) -> A__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.88 A__U93(tt, V1, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__U94(tt, V1, V2) -> A__U95(a__isNat(V1), V2) 42.17/12.88 A__U94(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.88 A__U95(tt, V2) -> A__U96(a__isNatList(V2)) 42.17/12.88 A__U95(tt, V2) -> A__ISNATLIST(V2) 42.17/12.88 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 42.17/12.88 A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) 42.17/12.88 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.17/12.88 A__ISNAT(s(V1)) -> A__ISNATKIND(V1) 42.17/12.88 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) 42.17/12.88 A__ISNATILIST(V) -> A__ISNATILISTKIND(V) 42.17/12.88 A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 42.17/12.88 A__ISNATILIST(cons(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.88 A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 42.17/12.88 A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.88 A__ISNATILISTKIND(take(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.17/12.88 A__ISNATILISTKIND(take(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.88 A__ISNATKIND(length(V1)) -> A__U71(a__isNatIListKind(V1)) 42.17/12.88 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 42.17/12.88 A__ISNATKIND(s(V1)) -> A__U81(a__isNatKind(V1)) 42.17/12.88 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.17/12.88 A__ISNATLIST(cons(V1, V2)) -> A__U91(a__isNatKind(V1), V1, V2) 42.17/12.88 A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.88 A__ISNATLIST(take(V1, V2)) -> A__U101(a__isNatKind(V1), V1, V2) 42.17/12.88 A__ISNATLIST(take(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.88 A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) 42.17/12.88 A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) 42.17/12.88 A__TAKE(0, IL) -> A__U121(a__isNatIList(IL), IL) 42.17/12.88 A__TAKE(0, IL) -> A__ISNATILIST(IL) 42.17/12.88 A__TAKE(s(M), cons(N, IL)) -> A__U131(a__isNatIList(IL), IL, M, N) 42.17/12.88 A__TAKE(s(M), cons(N, IL)) -> A__ISNATILIST(IL) 42.17/12.88 MARK(zeros) -> A__ZEROS 42.17/12.88 MARK(U101(X1, X2, X3)) -> A__U101(mark(X1), X2, X3) 42.17/12.88 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U102(X1, X2, X3)) -> A__U102(mark(X1), X2, X3) 42.17/12.88 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(isNatKind(X)) -> A__ISNATKIND(X) 42.17/12.88 MARK(U103(X1, X2, X3)) -> A__U103(mark(X1), X2, X3) 42.17/12.88 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) 42.17/12.88 MARK(U104(X1, X2, X3)) -> A__U104(mark(X1), X2, X3) 42.17/12.88 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U105(X1, X2)) -> A__U105(mark(X1), X2) 42.17/12.88 MARK(U105(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(isNat(X)) -> A__ISNAT(X) 42.17/12.88 MARK(U106(X)) -> A__U106(mark(X)) 42.17/12.88 MARK(U106(X)) -> MARK(X) 42.17/12.88 MARK(isNatIList(X)) -> A__ISNATILIST(X) 42.17/12.88 MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) 42.17/12.88 MARK(U11(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U12(X1, X2)) -> A__U12(mark(X1), X2) 42.17/12.88 MARK(U12(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) 42.17/12.88 MARK(U111(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) 42.17/12.88 MARK(U112(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) 42.17/12.88 MARK(U113(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) 42.17/12.88 MARK(U114(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(length(X)) -> A__LENGTH(mark(X)) 42.17/12.88 MARK(length(X)) -> MARK(X) 42.17/12.88 MARK(U13(X)) -> A__U13(mark(X)) 42.17/12.88 MARK(U13(X)) -> MARK(X) 42.17/12.88 MARK(isNatList(X)) -> A__ISNATLIST(X) 42.17/12.88 MARK(U121(X1, X2)) -> A__U121(mark(X1), X2) 42.17/12.88 MARK(U121(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U122(X)) -> A__U122(mark(X)) 42.17/12.88 MARK(U122(X)) -> MARK(X) 42.17/12.88 MARK(U131(X1, X2, X3, X4)) -> A__U131(mark(X1), X2, X3, X4) 42.17/12.88 MARK(U131(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.88 MARK(U132(X1, X2, X3, X4)) -> A__U132(mark(X1), X2, X3, X4) 42.17/12.88 MARK(U132(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.88 MARK(U133(X1, X2, X3, X4)) -> A__U133(mark(X1), X2, X3, X4) 42.17/12.88 MARK(U133(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.88 MARK(U134(X1, X2, X3, X4)) -> A__U134(mark(X1), X2, X3, X4) 42.17/12.88 MARK(U134(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.88 MARK(U135(X1, X2, X3, X4)) -> A__U135(mark(X1), X2, X3, X4) 42.17/12.88 MARK(U135(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.88 MARK(U136(X1, X2, X3, X4)) -> A__U136(mark(X1), X2, X3, X4) 42.17/12.88 MARK(U136(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.88 MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) 42.17/12.88 MARK(take(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(take(X1, X2)) -> MARK(X2) 42.17/12.88 MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) 42.17/12.88 MARK(U21(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U22(X1, X2)) -> A__U22(mark(X1), X2) 42.17/12.88 MARK(U22(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U23(X)) -> A__U23(mark(X)) 42.17/12.88 MARK(U23(X)) -> MARK(X) 42.17/12.88 MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) 42.17/12.88 MARK(U31(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U32(X1, X2)) -> A__U32(mark(X1), X2) 42.17/12.88 MARK(U32(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U33(X)) -> A__U33(mark(X)) 42.17/12.88 MARK(U33(X)) -> MARK(X) 42.17/12.88 MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) 42.17/12.88 MARK(U41(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U42(X1, X2, X3)) -> A__U42(mark(X1), X2, X3) 42.17/12.88 MARK(U42(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U43(X1, X2, X3)) -> A__U43(mark(X1), X2, X3) 42.17/12.88 MARK(U43(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U44(X1, X2, X3)) -> A__U44(mark(X1), X2, X3) 42.17/12.88 MARK(U44(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U45(X1, X2)) -> A__U45(mark(X1), X2) 42.17/12.88 MARK(U45(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U46(X)) -> A__U46(mark(X)) 42.17/12.88 MARK(U46(X)) -> MARK(X) 42.17/12.88 MARK(U51(X1, X2)) -> A__U51(mark(X1), X2) 42.17/12.88 MARK(U51(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U52(X)) -> A__U52(mark(X)) 42.17/12.88 MARK(U52(X)) -> MARK(X) 42.17/12.88 MARK(U61(X1, X2)) -> A__U61(mark(X1), X2) 42.17/12.88 MARK(U61(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U62(X)) -> A__U62(mark(X)) 42.17/12.88 MARK(U62(X)) -> MARK(X) 42.17/12.88 MARK(U71(X)) -> A__U71(mark(X)) 42.17/12.88 MARK(U71(X)) -> MARK(X) 42.17/12.88 MARK(U81(X)) -> A__U81(mark(X)) 42.17/12.88 MARK(U81(X)) -> MARK(X) 42.17/12.88 MARK(U91(X1, X2, X3)) -> A__U91(mark(X1), X2, X3) 42.17/12.88 MARK(U91(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U92(X1, X2, X3)) -> A__U92(mark(X1), X2, X3) 42.17/12.88 MARK(U92(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U93(X1, X2, X3)) -> A__U93(mark(X1), X2, X3) 42.17/12.88 MARK(U93(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U94(X1, X2, X3)) -> A__U94(mark(X1), X2, X3) 42.17/12.88 MARK(U94(X1, X2, X3)) -> MARK(X1) 42.17/12.88 MARK(U95(X1, X2)) -> A__U95(mark(X1), X2) 42.17/12.88 MARK(U95(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(U96(X)) -> A__U96(mark(X)) 42.17/12.88 MARK(U96(X)) -> MARK(X) 42.17/12.88 MARK(cons(X1, X2)) -> MARK(X1) 42.17/12.88 MARK(s(X)) -> MARK(X) 42.17/12.88 42.17/12.88 The TRS R consists of the following rules: 42.17/12.88 42.17/12.88 a__zeros -> cons(0, zeros) 42.17/12.88 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.88 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.88 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.88 a__U106(tt) -> tt 42.17/12.88 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.88 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.17/12.88 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.17/12.88 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.17/12.88 a__U114(tt, L) -> s(a__length(mark(L))) 42.17/12.88 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.88 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.17/12.88 a__U122(tt) -> nil 42.17/12.88 a__U13(tt) -> tt 42.17/12.88 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.88 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.17/12.88 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.17/12.88 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.17/12.88 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.17/12.88 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.17/12.88 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.88 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.88 a__U23(tt) -> tt 42.17/12.88 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.88 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.88 a__U33(tt) -> tt 42.17/12.88 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.88 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.88 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.88 a__U46(tt) -> tt 42.17/12.88 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.88 a__U52(tt) -> tt 42.17/12.88 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.88 a__U62(tt) -> tt 42.17/12.88 a__U71(tt) -> tt 42.17/12.88 a__U81(tt) -> tt 42.17/12.88 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.88 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.88 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.88 a__U96(tt) -> tt 42.17/12.88 a__isNat(0) -> tt 42.17/12.88 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.88 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.88 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.88 a__isNatIList(zeros) -> tt 42.17/12.88 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.88 a__isNatIListKind(nil) -> tt 42.17/12.88 a__isNatIListKind(zeros) -> tt 42.17/12.88 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.88 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.88 a__isNatKind(0) -> tt 42.17/12.88 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.88 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.88 a__isNatList(nil) -> tt 42.17/12.88 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.88 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.88 a__length(nil) -> 0 42.17/12.88 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.17/12.88 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.17/12.88 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.17/12.88 mark(zeros) -> a__zeros 42.17/12.88 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.17/12.88 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.17/12.88 mark(isNatKind(X)) -> a__isNatKind(X) 42.17/12.88 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.17/12.88 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.17/12.88 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.17/12.88 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.17/12.88 mark(isNat(X)) -> a__isNat(X) 42.17/12.88 mark(U106(X)) -> a__U106(mark(X)) 42.17/12.88 mark(isNatIList(X)) -> a__isNatIList(X) 42.17/12.88 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.17/12.88 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.17/12.88 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.17/12.88 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.17/12.88 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.17/12.88 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.17/12.88 mark(length(X)) -> a__length(mark(X)) 42.17/12.88 mark(U13(X)) -> a__U13(mark(X)) 42.17/12.88 mark(isNatList(X)) -> a__isNatList(X) 42.17/12.88 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.17/12.88 mark(U122(X)) -> a__U122(mark(X)) 42.17/12.88 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.17/12.88 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.17/12.88 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.17/12.88 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.17/12.88 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.17/12.88 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.17/12.88 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.17/12.88 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.17/12.88 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.17/12.88 mark(U23(X)) -> a__U23(mark(X)) 42.17/12.88 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.17/12.88 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.17/12.88 mark(U33(X)) -> a__U33(mark(X)) 42.17/12.88 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.17/12.88 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.17/12.88 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.17/12.88 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.17/12.88 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.17/12.88 mark(U46(X)) -> a__U46(mark(X)) 42.17/12.88 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.17/12.88 mark(U52(X)) -> a__U52(mark(X)) 42.17/12.88 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.17/12.88 mark(U62(X)) -> a__U62(mark(X)) 42.17/12.88 mark(U71(X)) -> a__U71(mark(X)) 42.17/12.88 mark(U81(X)) -> a__U81(mark(X)) 42.17/12.88 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.17/12.88 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.17/12.88 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.17/12.88 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.17/12.88 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.17/12.88 mark(U96(X)) -> a__U96(mark(X)) 42.17/12.88 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.17/12.88 mark(0) -> 0 42.17/12.88 mark(tt) -> tt 42.17/12.88 mark(s(X)) -> s(mark(X)) 42.17/12.88 mark(nil) -> nil 42.17/12.88 a__zeros -> zeros 42.17/12.88 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.88 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.88 a__isNatKind(X) -> isNatKind(X) 42.17/12.88 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.88 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.88 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.88 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.88 a__isNat(X) -> isNat(X) 42.17/12.88 a__U106(X) -> U106(X) 42.17/12.88 a__isNatIList(X) -> isNatIList(X) 42.17/12.88 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.88 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.88 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.17/12.88 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.17/12.88 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.17/12.88 a__U114(X1, X2) -> U114(X1, X2) 42.17/12.88 a__length(X) -> length(X) 42.17/12.88 a__U13(X) -> U13(X) 42.17/12.88 a__isNatList(X) -> isNatList(X) 42.17/12.88 a__U121(X1, X2) -> U121(X1, X2) 42.17/12.88 a__U122(X) -> U122(X) 42.17/12.88 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.17/12.88 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.17/12.88 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.17/12.88 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.17/12.88 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.17/12.88 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.17/12.88 a__take(X1, X2) -> take(X1, X2) 42.17/12.88 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.88 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.88 a__U23(X) -> U23(X) 42.17/12.88 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.88 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.88 a__U33(X) -> U33(X) 42.17/12.88 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.88 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.88 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.88 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.88 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.88 a__U46(X) -> U46(X) 42.17/12.88 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.88 a__U52(X) -> U52(X) 42.17/12.88 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.88 a__U62(X) -> U62(X) 42.17/12.88 a__U71(X) -> U71(X) 42.17/12.88 a__U81(X) -> U81(X) 42.17/12.88 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.88 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.88 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.88 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.88 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.88 a__U96(X) -> U96(X) 42.17/12.88 42.17/12.88 The set Q consists of the following terms: 42.17/12.88 42.17/12.88 a__zeros 42.17/12.88 a__isNatIList(x0) 42.17/12.88 mark(zeros) 42.17/12.88 mark(U101(x0, x1, x2)) 42.17/12.88 mark(U102(x0, x1, x2)) 42.17/12.88 mark(isNatKind(x0)) 42.17/12.88 mark(U103(x0, x1, x2)) 42.17/12.88 mark(isNatIListKind(x0)) 42.17/12.88 mark(U104(x0, x1, x2)) 42.17/12.88 mark(U105(x0, x1)) 42.17/12.88 mark(isNat(x0)) 42.17/12.88 mark(U106(x0)) 42.17/12.88 mark(isNatIList(x0)) 42.17/12.88 mark(U11(x0, x1)) 42.17/12.88 mark(U12(x0, x1)) 42.17/12.88 mark(U111(x0, x1, x2)) 42.17/12.88 mark(U112(x0, x1, x2)) 42.17/12.88 mark(U113(x0, x1, x2)) 42.17/12.88 mark(U114(x0, x1)) 42.17/12.88 mark(length(x0)) 42.17/12.88 mark(U13(x0)) 42.17/12.88 mark(isNatList(x0)) 42.17/12.88 mark(U121(x0, x1)) 42.17/12.88 mark(U122(x0)) 42.17/12.88 mark(U131(x0, x1, x2, x3)) 42.17/12.88 mark(U132(x0, x1, x2, x3)) 42.17/12.88 mark(U133(x0, x1, x2, x3)) 42.17/12.88 mark(U134(x0, x1, x2, x3)) 42.17/12.88 mark(U135(x0, x1, x2, x3)) 42.17/12.88 mark(U136(x0, x1, x2, x3)) 42.17/12.88 mark(take(x0, x1)) 42.17/12.88 mark(U21(x0, x1)) 42.17/12.88 mark(U22(x0, x1)) 42.17/12.88 mark(U23(x0)) 42.17/12.88 mark(U31(x0, x1)) 42.17/12.88 mark(U32(x0, x1)) 42.17/12.88 mark(U33(x0)) 42.17/12.88 mark(U41(x0, x1, x2)) 42.17/12.88 mark(U42(x0, x1, x2)) 42.17/12.88 mark(U43(x0, x1, x2)) 42.17/12.88 mark(U44(x0, x1, x2)) 42.17/12.88 mark(U45(x0, x1)) 42.17/12.88 mark(U46(x0)) 42.17/12.88 mark(U51(x0, x1)) 42.17/12.88 mark(U52(x0)) 42.17/12.88 mark(U61(x0, x1)) 42.17/12.88 mark(U62(x0)) 42.17/12.88 mark(U71(x0)) 42.17/12.88 mark(U81(x0)) 42.17/12.88 mark(U91(x0, x1, x2)) 42.17/12.88 mark(U92(x0, x1, x2)) 42.17/12.88 mark(U93(x0, x1, x2)) 42.17/12.88 mark(U94(x0, x1, x2)) 42.17/12.88 mark(U95(x0, x1)) 42.17/12.88 mark(U96(x0)) 42.17/12.88 mark(cons(x0, x1)) 42.17/12.88 mark(0) 42.17/12.88 mark(tt) 42.17/12.88 mark(s(x0)) 42.17/12.88 mark(nil) 42.17/12.88 a__U101(x0, x1, x2) 42.17/12.88 a__U102(x0, x1, x2) 42.17/12.88 a__isNatKind(x0) 42.17/12.88 a__U103(x0, x1, x2) 42.17/12.88 a__isNatIListKind(x0) 42.17/12.88 a__U104(x0, x1, x2) 42.17/12.88 a__U105(x0, x1) 42.17/12.88 a__isNat(x0) 42.17/12.88 a__U106(x0) 42.17/12.88 a__U11(x0, x1) 42.17/12.88 a__U12(x0, x1) 42.17/12.88 a__U111(x0, x1, x2) 42.17/12.88 a__U112(x0, x1, x2) 42.17/12.88 a__U113(x0, x1, x2) 42.17/12.88 a__U114(x0, x1) 42.17/12.88 a__length(x0) 42.17/12.88 a__U13(x0) 42.17/12.88 a__isNatList(x0) 42.17/12.88 a__U121(x0, x1) 42.17/12.88 a__U122(x0) 42.17/12.88 a__U131(x0, x1, x2, x3) 42.17/12.88 a__U132(x0, x1, x2, x3) 42.17/12.88 a__U133(x0, x1, x2, x3) 42.17/12.88 a__U134(x0, x1, x2, x3) 42.17/12.88 a__U135(x0, x1, x2, x3) 42.17/12.88 a__U136(x0, x1, x2, x3) 42.17/12.88 a__take(x0, x1) 42.17/12.88 a__U21(x0, x1) 42.17/12.88 a__U22(x0, x1) 42.17/12.88 a__U23(x0) 42.17/12.88 a__U31(x0, x1) 42.17/12.88 a__U32(x0, x1) 42.17/12.88 a__U33(x0) 42.17/12.88 a__U41(x0, x1, x2) 42.17/12.88 a__U42(x0, x1, x2) 42.17/12.88 a__U43(x0, x1, x2) 42.17/12.88 a__U44(x0, x1, x2) 42.17/12.88 a__U45(x0, x1) 42.17/12.88 a__U46(x0) 42.17/12.88 a__U51(x0, x1) 42.17/12.88 a__U52(x0) 42.17/12.88 a__U61(x0, x1) 42.17/12.88 a__U62(x0) 42.17/12.88 a__U71(x0) 42.17/12.88 a__U81(x0) 42.17/12.88 a__U91(x0, x1, x2) 42.17/12.88 a__U92(x0, x1, x2) 42.17/12.88 a__U93(x0, x1, x2) 42.17/12.88 a__U94(x0, x1, x2) 42.17/12.88 a__U95(x0, x1) 42.17/12.88 a__U96(x0) 42.17/12.88 42.17/12.88 We have to consider all minimal (P,Q,R)-chains. 42.17/12.88 ---------------------------------------- 42.17/12.88 42.17/12.88 (3) DependencyGraphProof (EQUIVALENT) 42.17/12.88 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 83 less nodes. 42.17/12.88 ---------------------------------------- 42.17/12.88 42.17/12.88 (4) 42.17/12.88 Complex Obligation (AND) 42.17/12.88 42.17/12.88 ---------------------------------------- 42.17/12.88 42.17/12.88 (5) 42.17/12.88 Obligation: 42.17/12.88 Q DP problem: 42.17/12.88 The TRS P consists of the following rules: 42.17/12.88 42.17/12.88 A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 42.17/12.88 A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.88 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 42.17/12.88 A__ISNATILISTKIND(take(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.17/12.88 A__U61(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.88 A__ISNATILISTKIND(take(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.88 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.17/12.88 42.17/12.88 The TRS R consists of the following rules: 42.17/12.88 42.17/12.88 a__zeros -> cons(0, zeros) 42.17/12.88 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.88 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.88 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.88 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.88 a__U106(tt) -> tt 42.17/12.88 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.88 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.17/12.88 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.17/12.88 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.17/12.88 a__U114(tt, L) -> s(a__length(mark(L))) 42.17/12.88 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.88 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.17/12.88 a__U122(tt) -> nil 42.17/12.88 a__U13(tt) -> tt 42.17/12.88 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.88 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.17/12.88 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.17/12.88 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.17/12.88 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.17/12.88 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.17/12.88 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.89 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.89 a__U23(tt) -> tt 42.17/12.89 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.89 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.89 a__U33(tt) -> tt 42.17/12.89 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.89 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.89 a__U46(tt) -> tt 42.17/12.89 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.89 a__U52(tt) -> tt 42.17/12.89 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.89 a__U62(tt) -> tt 42.17/12.89 a__U71(tt) -> tt 42.17/12.89 a__U81(tt) -> tt 42.17/12.89 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.89 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.89 a__U96(tt) -> tt 42.17/12.89 a__isNat(0) -> tt 42.17/12.89 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.89 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.89 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.89 a__isNatIList(zeros) -> tt 42.17/12.89 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatIListKind(nil) -> tt 42.17/12.89 a__isNatIListKind(zeros) -> tt 42.17/12.89 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.89 a__isNatKind(0) -> tt 42.17/12.89 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.89 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.89 a__isNatList(nil) -> tt 42.17/12.89 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.89 a__length(nil) -> 0 42.17/12.89 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.17/12.89 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.17/12.89 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.17/12.89 mark(zeros) -> a__zeros 42.17/12.89 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.17/12.89 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.17/12.89 mark(isNatKind(X)) -> a__isNatKind(X) 42.17/12.89 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.17/12.89 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.17/12.89 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.17/12.89 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.17/12.89 mark(isNat(X)) -> a__isNat(X) 42.17/12.89 mark(U106(X)) -> a__U106(mark(X)) 42.17/12.89 mark(isNatIList(X)) -> a__isNatIList(X) 42.17/12.89 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.17/12.89 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.17/12.89 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.17/12.89 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.17/12.89 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.17/12.89 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.17/12.89 mark(length(X)) -> a__length(mark(X)) 42.17/12.89 mark(U13(X)) -> a__U13(mark(X)) 42.17/12.89 mark(isNatList(X)) -> a__isNatList(X) 42.17/12.89 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.17/12.89 mark(U122(X)) -> a__U122(mark(X)) 42.17/12.89 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.17/12.89 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.17/12.89 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.17/12.89 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.17/12.89 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.17/12.89 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.17/12.89 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.17/12.89 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.17/12.89 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.17/12.89 mark(U23(X)) -> a__U23(mark(X)) 42.17/12.89 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.17/12.89 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.17/12.89 mark(U33(X)) -> a__U33(mark(X)) 42.17/12.89 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.17/12.89 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.17/12.89 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.17/12.89 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.17/12.89 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.17/12.89 mark(U46(X)) -> a__U46(mark(X)) 42.17/12.89 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.17/12.89 mark(U52(X)) -> a__U52(mark(X)) 42.17/12.89 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.17/12.89 mark(U62(X)) -> a__U62(mark(X)) 42.17/12.89 mark(U71(X)) -> a__U71(mark(X)) 42.17/12.89 mark(U81(X)) -> a__U81(mark(X)) 42.17/12.89 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.17/12.89 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.17/12.89 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.17/12.89 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.17/12.89 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.17/12.89 mark(U96(X)) -> a__U96(mark(X)) 42.17/12.89 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.17/12.89 mark(0) -> 0 42.17/12.89 mark(tt) -> tt 42.17/12.89 mark(s(X)) -> s(mark(X)) 42.17/12.89 mark(nil) -> nil 42.17/12.89 a__zeros -> zeros 42.17/12.89 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.89 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.89 a__isNatKind(X) -> isNatKind(X) 42.17/12.89 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.89 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.89 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.89 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.89 a__isNat(X) -> isNat(X) 42.17/12.89 a__U106(X) -> U106(X) 42.17/12.89 a__isNatIList(X) -> isNatIList(X) 42.17/12.89 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.89 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.89 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.17/12.89 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.17/12.89 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.17/12.89 a__U114(X1, X2) -> U114(X1, X2) 42.17/12.89 a__length(X) -> length(X) 42.17/12.89 a__U13(X) -> U13(X) 42.17/12.89 a__isNatList(X) -> isNatList(X) 42.17/12.89 a__U121(X1, X2) -> U121(X1, X2) 42.17/12.89 a__U122(X) -> U122(X) 42.17/12.89 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.17/12.89 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.17/12.89 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.17/12.89 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.17/12.89 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.17/12.89 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.17/12.89 a__take(X1, X2) -> take(X1, X2) 42.17/12.89 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.89 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.89 a__U23(X) -> U23(X) 42.17/12.89 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.89 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.89 a__U33(X) -> U33(X) 42.17/12.89 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.89 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.89 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.89 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.89 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.89 a__U46(X) -> U46(X) 42.17/12.89 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.89 a__U52(X) -> U52(X) 42.17/12.89 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.89 a__U62(X) -> U62(X) 42.17/12.89 a__U71(X) -> U71(X) 42.17/12.89 a__U81(X) -> U81(X) 42.17/12.89 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.89 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.89 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.89 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.89 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.89 a__U96(X) -> U96(X) 42.17/12.89 42.17/12.89 The set Q consists of the following terms: 42.17/12.89 42.17/12.89 a__zeros 42.17/12.89 a__isNatIList(x0) 42.17/12.89 mark(zeros) 42.17/12.89 mark(U101(x0, x1, x2)) 42.17/12.89 mark(U102(x0, x1, x2)) 42.17/12.89 mark(isNatKind(x0)) 42.17/12.89 mark(U103(x0, x1, x2)) 42.17/12.89 mark(isNatIListKind(x0)) 42.17/12.89 mark(U104(x0, x1, x2)) 42.17/12.89 mark(U105(x0, x1)) 42.17/12.89 mark(isNat(x0)) 42.17/12.89 mark(U106(x0)) 42.17/12.89 mark(isNatIList(x0)) 42.17/12.89 mark(U11(x0, x1)) 42.17/12.89 mark(U12(x0, x1)) 42.17/12.89 mark(U111(x0, x1, x2)) 42.17/12.89 mark(U112(x0, x1, x2)) 42.17/12.89 mark(U113(x0, x1, x2)) 42.17/12.89 mark(U114(x0, x1)) 42.17/12.89 mark(length(x0)) 42.17/12.89 mark(U13(x0)) 42.17/12.89 mark(isNatList(x0)) 42.17/12.89 mark(U121(x0, x1)) 42.17/12.89 mark(U122(x0)) 42.17/12.89 mark(U131(x0, x1, x2, x3)) 42.17/12.89 mark(U132(x0, x1, x2, x3)) 42.17/12.89 mark(U133(x0, x1, x2, x3)) 42.17/12.89 mark(U134(x0, x1, x2, x3)) 42.17/12.89 mark(U135(x0, x1, x2, x3)) 42.17/12.89 mark(U136(x0, x1, x2, x3)) 42.17/12.89 mark(take(x0, x1)) 42.17/12.89 mark(U21(x0, x1)) 42.17/12.89 mark(U22(x0, x1)) 42.17/12.89 mark(U23(x0)) 42.17/12.89 mark(U31(x0, x1)) 42.17/12.89 mark(U32(x0, x1)) 42.17/12.89 mark(U33(x0)) 42.17/12.89 mark(U41(x0, x1, x2)) 42.17/12.89 mark(U42(x0, x1, x2)) 42.17/12.89 mark(U43(x0, x1, x2)) 42.17/12.89 mark(U44(x0, x1, x2)) 42.17/12.89 mark(U45(x0, x1)) 42.17/12.89 mark(U46(x0)) 42.17/12.89 mark(U51(x0, x1)) 42.17/12.89 mark(U52(x0)) 42.17/12.89 mark(U61(x0, x1)) 42.17/12.89 mark(U62(x0)) 42.17/12.89 mark(U71(x0)) 42.17/12.89 mark(U81(x0)) 42.17/12.89 mark(U91(x0, x1, x2)) 42.17/12.89 mark(U92(x0, x1, x2)) 42.17/12.89 mark(U93(x0, x1, x2)) 42.17/12.89 mark(U94(x0, x1, x2)) 42.17/12.89 mark(U95(x0, x1)) 42.17/12.89 mark(U96(x0)) 42.17/12.89 mark(cons(x0, x1)) 42.17/12.89 mark(0) 42.17/12.89 mark(tt) 42.17/12.89 mark(s(x0)) 42.17/12.89 mark(nil) 42.17/12.89 a__U101(x0, x1, x2) 42.17/12.89 a__U102(x0, x1, x2) 42.17/12.89 a__isNatKind(x0) 42.17/12.89 a__U103(x0, x1, x2) 42.17/12.89 a__isNatIListKind(x0) 42.17/12.89 a__U104(x0, x1, x2) 42.17/12.89 a__U105(x0, x1) 42.17/12.89 a__isNat(x0) 42.17/12.89 a__U106(x0) 42.17/12.89 a__U11(x0, x1) 42.17/12.89 a__U12(x0, x1) 42.17/12.89 a__U111(x0, x1, x2) 42.17/12.89 a__U112(x0, x1, x2) 42.17/12.89 a__U113(x0, x1, x2) 42.17/12.89 a__U114(x0, x1) 42.17/12.89 a__length(x0) 42.17/12.89 a__U13(x0) 42.17/12.89 a__isNatList(x0) 42.17/12.89 a__U121(x0, x1) 42.17/12.89 a__U122(x0) 42.17/12.89 a__U131(x0, x1, x2, x3) 42.17/12.89 a__U132(x0, x1, x2, x3) 42.17/12.89 a__U133(x0, x1, x2, x3) 42.17/12.89 a__U134(x0, x1, x2, x3) 42.17/12.89 a__U135(x0, x1, x2, x3) 42.17/12.89 a__U136(x0, x1, x2, x3) 42.17/12.89 a__take(x0, x1) 42.17/12.89 a__U21(x0, x1) 42.17/12.89 a__U22(x0, x1) 42.17/12.89 a__U23(x0) 42.17/12.89 a__U31(x0, x1) 42.17/12.89 a__U32(x0, x1) 42.17/12.89 a__U33(x0) 42.17/12.89 a__U41(x0, x1, x2) 42.17/12.89 a__U42(x0, x1, x2) 42.17/12.89 a__U43(x0, x1, x2) 42.17/12.89 a__U44(x0, x1, x2) 42.17/12.89 a__U45(x0, x1) 42.17/12.89 a__U46(x0) 42.17/12.89 a__U51(x0, x1) 42.17/12.89 a__U52(x0) 42.17/12.89 a__U61(x0, x1) 42.17/12.89 a__U62(x0) 42.17/12.89 a__U71(x0) 42.17/12.89 a__U81(x0) 42.17/12.89 a__U91(x0, x1, x2) 42.17/12.89 a__U92(x0, x1, x2) 42.17/12.89 a__U93(x0, x1, x2) 42.17/12.89 a__U94(x0, x1, x2) 42.17/12.89 a__U95(x0, x1) 42.17/12.89 a__U96(x0) 42.17/12.89 42.17/12.89 We have to consider all minimal (P,Q,R)-chains. 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (6) UsableRulesProof (EQUIVALENT) 42.17/12.89 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (7) 42.17/12.89 Obligation: 42.17/12.89 Q DP problem: 42.17/12.89 The TRS P consists of the following rules: 42.17/12.89 42.17/12.89 A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.89 A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 42.17/12.89 A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.89 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 42.17/12.89 A__ISNATILISTKIND(take(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.17/12.89 A__U61(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.89 A__ISNATILISTKIND(take(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.89 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.17/12.89 42.17/12.89 The TRS R consists of the following rules: 42.17/12.89 42.17/12.89 a__isNatKind(0) -> tt 42.17/12.89 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.89 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.89 a__isNatKind(X) -> isNatKind(X) 42.17/12.89 a__U81(tt) -> tt 42.17/12.89 a__U81(X) -> U81(X) 42.17/12.89 a__isNatIListKind(nil) -> tt 42.17/12.89 a__isNatIListKind(zeros) -> tt 42.17/12.89 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.89 a__U71(tt) -> tt 42.17/12.89 a__U71(X) -> U71(X) 42.17/12.89 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.89 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.89 a__U62(tt) -> tt 42.17/12.89 a__U62(X) -> U62(X) 42.17/12.89 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.89 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.89 a__U52(tt) -> tt 42.17/12.89 a__U52(X) -> U52(X) 42.17/12.89 42.17/12.89 The set Q consists of the following terms: 42.17/12.89 42.17/12.89 a__zeros 42.17/12.89 a__isNatIList(x0) 42.17/12.89 mark(zeros) 42.17/12.89 mark(U101(x0, x1, x2)) 42.17/12.89 mark(U102(x0, x1, x2)) 42.17/12.89 mark(isNatKind(x0)) 42.17/12.89 mark(U103(x0, x1, x2)) 42.17/12.89 mark(isNatIListKind(x0)) 42.17/12.89 mark(U104(x0, x1, x2)) 42.17/12.89 mark(U105(x0, x1)) 42.17/12.89 mark(isNat(x0)) 42.17/12.89 mark(U106(x0)) 42.17/12.89 mark(isNatIList(x0)) 42.17/12.89 mark(U11(x0, x1)) 42.17/12.89 mark(U12(x0, x1)) 42.17/12.89 mark(U111(x0, x1, x2)) 42.17/12.89 mark(U112(x0, x1, x2)) 42.17/12.89 mark(U113(x0, x1, x2)) 42.17/12.89 mark(U114(x0, x1)) 42.17/12.89 mark(length(x0)) 42.17/12.89 mark(U13(x0)) 42.17/12.89 mark(isNatList(x0)) 42.17/12.89 mark(U121(x0, x1)) 42.17/12.89 mark(U122(x0)) 42.17/12.89 mark(U131(x0, x1, x2, x3)) 42.17/12.89 mark(U132(x0, x1, x2, x3)) 42.17/12.89 mark(U133(x0, x1, x2, x3)) 42.17/12.89 mark(U134(x0, x1, x2, x3)) 42.17/12.89 mark(U135(x0, x1, x2, x3)) 42.17/12.89 mark(U136(x0, x1, x2, x3)) 42.17/12.89 mark(take(x0, x1)) 42.17/12.89 mark(U21(x0, x1)) 42.17/12.89 mark(U22(x0, x1)) 42.17/12.89 mark(U23(x0)) 42.17/12.89 mark(U31(x0, x1)) 42.17/12.89 mark(U32(x0, x1)) 42.17/12.89 mark(U33(x0)) 42.17/12.89 mark(U41(x0, x1, x2)) 42.17/12.89 mark(U42(x0, x1, x2)) 42.17/12.89 mark(U43(x0, x1, x2)) 42.17/12.89 mark(U44(x0, x1, x2)) 42.17/12.89 mark(U45(x0, x1)) 42.17/12.89 mark(U46(x0)) 42.17/12.89 mark(U51(x0, x1)) 42.17/12.89 mark(U52(x0)) 42.17/12.89 mark(U61(x0, x1)) 42.17/12.89 mark(U62(x0)) 42.17/12.89 mark(U71(x0)) 42.17/12.89 mark(U81(x0)) 42.17/12.89 mark(U91(x0, x1, x2)) 42.17/12.89 mark(U92(x0, x1, x2)) 42.17/12.89 mark(U93(x0, x1, x2)) 42.17/12.89 mark(U94(x0, x1, x2)) 42.17/12.89 mark(U95(x0, x1)) 42.17/12.89 mark(U96(x0)) 42.17/12.89 mark(cons(x0, x1)) 42.17/12.89 mark(0) 42.17/12.89 mark(tt) 42.17/12.89 mark(s(x0)) 42.17/12.89 mark(nil) 42.17/12.89 a__U101(x0, x1, x2) 42.17/12.89 a__U102(x0, x1, x2) 42.17/12.89 a__isNatKind(x0) 42.17/12.89 a__U103(x0, x1, x2) 42.17/12.89 a__isNatIListKind(x0) 42.17/12.89 a__U104(x0, x1, x2) 42.17/12.89 a__U105(x0, x1) 42.17/12.89 a__isNat(x0) 42.17/12.89 a__U106(x0) 42.17/12.89 a__U11(x0, x1) 42.17/12.89 a__U12(x0, x1) 42.17/12.89 a__U111(x0, x1, x2) 42.17/12.89 a__U112(x0, x1, x2) 42.17/12.89 a__U113(x0, x1, x2) 42.17/12.89 a__U114(x0, x1) 42.17/12.89 a__length(x0) 42.17/12.89 a__U13(x0) 42.17/12.89 a__isNatList(x0) 42.17/12.89 a__U121(x0, x1) 42.17/12.89 a__U122(x0) 42.17/12.89 a__U131(x0, x1, x2, x3) 42.17/12.89 a__U132(x0, x1, x2, x3) 42.17/12.89 a__U133(x0, x1, x2, x3) 42.17/12.89 a__U134(x0, x1, x2, x3) 42.17/12.89 a__U135(x0, x1, x2, x3) 42.17/12.89 a__U136(x0, x1, x2, x3) 42.17/12.89 a__take(x0, x1) 42.17/12.89 a__U21(x0, x1) 42.17/12.89 a__U22(x0, x1) 42.17/12.89 a__U23(x0) 42.17/12.89 a__U31(x0, x1) 42.17/12.89 a__U32(x0, x1) 42.17/12.89 a__U33(x0) 42.17/12.89 a__U41(x0, x1, x2) 42.17/12.89 a__U42(x0, x1, x2) 42.17/12.89 a__U43(x0, x1, x2) 42.17/12.89 a__U44(x0, x1, x2) 42.17/12.89 a__U45(x0, x1) 42.17/12.89 a__U46(x0) 42.17/12.89 a__U51(x0, x1) 42.17/12.89 a__U52(x0) 42.17/12.89 a__U61(x0, x1) 42.17/12.89 a__U62(x0) 42.17/12.89 a__U71(x0) 42.17/12.89 a__U81(x0) 42.17/12.89 a__U91(x0, x1, x2) 42.17/12.89 a__U92(x0, x1, x2) 42.17/12.89 a__U93(x0, x1, x2) 42.17/12.89 a__U94(x0, x1, x2) 42.17/12.89 a__U95(x0, x1) 42.17/12.89 a__U96(x0) 42.17/12.89 42.17/12.89 We have to consider all minimal (P,Q,R)-chains. 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (8) QReductionProof (EQUIVALENT) 42.17/12.89 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 42.17/12.89 42.17/12.89 a__zeros 42.17/12.89 a__isNatIList(x0) 42.17/12.89 mark(zeros) 42.17/12.89 mark(U101(x0, x1, x2)) 42.17/12.89 mark(U102(x0, x1, x2)) 42.17/12.89 mark(isNatKind(x0)) 42.17/12.89 mark(U103(x0, x1, x2)) 42.17/12.89 mark(isNatIListKind(x0)) 42.17/12.89 mark(U104(x0, x1, x2)) 42.17/12.89 mark(U105(x0, x1)) 42.17/12.89 mark(isNat(x0)) 42.17/12.89 mark(U106(x0)) 42.17/12.89 mark(isNatIList(x0)) 42.17/12.89 mark(U11(x0, x1)) 42.17/12.89 mark(U12(x0, x1)) 42.17/12.89 mark(U111(x0, x1, x2)) 42.17/12.89 mark(U112(x0, x1, x2)) 42.17/12.89 mark(U113(x0, x1, x2)) 42.17/12.89 mark(U114(x0, x1)) 42.17/12.89 mark(length(x0)) 42.17/12.89 mark(U13(x0)) 42.17/12.89 mark(isNatList(x0)) 42.17/12.89 mark(U121(x0, x1)) 42.17/12.89 mark(U122(x0)) 42.17/12.89 mark(U131(x0, x1, x2, x3)) 42.17/12.89 mark(U132(x0, x1, x2, x3)) 42.17/12.89 mark(U133(x0, x1, x2, x3)) 42.17/12.89 mark(U134(x0, x1, x2, x3)) 42.17/12.89 mark(U135(x0, x1, x2, x3)) 42.17/12.89 mark(U136(x0, x1, x2, x3)) 42.17/12.89 mark(take(x0, x1)) 42.17/12.89 mark(U21(x0, x1)) 42.17/12.89 mark(U22(x0, x1)) 42.17/12.89 mark(U23(x0)) 42.17/12.89 mark(U31(x0, x1)) 42.17/12.89 mark(U32(x0, x1)) 42.17/12.89 mark(U33(x0)) 42.17/12.89 mark(U41(x0, x1, x2)) 42.17/12.89 mark(U42(x0, x1, x2)) 42.17/12.89 mark(U43(x0, x1, x2)) 42.17/12.89 mark(U44(x0, x1, x2)) 42.17/12.89 mark(U45(x0, x1)) 42.17/12.89 mark(U46(x0)) 42.17/12.89 mark(U51(x0, x1)) 42.17/12.89 mark(U52(x0)) 42.17/12.89 mark(U61(x0, x1)) 42.17/12.89 mark(U62(x0)) 42.17/12.89 mark(U71(x0)) 42.17/12.89 mark(U81(x0)) 42.17/12.89 mark(U91(x0, x1, x2)) 42.17/12.89 mark(U92(x0, x1, x2)) 42.17/12.89 mark(U93(x0, x1, x2)) 42.17/12.89 mark(U94(x0, x1, x2)) 42.17/12.89 mark(U95(x0, x1)) 42.17/12.89 mark(U96(x0)) 42.17/12.89 mark(cons(x0, x1)) 42.17/12.89 mark(0) 42.17/12.89 mark(tt) 42.17/12.89 mark(s(x0)) 42.17/12.89 mark(nil) 42.17/12.89 a__U101(x0, x1, x2) 42.17/12.89 a__U102(x0, x1, x2) 42.17/12.89 a__U103(x0, x1, x2) 42.17/12.89 a__U104(x0, x1, x2) 42.17/12.89 a__U105(x0, x1) 42.17/12.89 a__isNat(x0) 42.17/12.89 a__U106(x0) 42.17/12.89 a__U11(x0, x1) 42.17/12.89 a__U12(x0, x1) 42.17/12.89 a__U111(x0, x1, x2) 42.17/12.89 a__U112(x0, x1, x2) 42.17/12.89 a__U113(x0, x1, x2) 42.17/12.89 a__U114(x0, x1) 42.17/12.89 a__length(x0) 42.17/12.89 a__U13(x0) 42.17/12.89 a__isNatList(x0) 42.17/12.89 a__U121(x0, x1) 42.17/12.89 a__U122(x0) 42.17/12.89 a__U131(x0, x1, x2, x3) 42.17/12.89 a__U132(x0, x1, x2, x3) 42.17/12.89 a__U133(x0, x1, x2, x3) 42.17/12.89 a__U134(x0, x1, x2, x3) 42.17/12.89 a__U135(x0, x1, x2, x3) 42.17/12.89 a__U136(x0, x1, x2, x3) 42.17/12.89 a__take(x0, x1) 42.17/12.89 a__U21(x0, x1) 42.17/12.89 a__U22(x0, x1) 42.17/12.89 a__U23(x0) 42.17/12.89 a__U31(x0, x1) 42.17/12.89 a__U32(x0, x1) 42.17/12.89 a__U33(x0) 42.17/12.89 a__U41(x0, x1, x2) 42.17/12.89 a__U42(x0, x1, x2) 42.17/12.89 a__U43(x0, x1, x2) 42.17/12.89 a__U44(x0, x1, x2) 42.17/12.89 a__U45(x0, x1) 42.17/12.89 a__U46(x0) 42.17/12.89 a__U91(x0, x1, x2) 42.17/12.89 a__U92(x0, x1, x2) 42.17/12.89 a__U93(x0, x1, x2) 42.17/12.89 a__U94(x0, x1, x2) 42.17/12.89 a__U95(x0, x1) 42.17/12.89 a__U96(x0) 42.17/12.89 42.17/12.89 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (9) 42.17/12.89 Obligation: 42.17/12.89 Q DP problem: 42.17/12.89 The TRS P consists of the following rules: 42.17/12.89 42.17/12.89 A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.89 A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 42.17/12.89 A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.89 A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 42.17/12.89 A__ISNATILISTKIND(take(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.17/12.89 A__U61(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.89 A__ISNATILISTKIND(take(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.89 A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.17/12.89 42.17/12.89 The TRS R consists of the following rules: 42.17/12.89 42.17/12.89 a__isNatKind(0) -> tt 42.17/12.89 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.89 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.89 a__isNatKind(X) -> isNatKind(X) 42.17/12.89 a__U81(tt) -> tt 42.17/12.89 a__U81(X) -> U81(X) 42.17/12.89 a__isNatIListKind(nil) -> tt 42.17/12.89 a__isNatIListKind(zeros) -> tt 42.17/12.89 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.89 a__U71(tt) -> tt 42.17/12.89 a__U71(X) -> U71(X) 42.17/12.89 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.89 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.89 a__U62(tt) -> tt 42.17/12.89 a__U62(X) -> U62(X) 42.17/12.89 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.89 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.89 a__U52(tt) -> tt 42.17/12.89 a__U52(X) -> U52(X) 42.17/12.89 42.17/12.89 The set Q consists of the following terms: 42.17/12.89 42.17/12.89 a__isNatKind(x0) 42.17/12.89 a__isNatIListKind(x0) 42.17/12.89 a__U51(x0, x1) 42.17/12.89 a__U52(x0) 42.17/12.89 a__U61(x0, x1) 42.17/12.89 a__U62(x0) 42.17/12.89 a__U71(x0) 42.17/12.89 a__U81(x0) 42.17/12.89 42.17/12.89 We have to consider all minimal (P,Q,R)-chains. 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (10) QDPSizeChangeProof (EQUIVALENT) 42.17/12.89 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 42.17/12.89 42.17/12.89 From the DPs we obtained the following set of size-change graphs: 42.17/12.89 *A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) 42.17/12.89 The graph contains the following edges 1 > 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__U51(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) 42.17/12.89 The graph contains the following edges 1 > 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) 42.17/12.89 The graph contains the following edges 1 > 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__U61(tt, V2) -> A__ISNATILISTKIND(V2) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNATILISTKIND(take(V1, V2)) -> A__U61(a__isNatKind(V1), V2) 42.17/12.89 The graph contains the following edges 1 > 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.89 The graph contains the following edges 1 > 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNATILISTKIND(take(V1, V2)) -> A__ISNATKIND(V1) 42.17/12.89 The graph contains the following edges 1 > 1 42.17/12.89 42.17/12.89 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (11) 42.17/12.89 YES 42.17/12.89 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (12) 42.17/12.89 Obligation: 42.17/12.89 Q DP problem: 42.17/12.89 The TRS P consists of the following rules: 42.17/12.89 42.17/12.89 A__U102(tt, V1, V2) -> A__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U103(tt, V1, V2) -> A__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U104(tt, V1, V2) -> A__U105(a__isNat(V1), V2) 42.17/12.89 A__U105(tt, V2) -> A__ISNATILIST(V2) 42.17/12.89 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) 42.17/12.89 A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) 42.17/12.89 A__U32(tt, V) -> A__ISNATLIST(V) 42.17/12.89 A__ISNATLIST(cons(V1, V2)) -> A__U91(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U91(tt, V1, V2) -> A__U92(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U92(tt, V1, V2) -> A__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U93(tt, V1, V2) -> A__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U94(tt, V1, V2) -> A__U95(a__isNat(V1), V2) 42.17/12.89 A__U95(tt, V2) -> A__ISNATLIST(V2) 42.17/12.89 A__ISNATLIST(take(V1, V2)) -> A__U101(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U101(tt, V1, V2) -> A__U102(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U94(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 42.17/12.89 A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 42.17/12.89 A__U12(tt, V1) -> A__ISNATLIST(V1) 42.17/12.89 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.17/12.89 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.17/12.89 A__U22(tt, V1) -> A__ISNAT(V1) 42.17/12.89 A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 42.17/12.89 A__U45(tt, V2) -> A__ISNATILIST(V2) 42.17/12.89 A__U44(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 A__U104(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 42.17/12.89 The TRS R consists of the following rules: 42.17/12.89 42.17/12.89 a__zeros -> cons(0, zeros) 42.17/12.89 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.89 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.89 a__U106(tt) -> tt 42.17/12.89 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.89 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.17/12.89 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.17/12.89 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.17/12.89 a__U114(tt, L) -> s(a__length(mark(L))) 42.17/12.89 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.89 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.17/12.89 a__U122(tt) -> nil 42.17/12.89 a__U13(tt) -> tt 42.17/12.89 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.89 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.17/12.89 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.17/12.89 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.17/12.89 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.17/12.89 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.17/12.89 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.89 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.89 a__U23(tt) -> tt 42.17/12.89 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.89 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.89 a__U33(tt) -> tt 42.17/12.89 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.89 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.89 a__U46(tt) -> tt 42.17/12.89 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.89 a__U52(tt) -> tt 42.17/12.89 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.89 a__U62(tt) -> tt 42.17/12.89 a__U71(tt) -> tt 42.17/12.89 a__U81(tt) -> tt 42.17/12.89 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.89 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.89 a__U96(tt) -> tt 42.17/12.89 a__isNat(0) -> tt 42.17/12.89 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.89 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.89 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.89 a__isNatIList(zeros) -> tt 42.17/12.89 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatIListKind(nil) -> tt 42.17/12.89 a__isNatIListKind(zeros) -> tt 42.17/12.89 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.89 a__isNatKind(0) -> tt 42.17/12.89 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.89 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.89 a__isNatList(nil) -> tt 42.17/12.89 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.89 a__length(nil) -> 0 42.17/12.89 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.17/12.89 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.17/12.89 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.17/12.89 mark(zeros) -> a__zeros 42.17/12.89 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.17/12.89 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.17/12.89 mark(isNatKind(X)) -> a__isNatKind(X) 42.17/12.89 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.17/12.89 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.17/12.89 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.17/12.89 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.17/12.89 mark(isNat(X)) -> a__isNat(X) 42.17/12.89 mark(U106(X)) -> a__U106(mark(X)) 42.17/12.89 mark(isNatIList(X)) -> a__isNatIList(X) 42.17/12.89 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.17/12.89 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.17/12.89 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.17/12.89 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.17/12.89 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.17/12.89 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.17/12.89 mark(length(X)) -> a__length(mark(X)) 42.17/12.89 mark(U13(X)) -> a__U13(mark(X)) 42.17/12.89 mark(isNatList(X)) -> a__isNatList(X) 42.17/12.89 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.17/12.89 mark(U122(X)) -> a__U122(mark(X)) 42.17/12.89 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.17/12.89 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.17/12.89 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.17/12.89 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.17/12.89 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.17/12.89 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.17/12.89 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.17/12.89 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.17/12.89 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.17/12.89 mark(U23(X)) -> a__U23(mark(X)) 42.17/12.89 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.17/12.89 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.17/12.89 mark(U33(X)) -> a__U33(mark(X)) 42.17/12.89 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.17/12.89 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.17/12.89 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.17/12.89 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.17/12.89 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.17/12.89 mark(U46(X)) -> a__U46(mark(X)) 42.17/12.89 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.17/12.89 mark(U52(X)) -> a__U52(mark(X)) 42.17/12.89 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.17/12.89 mark(U62(X)) -> a__U62(mark(X)) 42.17/12.89 mark(U71(X)) -> a__U71(mark(X)) 42.17/12.89 mark(U81(X)) -> a__U81(mark(X)) 42.17/12.89 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.17/12.89 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.17/12.89 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.17/12.89 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.17/12.89 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.17/12.89 mark(U96(X)) -> a__U96(mark(X)) 42.17/12.89 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.17/12.89 mark(0) -> 0 42.17/12.89 mark(tt) -> tt 42.17/12.89 mark(s(X)) -> s(mark(X)) 42.17/12.89 mark(nil) -> nil 42.17/12.89 a__zeros -> zeros 42.17/12.89 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.89 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.89 a__isNatKind(X) -> isNatKind(X) 42.17/12.89 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.89 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.89 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.89 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.89 a__isNat(X) -> isNat(X) 42.17/12.89 a__U106(X) -> U106(X) 42.17/12.89 a__isNatIList(X) -> isNatIList(X) 42.17/12.89 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.89 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.89 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.17/12.89 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.17/12.89 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.17/12.89 a__U114(X1, X2) -> U114(X1, X2) 42.17/12.89 a__length(X) -> length(X) 42.17/12.89 a__U13(X) -> U13(X) 42.17/12.89 a__isNatList(X) -> isNatList(X) 42.17/12.89 a__U121(X1, X2) -> U121(X1, X2) 42.17/12.89 a__U122(X) -> U122(X) 42.17/12.89 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.17/12.89 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.17/12.89 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.17/12.89 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.17/12.89 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.17/12.89 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.17/12.89 a__take(X1, X2) -> take(X1, X2) 42.17/12.89 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.89 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.89 a__U23(X) -> U23(X) 42.17/12.89 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.89 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.89 a__U33(X) -> U33(X) 42.17/12.89 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.89 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.89 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.89 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.89 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.89 a__U46(X) -> U46(X) 42.17/12.89 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.89 a__U52(X) -> U52(X) 42.17/12.89 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.89 a__U62(X) -> U62(X) 42.17/12.89 a__U71(X) -> U71(X) 42.17/12.89 a__U81(X) -> U81(X) 42.17/12.89 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.89 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.89 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.89 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.89 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.89 a__U96(X) -> U96(X) 42.17/12.89 42.17/12.89 The set Q consists of the following terms: 42.17/12.89 42.17/12.89 a__zeros 42.17/12.89 a__isNatIList(x0) 42.17/12.89 mark(zeros) 42.17/12.89 mark(U101(x0, x1, x2)) 42.17/12.89 mark(U102(x0, x1, x2)) 42.17/12.89 mark(isNatKind(x0)) 42.17/12.89 mark(U103(x0, x1, x2)) 42.17/12.89 mark(isNatIListKind(x0)) 42.17/12.89 mark(U104(x0, x1, x2)) 42.17/12.89 mark(U105(x0, x1)) 42.17/12.89 mark(isNat(x0)) 42.17/12.89 mark(U106(x0)) 42.17/12.89 mark(isNatIList(x0)) 42.17/12.89 mark(U11(x0, x1)) 42.17/12.89 mark(U12(x0, x1)) 42.17/12.89 mark(U111(x0, x1, x2)) 42.17/12.89 mark(U112(x0, x1, x2)) 42.17/12.89 mark(U113(x0, x1, x2)) 42.17/12.89 mark(U114(x0, x1)) 42.17/12.89 mark(length(x0)) 42.17/12.89 mark(U13(x0)) 42.17/12.89 mark(isNatList(x0)) 42.17/12.89 mark(U121(x0, x1)) 42.17/12.89 mark(U122(x0)) 42.17/12.89 mark(U131(x0, x1, x2, x3)) 42.17/12.89 mark(U132(x0, x1, x2, x3)) 42.17/12.89 mark(U133(x0, x1, x2, x3)) 42.17/12.89 mark(U134(x0, x1, x2, x3)) 42.17/12.89 mark(U135(x0, x1, x2, x3)) 42.17/12.89 mark(U136(x0, x1, x2, x3)) 42.17/12.89 mark(take(x0, x1)) 42.17/12.89 mark(U21(x0, x1)) 42.17/12.89 mark(U22(x0, x1)) 42.17/12.89 mark(U23(x0)) 42.17/12.89 mark(U31(x0, x1)) 42.17/12.89 mark(U32(x0, x1)) 42.17/12.89 mark(U33(x0)) 42.17/12.89 mark(U41(x0, x1, x2)) 42.17/12.89 mark(U42(x0, x1, x2)) 42.17/12.89 mark(U43(x0, x1, x2)) 42.17/12.89 mark(U44(x0, x1, x2)) 42.17/12.89 mark(U45(x0, x1)) 42.17/12.89 mark(U46(x0)) 42.17/12.89 mark(U51(x0, x1)) 42.17/12.89 mark(U52(x0)) 42.17/12.89 mark(U61(x0, x1)) 42.17/12.89 mark(U62(x0)) 42.17/12.89 mark(U71(x0)) 42.17/12.89 mark(U81(x0)) 42.17/12.89 mark(U91(x0, x1, x2)) 42.17/12.89 mark(U92(x0, x1, x2)) 42.17/12.89 mark(U93(x0, x1, x2)) 42.17/12.89 mark(U94(x0, x1, x2)) 42.17/12.89 mark(U95(x0, x1)) 42.17/12.89 mark(U96(x0)) 42.17/12.89 mark(cons(x0, x1)) 42.17/12.89 mark(0) 42.17/12.89 mark(tt) 42.17/12.89 mark(s(x0)) 42.17/12.89 mark(nil) 42.17/12.89 a__U101(x0, x1, x2) 42.17/12.89 a__U102(x0, x1, x2) 42.17/12.89 a__isNatKind(x0) 42.17/12.89 a__U103(x0, x1, x2) 42.17/12.89 a__isNatIListKind(x0) 42.17/12.89 a__U104(x0, x1, x2) 42.17/12.89 a__U105(x0, x1) 42.17/12.89 a__isNat(x0) 42.17/12.89 a__U106(x0) 42.17/12.89 a__U11(x0, x1) 42.17/12.89 a__U12(x0, x1) 42.17/12.89 a__U111(x0, x1, x2) 42.17/12.89 a__U112(x0, x1, x2) 42.17/12.89 a__U113(x0, x1, x2) 42.17/12.89 a__U114(x0, x1) 42.17/12.89 a__length(x0) 42.17/12.89 a__U13(x0) 42.17/12.89 a__isNatList(x0) 42.17/12.89 a__U121(x0, x1) 42.17/12.89 a__U122(x0) 42.17/12.89 a__U131(x0, x1, x2, x3) 42.17/12.89 a__U132(x0, x1, x2, x3) 42.17/12.89 a__U133(x0, x1, x2, x3) 42.17/12.89 a__U134(x0, x1, x2, x3) 42.17/12.89 a__U135(x0, x1, x2, x3) 42.17/12.89 a__U136(x0, x1, x2, x3) 42.17/12.89 a__take(x0, x1) 42.17/12.89 a__U21(x0, x1) 42.17/12.89 a__U22(x0, x1) 42.17/12.89 a__U23(x0) 42.17/12.89 a__U31(x0, x1) 42.17/12.89 a__U32(x0, x1) 42.17/12.89 a__U33(x0) 42.17/12.89 a__U41(x0, x1, x2) 42.17/12.89 a__U42(x0, x1, x2) 42.17/12.89 a__U43(x0, x1, x2) 42.17/12.89 a__U44(x0, x1, x2) 42.17/12.89 a__U45(x0, x1) 42.17/12.89 a__U46(x0) 42.17/12.89 a__U51(x0, x1) 42.17/12.89 a__U52(x0) 42.17/12.89 a__U61(x0, x1) 42.17/12.89 a__U62(x0) 42.17/12.89 a__U71(x0) 42.17/12.89 a__U81(x0) 42.17/12.89 a__U91(x0, x1, x2) 42.17/12.89 a__U92(x0, x1, x2) 42.17/12.89 a__U93(x0, x1, x2) 42.17/12.89 a__U94(x0, x1, x2) 42.17/12.89 a__U95(x0, x1) 42.17/12.89 a__U96(x0) 42.17/12.89 42.17/12.89 We have to consider all minimal (P,Q,R)-chains. 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (13) UsableRulesProof (EQUIVALENT) 42.17/12.89 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (14) 42.17/12.89 Obligation: 42.17/12.89 Q DP problem: 42.17/12.89 The TRS P consists of the following rules: 42.17/12.89 42.17/12.89 A__U102(tt, V1, V2) -> A__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U103(tt, V1, V2) -> A__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U104(tt, V1, V2) -> A__U105(a__isNat(V1), V2) 42.17/12.89 A__U105(tt, V2) -> A__ISNATILIST(V2) 42.17/12.89 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) 42.17/12.89 A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) 42.17/12.89 A__U32(tt, V) -> A__ISNATLIST(V) 42.17/12.89 A__ISNATLIST(cons(V1, V2)) -> A__U91(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U91(tt, V1, V2) -> A__U92(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U92(tt, V1, V2) -> A__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U93(tt, V1, V2) -> A__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U94(tt, V1, V2) -> A__U95(a__isNat(V1), V2) 42.17/12.89 A__U95(tt, V2) -> A__ISNATLIST(V2) 42.17/12.89 A__ISNATLIST(take(V1, V2)) -> A__U101(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U101(tt, V1, V2) -> A__U102(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U94(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 42.17/12.89 A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 42.17/12.89 A__U12(tt, V1) -> A__ISNATLIST(V1) 42.17/12.89 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.17/12.89 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.17/12.89 A__U22(tt, V1) -> A__ISNAT(V1) 42.17/12.89 A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 42.17/12.89 A__U45(tt, V2) -> A__ISNATILIST(V2) 42.17/12.89 A__U44(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 A__U104(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 42.17/12.89 The TRS R consists of the following rules: 42.17/12.89 42.17/12.89 a__isNat(0) -> tt 42.17/12.89 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.89 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.89 a__isNat(X) -> isNat(X) 42.17/12.89 a__isNatKind(0) -> tt 42.17/12.89 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.89 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.89 a__isNatKind(X) -> isNatKind(X) 42.17/12.89 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.89 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.89 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.89 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.89 a__U23(tt) -> tt 42.17/12.89 a__U23(X) -> U23(X) 42.17/12.89 a__U81(tt) -> tt 42.17/12.89 a__U81(X) -> U81(X) 42.17/12.89 a__isNatIListKind(nil) -> tt 42.17/12.89 a__isNatIListKind(zeros) -> tt 42.17/12.89 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.89 a__U71(tt) -> tt 42.17/12.89 a__U71(X) -> U71(X) 42.17/12.89 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.89 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.89 a__U62(tt) -> tt 42.17/12.89 a__U62(X) -> U62(X) 42.17/12.89 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.89 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.89 a__U52(tt) -> tt 42.17/12.89 a__U52(X) -> U52(X) 42.17/12.89 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.89 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.89 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.89 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.89 a__isNatList(nil) -> tt 42.17/12.89 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatList(X) -> isNatList(X) 42.17/12.89 a__U13(tt) -> tt 42.17/12.89 a__U13(X) -> U13(X) 42.17/12.89 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.89 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.89 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.89 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.89 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.89 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.89 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.89 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.89 a__isNatIList(zeros) -> tt 42.17/12.89 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatIList(X) -> isNatIList(X) 42.17/12.89 a__U106(tt) -> tt 42.17/12.89 a__U106(X) -> U106(X) 42.17/12.89 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.89 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.89 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.89 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.89 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.89 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.89 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.89 a__U46(tt) -> tt 42.17/12.89 a__U46(X) -> U46(X) 42.17/12.89 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.89 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.89 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.89 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.89 a__U33(tt) -> tt 42.17/12.89 a__U33(X) -> U33(X) 42.17/12.89 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.89 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.89 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.89 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.89 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.89 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.89 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.89 a__U96(tt) -> tt 42.17/12.89 a__U96(X) -> U96(X) 42.17/12.89 42.17/12.89 The set Q consists of the following terms: 42.17/12.89 42.17/12.89 a__zeros 42.17/12.89 a__isNatIList(x0) 42.17/12.89 mark(zeros) 42.17/12.89 mark(U101(x0, x1, x2)) 42.17/12.89 mark(U102(x0, x1, x2)) 42.17/12.89 mark(isNatKind(x0)) 42.17/12.89 mark(U103(x0, x1, x2)) 42.17/12.89 mark(isNatIListKind(x0)) 42.17/12.89 mark(U104(x0, x1, x2)) 42.17/12.89 mark(U105(x0, x1)) 42.17/12.89 mark(isNat(x0)) 42.17/12.89 mark(U106(x0)) 42.17/12.89 mark(isNatIList(x0)) 42.17/12.89 mark(U11(x0, x1)) 42.17/12.89 mark(U12(x0, x1)) 42.17/12.89 mark(U111(x0, x1, x2)) 42.17/12.89 mark(U112(x0, x1, x2)) 42.17/12.89 mark(U113(x0, x1, x2)) 42.17/12.89 mark(U114(x0, x1)) 42.17/12.89 mark(length(x0)) 42.17/12.89 mark(U13(x0)) 42.17/12.89 mark(isNatList(x0)) 42.17/12.89 mark(U121(x0, x1)) 42.17/12.89 mark(U122(x0)) 42.17/12.89 mark(U131(x0, x1, x2, x3)) 42.17/12.89 mark(U132(x0, x1, x2, x3)) 42.17/12.89 mark(U133(x0, x1, x2, x3)) 42.17/12.89 mark(U134(x0, x1, x2, x3)) 42.17/12.89 mark(U135(x0, x1, x2, x3)) 42.17/12.89 mark(U136(x0, x1, x2, x3)) 42.17/12.89 mark(take(x0, x1)) 42.17/12.89 mark(U21(x0, x1)) 42.17/12.89 mark(U22(x0, x1)) 42.17/12.89 mark(U23(x0)) 42.17/12.89 mark(U31(x0, x1)) 42.17/12.89 mark(U32(x0, x1)) 42.17/12.89 mark(U33(x0)) 42.17/12.89 mark(U41(x0, x1, x2)) 42.17/12.89 mark(U42(x0, x1, x2)) 42.17/12.89 mark(U43(x0, x1, x2)) 42.17/12.89 mark(U44(x0, x1, x2)) 42.17/12.89 mark(U45(x0, x1)) 42.17/12.89 mark(U46(x0)) 42.17/12.89 mark(U51(x0, x1)) 42.17/12.89 mark(U52(x0)) 42.17/12.89 mark(U61(x0, x1)) 42.17/12.89 mark(U62(x0)) 42.17/12.89 mark(U71(x0)) 42.17/12.89 mark(U81(x0)) 42.17/12.89 mark(U91(x0, x1, x2)) 42.17/12.89 mark(U92(x0, x1, x2)) 42.17/12.89 mark(U93(x0, x1, x2)) 42.17/12.89 mark(U94(x0, x1, x2)) 42.17/12.89 mark(U95(x0, x1)) 42.17/12.89 mark(U96(x0)) 42.17/12.89 mark(cons(x0, x1)) 42.17/12.89 mark(0) 42.17/12.89 mark(tt) 42.17/12.89 mark(s(x0)) 42.17/12.89 mark(nil) 42.17/12.89 a__U101(x0, x1, x2) 42.17/12.89 a__U102(x0, x1, x2) 42.17/12.89 a__isNatKind(x0) 42.17/12.89 a__U103(x0, x1, x2) 42.17/12.89 a__isNatIListKind(x0) 42.17/12.89 a__U104(x0, x1, x2) 42.17/12.89 a__U105(x0, x1) 42.17/12.89 a__isNat(x0) 42.17/12.89 a__U106(x0) 42.17/12.89 a__U11(x0, x1) 42.17/12.89 a__U12(x0, x1) 42.17/12.89 a__U111(x0, x1, x2) 42.17/12.89 a__U112(x0, x1, x2) 42.17/12.89 a__U113(x0, x1, x2) 42.17/12.89 a__U114(x0, x1) 42.17/12.89 a__length(x0) 42.17/12.89 a__U13(x0) 42.17/12.89 a__isNatList(x0) 42.17/12.89 a__U121(x0, x1) 42.17/12.89 a__U122(x0) 42.17/12.89 a__U131(x0, x1, x2, x3) 42.17/12.89 a__U132(x0, x1, x2, x3) 42.17/12.89 a__U133(x0, x1, x2, x3) 42.17/12.89 a__U134(x0, x1, x2, x3) 42.17/12.89 a__U135(x0, x1, x2, x3) 42.17/12.89 a__U136(x0, x1, x2, x3) 42.17/12.89 a__take(x0, x1) 42.17/12.89 a__U21(x0, x1) 42.17/12.89 a__U22(x0, x1) 42.17/12.89 a__U23(x0) 42.17/12.89 a__U31(x0, x1) 42.17/12.89 a__U32(x0, x1) 42.17/12.89 a__U33(x0) 42.17/12.89 a__U41(x0, x1, x2) 42.17/12.89 a__U42(x0, x1, x2) 42.17/12.89 a__U43(x0, x1, x2) 42.17/12.89 a__U44(x0, x1, x2) 42.17/12.89 a__U45(x0, x1) 42.17/12.89 a__U46(x0) 42.17/12.89 a__U51(x0, x1) 42.17/12.89 a__U52(x0) 42.17/12.89 a__U61(x0, x1) 42.17/12.89 a__U62(x0) 42.17/12.89 a__U71(x0) 42.17/12.89 a__U81(x0) 42.17/12.89 a__U91(x0, x1, x2) 42.17/12.89 a__U92(x0, x1, x2) 42.17/12.89 a__U93(x0, x1, x2) 42.17/12.89 a__U94(x0, x1, x2) 42.17/12.89 a__U95(x0, x1) 42.17/12.89 a__U96(x0) 42.17/12.89 42.17/12.89 We have to consider all minimal (P,Q,R)-chains. 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (15) QReductionProof (EQUIVALENT) 42.17/12.89 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 42.17/12.89 42.17/12.89 a__zeros 42.17/12.89 mark(zeros) 42.17/12.89 mark(U101(x0, x1, x2)) 42.17/12.89 mark(U102(x0, x1, x2)) 42.17/12.89 mark(isNatKind(x0)) 42.17/12.89 mark(U103(x0, x1, x2)) 42.17/12.89 mark(isNatIListKind(x0)) 42.17/12.89 mark(U104(x0, x1, x2)) 42.17/12.89 mark(U105(x0, x1)) 42.17/12.89 mark(isNat(x0)) 42.17/12.89 mark(U106(x0)) 42.17/12.89 mark(isNatIList(x0)) 42.17/12.89 mark(U11(x0, x1)) 42.17/12.89 mark(U12(x0, x1)) 42.17/12.89 mark(U111(x0, x1, x2)) 42.17/12.89 mark(U112(x0, x1, x2)) 42.17/12.89 mark(U113(x0, x1, x2)) 42.17/12.89 mark(U114(x0, x1)) 42.17/12.89 mark(length(x0)) 42.17/12.89 mark(U13(x0)) 42.17/12.89 mark(isNatList(x0)) 42.17/12.89 mark(U121(x0, x1)) 42.17/12.89 mark(U122(x0)) 42.17/12.89 mark(U131(x0, x1, x2, x3)) 42.17/12.89 mark(U132(x0, x1, x2, x3)) 42.17/12.89 mark(U133(x0, x1, x2, x3)) 42.17/12.89 mark(U134(x0, x1, x2, x3)) 42.17/12.89 mark(U135(x0, x1, x2, x3)) 42.17/12.89 mark(U136(x0, x1, x2, x3)) 42.17/12.89 mark(take(x0, x1)) 42.17/12.89 mark(U21(x0, x1)) 42.17/12.89 mark(U22(x0, x1)) 42.17/12.89 mark(U23(x0)) 42.17/12.89 mark(U31(x0, x1)) 42.17/12.89 mark(U32(x0, x1)) 42.17/12.89 mark(U33(x0)) 42.17/12.89 mark(U41(x0, x1, x2)) 42.17/12.89 mark(U42(x0, x1, x2)) 42.17/12.89 mark(U43(x0, x1, x2)) 42.17/12.89 mark(U44(x0, x1, x2)) 42.17/12.89 mark(U45(x0, x1)) 42.17/12.89 mark(U46(x0)) 42.17/12.89 mark(U51(x0, x1)) 42.17/12.89 mark(U52(x0)) 42.17/12.89 mark(U61(x0, x1)) 42.17/12.89 mark(U62(x0)) 42.17/12.89 mark(U71(x0)) 42.17/12.89 mark(U81(x0)) 42.17/12.89 mark(U91(x0, x1, x2)) 42.17/12.89 mark(U92(x0, x1, x2)) 42.17/12.89 mark(U93(x0, x1, x2)) 42.17/12.89 mark(U94(x0, x1, x2)) 42.17/12.89 mark(U95(x0, x1)) 42.17/12.89 mark(U96(x0)) 42.17/12.89 mark(cons(x0, x1)) 42.17/12.89 mark(0) 42.17/12.89 mark(tt) 42.17/12.89 mark(s(x0)) 42.17/12.89 mark(nil) 42.17/12.89 a__U111(x0, x1, x2) 42.17/12.89 a__U112(x0, x1, x2) 42.17/12.89 a__U113(x0, x1, x2) 42.17/12.89 a__U114(x0, x1) 42.17/12.89 a__length(x0) 42.17/12.89 a__U121(x0, x1) 42.17/12.89 a__U122(x0) 42.17/12.89 a__U131(x0, x1, x2, x3) 42.17/12.89 a__U132(x0, x1, x2, x3) 42.17/12.89 a__U133(x0, x1, x2, x3) 42.17/12.89 a__U134(x0, x1, x2, x3) 42.17/12.89 a__U135(x0, x1, x2, x3) 42.17/12.89 a__U136(x0, x1, x2, x3) 42.17/12.89 a__take(x0, x1) 42.17/12.89 42.17/12.89 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (16) 42.17/12.89 Obligation: 42.17/12.89 Q DP problem: 42.17/12.89 The TRS P consists of the following rules: 42.17/12.89 42.17/12.89 A__U102(tt, V1, V2) -> A__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U103(tt, V1, V2) -> A__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U104(tt, V1, V2) -> A__U105(a__isNat(V1), V2) 42.17/12.89 A__U105(tt, V2) -> A__ISNATILIST(V2) 42.17/12.89 A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) 42.17/12.89 A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) 42.17/12.89 A__U32(tt, V) -> A__ISNATLIST(V) 42.17/12.89 A__ISNATLIST(cons(V1, V2)) -> A__U91(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U91(tt, V1, V2) -> A__U92(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U92(tt, V1, V2) -> A__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U93(tt, V1, V2) -> A__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U94(tt, V1, V2) -> A__U95(a__isNat(V1), V2) 42.17/12.89 A__U95(tt, V2) -> A__ISNATLIST(V2) 42.17/12.89 A__ISNATLIST(take(V1, V2)) -> A__U101(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U101(tt, V1, V2) -> A__U102(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U94(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 42.17/12.89 A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 42.17/12.89 A__U12(tt, V1) -> A__ISNATLIST(V1) 42.17/12.89 A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.17/12.89 A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.17/12.89 A__U22(tt, V1) -> A__ISNAT(V1) 42.17/12.89 A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 42.17/12.89 A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.89 A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 42.17/12.89 A__U45(tt, V2) -> A__ISNATILIST(V2) 42.17/12.89 A__U44(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 A__U104(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 42.17/12.89 The TRS R consists of the following rules: 42.17/12.89 42.17/12.89 a__isNat(0) -> tt 42.17/12.89 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.89 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.89 a__isNat(X) -> isNat(X) 42.17/12.89 a__isNatKind(0) -> tt 42.17/12.89 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.89 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.89 a__isNatKind(X) -> isNatKind(X) 42.17/12.89 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.89 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.89 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.89 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.89 a__U23(tt) -> tt 42.17/12.89 a__U23(X) -> U23(X) 42.17/12.89 a__U81(tt) -> tt 42.17/12.89 a__U81(X) -> U81(X) 42.17/12.89 a__isNatIListKind(nil) -> tt 42.17/12.89 a__isNatIListKind(zeros) -> tt 42.17/12.89 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.89 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.89 a__U71(tt) -> tt 42.17/12.89 a__U71(X) -> U71(X) 42.17/12.89 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.89 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.89 a__U62(tt) -> tt 42.17/12.89 a__U62(X) -> U62(X) 42.17/12.89 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.89 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.89 a__U52(tt) -> tt 42.17/12.89 a__U52(X) -> U52(X) 42.17/12.89 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.89 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.89 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.89 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.89 a__isNatList(nil) -> tt 42.17/12.89 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatList(X) -> isNatList(X) 42.17/12.89 a__U13(tt) -> tt 42.17/12.89 a__U13(X) -> U13(X) 42.17/12.89 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.89 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.89 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.89 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.89 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.89 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.89 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.89 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.89 a__isNatIList(zeros) -> tt 42.17/12.89 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.89 a__isNatIList(X) -> isNatIList(X) 42.17/12.89 a__U106(tt) -> tt 42.17/12.89 a__U106(X) -> U106(X) 42.17/12.89 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.89 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.89 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.89 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.89 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.89 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.89 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.89 a__U46(tt) -> tt 42.17/12.89 a__U46(X) -> U46(X) 42.17/12.89 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.89 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.89 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.89 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.89 a__U33(tt) -> tt 42.17/12.89 a__U33(X) -> U33(X) 42.17/12.89 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.89 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.89 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.89 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.89 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.89 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.89 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.89 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.89 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.89 a__U96(tt) -> tt 42.17/12.89 a__U96(X) -> U96(X) 42.17/12.89 42.17/12.89 The set Q consists of the following terms: 42.17/12.89 42.17/12.89 a__isNatIList(x0) 42.17/12.89 a__U101(x0, x1, x2) 42.17/12.89 a__U102(x0, x1, x2) 42.17/12.89 a__isNatKind(x0) 42.17/12.89 a__U103(x0, x1, x2) 42.17/12.89 a__isNatIListKind(x0) 42.17/12.89 a__U104(x0, x1, x2) 42.17/12.89 a__U105(x0, x1) 42.17/12.89 a__isNat(x0) 42.17/12.89 a__U106(x0) 42.17/12.89 a__U11(x0, x1) 42.17/12.89 a__U12(x0, x1) 42.17/12.89 a__U13(x0) 42.17/12.89 a__isNatList(x0) 42.17/12.89 a__U21(x0, x1) 42.17/12.89 a__U22(x0, x1) 42.17/12.89 a__U23(x0) 42.17/12.89 a__U31(x0, x1) 42.17/12.89 a__U32(x0, x1) 42.17/12.89 a__U33(x0) 42.17/12.89 a__U41(x0, x1, x2) 42.17/12.89 a__U42(x0, x1, x2) 42.17/12.89 a__U43(x0, x1, x2) 42.17/12.89 a__U44(x0, x1, x2) 42.17/12.89 a__U45(x0, x1) 42.17/12.89 a__U46(x0) 42.17/12.89 a__U51(x0, x1) 42.17/12.89 a__U52(x0) 42.17/12.89 a__U61(x0, x1) 42.17/12.89 a__U62(x0) 42.17/12.89 a__U71(x0) 42.17/12.89 a__U81(x0) 42.17/12.89 a__U91(x0, x1, x2) 42.17/12.89 a__U92(x0, x1, x2) 42.17/12.89 a__U93(x0, x1, x2) 42.17/12.89 a__U94(x0, x1, x2) 42.17/12.89 a__U95(x0, x1) 42.17/12.89 a__U96(x0) 42.17/12.89 42.17/12.89 We have to consider all minimal (P,Q,R)-chains. 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (17) QDPSizeChangeProof (EQUIVALENT) 42.17/12.89 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 42.17/12.89 42.17/12.89 From the DPs we obtained the following set of size-change graphs: 42.17/12.89 *A__U103(tt, V1, V2) -> A__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.89 The graph contains the following edges 2 >= 2, 3 >= 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U101(tt, V1, V2) -> A__U102(a__isNatKind(V1), V1, V2) 42.17/12.89 The graph contains the following edges 2 >= 2, 3 >= 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U102(tt, V1, V2) -> A__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.89 The graph contains the following edges 2 >= 2, 3 >= 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U105(tt, V2) -> A__ISNATILIST(V2) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__U104(tt, V1, V2) -> A__U105(a__isNat(V1), V2) 42.17/12.89 The graph contains the following edges 3 >= 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__U104(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) 42.17/12.89 The graph contains the following edges 2 >= 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__U45(tt, V2) -> A__ISNATILIST(V2) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) 42.17/12.89 The graph contains the following edges 1 >= 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) 42.17/12.89 The graph contains the following edges 1 > 2, 1 > 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U32(tt, V) -> A__ISNATLIST(V) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__U91(tt, V1, V2) -> A__U92(a__isNatKind(V1), V1, V2) 42.17/12.89 The graph contains the following edges 2 >= 2, 3 >= 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNATLIST(cons(V1, V2)) -> A__U91(a__isNatKind(V1), V1, V2) 42.17/12.89 The graph contains the following edges 1 > 2, 1 > 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNATLIST(take(V1, V2)) -> A__U101(a__isNatKind(V1), V1, V2) 42.17/12.89 The graph contains the following edges 1 > 2, 1 > 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U92(tt, V1, V2) -> A__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.89 The graph contains the following edges 2 >= 2, 3 >= 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U93(tt, V1, V2) -> A__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.89 The graph contains the following edges 2 >= 2, 3 >= 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U95(tt, V2) -> A__ISNATLIST(V2) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__U12(tt, V1) -> A__ISNATLIST(V1) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__U94(tt, V1, V2) -> A__U95(a__isNat(V1), V2) 42.17/12.89 The graph contains the following edges 3 >= 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__U94(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) 42.17/12.89 The graph contains the following edges 2 >= 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) 42.17/12.89 The graph contains the following edges 1 > 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) 42.17/12.89 The graph contains the following edges 1 > 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) 42.17/12.89 The graph contains the following edges 2 >= 2 42.17/12.89 42.17/12.89 42.17/12.89 *A__U22(tt, V1) -> A__ISNAT(V1) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__U44(tt, V1, V2) -> A__ISNAT(V1) 42.17/12.89 The graph contains the following edges 2 >= 1 42.17/12.89 42.17/12.89 42.17/12.89 *A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) 42.17/12.89 The graph contains the following edges 2 >= 2, 3 >= 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.89 The graph contains the following edges 2 >= 2, 3 >= 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.89 The graph contains the following edges 2 >= 2, 3 >= 3 42.17/12.89 42.17/12.89 42.17/12.89 *A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) 42.17/12.89 The graph contains the following edges 3 >= 2 42.17/12.89 42.17/12.89 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (18) 42.17/12.89 YES 42.17/12.89 42.17/12.89 ---------------------------------------- 42.17/12.89 42.17/12.89 (19) 42.17/12.89 Obligation: 42.17/12.89 Q DP problem: 42.17/12.89 The TRS P consists of the following rules: 42.17/12.89 42.17/12.89 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.17/12.89 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.17/12.89 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U105(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U106(X)) -> MARK(X) 42.17/12.90 MARK(U11(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U12(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) 42.17/12.90 A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) 42.17/12.90 A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) 42.17/12.90 A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) 42.17/12.90 A__U114(tt, L) -> A__LENGTH(mark(L)) 42.17/12.90 A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) 42.17/12.90 A__U114(tt, L) -> MARK(L) 42.17/12.90 MARK(U111(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) 42.17/12.90 MARK(U112(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) 42.17/12.90 MARK(U113(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) 42.17/12.90 MARK(U114(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(length(X)) -> A__LENGTH(mark(X)) 42.17/12.90 MARK(length(X)) -> MARK(X) 42.17/12.90 MARK(U13(X)) -> MARK(X) 42.17/12.90 MARK(U121(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U122(X)) -> MARK(X) 42.17/12.90 MARK(U131(X1, X2, X3, X4)) -> A__U131(mark(X1), X2, X3, X4) 42.17/12.90 A__U131(tt, IL, M, N) -> A__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.90 A__U132(tt, IL, M, N) -> A__U133(a__isNat(M), IL, M, N) 42.17/12.90 A__U133(tt, IL, M, N) -> A__U134(a__isNatKind(M), IL, M, N) 42.17/12.90 A__U134(tt, IL, M, N) -> A__U135(a__isNat(N), IL, M, N) 42.17/12.90 A__U135(tt, IL, M, N) -> A__U136(a__isNatKind(N), IL, M, N) 42.17/12.90 A__U136(tt, IL, M, N) -> MARK(N) 42.17/12.90 MARK(U131(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U132(X1, X2, X3, X4)) -> A__U132(mark(X1), X2, X3, X4) 42.17/12.90 MARK(U132(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U133(X1, X2, X3, X4)) -> A__U133(mark(X1), X2, X3, X4) 42.17/12.90 MARK(U133(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U134(X1, X2, X3, X4)) -> A__U134(mark(X1), X2, X3, X4) 42.17/12.90 MARK(U134(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U135(X1, X2, X3, X4)) -> A__U135(mark(X1), X2, X3, X4) 42.17/12.90 MARK(U135(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U136(X1, X2, X3, X4)) -> A__U136(mark(X1), X2, X3, X4) 42.17/12.90 MARK(U136(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) 42.17/12.90 A__TAKE(s(M), cons(N, IL)) -> A__U131(a__isNatIList(IL), IL, M, N) 42.17/12.90 MARK(take(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(take(X1, X2)) -> MARK(X2) 42.17/12.90 MARK(U21(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U22(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U23(X)) -> MARK(X) 42.17/12.90 MARK(U31(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U32(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U33(X)) -> MARK(X) 42.17/12.90 MARK(U41(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U42(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U43(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U44(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U45(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U46(X)) -> MARK(X) 42.17/12.90 MARK(U51(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U52(X)) -> MARK(X) 42.17/12.90 MARK(U61(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U62(X)) -> MARK(X) 42.17/12.90 MARK(U71(X)) -> MARK(X) 42.17/12.90 MARK(U81(X)) -> MARK(X) 42.17/12.90 MARK(U91(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U92(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U93(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U94(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U95(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U96(X)) -> MARK(X) 42.17/12.90 MARK(cons(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(s(X)) -> MARK(X) 42.17/12.90 42.17/12.90 The TRS R consists of the following rules: 42.17/12.90 42.17/12.90 a__zeros -> cons(0, zeros) 42.17/12.90 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.90 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.90 a__U106(tt) -> tt 42.17/12.90 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.90 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.17/12.90 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.17/12.90 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.17/12.90 a__U114(tt, L) -> s(a__length(mark(L))) 42.17/12.90 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.90 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.17/12.90 a__U122(tt) -> nil 42.17/12.90 a__U13(tt) -> tt 42.17/12.90 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.90 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.17/12.90 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.17/12.90 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.17/12.90 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.17/12.90 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.17/12.90 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.90 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.90 a__U23(tt) -> tt 42.17/12.90 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.90 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.90 a__U33(tt) -> tt 42.17/12.90 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.90 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.90 a__U46(tt) -> tt 42.17/12.90 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.90 a__U52(tt) -> tt 42.17/12.90 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.90 a__U62(tt) -> tt 42.17/12.90 a__U71(tt) -> tt 42.17/12.90 a__U81(tt) -> tt 42.17/12.90 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.90 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.90 a__U96(tt) -> tt 42.17/12.90 a__isNat(0) -> tt 42.17/12.90 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.90 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.90 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.90 a__isNatIList(zeros) -> tt 42.17/12.90 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.90 a__isNatIListKind(nil) -> tt 42.17/12.90 a__isNatIListKind(zeros) -> tt 42.17/12.90 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.90 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.90 a__isNatKind(0) -> tt 42.17/12.90 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.90 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.90 a__isNatList(nil) -> tt 42.17/12.90 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.90 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.90 a__length(nil) -> 0 42.17/12.90 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.17/12.90 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.17/12.90 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.17/12.90 mark(zeros) -> a__zeros 42.17/12.90 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.17/12.90 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.17/12.90 mark(isNatKind(X)) -> a__isNatKind(X) 42.17/12.90 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.17/12.90 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.17/12.90 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.17/12.90 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.17/12.90 mark(isNat(X)) -> a__isNat(X) 42.17/12.90 mark(U106(X)) -> a__U106(mark(X)) 42.17/12.90 mark(isNatIList(X)) -> a__isNatIList(X) 42.17/12.90 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.17/12.90 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.17/12.90 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.17/12.90 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.17/12.90 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.17/12.90 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.17/12.90 mark(length(X)) -> a__length(mark(X)) 42.17/12.90 mark(U13(X)) -> a__U13(mark(X)) 42.17/12.90 mark(isNatList(X)) -> a__isNatList(X) 42.17/12.90 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.17/12.90 mark(U122(X)) -> a__U122(mark(X)) 42.17/12.90 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.17/12.90 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.17/12.90 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.17/12.90 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.17/12.90 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.17/12.90 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.17/12.90 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.17/12.90 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.17/12.90 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.17/12.90 mark(U23(X)) -> a__U23(mark(X)) 42.17/12.90 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.17/12.90 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.17/12.90 mark(U33(X)) -> a__U33(mark(X)) 42.17/12.90 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.17/12.90 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.17/12.90 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.17/12.90 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.17/12.90 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.17/12.90 mark(U46(X)) -> a__U46(mark(X)) 42.17/12.90 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.17/12.90 mark(U52(X)) -> a__U52(mark(X)) 42.17/12.90 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.17/12.90 mark(U62(X)) -> a__U62(mark(X)) 42.17/12.90 mark(U71(X)) -> a__U71(mark(X)) 42.17/12.90 mark(U81(X)) -> a__U81(mark(X)) 42.17/12.90 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.17/12.90 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.17/12.90 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.17/12.90 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.17/12.90 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.17/12.90 mark(U96(X)) -> a__U96(mark(X)) 42.17/12.90 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.17/12.90 mark(0) -> 0 42.17/12.90 mark(tt) -> tt 42.17/12.90 mark(s(X)) -> s(mark(X)) 42.17/12.90 mark(nil) -> nil 42.17/12.90 a__zeros -> zeros 42.17/12.90 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.90 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.90 a__isNatKind(X) -> isNatKind(X) 42.17/12.90 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.90 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.90 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.90 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.90 a__isNat(X) -> isNat(X) 42.17/12.90 a__U106(X) -> U106(X) 42.17/12.90 a__isNatIList(X) -> isNatIList(X) 42.17/12.90 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.90 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.90 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.17/12.90 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.17/12.90 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.17/12.90 a__U114(X1, X2) -> U114(X1, X2) 42.17/12.90 a__length(X) -> length(X) 42.17/12.90 a__U13(X) -> U13(X) 42.17/12.90 a__isNatList(X) -> isNatList(X) 42.17/12.90 a__U121(X1, X2) -> U121(X1, X2) 42.17/12.90 a__U122(X) -> U122(X) 42.17/12.90 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.17/12.90 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.17/12.90 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.17/12.90 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.17/12.90 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.17/12.90 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.17/12.90 a__take(X1, X2) -> take(X1, X2) 42.17/12.90 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.90 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.90 a__U23(X) -> U23(X) 42.17/12.90 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.90 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.90 a__U33(X) -> U33(X) 42.17/12.90 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.90 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.90 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.90 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.90 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.90 a__U46(X) -> U46(X) 42.17/12.90 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.90 a__U52(X) -> U52(X) 42.17/12.90 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.90 a__U62(X) -> U62(X) 42.17/12.90 a__U71(X) -> U71(X) 42.17/12.90 a__U81(X) -> U81(X) 42.17/12.90 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.90 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.90 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.90 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.90 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.90 a__U96(X) -> U96(X) 42.17/12.90 42.17/12.90 The set Q consists of the following terms: 42.17/12.90 42.17/12.90 a__zeros 42.17/12.90 a__isNatIList(x0) 42.17/12.90 mark(zeros) 42.17/12.90 mark(U101(x0, x1, x2)) 42.17/12.90 mark(U102(x0, x1, x2)) 42.17/12.90 mark(isNatKind(x0)) 42.17/12.90 mark(U103(x0, x1, x2)) 42.17/12.90 mark(isNatIListKind(x0)) 42.17/12.90 mark(U104(x0, x1, x2)) 42.17/12.90 mark(U105(x0, x1)) 42.17/12.90 mark(isNat(x0)) 42.17/12.90 mark(U106(x0)) 42.17/12.90 mark(isNatIList(x0)) 42.17/12.90 mark(U11(x0, x1)) 42.17/12.90 mark(U12(x0, x1)) 42.17/12.90 mark(U111(x0, x1, x2)) 42.17/12.90 mark(U112(x0, x1, x2)) 42.17/12.90 mark(U113(x0, x1, x2)) 42.17/12.90 mark(U114(x0, x1)) 42.17/12.90 mark(length(x0)) 42.17/12.90 mark(U13(x0)) 42.17/12.90 mark(isNatList(x0)) 42.17/12.90 mark(U121(x0, x1)) 42.17/12.90 mark(U122(x0)) 42.17/12.90 mark(U131(x0, x1, x2, x3)) 42.17/12.90 mark(U132(x0, x1, x2, x3)) 42.17/12.90 mark(U133(x0, x1, x2, x3)) 42.17/12.90 mark(U134(x0, x1, x2, x3)) 42.17/12.90 mark(U135(x0, x1, x2, x3)) 42.17/12.90 mark(U136(x0, x1, x2, x3)) 42.17/12.90 mark(take(x0, x1)) 42.17/12.90 mark(U21(x0, x1)) 42.17/12.90 mark(U22(x0, x1)) 42.17/12.90 mark(U23(x0)) 42.17/12.90 mark(U31(x0, x1)) 42.17/12.90 mark(U32(x0, x1)) 42.17/12.90 mark(U33(x0)) 42.17/12.90 mark(U41(x0, x1, x2)) 42.17/12.90 mark(U42(x0, x1, x2)) 42.17/12.90 mark(U43(x0, x1, x2)) 42.17/12.90 mark(U44(x0, x1, x2)) 42.17/12.90 mark(U45(x0, x1)) 42.17/12.90 mark(U46(x0)) 42.17/12.90 mark(U51(x0, x1)) 42.17/12.90 mark(U52(x0)) 42.17/12.90 mark(U61(x0, x1)) 42.17/12.90 mark(U62(x0)) 42.17/12.90 mark(U71(x0)) 42.17/12.90 mark(U81(x0)) 42.17/12.90 mark(U91(x0, x1, x2)) 42.17/12.90 mark(U92(x0, x1, x2)) 42.17/12.90 mark(U93(x0, x1, x2)) 42.17/12.90 mark(U94(x0, x1, x2)) 42.17/12.90 mark(U95(x0, x1)) 42.17/12.90 mark(U96(x0)) 42.17/12.90 mark(cons(x0, x1)) 42.17/12.90 mark(0) 42.17/12.90 mark(tt) 42.17/12.90 mark(s(x0)) 42.17/12.90 mark(nil) 42.17/12.90 a__U101(x0, x1, x2) 42.17/12.90 a__U102(x0, x1, x2) 42.17/12.90 a__isNatKind(x0) 42.17/12.90 a__U103(x0, x1, x2) 42.17/12.90 a__isNatIListKind(x0) 42.17/12.90 a__U104(x0, x1, x2) 42.17/12.90 a__U105(x0, x1) 42.17/12.90 a__isNat(x0) 42.17/12.90 a__U106(x0) 42.17/12.90 a__U11(x0, x1) 42.17/12.90 a__U12(x0, x1) 42.17/12.90 a__U111(x0, x1, x2) 42.17/12.90 a__U112(x0, x1, x2) 42.17/12.90 a__U113(x0, x1, x2) 42.17/12.90 a__U114(x0, x1) 42.17/12.90 a__length(x0) 42.17/12.90 a__U13(x0) 42.17/12.90 a__isNatList(x0) 42.17/12.90 a__U121(x0, x1) 42.17/12.90 a__U122(x0) 42.17/12.90 a__U131(x0, x1, x2, x3) 42.17/12.90 a__U132(x0, x1, x2, x3) 42.17/12.90 a__U133(x0, x1, x2, x3) 42.17/12.90 a__U134(x0, x1, x2, x3) 42.17/12.90 a__U135(x0, x1, x2, x3) 42.17/12.90 a__U136(x0, x1, x2, x3) 42.17/12.90 a__take(x0, x1) 42.17/12.90 a__U21(x0, x1) 42.17/12.90 a__U22(x0, x1) 42.17/12.90 a__U23(x0) 42.17/12.90 a__U31(x0, x1) 42.17/12.90 a__U32(x0, x1) 42.17/12.90 a__U33(x0) 42.17/12.90 a__U41(x0, x1, x2) 42.17/12.90 a__U42(x0, x1, x2) 42.17/12.90 a__U43(x0, x1, x2) 42.17/12.90 a__U44(x0, x1, x2) 42.17/12.90 a__U45(x0, x1) 42.17/12.90 a__U46(x0) 42.17/12.90 a__U51(x0, x1) 42.17/12.90 a__U52(x0) 42.17/12.90 a__U61(x0, x1) 42.17/12.90 a__U62(x0) 42.17/12.90 a__U71(x0) 42.17/12.90 a__U81(x0) 42.17/12.90 a__U91(x0, x1, x2) 42.17/12.90 a__U92(x0, x1, x2) 42.17/12.90 a__U93(x0, x1, x2) 42.17/12.90 a__U94(x0, x1, x2) 42.17/12.90 a__U95(x0, x1) 42.17/12.90 a__U96(x0) 42.17/12.90 42.17/12.90 We have to consider all minimal (P,Q,R)-chains. 42.17/12.90 ---------------------------------------- 42.17/12.90 42.17/12.90 (20) QDPOrderProof (EQUIVALENT) 42.17/12.90 We use the reduction pair processor [LPAR04,JAR06]. 42.17/12.90 42.17/12.90 42.17/12.90 The following pairs can be oriented strictly and are deleted. 42.17/12.90 42.17/12.90 MARK(U121(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U122(X)) -> MARK(X) 42.17/12.90 A__U133(tt, IL, M, N) -> A__U134(a__isNatKind(M), IL, M, N) 42.17/12.90 MARK(U131(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U132(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U133(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U134(X1, X2, X3, X4)) -> A__U134(mark(X1), X2, X3, X4) 42.17/12.90 MARK(U134(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U135(X1, X2, X3, X4)) -> A__U135(mark(X1), X2, X3, X4) 42.17/12.90 MARK(U135(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(U136(X1, X2, X3, X4)) -> A__U136(mark(X1), X2, X3, X4) 42.17/12.90 MARK(U136(X1, X2, X3, X4)) -> MARK(X1) 42.17/12.90 MARK(take(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(take(X1, X2)) -> MARK(X2) 42.17/12.90 The remaining pairs can at least be oriented weakly. 42.17/12.90 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 42.17/12.90 42.17/12.90 POL( A__LENGTH_1(x_1) ) = x_1 42.17/12.90 POL( A__TAKE_2(x_1, x_2) ) = x_2 + 1 42.17/12.90 POL( A__U111_3(x_1, ..., x_3) ) = x_2 + x_3 42.17/12.90 POL( A__U112_3(x_1, ..., x_3) ) = x_2 42.17/12.90 POL( A__U113_3(x_1, ..., x_3) ) = x_2 42.17/12.90 POL( A__U114_2(x_1, x_2) ) = x_2 42.17/12.90 POL( A__U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 42.17/12.90 POL( A__U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 42.17/12.90 POL( A__U133_4(x_1, ..., x_4) ) = 2x_4 + 1 42.17/12.90 POL( A__U134_4(x_1, ..., x_4) ) = x_4 42.17/12.90 POL( A__U135_4(x_1, ..., x_4) ) = x_4 42.17/12.90 POL( A__U136_4(x_1, ..., x_4) ) = x_4 42.17/12.90 POL( mark_1(x_1) ) = x_1 42.17/12.90 POL( zeros ) = 0 42.17/12.90 POL( a__zeros ) = 0 42.17/12.90 POL( U101_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U101_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U102_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U102_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( isNatKind_1(x_1) ) = 0 42.17/12.90 POL( a__isNatKind_1(x_1) ) = 0 42.17/12.90 POL( U103_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U103_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( isNatIListKind_1(x_1) ) = 0 42.17/12.90 POL( a__isNatIListKind_1(x_1) ) = 0 42.17/12.90 POL( U104_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U104_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U105_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U105_2(x_1, x_2) ) = x_1 42.17/12.90 POL( isNat_1(x_1) ) = 0 42.17/12.90 POL( a__isNat_1(x_1) ) = 0 42.17/12.90 POL( U106_1(x_1) ) = x_1 42.17/12.90 POL( a__U106_1(x_1) ) = x_1 42.17/12.90 POL( isNatIList_1(x_1) ) = 0 42.17/12.90 POL( a__isNatIList_1(x_1) ) = 0 42.17/12.90 POL( U11_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U11_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( U12_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U12_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( U111_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 42.17/12.90 POL( a__U111_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 42.17/12.90 POL( U112_3(x_1, ..., x_3) ) = 2x_1 + x_2 42.17/12.90 POL( a__U112_3(x_1, ..., x_3) ) = 2x_1 + x_2 42.17/12.90 POL( U113_3(x_1, ..., x_3) ) = x_1 + x_2 42.17/12.90 POL( a__U113_3(x_1, ..., x_3) ) = x_1 + x_2 42.17/12.90 POL( U114_2(x_1, x_2) ) = x_1 + x_2 42.17/12.90 POL( a__U114_2(x_1, x_2) ) = x_1 + x_2 42.17/12.90 POL( length_1(x_1) ) = x_1 42.17/12.90 POL( a__length_1(x_1) ) = x_1 42.17/12.90 POL( U13_1(x_1) ) = x_1 42.17/12.90 POL( a__U13_1(x_1) ) = x_1 42.17/12.90 POL( isNatList_1(x_1) ) = 0 42.17/12.90 POL( a__isNatList_1(x_1) ) = 0 42.17/12.90 POL( U121_2(x_1, x_2) ) = x_1 + 1 42.17/12.90 POL( a__U121_2(x_1, x_2) ) = x_1 + 1 42.17/12.90 POL( U122_1(x_1) ) = 2x_1 + 1 42.17/12.90 POL( a__U122_1(x_1) ) = 2x_1 + 1 42.17/12.90 POL( U131_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( a__U131_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( U132_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( a__U132_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( U133_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( a__U133_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( U134_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( a__U134_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( U135_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( a__U135_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( U136_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( a__U136_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 1 42.17/12.90 POL( take_2(x_1, x_2) ) = x_1 + x_2 + 1 42.17/12.90 POL( a__take_2(x_1, x_2) ) = x_1 + x_2 + 1 42.17/12.90 POL( U21_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U21_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U22_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U22_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( U23_1(x_1) ) = x_1 42.17/12.90 POL( a__U23_1(x_1) ) = x_1 42.17/12.90 POL( U31_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U31_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U32_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U32_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U33_1(x_1) ) = x_1 42.17/12.90 POL( a__U33_1(x_1) ) = x_1 42.17/12.90 POL( U41_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U41_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U42_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U42_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U43_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U43_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U44_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U44_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U45_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U45_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( U46_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U46_1(x_1) ) = 2x_1 42.17/12.90 POL( U51_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U51_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U52_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U52_1(x_1) ) = 2x_1 42.17/12.90 POL( U61_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U61_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U62_1(x_1) ) = x_1 42.17/12.90 POL( a__U62_1(x_1) ) = x_1 42.17/12.90 POL( U71_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U71_1(x_1) ) = 2x_1 42.17/12.90 POL( U81_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U81_1(x_1) ) = 2x_1 42.17/12.90 POL( U91_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U91_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U92_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U92_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U93_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U93_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U94_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U94_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U95_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U95_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( U96_1(x_1) ) = x_1 42.17/12.90 POL( a__U96_1(x_1) ) = x_1 42.17/12.90 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 42.17/12.90 POL( 0 ) = 0 42.17/12.90 POL( tt ) = 0 42.17/12.90 POL( s_1(x_1) ) = x_1 42.17/12.90 POL( nil ) = 0 42.17/12.90 POL( MARK_1(x_1) ) = x_1 42.17/12.90 42.17/12.90 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.17/12.90 42.17/12.90 mark(zeros) -> a__zeros 42.17/12.90 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.17/12.90 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.17/12.90 mark(isNatKind(X)) -> a__isNatKind(X) 42.17/12.90 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.17/12.90 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.17/12.90 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.17/12.90 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.17/12.90 mark(isNat(X)) -> a__isNat(X) 42.17/12.90 mark(U106(X)) -> a__U106(mark(X)) 42.17/12.90 mark(isNatIList(X)) -> a__isNatIList(X) 42.17/12.90 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.17/12.90 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.17/12.90 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.17/12.90 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.17/12.90 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.17/12.90 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.17/12.90 mark(length(X)) -> a__length(mark(X)) 42.17/12.90 mark(U13(X)) -> a__U13(mark(X)) 42.17/12.90 mark(isNatList(X)) -> a__isNatList(X) 42.17/12.90 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.17/12.90 mark(U122(X)) -> a__U122(mark(X)) 42.17/12.90 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.17/12.90 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.17/12.90 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.17/12.90 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.17/12.90 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.17/12.90 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.17/12.90 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.17/12.90 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.17/12.90 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.17/12.90 mark(U23(X)) -> a__U23(mark(X)) 42.17/12.90 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.17/12.90 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.17/12.90 mark(U33(X)) -> a__U33(mark(X)) 42.17/12.90 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.17/12.90 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.17/12.90 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.17/12.90 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.17/12.90 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.17/12.90 mark(U46(X)) -> a__U46(mark(X)) 42.17/12.90 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.17/12.90 mark(U52(X)) -> a__U52(mark(X)) 42.17/12.90 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.17/12.90 mark(U62(X)) -> a__U62(mark(X)) 42.17/12.90 mark(U71(X)) -> a__U71(mark(X)) 42.17/12.90 mark(U81(X)) -> a__U81(mark(X)) 42.17/12.90 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.17/12.90 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.17/12.90 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.17/12.90 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.17/12.90 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.17/12.90 mark(U96(X)) -> a__U96(mark(X)) 42.17/12.90 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.17/12.90 mark(0) -> 0 42.17/12.90 mark(tt) -> tt 42.17/12.90 mark(s(X)) -> s(mark(X)) 42.17/12.90 mark(nil) -> nil 42.17/12.90 a__isNatIListKind(nil) -> tt 42.17/12.90 a__isNatIListKind(zeros) -> tt 42.17/12.90 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.90 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.90 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.90 a__isNat(0) -> tt 42.17/12.90 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.90 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.90 a__isNat(X) -> isNat(X) 42.17/12.90 a__isNatKind(0) -> tt 42.17/12.90 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.90 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.90 a__isNatKind(X) -> isNatKind(X) 42.17/12.90 a__isNatList(nil) -> tt 42.17/12.90 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.90 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.90 a__isNatList(X) -> isNatList(X) 42.17/12.90 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.90 a__isNatIList(zeros) -> tt 42.17/12.90 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.90 a__isNatIList(X) -> isNatIList(X) 42.17/12.90 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.90 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.90 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.90 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.90 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.90 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.90 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.90 a__U106(tt) -> tt 42.17/12.90 a__U106(X) -> U106(X) 42.17/12.90 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.90 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.90 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.90 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.90 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.17/12.90 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.17/12.90 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.17/12.90 a__U114(X1, X2) -> U114(X1, X2) 42.17/12.90 a__length(nil) -> 0 42.17/12.90 a__length(X) -> length(X) 42.17/12.90 a__U13(tt) -> tt 42.17/12.90 a__U13(X) -> U13(X) 42.17/12.90 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.17/12.90 a__U121(X1, X2) -> U121(X1, X2) 42.17/12.90 a__U122(tt) -> nil 42.17/12.90 a__U122(X) -> U122(X) 42.17/12.90 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.17/12.90 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.17/12.90 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.17/12.90 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.17/12.90 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.17/12.90 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.17/12.90 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.17/12.90 a__take(X1, X2) -> take(X1, X2) 42.17/12.90 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.90 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.90 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.90 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.90 a__U23(tt) -> tt 42.17/12.90 a__U23(X) -> U23(X) 42.17/12.90 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.90 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.90 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.90 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.90 a__U33(tt) -> tt 42.17/12.90 a__U33(X) -> U33(X) 42.17/12.90 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.90 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.90 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.90 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.90 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.90 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.90 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.90 a__U46(tt) -> tt 42.17/12.90 a__U46(X) -> U46(X) 42.17/12.90 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.90 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.90 a__U52(tt) -> tt 42.17/12.90 a__U52(X) -> U52(X) 42.17/12.90 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.90 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.90 a__U62(tt) -> tt 42.17/12.90 a__U62(X) -> U62(X) 42.17/12.90 a__U71(tt) -> tt 42.17/12.90 a__U71(X) -> U71(X) 42.17/12.90 a__U81(tt) -> tt 42.17/12.90 a__U81(X) -> U81(X) 42.17/12.90 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.90 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.90 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.90 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.90 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.90 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.90 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.90 a__U96(tt) -> tt 42.17/12.90 a__U96(X) -> U96(X) 42.17/12.90 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.17/12.90 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.90 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.17/12.90 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.17/12.90 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.17/12.90 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.17/12.90 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.17/12.90 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.17/12.90 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.17/12.90 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.17/12.90 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.17/12.90 a__U114(tt, L) -> s(a__length(mark(L))) 42.17/12.90 a__zeros -> cons(0, zeros) 42.17/12.90 a__zeros -> zeros 42.17/12.90 42.17/12.90 42.17/12.90 ---------------------------------------- 42.17/12.90 42.17/12.90 (21) 42.17/12.90 Obligation: 42.17/12.90 Q DP problem: 42.17/12.90 The TRS P consists of the following rules: 42.17/12.90 42.17/12.90 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U105(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U106(X)) -> MARK(X) 42.17/12.90 MARK(U11(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U12(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) 42.17/12.90 A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) 42.17/12.90 A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) 42.17/12.90 A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) 42.17/12.90 A__U114(tt, L) -> A__LENGTH(mark(L)) 42.17/12.90 A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) 42.17/12.90 A__U114(tt, L) -> MARK(L) 42.17/12.90 MARK(U111(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) 42.17/12.90 MARK(U112(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) 42.17/12.90 MARK(U113(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) 42.17/12.90 MARK(U114(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(length(X)) -> A__LENGTH(mark(X)) 42.17/12.90 MARK(length(X)) -> MARK(X) 42.17/12.90 MARK(U13(X)) -> MARK(X) 42.17/12.90 MARK(U131(X1, X2, X3, X4)) -> A__U131(mark(X1), X2, X3, X4) 42.17/12.90 A__U131(tt, IL, M, N) -> A__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.90 A__U132(tt, IL, M, N) -> A__U133(a__isNat(M), IL, M, N) 42.17/12.90 A__U134(tt, IL, M, N) -> A__U135(a__isNat(N), IL, M, N) 42.17/12.90 A__U135(tt, IL, M, N) -> A__U136(a__isNatKind(N), IL, M, N) 42.17/12.90 A__U136(tt, IL, M, N) -> MARK(N) 42.17/12.90 MARK(U132(X1, X2, X3, X4)) -> A__U132(mark(X1), X2, X3, X4) 42.17/12.90 MARK(U133(X1, X2, X3, X4)) -> A__U133(mark(X1), X2, X3, X4) 42.17/12.90 MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) 42.17/12.90 A__TAKE(s(M), cons(N, IL)) -> A__U131(a__isNatIList(IL), IL, M, N) 42.17/12.90 MARK(U21(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U22(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U23(X)) -> MARK(X) 42.17/12.90 MARK(U31(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U32(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U33(X)) -> MARK(X) 42.17/12.90 MARK(U41(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U42(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U43(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U44(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U45(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U46(X)) -> MARK(X) 42.17/12.90 MARK(U51(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U52(X)) -> MARK(X) 42.17/12.90 MARK(U61(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U62(X)) -> MARK(X) 42.17/12.90 MARK(U71(X)) -> MARK(X) 42.17/12.90 MARK(U81(X)) -> MARK(X) 42.17/12.90 MARK(U91(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U92(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U93(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U94(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U95(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U96(X)) -> MARK(X) 42.17/12.90 MARK(cons(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(s(X)) -> MARK(X) 42.17/12.90 42.17/12.90 The TRS R consists of the following rules: 42.17/12.90 42.17/12.90 a__zeros -> cons(0, zeros) 42.17/12.90 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.90 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.90 a__U106(tt) -> tt 42.17/12.90 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.90 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.17/12.90 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.17/12.90 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.17/12.90 a__U114(tt, L) -> s(a__length(mark(L))) 42.17/12.90 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.90 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.17/12.90 a__U122(tt) -> nil 42.17/12.90 a__U13(tt) -> tt 42.17/12.90 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.90 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.17/12.90 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.17/12.90 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.17/12.90 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.17/12.90 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.17/12.90 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.90 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.90 a__U23(tt) -> tt 42.17/12.90 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.90 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.90 a__U33(tt) -> tt 42.17/12.90 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.90 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.90 a__U46(tt) -> tt 42.17/12.90 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.90 a__U52(tt) -> tt 42.17/12.90 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.90 a__U62(tt) -> tt 42.17/12.90 a__U71(tt) -> tt 42.17/12.90 a__U81(tt) -> tt 42.17/12.90 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.90 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.90 a__U96(tt) -> tt 42.17/12.90 a__isNat(0) -> tt 42.17/12.90 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.90 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.90 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.90 a__isNatIList(zeros) -> tt 42.17/12.90 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.90 a__isNatIListKind(nil) -> tt 42.17/12.90 a__isNatIListKind(zeros) -> tt 42.17/12.90 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.90 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.90 a__isNatKind(0) -> tt 42.17/12.90 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.90 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.90 a__isNatList(nil) -> tt 42.17/12.90 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.90 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.90 a__length(nil) -> 0 42.17/12.90 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.17/12.90 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.17/12.90 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.17/12.90 mark(zeros) -> a__zeros 42.17/12.90 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.17/12.90 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.17/12.90 mark(isNatKind(X)) -> a__isNatKind(X) 42.17/12.90 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.17/12.90 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.17/12.90 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.17/12.90 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.17/12.90 mark(isNat(X)) -> a__isNat(X) 42.17/12.90 mark(U106(X)) -> a__U106(mark(X)) 42.17/12.90 mark(isNatIList(X)) -> a__isNatIList(X) 42.17/12.90 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.17/12.90 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.17/12.90 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.17/12.90 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.17/12.90 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.17/12.90 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.17/12.90 mark(length(X)) -> a__length(mark(X)) 42.17/12.90 mark(U13(X)) -> a__U13(mark(X)) 42.17/12.90 mark(isNatList(X)) -> a__isNatList(X) 42.17/12.90 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.17/12.90 mark(U122(X)) -> a__U122(mark(X)) 42.17/12.90 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.17/12.90 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.17/12.90 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.17/12.90 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.17/12.90 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.17/12.90 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.17/12.90 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.17/12.90 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.17/12.90 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.17/12.90 mark(U23(X)) -> a__U23(mark(X)) 42.17/12.90 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.17/12.90 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.17/12.90 mark(U33(X)) -> a__U33(mark(X)) 42.17/12.90 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.17/12.90 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.17/12.90 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.17/12.90 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.17/12.90 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.17/12.90 mark(U46(X)) -> a__U46(mark(X)) 42.17/12.90 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.17/12.90 mark(U52(X)) -> a__U52(mark(X)) 42.17/12.90 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.17/12.90 mark(U62(X)) -> a__U62(mark(X)) 42.17/12.90 mark(U71(X)) -> a__U71(mark(X)) 42.17/12.90 mark(U81(X)) -> a__U81(mark(X)) 42.17/12.90 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.17/12.90 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.17/12.90 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.17/12.90 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.17/12.90 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.17/12.90 mark(U96(X)) -> a__U96(mark(X)) 42.17/12.90 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.17/12.90 mark(0) -> 0 42.17/12.90 mark(tt) -> tt 42.17/12.90 mark(s(X)) -> s(mark(X)) 42.17/12.90 mark(nil) -> nil 42.17/12.90 a__zeros -> zeros 42.17/12.90 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.90 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.90 a__isNatKind(X) -> isNatKind(X) 42.17/12.90 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.90 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.90 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.90 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.90 a__isNat(X) -> isNat(X) 42.17/12.90 a__U106(X) -> U106(X) 42.17/12.90 a__isNatIList(X) -> isNatIList(X) 42.17/12.90 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.90 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.90 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.17/12.90 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.17/12.90 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.17/12.90 a__U114(X1, X2) -> U114(X1, X2) 42.17/12.90 a__length(X) -> length(X) 42.17/12.90 a__U13(X) -> U13(X) 42.17/12.90 a__isNatList(X) -> isNatList(X) 42.17/12.90 a__U121(X1, X2) -> U121(X1, X2) 42.17/12.90 a__U122(X) -> U122(X) 42.17/12.90 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.17/12.90 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.17/12.90 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.17/12.90 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.17/12.90 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.17/12.90 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.17/12.90 a__take(X1, X2) -> take(X1, X2) 42.17/12.90 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.90 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.90 a__U23(X) -> U23(X) 42.17/12.90 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.90 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.90 a__U33(X) -> U33(X) 42.17/12.90 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.90 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.90 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.90 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.90 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.90 a__U46(X) -> U46(X) 42.17/12.90 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.90 a__U52(X) -> U52(X) 42.17/12.90 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.90 a__U62(X) -> U62(X) 42.17/12.90 a__U71(X) -> U71(X) 42.17/12.90 a__U81(X) -> U81(X) 42.17/12.90 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.90 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.90 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.90 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.90 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.90 a__U96(X) -> U96(X) 42.17/12.90 42.17/12.90 The set Q consists of the following terms: 42.17/12.90 42.17/12.90 a__zeros 42.17/12.90 a__isNatIList(x0) 42.17/12.90 mark(zeros) 42.17/12.90 mark(U101(x0, x1, x2)) 42.17/12.90 mark(U102(x0, x1, x2)) 42.17/12.90 mark(isNatKind(x0)) 42.17/12.90 mark(U103(x0, x1, x2)) 42.17/12.90 mark(isNatIListKind(x0)) 42.17/12.90 mark(U104(x0, x1, x2)) 42.17/12.90 mark(U105(x0, x1)) 42.17/12.90 mark(isNat(x0)) 42.17/12.90 mark(U106(x0)) 42.17/12.90 mark(isNatIList(x0)) 42.17/12.90 mark(U11(x0, x1)) 42.17/12.90 mark(U12(x0, x1)) 42.17/12.90 mark(U111(x0, x1, x2)) 42.17/12.90 mark(U112(x0, x1, x2)) 42.17/12.90 mark(U113(x0, x1, x2)) 42.17/12.90 mark(U114(x0, x1)) 42.17/12.90 mark(length(x0)) 42.17/12.90 mark(U13(x0)) 42.17/12.90 mark(isNatList(x0)) 42.17/12.90 mark(U121(x0, x1)) 42.17/12.90 mark(U122(x0)) 42.17/12.90 mark(U131(x0, x1, x2, x3)) 42.17/12.90 mark(U132(x0, x1, x2, x3)) 42.17/12.90 mark(U133(x0, x1, x2, x3)) 42.17/12.90 mark(U134(x0, x1, x2, x3)) 42.17/12.90 mark(U135(x0, x1, x2, x3)) 42.17/12.90 mark(U136(x0, x1, x2, x3)) 42.17/12.90 mark(take(x0, x1)) 42.17/12.90 mark(U21(x0, x1)) 42.17/12.90 mark(U22(x0, x1)) 42.17/12.90 mark(U23(x0)) 42.17/12.90 mark(U31(x0, x1)) 42.17/12.90 mark(U32(x0, x1)) 42.17/12.90 mark(U33(x0)) 42.17/12.90 mark(U41(x0, x1, x2)) 42.17/12.90 mark(U42(x0, x1, x2)) 42.17/12.90 mark(U43(x0, x1, x2)) 42.17/12.90 mark(U44(x0, x1, x2)) 42.17/12.90 mark(U45(x0, x1)) 42.17/12.90 mark(U46(x0)) 42.17/12.90 mark(U51(x0, x1)) 42.17/12.90 mark(U52(x0)) 42.17/12.90 mark(U61(x0, x1)) 42.17/12.90 mark(U62(x0)) 42.17/12.90 mark(U71(x0)) 42.17/12.90 mark(U81(x0)) 42.17/12.90 mark(U91(x0, x1, x2)) 42.17/12.90 mark(U92(x0, x1, x2)) 42.17/12.90 mark(U93(x0, x1, x2)) 42.17/12.90 mark(U94(x0, x1, x2)) 42.17/12.90 mark(U95(x0, x1)) 42.17/12.90 mark(U96(x0)) 42.17/12.90 mark(cons(x0, x1)) 42.17/12.90 mark(0) 42.17/12.90 mark(tt) 42.17/12.90 mark(s(x0)) 42.17/12.90 mark(nil) 42.17/12.90 a__U101(x0, x1, x2) 42.17/12.90 a__U102(x0, x1, x2) 42.17/12.90 a__isNatKind(x0) 42.17/12.90 a__U103(x0, x1, x2) 42.17/12.90 a__isNatIListKind(x0) 42.17/12.90 a__U104(x0, x1, x2) 42.17/12.90 a__U105(x0, x1) 42.17/12.90 a__isNat(x0) 42.17/12.90 a__U106(x0) 42.17/12.90 a__U11(x0, x1) 42.17/12.90 a__U12(x0, x1) 42.17/12.90 a__U111(x0, x1, x2) 42.17/12.90 a__U112(x0, x1, x2) 42.17/12.90 a__U113(x0, x1, x2) 42.17/12.90 a__U114(x0, x1) 42.17/12.90 a__length(x0) 42.17/12.90 a__U13(x0) 42.17/12.90 a__isNatList(x0) 42.17/12.90 a__U121(x0, x1) 42.17/12.90 a__U122(x0) 42.17/12.90 a__U131(x0, x1, x2, x3) 42.17/12.90 a__U132(x0, x1, x2, x3) 42.17/12.90 a__U133(x0, x1, x2, x3) 42.17/12.90 a__U134(x0, x1, x2, x3) 42.17/12.90 a__U135(x0, x1, x2, x3) 42.17/12.90 a__U136(x0, x1, x2, x3) 42.17/12.90 a__take(x0, x1) 42.17/12.90 a__U21(x0, x1) 42.17/12.90 a__U22(x0, x1) 42.17/12.90 a__U23(x0) 42.17/12.90 a__U31(x0, x1) 42.17/12.90 a__U32(x0, x1) 42.17/12.90 a__U33(x0) 42.17/12.90 a__U41(x0, x1, x2) 42.17/12.90 a__U42(x0, x1, x2) 42.17/12.90 a__U43(x0, x1, x2) 42.17/12.90 a__U44(x0, x1, x2) 42.17/12.90 a__U45(x0, x1) 42.17/12.90 a__U46(x0) 42.17/12.90 a__U51(x0, x1) 42.17/12.90 a__U52(x0) 42.17/12.90 a__U61(x0, x1) 42.17/12.90 a__U62(x0) 42.17/12.90 a__U71(x0) 42.17/12.90 a__U81(x0) 42.17/12.90 a__U91(x0, x1, x2) 42.17/12.90 a__U92(x0, x1, x2) 42.17/12.90 a__U93(x0, x1, x2) 42.17/12.90 a__U94(x0, x1, x2) 42.17/12.90 a__U95(x0, x1) 42.17/12.90 a__U96(x0) 42.17/12.90 42.17/12.90 We have to consider all minimal (P,Q,R)-chains. 42.17/12.90 ---------------------------------------- 42.17/12.90 42.17/12.90 (22) DependencyGraphProof (EQUIVALENT) 42.17/12.90 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes. 42.17/12.90 ---------------------------------------- 42.17/12.90 42.17/12.90 (23) 42.17/12.90 Obligation: 42.17/12.90 Q DP problem: 42.17/12.90 The TRS P consists of the following rules: 42.17/12.90 42.17/12.90 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U105(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U106(X)) -> MARK(X) 42.17/12.90 MARK(U11(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U12(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) 42.17/12.90 A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) 42.17/12.90 A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) 42.17/12.90 A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) 42.17/12.90 A__U114(tt, L) -> A__LENGTH(mark(L)) 42.17/12.90 A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) 42.17/12.90 A__U114(tt, L) -> MARK(L) 42.17/12.90 MARK(U111(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) 42.17/12.90 MARK(U112(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) 42.17/12.90 MARK(U113(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) 42.17/12.90 MARK(U114(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(length(X)) -> A__LENGTH(mark(X)) 42.17/12.90 MARK(length(X)) -> MARK(X) 42.17/12.90 MARK(U13(X)) -> MARK(X) 42.17/12.90 MARK(U21(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U22(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U23(X)) -> MARK(X) 42.17/12.90 MARK(U31(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U32(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U33(X)) -> MARK(X) 42.17/12.90 MARK(U41(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U42(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U43(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U44(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U45(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U46(X)) -> MARK(X) 42.17/12.90 MARK(U51(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U52(X)) -> MARK(X) 42.17/12.90 MARK(U61(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U62(X)) -> MARK(X) 42.17/12.90 MARK(U71(X)) -> MARK(X) 42.17/12.90 MARK(U81(X)) -> MARK(X) 42.17/12.90 MARK(U91(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U92(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U93(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U94(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U95(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(U96(X)) -> MARK(X) 42.17/12.90 MARK(cons(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(s(X)) -> MARK(X) 42.17/12.90 42.17/12.90 The TRS R consists of the following rules: 42.17/12.90 42.17/12.90 a__zeros -> cons(0, zeros) 42.17/12.90 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.17/12.90 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.17/12.90 a__U106(tt) -> tt 42.17/12.90 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.17/12.90 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.17/12.90 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.17/12.90 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.17/12.90 a__U114(tt, L) -> s(a__length(mark(L))) 42.17/12.90 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.17/12.90 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.17/12.90 a__U122(tt) -> nil 42.17/12.90 a__U13(tt) -> tt 42.17/12.90 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.17/12.90 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.17/12.90 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.17/12.90 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.17/12.90 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.17/12.90 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.17/12.90 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.17/12.90 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.17/12.90 a__U23(tt) -> tt 42.17/12.90 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.17/12.90 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.17/12.90 a__U33(tt) -> tt 42.17/12.90 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.17/12.90 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.17/12.90 a__U46(tt) -> tt 42.17/12.90 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.17/12.90 a__U52(tt) -> tt 42.17/12.90 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.17/12.90 a__U62(tt) -> tt 42.17/12.90 a__U71(tt) -> tt 42.17/12.90 a__U81(tt) -> tt 42.17/12.90 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.17/12.90 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.17/12.90 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.17/12.90 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.17/12.90 a__U96(tt) -> tt 42.17/12.90 a__isNat(0) -> tt 42.17/12.90 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.17/12.90 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.17/12.90 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.17/12.90 a__isNatIList(zeros) -> tt 42.17/12.90 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.17/12.90 a__isNatIListKind(nil) -> tt 42.17/12.90 a__isNatIListKind(zeros) -> tt 42.17/12.90 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.17/12.90 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.17/12.90 a__isNatKind(0) -> tt 42.17/12.90 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.17/12.90 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.17/12.90 a__isNatList(nil) -> tt 42.17/12.90 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.17/12.90 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.17/12.90 a__length(nil) -> 0 42.17/12.90 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.17/12.90 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.17/12.90 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.17/12.90 mark(zeros) -> a__zeros 42.17/12.90 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.17/12.90 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.17/12.90 mark(isNatKind(X)) -> a__isNatKind(X) 42.17/12.90 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.17/12.90 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.17/12.90 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.17/12.90 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.17/12.90 mark(isNat(X)) -> a__isNat(X) 42.17/12.90 mark(U106(X)) -> a__U106(mark(X)) 42.17/12.90 mark(isNatIList(X)) -> a__isNatIList(X) 42.17/12.90 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.17/12.90 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.17/12.90 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.17/12.90 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.17/12.90 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.17/12.90 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.17/12.90 mark(length(X)) -> a__length(mark(X)) 42.17/12.90 mark(U13(X)) -> a__U13(mark(X)) 42.17/12.90 mark(isNatList(X)) -> a__isNatList(X) 42.17/12.90 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.17/12.90 mark(U122(X)) -> a__U122(mark(X)) 42.17/12.90 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.17/12.90 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.17/12.90 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.17/12.90 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.17/12.90 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.17/12.90 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.17/12.90 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.17/12.90 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.17/12.90 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.17/12.90 mark(U23(X)) -> a__U23(mark(X)) 42.17/12.90 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.17/12.90 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.17/12.90 mark(U33(X)) -> a__U33(mark(X)) 42.17/12.90 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.17/12.90 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.17/12.90 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.17/12.90 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.17/12.90 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.17/12.90 mark(U46(X)) -> a__U46(mark(X)) 42.17/12.90 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.17/12.90 mark(U52(X)) -> a__U52(mark(X)) 42.17/12.90 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.17/12.90 mark(U62(X)) -> a__U62(mark(X)) 42.17/12.90 mark(U71(X)) -> a__U71(mark(X)) 42.17/12.90 mark(U81(X)) -> a__U81(mark(X)) 42.17/12.90 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.17/12.90 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.17/12.90 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.17/12.90 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.17/12.90 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.17/12.90 mark(U96(X)) -> a__U96(mark(X)) 42.17/12.90 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.17/12.90 mark(0) -> 0 42.17/12.90 mark(tt) -> tt 42.17/12.90 mark(s(X)) -> s(mark(X)) 42.17/12.90 mark(nil) -> nil 42.17/12.90 a__zeros -> zeros 42.17/12.90 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.17/12.90 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.17/12.90 a__isNatKind(X) -> isNatKind(X) 42.17/12.90 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.17/12.90 a__isNatIListKind(X) -> isNatIListKind(X) 42.17/12.90 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.17/12.90 a__U105(X1, X2) -> U105(X1, X2) 42.17/12.90 a__isNat(X) -> isNat(X) 42.17/12.90 a__U106(X) -> U106(X) 42.17/12.90 a__isNatIList(X) -> isNatIList(X) 42.17/12.90 a__U11(X1, X2) -> U11(X1, X2) 42.17/12.90 a__U12(X1, X2) -> U12(X1, X2) 42.17/12.90 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.17/12.90 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.17/12.90 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.17/12.90 a__U114(X1, X2) -> U114(X1, X2) 42.17/12.90 a__length(X) -> length(X) 42.17/12.90 a__U13(X) -> U13(X) 42.17/12.90 a__isNatList(X) -> isNatList(X) 42.17/12.90 a__U121(X1, X2) -> U121(X1, X2) 42.17/12.90 a__U122(X) -> U122(X) 42.17/12.90 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.17/12.90 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.17/12.90 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.17/12.90 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.17/12.90 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.17/12.90 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.17/12.90 a__take(X1, X2) -> take(X1, X2) 42.17/12.90 a__U21(X1, X2) -> U21(X1, X2) 42.17/12.90 a__U22(X1, X2) -> U22(X1, X2) 42.17/12.90 a__U23(X) -> U23(X) 42.17/12.90 a__U31(X1, X2) -> U31(X1, X2) 42.17/12.90 a__U32(X1, X2) -> U32(X1, X2) 42.17/12.90 a__U33(X) -> U33(X) 42.17/12.90 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.17/12.90 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.17/12.90 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.17/12.90 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.17/12.90 a__U45(X1, X2) -> U45(X1, X2) 42.17/12.90 a__U46(X) -> U46(X) 42.17/12.90 a__U51(X1, X2) -> U51(X1, X2) 42.17/12.90 a__U52(X) -> U52(X) 42.17/12.90 a__U61(X1, X2) -> U61(X1, X2) 42.17/12.90 a__U62(X) -> U62(X) 42.17/12.90 a__U71(X) -> U71(X) 42.17/12.90 a__U81(X) -> U81(X) 42.17/12.90 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.17/12.90 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.17/12.90 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.17/12.90 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.17/12.90 a__U95(X1, X2) -> U95(X1, X2) 42.17/12.90 a__U96(X) -> U96(X) 42.17/12.90 42.17/12.90 The set Q consists of the following terms: 42.17/12.90 42.17/12.90 a__zeros 42.17/12.90 a__isNatIList(x0) 42.17/12.90 mark(zeros) 42.17/12.90 mark(U101(x0, x1, x2)) 42.17/12.90 mark(U102(x0, x1, x2)) 42.17/12.90 mark(isNatKind(x0)) 42.17/12.90 mark(U103(x0, x1, x2)) 42.17/12.90 mark(isNatIListKind(x0)) 42.17/12.90 mark(U104(x0, x1, x2)) 42.17/12.90 mark(U105(x0, x1)) 42.17/12.90 mark(isNat(x0)) 42.17/12.90 mark(U106(x0)) 42.17/12.90 mark(isNatIList(x0)) 42.17/12.90 mark(U11(x0, x1)) 42.17/12.90 mark(U12(x0, x1)) 42.17/12.90 mark(U111(x0, x1, x2)) 42.17/12.90 mark(U112(x0, x1, x2)) 42.17/12.90 mark(U113(x0, x1, x2)) 42.17/12.90 mark(U114(x0, x1)) 42.17/12.90 mark(length(x0)) 42.17/12.90 mark(U13(x0)) 42.17/12.90 mark(isNatList(x0)) 42.17/12.90 mark(U121(x0, x1)) 42.17/12.90 mark(U122(x0)) 42.17/12.90 mark(U131(x0, x1, x2, x3)) 42.17/12.90 mark(U132(x0, x1, x2, x3)) 42.17/12.90 mark(U133(x0, x1, x2, x3)) 42.17/12.90 mark(U134(x0, x1, x2, x3)) 42.17/12.90 mark(U135(x0, x1, x2, x3)) 42.17/12.90 mark(U136(x0, x1, x2, x3)) 42.17/12.90 mark(take(x0, x1)) 42.17/12.90 mark(U21(x0, x1)) 42.17/12.90 mark(U22(x0, x1)) 42.17/12.90 mark(U23(x0)) 42.17/12.90 mark(U31(x0, x1)) 42.17/12.90 mark(U32(x0, x1)) 42.17/12.90 mark(U33(x0)) 42.17/12.90 mark(U41(x0, x1, x2)) 42.17/12.90 mark(U42(x0, x1, x2)) 42.17/12.90 mark(U43(x0, x1, x2)) 42.17/12.90 mark(U44(x0, x1, x2)) 42.17/12.90 mark(U45(x0, x1)) 42.17/12.90 mark(U46(x0)) 42.17/12.90 mark(U51(x0, x1)) 42.17/12.90 mark(U52(x0)) 42.17/12.90 mark(U61(x0, x1)) 42.17/12.90 mark(U62(x0)) 42.17/12.90 mark(U71(x0)) 42.17/12.90 mark(U81(x0)) 42.17/12.90 mark(U91(x0, x1, x2)) 42.17/12.90 mark(U92(x0, x1, x2)) 42.17/12.90 mark(U93(x0, x1, x2)) 42.17/12.90 mark(U94(x0, x1, x2)) 42.17/12.90 mark(U95(x0, x1)) 42.17/12.90 mark(U96(x0)) 42.17/12.90 mark(cons(x0, x1)) 42.17/12.90 mark(0) 42.17/12.90 mark(tt) 42.17/12.90 mark(s(x0)) 42.17/12.90 mark(nil) 42.17/12.90 a__U101(x0, x1, x2) 42.17/12.90 a__U102(x0, x1, x2) 42.17/12.90 a__isNatKind(x0) 42.17/12.90 a__U103(x0, x1, x2) 42.17/12.90 a__isNatIListKind(x0) 42.17/12.90 a__U104(x0, x1, x2) 42.17/12.90 a__U105(x0, x1) 42.17/12.90 a__isNat(x0) 42.17/12.90 a__U106(x0) 42.17/12.90 a__U11(x0, x1) 42.17/12.90 a__U12(x0, x1) 42.17/12.90 a__U111(x0, x1, x2) 42.17/12.90 a__U112(x0, x1, x2) 42.17/12.90 a__U113(x0, x1, x2) 42.17/12.90 a__U114(x0, x1) 42.17/12.90 a__length(x0) 42.17/12.90 a__U13(x0) 42.17/12.90 a__isNatList(x0) 42.17/12.90 a__U121(x0, x1) 42.17/12.90 a__U122(x0) 42.17/12.90 a__U131(x0, x1, x2, x3) 42.17/12.90 a__U132(x0, x1, x2, x3) 42.17/12.90 a__U133(x0, x1, x2, x3) 42.17/12.90 a__U134(x0, x1, x2, x3) 42.17/12.90 a__U135(x0, x1, x2, x3) 42.17/12.90 a__U136(x0, x1, x2, x3) 42.17/12.90 a__take(x0, x1) 42.17/12.90 a__U21(x0, x1) 42.17/12.90 a__U22(x0, x1) 42.17/12.90 a__U23(x0) 42.17/12.90 a__U31(x0, x1) 42.17/12.90 a__U32(x0, x1) 42.17/12.90 a__U33(x0) 42.17/12.90 a__U41(x0, x1, x2) 42.17/12.90 a__U42(x0, x1, x2) 42.17/12.90 a__U43(x0, x1, x2) 42.17/12.90 a__U44(x0, x1, x2) 42.17/12.90 a__U45(x0, x1) 42.17/12.90 a__U46(x0) 42.17/12.90 a__U51(x0, x1) 42.17/12.90 a__U52(x0) 42.17/12.90 a__U61(x0, x1) 42.17/12.90 a__U62(x0) 42.17/12.90 a__U71(x0) 42.17/12.90 a__U81(x0) 42.17/12.90 a__U91(x0, x1, x2) 42.17/12.90 a__U92(x0, x1, x2) 42.17/12.90 a__U93(x0, x1, x2) 42.17/12.90 a__U94(x0, x1, x2) 42.17/12.90 a__U95(x0, x1) 42.17/12.90 a__U96(x0) 42.17/12.90 42.17/12.90 We have to consider all minimal (P,Q,R)-chains. 42.17/12.90 ---------------------------------------- 42.17/12.90 42.17/12.90 (24) QDPOrderProof (EQUIVALENT) 42.17/12.90 We use the reduction pair processor [LPAR04,JAR06]. 42.17/12.90 42.17/12.90 42.17/12.90 The following pairs can be oriented strictly and are deleted. 42.17/12.90 42.17/12.90 MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) 42.17/12.90 MARK(U111(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) 42.17/12.90 MARK(U112(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) 42.17/12.90 MARK(U113(X1, X2, X3)) -> MARK(X1) 42.17/12.90 MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) 42.17/12.90 MARK(U114(X1, X2)) -> MARK(X1) 42.17/12.90 MARK(length(X)) -> A__LENGTH(mark(X)) 42.17/12.90 MARK(length(X)) -> MARK(X) 42.17/12.90 The remaining pairs can at least be oriented weakly. 42.17/12.90 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 42.17/12.90 42.17/12.90 POL( A__LENGTH_1(x_1) ) = x_1 42.17/12.90 POL( A__U111_3(x_1, ..., x_3) ) = x_2 + 2x_3 42.17/12.90 POL( A__U112_3(x_1, ..., x_3) ) = x_2 42.17/12.90 POL( A__U113_3(x_1, ..., x_3) ) = x_2 42.17/12.90 POL( A__U114_2(x_1, x_2) ) = x_2 42.17/12.90 POL( mark_1(x_1) ) = x_1 42.17/12.90 POL( zeros ) = 0 42.17/12.90 POL( a__zeros ) = 0 42.17/12.90 POL( U101_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U101_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U102_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U102_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( isNatKind_1(x_1) ) = 0 42.17/12.90 POL( a__isNatKind_1(x_1) ) = 0 42.17/12.90 POL( U103_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U103_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( isNatIListKind_1(x_1) ) = 0 42.17/12.90 POL( a__isNatIListKind_1(x_1) ) = 0 42.17/12.90 POL( U104_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U104_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U105_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U105_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( isNat_1(x_1) ) = 0 42.17/12.90 POL( a__isNat_1(x_1) ) = 0 42.17/12.90 POL( U106_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U106_1(x_1) ) = 2x_1 42.17/12.90 POL( isNatIList_1(x_1) ) = 0 42.17/12.90 POL( a__isNatIList_1(x_1) ) = 0 42.17/12.90 POL( U11_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U11_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U12_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U12_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U111_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2 42.17/12.90 POL( a__U111_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2 42.17/12.90 POL( U112_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 42.17/12.90 POL( a__U112_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 42.17/12.90 POL( U113_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 42.17/12.90 POL( a__U113_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 42.17/12.90 POL( U114_2(x_1, x_2) ) = x_1 + x_2 + 2 42.17/12.90 POL( a__U114_2(x_1, x_2) ) = x_1 + x_2 + 2 42.17/12.90 POL( length_1(x_1) ) = x_1 + 2 42.17/12.90 POL( a__length_1(x_1) ) = x_1 + 2 42.17/12.90 POL( U13_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U13_1(x_1) ) = 2x_1 42.17/12.90 POL( isNatList_1(x_1) ) = 0 42.17/12.90 POL( a__isNatList_1(x_1) ) = 0 42.17/12.90 POL( U121_2(x_1, x_2) ) = 0 42.17/12.90 POL( a__U121_2(x_1, x_2) ) = max{0, -2} 42.17/12.90 POL( U122_1(x_1) ) = 0 42.17/12.90 POL( a__U122_1(x_1) ) = max{0, -2} 42.17/12.90 POL( U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( a__U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( a__U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( U133_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( a__U133_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( U134_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( a__U134_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( U135_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( a__U135_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( U136_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( a__U136_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 42.17/12.90 POL( take_2(x_1, x_2) ) = x_2 + 2 42.17/12.90 POL( a__take_2(x_1, x_2) ) = x_2 + 2 42.17/12.90 POL( U21_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U21_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U22_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U22_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( U23_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U23_1(x_1) ) = 2x_1 42.17/12.90 POL( U31_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U31_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U32_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U32_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( U33_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U33_1(x_1) ) = 2x_1 42.17/12.90 POL( U41_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U41_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U42_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U42_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U43_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U43_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U44_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U44_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U45_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U45_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U46_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U46_1(x_1) ) = 2x_1 42.17/12.90 POL( U51_2(x_1, x_2) ) = x_1 42.17/12.90 POL( a__U51_2(x_1, x_2) ) = x_1 42.17/12.90 POL( U52_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U52_1(x_1) ) = 2x_1 42.17/12.90 POL( U61_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U61_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( U62_1(x_1) ) = x_1 42.17/12.90 POL( a__U62_1(x_1) ) = x_1 42.17/12.90 POL( U71_1(x_1) ) = x_1 42.17/12.90 POL( a__U71_1(x_1) ) = x_1 42.17/12.90 POL( U81_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U81_1(x_1) ) = 2x_1 42.17/12.90 POL( U91_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U91_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U92_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( a__U92_3(x_1, ..., x_3) ) = 2x_1 42.17/12.90 POL( U93_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U93_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U94_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( a__U94_3(x_1, ..., x_3) ) = x_1 42.17/12.90 POL( U95_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( a__U95_2(x_1, x_2) ) = 2x_1 42.17/12.90 POL( U96_1(x_1) ) = 2x_1 42.17/12.90 POL( a__U96_1(x_1) ) = 2x_1 42.17/12.90 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 42.17/12.90 POL( 0 ) = 0 42.17/12.90 POL( tt ) = 0 42.17/12.90 POL( s_1(x_1) ) = x_1 42.17/12.90 POL( nil ) = 0 42.17/12.90 POL( MARK_1(x_1) ) = x_1 42.17/12.90 42.17/12.90 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.17/12.90 42.17/12.90 mark(zeros) -> a__zeros 42.17/12.90 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.17/12.90 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.17/12.90 mark(isNatKind(X)) -> a__isNatKind(X) 42.17/12.90 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.17/12.90 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.17/12.90 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.17/12.90 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.17/12.90 mark(isNat(X)) -> a__isNat(X) 42.17/12.90 mark(U106(X)) -> a__U106(mark(X)) 42.17/12.90 mark(isNatIList(X)) -> a__isNatIList(X) 42.17/12.90 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.17/12.90 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.17/12.90 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.17/12.90 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.17/12.90 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.17/12.90 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.17/12.90 mark(length(X)) -> a__length(mark(X)) 42.17/12.90 mark(U13(X)) -> a__U13(mark(X)) 42.17/12.90 mark(isNatList(X)) -> a__isNatList(X) 42.17/12.90 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.17/12.90 mark(U122(X)) -> a__U122(mark(X)) 42.17/12.90 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.17/12.90 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.17/12.90 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.17/12.90 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.17/12.90 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.17/12.90 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.17/12.90 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.17/12.90 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.17/12.90 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.17/12.90 mark(U23(X)) -> a__U23(mark(X)) 42.17/12.90 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.17/12.90 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.17/12.90 mark(U33(X)) -> a__U33(mark(X)) 42.17/12.90 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.17/12.90 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.17/12.90 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.17/12.90 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.17/12.90 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.17/12.90 mark(U46(X)) -> a__U46(mark(X)) 42.17/12.90 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.17/12.90 mark(U52(X)) -> a__U52(mark(X)) 42.17/12.90 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.17/12.90 mark(U62(X)) -> a__U62(mark(X)) 42.17/12.90 mark(U71(X)) -> a__U71(mark(X)) 42.17/12.90 mark(U81(X)) -> a__U81(mark(X)) 42.17/12.90 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.17/12.90 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.17/12.90 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.17/12.90 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.17/12.90 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.17/12.90 mark(U96(X)) -> a__U96(mark(X)) 42.27/12.90 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.27/12.90 mark(0) -> 0 42.27/12.90 mark(tt) -> tt 42.27/12.90 mark(s(X)) -> s(mark(X)) 42.27/12.90 mark(nil) -> nil 42.27/12.90 a__isNatIListKind(nil) -> tt 42.27/12.90 a__isNatIListKind(zeros) -> tt 42.27/12.90 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.27/12.90 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.27/12.90 a__isNatIListKind(X) -> isNatIListKind(X) 42.27/12.90 a__isNat(0) -> tt 42.27/12.90 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.27/12.90 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.27/12.90 a__isNat(X) -> isNat(X) 42.27/12.90 a__isNatKind(0) -> tt 42.27/12.90 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.27/12.90 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.27/12.90 a__isNatKind(X) -> isNatKind(X) 42.27/12.90 a__isNatList(nil) -> tt 42.27/12.90 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.27/12.90 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.27/12.90 a__isNatList(X) -> isNatList(X) 42.27/12.90 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.27/12.90 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.27/12.90 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.27/12.90 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.27/12.90 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.27/12.90 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.27/12.90 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.27/12.90 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.27/12.90 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.27/12.90 a__U105(X1, X2) -> U105(X1, X2) 42.27/12.90 a__U106(tt) -> tt 42.27/12.90 a__U106(X) -> U106(X) 42.27/12.90 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.27/12.90 a__U11(X1, X2) -> U11(X1, X2) 42.27/12.90 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.27/12.90 a__U12(X1, X2) -> U12(X1, X2) 42.27/12.90 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.27/12.90 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.27/12.90 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.27/12.90 a__U114(X1, X2) -> U114(X1, X2) 42.27/12.90 a__length(nil) -> 0 42.27/12.90 a__length(X) -> length(X) 42.27/12.90 a__U13(tt) -> tt 42.27/12.90 a__U13(X) -> U13(X) 42.27/12.90 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.27/12.90 a__U121(X1, X2) -> U121(X1, X2) 42.27/12.90 a__U122(tt) -> nil 42.27/12.90 a__U122(X) -> U122(X) 42.27/12.90 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.27/12.90 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.27/12.90 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.27/12.90 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.27/12.90 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.27/12.90 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.27/12.90 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.27/12.90 a__take(X1, X2) -> take(X1, X2) 42.27/12.90 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.27/12.90 a__U21(X1, X2) -> U21(X1, X2) 42.27/12.90 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.27/12.90 a__U22(X1, X2) -> U22(X1, X2) 42.27/12.90 a__U23(tt) -> tt 42.27/12.90 a__U23(X) -> U23(X) 42.27/12.90 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.27/12.90 a__U31(X1, X2) -> U31(X1, X2) 42.27/12.90 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.27/12.90 a__U32(X1, X2) -> U32(X1, X2) 42.27/12.90 a__U33(tt) -> tt 42.27/12.90 a__U33(X) -> U33(X) 42.27/12.90 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.27/12.90 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.27/12.90 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.27/12.90 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.27/12.90 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.27/12.90 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.27/12.90 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.27/12.90 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.27/12.90 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.27/12.90 a__U45(X1, X2) -> U45(X1, X2) 42.27/12.90 a__U46(tt) -> tt 42.27/12.90 a__U46(X) -> U46(X) 42.27/12.90 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.27/12.90 a__U51(X1, X2) -> U51(X1, X2) 42.27/12.90 a__U52(tt) -> tt 42.27/12.90 a__U52(X) -> U52(X) 42.27/12.90 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.27/12.90 a__U61(X1, X2) -> U61(X1, X2) 42.27/12.90 a__U62(tt) -> tt 42.27/12.90 a__U62(X) -> U62(X) 42.27/12.90 a__U71(tt) -> tt 42.27/12.90 a__U71(X) -> U71(X) 42.27/12.90 a__U81(tt) -> tt 42.27/12.90 a__U81(X) -> U81(X) 42.27/12.90 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.27/12.90 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.27/12.90 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.27/12.90 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.27/12.90 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.27/12.90 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.27/12.90 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.27/12.90 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.27/12.90 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.27/12.90 a__U95(X1, X2) -> U95(X1, X2) 42.27/12.90 a__U96(tt) -> tt 42.27/12.90 a__U96(X) -> U96(X) 42.27/12.90 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.27/12.90 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.27/12.90 a__isNatIList(zeros) -> tt 42.27/12.90 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.27/12.90 a__isNatIList(X) -> isNatIList(X) 42.27/12.90 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.27/12.90 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.27/12.90 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.27/12.90 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.27/12.90 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.27/12.90 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.27/12.90 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.27/12.90 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.27/12.90 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.27/12.90 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.27/12.90 a__U114(tt, L) -> s(a__length(mark(L))) 42.27/12.90 a__zeros -> cons(0, zeros) 42.27/12.90 a__zeros -> zeros 42.27/12.90 42.27/12.90 42.27/12.90 ---------------------------------------- 42.27/12.90 42.27/12.90 (25) 42.27/12.90 Obligation: 42.27/12.90 Q DP problem: 42.27/12.90 The TRS P consists of the following rules: 42.27/12.90 42.27/12.90 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.27/12.90 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.27/12.90 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.27/12.90 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.27/12.90 MARK(U105(X1, X2)) -> MARK(X1) 42.27/12.90 MARK(U106(X)) -> MARK(X) 42.27/12.90 MARK(U11(X1, X2)) -> MARK(X1) 42.27/12.90 MARK(U12(X1, X2)) -> MARK(X1) 42.27/12.90 A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) 42.27/12.90 A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) 42.27/12.90 A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) 42.27/12.90 A__U114(tt, L) -> A__LENGTH(mark(L)) 42.27/12.90 A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) 42.27/12.90 A__U114(tt, L) -> MARK(L) 42.27/12.90 MARK(U13(X)) -> MARK(X) 42.27/12.90 MARK(U21(X1, X2)) -> MARK(X1) 42.27/12.90 MARK(U22(X1, X2)) -> MARK(X1) 42.27/12.90 MARK(U23(X)) -> MARK(X) 42.27/12.90 MARK(U31(X1, X2)) -> MARK(X1) 42.27/12.90 MARK(U32(X1, X2)) -> MARK(X1) 42.27/12.90 MARK(U33(X)) -> MARK(X) 42.27/12.90 MARK(U41(X1, X2, X3)) -> MARK(X1) 42.27/12.90 MARK(U42(X1, X2, X3)) -> MARK(X1) 42.27/12.90 MARK(U43(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U44(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U45(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U46(X)) -> MARK(X) 42.27/12.91 MARK(U51(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U52(X)) -> MARK(X) 42.27/12.91 MARK(U61(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U62(X)) -> MARK(X) 42.27/12.91 MARK(U71(X)) -> MARK(X) 42.27/12.91 MARK(U81(X)) -> MARK(X) 42.27/12.91 MARK(U91(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U92(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U93(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U94(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U95(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U96(X)) -> MARK(X) 42.27/12.91 MARK(cons(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(s(X)) -> MARK(X) 42.27/12.91 42.27/12.91 The TRS R consists of the following rules: 42.27/12.91 42.27/12.91 a__zeros -> cons(0, zeros) 42.27/12.91 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.27/12.91 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.27/12.91 a__U106(tt) -> tt 42.27/12.91 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.27/12.91 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.27/12.91 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.27/12.91 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.27/12.91 a__U114(tt, L) -> s(a__length(mark(L))) 42.27/12.91 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.27/12.91 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.27/12.91 a__U122(tt) -> nil 42.27/12.91 a__U13(tt) -> tt 42.27/12.91 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.27/12.91 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.27/12.91 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.27/12.91 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.27/12.91 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.27/12.91 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.27/12.91 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.27/12.91 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.27/12.91 a__U23(tt) -> tt 42.27/12.91 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.27/12.91 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.27/12.91 a__U33(tt) -> tt 42.27/12.91 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.27/12.91 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.27/12.91 a__U46(tt) -> tt 42.27/12.91 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.27/12.91 a__U52(tt) -> tt 42.27/12.91 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.27/12.91 a__U62(tt) -> tt 42.27/12.91 a__U71(tt) -> tt 42.27/12.91 a__U81(tt) -> tt 42.27/12.91 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.27/12.91 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.27/12.91 a__U96(tt) -> tt 42.27/12.91 a__isNat(0) -> tt 42.27/12.91 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.27/12.91 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.27/12.91 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.27/12.91 a__isNatIList(zeros) -> tt 42.27/12.91 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatIListKind(nil) -> tt 42.27/12.91 a__isNatIListKind(zeros) -> tt 42.27/12.91 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.27/12.91 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.27/12.91 a__isNatKind(0) -> tt 42.27/12.91 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.27/12.91 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.27/12.91 a__isNatList(nil) -> tt 42.27/12.91 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.27/12.91 a__length(nil) -> 0 42.27/12.91 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.27/12.91 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.27/12.91 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.27/12.91 mark(zeros) -> a__zeros 42.27/12.91 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.27/12.91 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.27/12.91 mark(isNatKind(X)) -> a__isNatKind(X) 42.27/12.91 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.27/12.91 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.27/12.91 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.27/12.91 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.27/12.91 mark(isNat(X)) -> a__isNat(X) 42.27/12.91 mark(U106(X)) -> a__U106(mark(X)) 42.27/12.91 mark(isNatIList(X)) -> a__isNatIList(X) 42.27/12.91 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.27/12.91 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.27/12.91 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.27/12.91 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.27/12.91 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.27/12.91 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.27/12.91 mark(length(X)) -> a__length(mark(X)) 42.27/12.91 mark(U13(X)) -> a__U13(mark(X)) 42.27/12.91 mark(isNatList(X)) -> a__isNatList(X) 42.27/12.91 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.27/12.91 mark(U122(X)) -> a__U122(mark(X)) 42.27/12.91 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.27/12.91 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.27/12.91 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.27/12.91 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.27/12.91 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.27/12.91 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.27/12.91 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.27/12.91 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.27/12.91 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.27/12.91 mark(U23(X)) -> a__U23(mark(X)) 42.27/12.91 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.27/12.91 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.27/12.91 mark(U33(X)) -> a__U33(mark(X)) 42.27/12.91 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.27/12.91 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.27/12.91 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.27/12.91 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.27/12.91 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.27/12.91 mark(U46(X)) -> a__U46(mark(X)) 42.27/12.91 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.27/12.91 mark(U52(X)) -> a__U52(mark(X)) 42.27/12.91 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.27/12.91 mark(U62(X)) -> a__U62(mark(X)) 42.27/12.91 mark(U71(X)) -> a__U71(mark(X)) 42.27/12.91 mark(U81(X)) -> a__U81(mark(X)) 42.27/12.91 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.27/12.91 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.27/12.91 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.27/12.91 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.27/12.91 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.27/12.91 mark(U96(X)) -> a__U96(mark(X)) 42.27/12.91 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.27/12.91 mark(0) -> 0 42.27/12.91 mark(tt) -> tt 42.27/12.91 mark(s(X)) -> s(mark(X)) 42.27/12.91 mark(nil) -> nil 42.27/12.91 a__zeros -> zeros 42.27/12.91 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.27/12.91 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.27/12.91 a__isNatKind(X) -> isNatKind(X) 42.27/12.91 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.27/12.91 a__isNatIListKind(X) -> isNatIListKind(X) 42.27/12.91 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.27/12.91 a__U105(X1, X2) -> U105(X1, X2) 42.27/12.91 a__isNat(X) -> isNat(X) 42.27/12.91 a__U106(X) -> U106(X) 42.27/12.91 a__isNatIList(X) -> isNatIList(X) 42.27/12.91 a__U11(X1, X2) -> U11(X1, X2) 42.27/12.91 a__U12(X1, X2) -> U12(X1, X2) 42.27/12.91 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.27/12.91 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.27/12.91 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.27/12.91 a__U114(X1, X2) -> U114(X1, X2) 42.27/12.91 a__length(X) -> length(X) 42.27/12.91 a__U13(X) -> U13(X) 42.27/12.91 a__isNatList(X) -> isNatList(X) 42.27/12.91 a__U121(X1, X2) -> U121(X1, X2) 42.27/12.91 a__U122(X) -> U122(X) 42.27/12.91 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.27/12.91 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.27/12.91 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.27/12.91 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.27/12.91 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.27/12.91 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.27/12.91 a__take(X1, X2) -> take(X1, X2) 42.27/12.91 a__U21(X1, X2) -> U21(X1, X2) 42.27/12.91 a__U22(X1, X2) -> U22(X1, X2) 42.27/12.91 a__U23(X) -> U23(X) 42.27/12.91 a__U31(X1, X2) -> U31(X1, X2) 42.27/12.91 a__U32(X1, X2) -> U32(X1, X2) 42.27/12.91 a__U33(X) -> U33(X) 42.27/12.91 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.27/12.91 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.27/12.91 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.27/12.91 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.27/12.91 a__U45(X1, X2) -> U45(X1, X2) 42.27/12.91 a__U46(X) -> U46(X) 42.27/12.91 a__U51(X1, X2) -> U51(X1, X2) 42.27/12.91 a__U52(X) -> U52(X) 42.27/12.91 a__U61(X1, X2) -> U61(X1, X2) 42.27/12.91 a__U62(X) -> U62(X) 42.27/12.91 a__U71(X) -> U71(X) 42.27/12.91 a__U81(X) -> U81(X) 42.27/12.91 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.27/12.91 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.27/12.91 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.27/12.91 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.27/12.91 a__U95(X1, X2) -> U95(X1, X2) 42.27/12.91 a__U96(X) -> U96(X) 42.27/12.91 42.27/12.91 The set Q consists of the following terms: 42.27/12.91 42.27/12.91 a__zeros 42.27/12.91 a__isNatIList(x0) 42.27/12.91 mark(zeros) 42.27/12.91 mark(U101(x0, x1, x2)) 42.27/12.91 mark(U102(x0, x1, x2)) 42.27/12.91 mark(isNatKind(x0)) 42.27/12.91 mark(U103(x0, x1, x2)) 42.27/12.91 mark(isNatIListKind(x0)) 42.27/12.91 mark(U104(x0, x1, x2)) 42.27/12.91 mark(U105(x0, x1)) 42.27/12.91 mark(isNat(x0)) 42.27/12.91 mark(U106(x0)) 42.27/12.91 mark(isNatIList(x0)) 42.27/12.91 mark(U11(x0, x1)) 42.27/12.91 mark(U12(x0, x1)) 42.27/12.91 mark(U111(x0, x1, x2)) 42.27/12.91 mark(U112(x0, x1, x2)) 42.27/12.91 mark(U113(x0, x1, x2)) 42.27/12.91 mark(U114(x0, x1)) 42.27/12.91 mark(length(x0)) 42.27/12.91 mark(U13(x0)) 42.27/12.91 mark(isNatList(x0)) 42.27/12.91 mark(U121(x0, x1)) 42.27/12.91 mark(U122(x0)) 42.27/12.91 mark(U131(x0, x1, x2, x3)) 42.27/12.91 mark(U132(x0, x1, x2, x3)) 42.27/12.91 mark(U133(x0, x1, x2, x3)) 42.27/12.91 mark(U134(x0, x1, x2, x3)) 42.27/12.91 mark(U135(x0, x1, x2, x3)) 42.27/12.91 mark(U136(x0, x1, x2, x3)) 42.27/12.91 mark(take(x0, x1)) 42.27/12.91 mark(U21(x0, x1)) 42.27/12.91 mark(U22(x0, x1)) 42.27/12.91 mark(U23(x0)) 42.27/12.91 mark(U31(x0, x1)) 42.27/12.91 mark(U32(x0, x1)) 42.27/12.91 mark(U33(x0)) 42.27/12.91 mark(U41(x0, x1, x2)) 42.27/12.91 mark(U42(x0, x1, x2)) 42.27/12.91 mark(U43(x0, x1, x2)) 42.27/12.91 mark(U44(x0, x1, x2)) 42.27/12.91 mark(U45(x0, x1)) 42.27/12.91 mark(U46(x0)) 42.27/12.91 mark(U51(x0, x1)) 42.27/12.91 mark(U52(x0)) 42.27/12.91 mark(U61(x0, x1)) 42.27/12.91 mark(U62(x0)) 42.27/12.91 mark(U71(x0)) 42.27/12.91 mark(U81(x0)) 42.27/12.91 mark(U91(x0, x1, x2)) 42.27/12.91 mark(U92(x0, x1, x2)) 42.27/12.91 mark(U93(x0, x1, x2)) 42.27/12.91 mark(U94(x0, x1, x2)) 42.27/12.91 mark(U95(x0, x1)) 42.27/12.91 mark(U96(x0)) 42.27/12.91 mark(cons(x0, x1)) 42.27/12.91 mark(0) 42.27/12.91 mark(tt) 42.27/12.91 mark(s(x0)) 42.27/12.91 mark(nil) 42.27/12.91 a__U101(x0, x1, x2) 42.27/12.91 a__U102(x0, x1, x2) 42.27/12.91 a__isNatKind(x0) 42.27/12.91 a__U103(x0, x1, x2) 42.27/12.91 a__isNatIListKind(x0) 42.27/12.91 a__U104(x0, x1, x2) 42.27/12.91 a__U105(x0, x1) 42.27/12.91 a__isNat(x0) 42.27/12.91 a__U106(x0) 42.27/12.91 a__U11(x0, x1) 42.27/12.91 a__U12(x0, x1) 42.27/12.91 a__U111(x0, x1, x2) 42.27/12.91 a__U112(x0, x1, x2) 42.27/12.91 a__U113(x0, x1, x2) 42.27/12.91 a__U114(x0, x1) 42.27/12.91 a__length(x0) 42.27/12.91 a__U13(x0) 42.27/12.91 a__isNatList(x0) 42.27/12.91 a__U121(x0, x1) 42.27/12.91 a__U122(x0) 42.27/12.91 a__U131(x0, x1, x2, x3) 42.27/12.91 a__U132(x0, x1, x2, x3) 42.27/12.91 a__U133(x0, x1, x2, x3) 42.27/12.91 a__U134(x0, x1, x2, x3) 42.27/12.91 a__U135(x0, x1, x2, x3) 42.27/12.91 a__U136(x0, x1, x2, x3) 42.27/12.91 a__take(x0, x1) 42.27/12.91 a__U21(x0, x1) 42.27/12.91 a__U22(x0, x1) 42.27/12.91 a__U23(x0) 42.27/12.91 a__U31(x0, x1) 42.27/12.91 a__U32(x0, x1) 42.27/12.91 a__U33(x0) 42.27/12.91 a__U41(x0, x1, x2) 42.27/12.91 a__U42(x0, x1, x2) 42.27/12.91 a__U43(x0, x1, x2) 42.27/12.91 a__U44(x0, x1, x2) 42.27/12.91 a__U45(x0, x1) 42.27/12.91 a__U46(x0) 42.27/12.91 a__U51(x0, x1) 42.27/12.91 a__U52(x0) 42.27/12.91 a__U61(x0, x1) 42.27/12.91 a__U62(x0) 42.27/12.91 a__U71(x0) 42.27/12.91 a__U81(x0) 42.27/12.91 a__U91(x0, x1, x2) 42.27/12.91 a__U92(x0, x1, x2) 42.27/12.91 a__U93(x0, x1, x2) 42.27/12.91 a__U94(x0, x1, x2) 42.27/12.91 a__U95(x0, x1) 42.27/12.91 a__U96(x0) 42.27/12.91 42.27/12.91 We have to consider all minimal (P,Q,R)-chains. 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (26) DependencyGraphProof (EQUIVALENT) 42.27/12.91 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (27) 42.27/12.91 Complex Obligation (AND) 42.27/12.91 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (28) 42.27/12.91 Obligation: 42.27/12.91 Q DP problem: 42.27/12.91 The TRS P consists of the following rules: 42.27/12.91 42.27/12.91 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U105(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U106(X)) -> MARK(X) 42.27/12.91 MARK(U11(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U12(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U13(X)) -> MARK(X) 42.27/12.91 MARK(U21(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U22(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U23(X)) -> MARK(X) 42.27/12.91 MARK(U31(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U32(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U33(X)) -> MARK(X) 42.27/12.91 MARK(U41(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U42(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U43(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U44(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U45(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U46(X)) -> MARK(X) 42.27/12.91 MARK(U51(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U52(X)) -> MARK(X) 42.27/12.91 MARK(U61(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U62(X)) -> MARK(X) 42.27/12.91 MARK(U71(X)) -> MARK(X) 42.27/12.91 MARK(U81(X)) -> MARK(X) 42.27/12.91 MARK(U91(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U92(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U93(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U94(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U95(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U96(X)) -> MARK(X) 42.27/12.91 MARK(cons(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(s(X)) -> MARK(X) 42.27/12.91 42.27/12.91 The TRS R consists of the following rules: 42.27/12.91 42.27/12.91 a__zeros -> cons(0, zeros) 42.27/12.91 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.27/12.91 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.27/12.91 a__U106(tt) -> tt 42.27/12.91 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.27/12.91 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.27/12.91 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.27/12.91 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.27/12.91 a__U114(tt, L) -> s(a__length(mark(L))) 42.27/12.91 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.27/12.91 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.27/12.91 a__U122(tt) -> nil 42.27/12.91 a__U13(tt) -> tt 42.27/12.91 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.27/12.91 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.27/12.91 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.27/12.91 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.27/12.91 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.27/12.91 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.27/12.91 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.27/12.91 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.27/12.91 a__U23(tt) -> tt 42.27/12.91 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.27/12.91 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.27/12.91 a__U33(tt) -> tt 42.27/12.91 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.27/12.91 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.27/12.91 a__U46(tt) -> tt 42.27/12.91 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.27/12.91 a__U52(tt) -> tt 42.27/12.91 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.27/12.91 a__U62(tt) -> tt 42.27/12.91 a__U71(tt) -> tt 42.27/12.91 a__U81(tt) -> tt 42.27/12.91 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.27/12.91 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.27/12.91 a__U96(tt) -> tt 42.27/12.91 a__isNat(0) -> tt 42.27/12.91 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.27/12.91 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.27/12.91 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.27/12.91 a__isNatIList(zeros) -> tt 42.27/12.91 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatIListKind(nil) -> tt 42.27/12.91 a__isNatIListKind(zeros) -> tt 42.27/12.91 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.27/12.91 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.27/12.91 a__isNatKind(0) -> tt 42.27/12.91 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.27/12.91 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.27/12.91 a__isNatList(nil) -> tt 42.27/12.91 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.27/12.91 a__length(nil) -> 0 42.27/12.91 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.27/12.91 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.27/12.91 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.27/12.91 mark(zeros) -> a__zeros 42.27/12.91 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.27/12.91 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.27/12.91 mark(isNatKind(X)) -> a__isNatKind(X) 42.27/12.91 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.27/12.91 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.27/12.91 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.27/12.91 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.27/12.91 mark(isNat(X)) -> a__isNat(X) 42.27/12.91 mark(U106(X)) -> a__U106(mark(X)) 42.27/12.91 mark(isNatIList(X)) -> a__isNatIList(X) 42.27/12.91 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.27/12.91 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.27/12.91 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.27/12.91 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.27/12.91 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.27/12.91 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.27/12.91 mark(length(X)) -> a__length(mark(X)) 42.27/12.91 mark(U13(X)) -> a__U13(mark(X)) 42.27/12.91 mark(isNatList(X)) -> a__isNatList(X) 42.27/12.91 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.27/12.91 mark(U122(X)) -> a__U122(mark(X)) 42.27/12.91 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.27/12.91 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.27/12.91 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.27/12.91 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.27/12.91 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.27/12.91 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.27/12.91 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.27/12.91 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.27/12.91 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.27/12.91 mark(U23(X)) -> a__U23(mark(X)) 42.27/12.91 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.27/12.91 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.27/12.91 mark(U33(X)) -> a__U33(mark(X)) 42.27/12.91 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.27/12.91 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.27/12.91 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.27/12.91 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.27/12.91 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.27/12.91 mark(U46(X)) -> a__U46(mark(X)) 42.27/12.91 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.27/12.91 mark(U52(X)) -> a__U52(mark(X)) 42.27/12.91 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.27/12.91 mark(U62(X)) -> a__U62(mark(X)) 42.27/12.91 mark(U71(X)) -> a__U71(mark(X)) 42.27/12.91 mark(U81(X)) -> a__U81(mark(X)) 42.27/12.91 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.27/12.91 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.27/12.91 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.27/12.91 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.27/12.91 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.27/12.91 mark(U96(X)) -> a__U96(mark(X)) 42.27/12.91 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.27/12.91 mark(0) -> 0 42.27/12.91 mark(tt) -> tt 42.27/12.91 mark(s(X)) -> s(mark(X)) 42.27/12.91 mark(nil) -> nil 42.27/12.91 a__zeros -> zeros 42.27/12.91 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.27/12.91 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.27/12.91 a__isNatKind(X) -> isNatKind(X) 42.27/12.91 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.27/12.91 a__isNatIListKind(X) -> isNatIListKind(X) 42.27/12.91 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.27/12.91 a__U105(X1, X2) -> U105(X1, X2) 42.27/12.91 a__isNat(X) -> isNat(X) 42.27/12.91 a__U106(X) -> U106(X) 42.27/12.91 a__isNatIList(X) -> isNatIList(X) 42.27/12.91 a__U11(X1, X2) -> U11(X1, X2) 42.27/12.91 a__U12(X1, X2) -> U12(X1, X2) 42.27/12.91 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.27/12.91 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.27/12.91 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.27/12.91 a__U114(X1, X2) -> U114(X1, X2) 42.27/12.91 a__length(X) -> length(X) 42.27/12.91 a__U13(X) -> U13(X) 42.27/12.91 a__isNatList(X) -> isNatList(X) 42.27/12.91 a__U121(X1, X2) -> U121(X1, X2) 42.27/12.91 a__U122(X) -> U122(X) 42.27/12.91 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.27/12.91 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.27/12.91 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.27/12.91 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.27/12.91 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.27/12.91 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.27/12.91 a__take(X1, X2) -> take(X1, X2) 42.27/12.91 a__U21(X1, X2) -> U21(X1, X2) 42.27/12.91 a__U22(X1, X2) -> U22(X1, X2) 42.27/12.91 a__U23(X) -> U23(X) 42.27/12.91 a__U31(X1, X2) -> U31(X1, X2) 42.27/12.91 a__U32(X1, X2) -> U32(X1, X2) 42.27/12.91 a__U33(X) -> U33(X) 42.27/12.91 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.27/12.91 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.27/12.91 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.27/12.91 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.27/12.91 a__U45(X1, X2) -> U45(X1, X2) 42.27/12.91 a__U46(X) -> U46(X) 42.27/12.91 a__U51(X1, X2) -> U51(X1, X2) 42.27/12.91 a__U52(X) -> U52(X) 42.27/12.91 a__U61(X1, X2) -> U61(X1, X2) 42.27/12.91 a__U62(X) -> U62(X) 42.27/12.91 a__U71(X) -> U71(X) 42.27/12.91 a__U81(X) -> U81(X) 42.27/12.91 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.27/12.91 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.27/12.91 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.27/12.91 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.27/12.91 a__U95(X1, X2) -> U95(X1, X2) 42.27/12.91 a__U96(X) -> U96(X) 42.27/12.91 42.27/12.91 The set Q consists of the following terms: 42.27/12.91 42.27/12.91 a__zeros 42.27/12.91 a__isNatIList(x0) 42.27/12.91 mark(zeros) 42.27/12.91 mark(U101(x0, x1, x2)) 42.27/12.91 mark(U102(x0, x1, x2)) 42.27/12.91 mark(isNatKind(x0)) 42.27/12.91 mark(U103(x0, x1, x2)) 42.27/12.91 mark(isNatIListKind(x0)) 42.27/12.91 mark(U104(x0, x1, x2)) 42.27/12.91 mark(U105(x0, x1)) 42.27/12.91 mark(isNat(x0)) 42.27/12.91 mark(U106(x0)) 42.27/12.91 mark(isNatIList(x0)) 42.27/12.91 mark(U11(x0, x1)) 42.27/12.91 mark(U12(x0, x1)) 42.27/12.91 mark(U111(x0, x1, x2)) 42.27/12.91 mark(U112(x0, x1, x2)) 42.27/12.91 mark(U113(x0, x1, x2)) 42.27/12.91 mark(U114(x0, x1)) 42.27/12.91 mark(length(x0)) 42.27/12.91 mark(U13(x0)) 42.27/12.91 mark(isNatList(x0)) 42.27/12.91 mark(U121(x0, x1)) 42.27/12.91 mark(U122(x0)) 42.27/12.91 mark(U131(x0, x1, x2, x3)) 42.27/12.91 mark(U132(x0, x1, x2, x3)) 42.27/12.91 mark(U133(x0, x1, x2, x3)) 42.27/12.91 mark(U134(x0, x1, x2, x3)) 42.27/12.91 mark(U135(x0, x1, x2, x3)) 42.27/12.91 mark(U136(x0, x1, x2, x3)) 42.27/12.91 mark(take(x0, x1)) 42.27/12.91 mark(U21(x0, x1)) 42.27/12.91 mark(U22(x0, x1)) 42.27/12.91 mark(U23(x0)) 42.27/12.91 mark(U31(x0, x1)) 42.27/12.91 mark(U32(x0, x1)) 42.27/12.91 mark(U33(x0)) 42.27/12.91 mark(U41(x0, x1, x2)) 42.27/12.91 mark(U42(x0, x1, x2)) 42.27/12.91 mark(U43(x0, x1, x2)) 42.27/12.91 mark(U44(x0, x1, x2)) 42.27/12.91 mark(U45(x0, x1)) 42.27/12.91 mark(U46(x0)) 42.27/12.91 mark(U51(x0, x1)) 42.27/12.91 mark(U52(x0)) 42.27/12.91 mark(U61(x0, x1)) 42.27/12.91 mark(U62(x0)) 42.27/12.91 mark(U71(x0)) 42.27/12.91 mark(U81(x0)) 42.27/12.91 mark(U91(x0, x1, x2)) 42.27/12.91 mark(U92(x0, x1, x2)) 42.27/12.91 mark(U93(x0, x1, x2)) 42.27/12.91 mark(U94(x0, x1, x2)) 42.27/12.91 mark(U95(x0, x1)) 42.27/12.91 mark(U96(x0)) 42.27/12.91 mark(cons(x0, x1)) 42.27/12.91 mark(0) 42.27/12.91 mark(tt) 42.27/12.91 mark(s(x0)) 42.27/12.91 mark(nil) 42.27/12.91 a__U101(x0, x1, x2) 42.27/12.91 a__U102(x0, x1, x2) 42.27/12.91 a__isNatKind(x0) 42.27/12.91 a__U103(x0, x1, x2) 42.27/12.91 a__isNatIListKind(x0) 42.27/12.91 a__U104(x0, x1, x2) 42.27/12.91 a__U105(x0, x1) 42.27/12.91 a__isNat(x0) 42.27/12.91 a__U106(x0) 42.27/12.91 a__U11(x0, x1) 42.27/12.91 a__U12(x0, x1) 42.27/12.91 a__U111(x0, x1, x2) 42.27/12.91 a__U112(x0, x1, x2) 42.27/12.91 a__U113(x0, x1, x2) 42.27/12.91 a__U114(x0, x1) 42.27/12.91 a__length(x0) 42.27/12.91 a__U13(x0) 42.27/12.91 a__isNatList(x0) 42.27/12.91 a__U121(x0, x1) 42.27/12.91 a__U122(x0) 42.27/12.91 a__U131(x0, x1, x2, x3) 42.27/12.91 a__U132(x0, x1, x2, x3) 42.27/12.91 a__U133(x0, x1, x2, x3) 42.27/12.91 a__U134(x0, x1, x2, x3) 42.27/12.91 a__U135(x0, x1, x2, x3) 42.27/12.91 a__U136(x0, x1, x2, x3) 42.27/12.91 a__take(x0, x1) 42.27/12.91 a__U21(x0, x1) 42.27/12.91 a__U22(x0, x1) 42.27/12.91 a__U23(x0) 42.27/12.91 a__U31(x0, x1) 42.27/12.91 a__U32(x0, x1) 42.27/12.91 a__U33(x0) 42.27/12.91 a__U41(x0, x1, x2) 42.27/12.91 a__U42(x0, x1, x2) 42.27/12.91 a__U43(x0, x1, x2) 42.27/12.91 a__U44(x0, x1, x2) 42.27/12.91 a__U45(x0, x1) 42.27/12.91 a__U46(x0) 42.27/12.91 a__U51(x0, x1) 42.27/12.91 a__U52(x0) 42.27/12.91 a__U61(x0, x1) 42.27/12.91 a__U62(x0) 42.27/12.91 a__U71(x0) 42.27/12.91 a__U81(x0) 42.27/12.91 a__U91(x0, x1, x2) 42.27/12.91 a__U92(x0, x1, x2) 42.27/12.91 a__U93(x0, x1, x2) 42.27/12.91 a__U94(x0, x1, x2) 42.27/12.91 a__U95(x0, x1) 42.27/12.91 a__U96(x0) 42.27/12.91 42.27/12.91 We have to consider all minimal (P,Q,R)-chains. 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (29) UsableRulesProof (EQUIVALENT) 42.27/12.91 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (30) 42.27/12.91 Obligation: 42.27/12.91 Q DP problem: 42.27/12.91 The TRS P consists of the following rules: 42.27/12.91 42.27/12.91 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U105(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U106(X)) -> MARK(X) 42.27/12.91 MARK(U11(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U12(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U13(X)) -> MARK(X) 42.27/12.91 MARK(U21(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U22(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U23(X)) -> MARK(X) 42.27/12.91 MARK(U31(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U32(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U33(X)) -> MARK(X) 42.27/12.91 MARK(U41(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U42(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U43(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U44(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U45(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U46(X)) -> MARK(X) 42.27/12.91 MARK(U51(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U52(X)) -> MARK(X) 42.27/12.91 MARK(U61(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U62(X)) -> MARK(X) 42.27/12.91 MARK(U71(X)) -> MARK(X) 42.27/12.91 MARK(U81(X)) -> MARK(X) 42.27/12.91 MARK(U91(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U92(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U93(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U94(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U95(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U96(X)) -> MARK(X) 42.27/12.91 MARK(cons(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(s(X)) -> MARK(X) 42.27/12.91 42.27/12.91 R is empty. 42.27/12.91 The set Q consists of the following terms: 42.27/12.91 42.27/12.91 a__zeros 42.27/12.91 a__isNatIList(x0) 42.27/12.91 mark(zeros) 42.27/12.91 mark(U101(x0, x1, x2)) 42.27/12.91 mark(U102(x0, x1, x2)) 42.27/12.91 mark(isNatKind(x0)) 42.27/12.91 mark(U103(x0, x1, x2)) 42.27/12.91 mark(isNatIListKind(x0)) 42.27/12.91 mark(U104(x0, x1, x2)) 42.27/12.91 mark(U105(x0, x1)) 42.27/12.91 mark(isNat(x0)) 42.27/12.91 mark(U106(x0)) 42.27/12.91 mark(isNatIList(x0)) 42.27/12.91 mark(U11(x0, x1)) 42.27/12.91 mark(U12(x0, x1)) 42.27/12.91 mark(U111(x0, x1, x2)) 42.27/12.91 mark(U112(x0, x1, x2)) 42.27/12.91 mark(U113(x0, x1, x2)) 42.27/12.91 mark(U114(x0, x1)) 42.27/12.91 mark(length(x0)) 42.27/12.91 mark(U13(x0)) 42.27/12.91 mark(isNatList(x0)) 42.27/12.91 mark(U121(x0, x1)) 42.27/12.91 mark(U122(x0)) 42.27/12.91 mark(U131(x0, x1, x2, x3)) 42.27/12.91 mark(U132(x0, x1, x2, x3)) 42.27/12.91 mark(U133(x0, x1, x2, x3)) 42.27/12.91 mark(U134(x0, x1, x2, x3)) 42.27/12.91 mark(U135(x0, x1, x2, x3)) 42.27/12.91 mark(U136(x0, x1, x2, x3)) 42.27/12.91 mark(take(x0, x1)) 42.27/12.91 mark(U21(x0, x1)) 42.27/12.91 mark(U22(x0, x1)) 42.27/12.91 mark(U23(x0)) 42.27/12.91 mark(U31(x0, x1)) 42.27/12.91 mark(U32(x0, x1)) 42.27/12.91 mark(U33(x0)) 42.27/12.91 mark(U41(x0, x1, x2)) 42.27/12.91 mark(U42(x0, x1, x2)) 42.27/12.91 mark(U43(x0, x1, x2)) 42.27/12.91 mark(U44(x0, x1, x2)) 42.27/12.91 mark(U45(x0, x1)) 42.27/12.91 mark(U46(x0)) 42.27/12.91 mark(U51(x0, x1)) 42.27/12.91 mark(U52(x0)) 42.27/12.91 mark(U61(x0, x1)) 42.27/12.91 mark(U62(x0)) 42.27/12.91 mark(U71(x0)) 42.27/12.91 mark(U81(x0)) 42.27/12.91 mark(U91(x0, x1, x2)) 42.27/12.91 mark(U92(x0, x1, x2)) 42.27/12.91 mark(U93(x0, x1, x2)) 42.27/12.91 mark(U94(x0, x1, x2)) 42.27/12.91 mark(U95(x0, x1)) 42.27/12.91 mark(U96(x0)) 42.27/12.91 mark(cons(x0, x1)) 42.27/12.91 mark(0) 42.27/12.91 mark(tt) 42.27/12.91 mark(s(x0)) 42.27/12.91 mark(nil) 42.27/12.91 a__U101(x0, x1, x2) 42.27/12.91 a__U102(x0, x1, x2) 42.27/12.91 a__isNatKind(x0) 42.27/12.91 a__U103(x0, x1, x2) 42.27/12.91 a__isNatIListKind(x0) 42.27/12.91 a__U104(x0, x1, x2) 42.27/12.91 a__U105(x0, x1) 42.27/12.91 a__isNat(x0) 42.27/12.91 a__U106(x0) 42.27/12.91 a__U11(x0, x1) 42.27/12.91 a__U12(x0, x1) 42.27/12.91 a__U111(x0, x1, x2) 42.27/12.91 a__U112(x0, x1, x2) 42.27/12.91 a__U113(x0, x1, x2) 42.27/12.91 a__U114(x0, x1) 42.27/12.91 a__length(x0) 42.27/12.91 a__U13(x0) 42.27/12.91 a__isNatList(x0) 42.27/12.91 a__U121(x0, x1) 42.27/12.91 a__U122(x0) 42.27/12.91 a__U131(x0, x1, x2, x3) 42.27/12.91 a__U132(x0, x1, x2, x3) 42.27/12.91 a__U133(x0, x1, x2, x3) 42.27/12.91 a__U134(x0, x1, x2, x3) 42.27/12.91 a__U135(x0, x1, x2, x3) 42.27/12.91 a__U136(x0, x1, x2, x3) 42.27/12.91 a__take(x0, x1) 42.27/12.91 a__U21(x0, x1) 42.27/12.91 a__U22(x0, x1) 42.27/12.91 a__U23(x0) 42.27/12.91 a__U31(x0, x1) 42.27/12.91 a__U32(x0, x1) 42.27/12.91 a__U33(x0) 42.27/12.91 a__U41(x0, x1, x2) 42.27/12.91 a__U42(x0, x1, x2) 42.27/12.91 a__U43(x0, x1, x2) 42.27/12.91 a__U44(x0, x1, x2) 42.27/12.91 a__U45(x0, x1) 42.27/12.91 a__U46(x0) 42.27/12.91 a__U51(x0, x1) 42.27/12.91 a__U52(x0) 42.27/12.91 a__U61(x0, x1) 42.27/12.91 a__U62(x0) 42.27/12.91 a__U71(x0) 42.27/12.91 a__U81(x0) 42.27/12.91 a__U91(x0, x1, x2) 42.27/12.91 a__U92(x0, x1, x2) 42.27/12.91 a__U93(x0, x1, x2) 42.27/12.91 a__U94(x0, x1, x2) 42.27/12.91 a__U95(x0, x1) 42.27/12.91 a__U96(x0) 42.27/12.91 42.27/12.91 We have to consider all minimal (P,Q,R)-chains. 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (31) QReductionProof (EQUIVALENT) 42.27/12.91 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 42.27/12.91 42.27/12.91 a__zeros 42.27/12.91 a__isNatIList(x0) 42.27/12.91 mark(zeros) 42.27/12.91 mark(U101(x0, x1, x2)) 42.27/12.91 mark(U102(x0, x1, x2)) 42.27/12.91 mark(isNatKind(x0)) 42.27/12.91 mark(U103(x0, x1, x2)) 42.27/12.91 mark(isNatIListKind(x0)) 42.27/12.91 mark(U104(x0, x1, x2)) 42.27/12.91 mark(U105(x0, x1)) 42.27/12.91 mark(isNat(x0)) 42.27/12.91 mark(U106(x0)) 42.27/12.91 mark(isNatIList(x0)) 42.27/12.91 mark(U11(x0, x1)) 42.27/12.91 mark(U12(x0, x1)) 42.27/12.91 mark(U111(x0, x1, x2)) 42.27/12.91 mark(U112(x0, x1, x2)) 42.27/12.91 mark(U113(x0, x1, x2)) 42.27/12.91 mark(U114(x0, x1)) 42.27/12.91 mark(length(x0)) 42.27/12.91 mark(U13(x0)) 42.27/12.91 mark(isNatList(x0)) 42.27/12.91 mark(U121(x0, x1)) 42.27/12.91 mark(U122(x0)) 42.27/12.91 mark(U131(x0, x1, x2, x3)) 42.27/12.91 mark(U132(x0, x1, x2, x3)) 42.27/12.91 mark(U133(x0, x1, x2, x3)) 42.27/12.91 mark(U134(x0, x1, x2, x3)) 42.27/12.91 mark(U135(x0, x1, x2, x3)) 42.27/12.91 mark(U136(x0, x1, x2, x3)) 42.27/12.91 mark(take(x0, x1)) 42.27/12.91 mark(U21(x0, x1)) 42.27/12.91 mark(U22(x0, x1)) 42.27/12.91 mark(U23(x0)) 42.27/12.91 mark(U31(x0, x1)) 42.27/12.91 mark(U32(x0, x1)) 42.27/12.91 mark(U33(x0)) 42.27/12.91 mark(U41(x0, x1, x2)) 42.27/12.91 mark(U42(x0, x1, x2)) 42.27/12.91 mark(U43(x0, x1, x2)) 42.27/12.91 mark(U44(x0, x1, x2)) 42.27/12.91 mark(U45(x0, x1)) 42.27/12.91 mark(U46(x0)) 42.27/12.91 mark(U51(x0, x1)) 42.27/12.91 mark(U52(x0)) 42.27/12.91 mark(U61(x0, x1)) 42.27/12.91 mark(U62(x0)) 42.27/12.91 mark(U71(x0)) 42.27/12.91 mark(U81(x0)) 42.27/12.91 mark(U91(x0, x1, x2)) 42.27/12.91 mark(U92(x0, x1, x2)) 42.27/12.91 mark(U93(x0, x1, x2)) 42.27/12.91 mark(U94(x0, x1, x2)) 42.27/12.91 mark(U95(x0, x1)) 42.27/12.91 mark(U96(x0)) 42.27/12.91 mark(cons(x0, x1)) 42.27/12.91 mark(0) 42.27/12.91 mark(tt) 42.27/12.91 mark(s(x0)) 42.27/12.91 mark(nil) 42.27/12.91 a__U101(x0, x1, x2) 42.27/12.91 a__U102(x0, x1, x2) 42.27/12.91 a__isNatKind(x0) 42.27/12.91 a__U103(x0, x1, x2) 42.27/12.91 a__isNatIListKind(x0) 42.27/12.91 a__U104(x0, x1, x2) 42.27/12.91 a__U105(x0, x1) 42.27/12.91 a__isNat(x0) 42.27/12.91 a__U106(x0) 42.27/12.91 a__U11(x0, x1) 42.27/12.91 a__U12(x0, x1) 42.27/12.91 a__U111(x0, x1, x2) 42.27/12.91 a__U112(x0, x1, x2) 42.27/12.91 a__U113(x0, x1, x2) 42.27/12.91 a__U114(x0, x1) 42.27/12.91 a__length(x0) 42.27/12.91 a__U13(x0) 42.27/12.91 a__isNatList(x0) 42.27/12.91 a__U121(x0, x1) 42.27/12.91 a__U122(x0) 42.27/12.91 a__U131(x0, x1, x2, x3) 42.27/12.91 a__U132(x0, x1, x2, x3) 42.27/12.91 a__U133(x0, x1, x2, x3) 42.27/12.91 a__U134(x0, x1, x2, x3) 42.27/12.91 a__U135(x0, x1, x2, x3) 42.27/12.91 a__U136(x0, x1, x2, x3) 42.27/12.91 a__take(x0, x1) 42.27/12.91 a__U21(x0, x1) 42.27/12.91 a__U22(x0, x1) 42.27/12.91 a__U23(x0) 42.27/12.91 a__U31(x0, x1) 42.27/12.91 a__U32(x0, x1) 42.27/12.91 a__U33(x0) 42.27/12.91 a__U41(x0, x1, x2) 42.27/12.91 a__U42(x0, x1, x2) 42.27/12.91 a__U43(x0, x1, x2) 42.27/12.91 a__U44(x0, x1, x2) 42.27/12.91 a__U45(x0, x1) 42.27/12.91 a__U46(x0) 42.27/12.91 a__U51(x0, x1) 42.27/12.91 a__U52(x0) 42.27/12.91 a__U61(x0, x1) 42.27/12.91 a__U62(x0) 42.27/12.91 a__U71(x0) 42.27/12.91 a__U81(x0) 42.27/12.91 a__U91(x0, x1, x2) 42.27/12.91 a__U92(x0, x1, x2) 42.27/12.91 a__U93(x0, x1, x2) 42.27/12.91 a__U94(x0, x1, x2) 42.27/12.91 a__U95(x0, x1) 42.27/12.91 a__U96(x0) 42.27/12.91 42.27/12.91 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (32) 42.27/12.91 Obligation: 42.27/12.91 Q DP problem: 42.27/12.91 The TRS P consists of the following rules: 42.27/12.91 42.27/12.91 MARK(U101(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U102(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U103(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U104(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U105(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U106(X)) -> MARK(X) 42.27/12.91 MARK(U11(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U12(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U13(X)) -> MARK(X) 42.27/12.91 MARK(U21(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U22(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U23(X)) -> MARK(X) 42.27/12.91 MARK(U31(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U32(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U33(X)) -> MARK(X) 42.27/12.91 MARK(U41(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U42(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U43(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U44(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U45(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U46(X)) -> MARK(X) 42.27/12.91 MARK(U51(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U52(X)) -> MARK(X) 42.27/12.91 MARK(U61(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U62(X)) -> MARK(X) 42.27/12.91 MARK(U71(X)) -> MARK(X) 42.27/12.91 MARK(U81(X)) -> MARK(X) 42.27/12.91 MARK(U91(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U92(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U93(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U94(X1, X2, X3)) -> MARK(X1) 42.27/12.91 MARK(U95(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(U96(X)) -> MARK(X) 42.27/12.91 MARK(cons(X1, X2)) -> MARK(X1) 42.27/12.91 MARK(s(X)) -> MARK(X) 42.27/12.91 42.27/12.91 R is empty. 42.27/12.91 Q is empty. 42.27/12.91 We have to consider all minimal (P,Q,R)-chains. 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (33) QDPSizeChangeProof (EQUIVALENT) 42.27/12.91 By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 42.27/12.91 42.27/12.91 From the DPs we obtained the following set of size-change graphs: 42.27/12.91 *MARK(U101(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U102(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U103(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U104(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U105(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U106(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U11(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U12(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U13(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U21(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U22(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U23(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U31(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U32(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U33(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U41(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U42(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U43(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U44(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U45(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U46(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U51(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U52(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U61(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U62(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U71(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U81(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U91(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U92(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U93(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U94(X1, X2, X3)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U95(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(U96(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(cons(X1, X2)) -> MARK(X1) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 *MARK(s(X)) -> MARK(X) 42.27/12.91 The graph contains the following edges 1 > 1 42.27/12.91 42.27/12.91 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (34) 42.27/12.91 YES 42.27/12.91 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (35) 42.27/12.91 Obligation: 42.27/12.91 Q DP problem: 42.27/12.91 The TRS P consists of the following rules: 42.27/12.91 42.27/12.91 A__U114(tt, L) -> A__LENGTH(mark(L)) 42.27/12.91 A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) 42.27/12.91 A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) 42.27/12.91 A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) 42.27/12.91 A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) 42.27/12.91 42.27/12.91 The TRS R consists of the following rules: 42.27/12.91 42.27/12.91 a__zeros -> cons(0, zeros) 42.27/12.91 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.27/12.91 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.27/12.91 a__U106(tt) -> tt 42.27/12.91 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.27/12.91 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.27/12.91 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.27/12.91 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.27/12.91 a__U114(tt, L) -> s(a__length(mark(L))) 42.27/12.91 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.27/12.91 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.27/12.91 a__U122(tt) -> nil 42.27/12.91 a__U13(tt) -> tt 42.27/12.91 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.27/12.91 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.27/12.91 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.27/12.91 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.27/12.91 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.27/12.91 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.27/12.91 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.27/12.91 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.27/12.91 a__U23(tt) -> tt 42.27/12.91 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.27/12.91 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.27/12.91 a__U33(tt) -> tt 42.27/12.91 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.27/12.91 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.27/12.91 a__U46(tt) -> tt 42.27/12.91 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.27/12.91 a__U52(tt) -> tt 42.27/12.91 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.27/12.91 a__U62(tt) -> tt 42.27/12.91 a__U71(tt) -> tt 42.27/12.91 a__U81(tt) -> tt 42.27/12.91 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.27/12.91 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.27/12.91 a__U96(tt) -> tt 42.27/12.91 a__isNat(0) -> tt 42.27/12.91 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.27/12.91 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.27/12.91 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.27/12.91 a__isNatIList(zeros) -> tt 42.27/12.91 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatIListKind(nil) -> tt 42.27/12.91 a__isNatIListKind(zeros) -> tt 42.27/12.91 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.27/12.91 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.27/12.91 a__isNatKind(0) -> tt 42.27/12.91 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.27/12.91 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.27/12.91 a__isNatList(nil) -> tt 42.27/12.91 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.27/12.91 a__length(nil) -> 0 42.27/12.91 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.27/12.91 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.27/12.91 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.27/12.91 mark(zeros) -> a__zeros 42.27/12.91 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.27/12.91 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.27/12.91 mark(isNatKind(X)) -> a__isNatKind(X) 42.27/12.91 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.27/12.91 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.27/12.91 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.27/12.91 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.27/12.91 mark(isNat(X)) -> a__isNat(X) 42.27/12.91 mark(U106(X)) -> a__U106(mark(X)) 42.27/12.91 mark(isNatIList(X)) -> a__isNatIList(X) 42.27/12.91 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.27/12.91 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.27/12.91 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.27/12.91 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.27/12.91 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.27/12.91 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.27/12.91 mark(length(X)) -> a__length(mark(X)) 42.27/12.91 mark(U13(X)) -> a__U13(mark(X)) 42.27/12.91 mark(isNatList(X)) -> a__isNatList(X) 42.27/12.91 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.27/12.91 mark(U122(X)) -> a__U122(mark(X)) 42.27/12.91 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.27/12.91 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.27/12.91 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.27/12.91 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.27/12.91 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.27/12.91 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.27/12.91 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.27/12.91 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.27/12.91 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.27/12.91 mark(U23(X)) -> a__U23(mark(X)) 42.27/12.91 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.27/12.91 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.27/12.91 mark(U33(X)) -> a__U33(mark(X)) 42.27/12.91 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.27/12.91 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.27/12.91 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.27/12.91 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.27/12.91 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.27/12.91 mark(U46(X)) -> a__U46(mark(X)) 42.27/12.91 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.27/12.91 mark(U52(X)) -> a__U52(mark(X)) 42.27/12.91 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.27/12.91 mark(U62(X)) -> a__U62(mark(X)) 42.27/12.91 mark(U71(X)) -> a__U71(mark(X)) 42.27/12.91 mark(U81(X)) -> a__U81(mark(X)) 42.27/12.91 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.27/12.91 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.27/12.91 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.27/12.91 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.27/12.91 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.27/12.91 mark(U96(X)) -> a__U96(mark(X)) 42.27/12.91 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.27/12.91 mark(0) -> 0 42.27/12.91 mark(tt) -> tt 42.27/12.91 mark(s(X)) -> s(mark(X)) 42.27/12.91 mark(nil) -> nil 42.27/12.91 a__zeros -> zeros 42.27/12.91 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.27/12.91 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.27/12.91 a__isNatKind(X) -> isNatKind(X) 42.27/12.91 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.27/12.91 a__isNatIListKind(X) -> isNatIListKind(X) 42.27/12.91 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.27/12.91 a__U105(X1, X2) -> U105(X1, X2) 42.27/12.91 a__isNat(X) -> isNat(X) 42.27/12.91 a__U106(X) -> U106(X) 42.27/12.91 a__isNatIList(X) -> isNatIList(X) 42.27/12.91 a__U11(X1, X2) -> U11(X1, X2) 42.27/12.91 a__U12(X1, X2) -> U12(X1, X2) 42.27/12.91 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.27/12.91 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.27/12.91 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.27/12.91 a__U114(X1, X2) -> U114(X1, X2) 42.27/12.91 a__length(X) -> length(X) 42.27/12.91 a__U13(X) -> U13(X) 42.27/12.91 a__isNatList(X) -> isNatList(X) 42.27/12.91 a__U121(X1, X2) -> U121(X1, X2) 42.27/12.91 a__U122(X) -> U122(X) 42.27/12.91 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.27/12.91 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.27/12.91 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.27/12.91 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.27/12.91 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.27/12.91 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.27/12.91 a__take(X1, X2) -> take(X1, X2) 42.27/12.91 a__U21(X1, X2) -> U21(X1, X2) 42.27/12.91 a__U22(X1, X2) -> U22(X1, X2) 42.27/12.91 a__U23(X) -> U23(X) 42.27/12.91 a__U31(X1, X2) -> U31(X1, X2) 42.27/12.91 a__U32(X1, X2) -> U32(X1, X2) 42.27/12.91 a__U33(X) -> U33(X) 42.27/12.91 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.27/12.91 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.27/12.91 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.27/12.91 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.27/12.91 a__U45(X1, X2) -> U45(X1, X2) 42.27/12.91 a__U46(X) -> U46(X) 42.27/12.91 a__U51(X1, X2) -> U51(X1, X2) 42.27/12.91 a__U52(X) -> U52(X) 42.27/12.91 a__U61(X1, X2) -> U61(X1, X2) 42.27/12.91 a__U62(X) -> U62(X) 42.27/12.91 a__U71(X) -> U71(X) 42.27/12.91 a__U81(X) -> U81(X) 42.27/12.91 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.27/12.91 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.27/12.91 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.27/12.91 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.27/12.91 a__U95(X1, X2) -> U95(X1, X2) 42.27/12.91 a__U96(X) -> U96(X) 42.27/12.91 42.27/12.91 The set Q consists of the following terms: 42.27/12.91 42.27/12.91 a__zeros 42.27/12.91 a__isNatIList(x0) 42.27/12.91 mark(zeros) 42.27/12.91 mark(U101(x0, x1, x2)) 42.27/12.91 mark(U102(x0, x1, x2)) 42.27/12.91 mark(isNatKind(x0)) 42.27/12.91 mark(U103(x0, x1, x2)) 42.27/12.91 mark(isNatIListKind(x0)) 42.27/12.91 mark(U104(x0, x1, x2)) 42.27/12.91 mark(U105(x0, x1)) 42.27/12.91 mark(isNat(x0)) 42.27/12.91 mark(U106(x0)) 42.27/12.91 mark(isNatIList(x0)) 42.27/12.91 mark(U11(x0, x1)) 42.27/12.91 mark(U12(x0, x1)) 42.27/12.91 mark(U111(x0, x1, x2)) 42.27/12.91 mark(U112(x0, x1, x2)) 42.27/12.91 mark(U113(x0, x1, x2)) 42.27/12.91 mark(U114(x0, x1)) 42.27/12.91 mark(length(x0)) 42.27/12.91 mark(U13(x0)) 42.27/12.91 mark(isNatList(x0)) 42.27/12.91 mark(U121(x0, x1)) 42.27/12.91 mark(U122(x0)) 42.27/12.91 mark(U131(x0, x1, x2, x3)) 42.27/12.91 mark(U132(x0, x1, x2, x3)) 42.27/12.91 mark(U133(x0, x1, x2, x3)) 42.27/12.91 mark(U134(x0, x1, x2, x3)) 42.27/12.91 mark(U135(x0, x1, x2, x3)) 42.27/12.91 mark(U136(x0, x1, x2, x3)) 42.27/12.91 mark(take(x0, x1)) 42.27/12.91 mark(U21(x0, x1)) 42.27/12.91 mark(U22(x0, x1)) 42.27/12.91 mark(U23(x0)) 42.27/12.91 mark(U31(x0, x1)) 42.27/12.91 mark(U32(x0, x1)) 42.27/12.91 mark(U33(x0)) 42.27/12.91 mark(U41(x0, x1, x2)) 42.27/12.91 mark(U42(x0, x1, x2)) 42.27/12.91 mark(U43(x0, x1, x2)) 42.27/12.91 mark(U44(x0, x1, x2)) 42.27/12.91 mark(U45(x0, x1)) 42.27/12.91 mark(U46(x0)) 42.27/12.91 mark(U51(x0, x1)) 42.27/12.91 mark(U52(x0)) 42.27/12.91 mark(U61(x0, x1)) 42.27/12.91 mark(U62(x0)) 42.27/12.91 mark(U71(x0)) 42.27/12.91 mark(U81(x0)) 42.27/12.91 mark(U91(x0, x1, x2)) 42.27/12.91 mark(U92(x0, x1, x2)) 42.27/12.91 mark(U93(x0, x1, x2)) 42.27/12.91 mark(U94(x0, x1, x2)) 42.27/12.91 mark(U95(x0, x1)) 42.27/12.91 mark(U96(x0)) 42.27/12.91 mark(cons(x0, x1)) 42.27/12.91 mark(0) 42.27/12.91 mark(tt) 42.27/12.91 mark(s(x0)) 42.27/12.91 mark(nil) 42.27/12.91 a__U101(x0, x1, x2) 42.27/12.91 a__U102(x0, x1, x2) 42.27/12.91 a__isNatKind(x0) 42.27/12.91 a__U103(x0, x1, x2) 42.27/12.91 a__isNatIListKind(x0) 42.27/12.91 a__U104(x0, x1, x2) 42.27/12.91 a__U105(x0, x1) 42.27/12.91 a__isNat(x0) 42.27/12.91 a__U106(x0) 42.27/12.91 a__U11(x0, x1) 42.27/12.91 a__U12(x0, x1) 42.27/12.91 a__U111(x0, x1, x2) 42.27/12.91 a__U112(x0, x1, x2) 42.27/12.91 a__U113(x0, x1, x2) 42.27/12.91 a__U114(x0, x1) 42.27/12.91 a__length(x0) 42.27/12.91 a__U13(x0) 42.27/12.91 a__isNatList(x0) 42.27/12.91 a__U121(x0, x1) 42.27/12.91 a__U122(x0) 42.27/12.91 a__U131(x0, x1, x2, x3) 42.27/12.91 a__U132(x0, x1, x2, x3) 42.27/12.91 a__U133(x0, x1, x2, x3) 42.27/12.91 a__U134(x0, x1, x2, x3) 42.27/12.91 a__U135(x0, x1, x2, x3) 42.27/12.91 a__U136(x0, x1, x2, x3) 42.27/12.91 a__take(x0, x1) 42.27/12.91 a__U21(x0, x1) 42.27/12.91 a__U22(x0, x1) 42.27/12.91 a__U23(x0) 42.27/12.91 a__U31(x0, x1) 42.27/12.91 a__U32(x0, x1) 42.27/12.91 a__U33(x0) 42.27/12.91 a__U41(x0, x1, x2) 42.27/12.91 a__U42(x0, x1, x2) 42.27/12.91 a__U43(x0, x1, x2) 42.27/12.91 a__U44(x0, x1, x2) 42.27/12.91 a__U45(x0, x1) 42.27/12.91 a__U46(x0) 42.27/12.91 a__U51(x0, x1) 42.27/12.91 a__U52(x0) 42.27/12.91 a__U61(x0, x1) 42.27/12.91 a__U62(x0) 42.27/12.91 a__U71(x0) 42.27/12.91 a__U81(x0) 42.27/12.91 a__U91(x0, x1, x2) 42.27/12.91 a__U92(x0, x1, x2) 42.27/12.91 a__U93(x0, x1, x2) 42.27/12.91 a__U94(x0, x1, x2) 42.27/12.91 a__U95(x0, x1) 42.27/12.91 a__U96(x0) 42.27/12.91 42.27/12.91 We have to consider all minimal (P,Q,R)-chains. 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (36) QDPOrderProof (EQUIVALENT) 42.27/12.91 We use the reduction pair processor [LPAR04,JAR06]. 42.27/12.91 42.27/12.91 42.27/12.91 The following pairs can be oriented strictly and are deleted. 42.27/12.91 42.27/12.91 A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) 42.27/12.91 The remaining pairs can at least be oriented weakly. 42.27/12.91 Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: 42.27/12.91 42.27/12.91 POL( A__LENGTH_1(x_1) ) = 2x_1 42.27/12.91 POL( mark_1(x_1) ) = x_1 42.27/12.91 POL( zeros ) = 0 42.27/12.91 POL( a__zeros ) = 0 42.27/12.91 POL( U101_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( a__U101_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( U102_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( a__U102_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( isNatKind_1(x_1) ) = 2 42.27/12.91 POL( a__isNatKind_1(x_1) ) = 2 42.27/12.91 POL( U103_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( a__U103_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( isNatIListKind_1(x_1) ) = 1 42.27/12.91 POL( a__isNatIListKind_1(x_1) ) = 1 42.27/12.91 POL( U104_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( a__U104_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( U105_2(x_1, x_2) ) = x_1 42.27/12.91 POL( a__U105_2(x_1, x_2) ) = x_1 42.27/12.91 POL( isNat_1(x_1) ) = 2x_1 42.27/12.91 POL( a__isNat_1(x_1) ) = 2x_1 42.27/12.91 POL( U106_1(x_1) ) = 1 42.27/12.91 POL( a__U106_1(x_1) ) = 1 42.27/12.91 POL( isNatIList_1(x_1) ) = 2 42.27/12.91 POL( a__isNatIList_1(x_1) ) = 2 42.27/12.91 POL( U11_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( a__U11_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( U12_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( a__U12_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( U111_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( a__U111_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( U112_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( a__U112_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( U113_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( a__U113_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( U114_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( a__U114_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( length_1(x_1) ) = x_1 42.27/12.91 POL( a__length_1(x_1) ) = x_1 42.27/12.91 POL( U13_1(x_1) ) = x_1 42.27/12.91 POL( a__U13_1(x_1) ) = x_1 42.27/12.91 POL( isNatList_1(x_1) ) = 2x_1 42.27/12.91 POL( a__isNatList_1(x_1) ) = 2x_1 42.27/12.91 POL( U121_2(x_1, x_2) ) = 2 42.27/12.91 POL( a__U121_2(x_1, x_2) ) = 2 42.27/12.91 POL( U122_1(x_1) ) = x_1 + 1 42.27/12.91 POL( a__U122_1(x_1) ) = x_1 + 1 42.27/12.91 POL( U131_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( a__U131_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( U132_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( a__U132_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( U133_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( a__U133_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( U134_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( a__U134_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( U135_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( a__U135_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( U136_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( a__U136_4(x_1, ..., x_4) ) = 2x_3 42.27/12.91 POL( take_2(x_1, x_2) ) = x_1 42.27/12.91 POL( a__take_2(x_1, x_2) ) = x_1 42.27/12.91 POL( U21_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( a__U21_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( U22_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( a__U22_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( U23_1(x_1) ) = x_1 42.27/12.91 POL( a__U23_1(x_1) ) = x_1 42.27/12.91 POL( U31_2(x_1, x_2) ) = 1 42.27/12.91 POL( a__U31_2(x_1, x_2) ) = 1 42.27/12.91 POL( U32_2(x_1, x_2) ) = 1 42.27/12.91 POL( a__U32_2(x_1, x_2) ) = 1 42.27/12.91 POL( U33_1(x_1) ) = 1 42.27/12.91 POL( a__U33_1(x_1) ) = 1 42.27/12.91 POL( U41_3(x_1, ..., x_3) ) = 2 42.27/12.91 POL( a__U41_3(x_1, ..., x_3) ) = 2 42.27/12.91 POL( U42_3(x_1, ..., x_3) ) = 2 42.27/12.91 POL( a__U42_3(x_1, ..., x_3) ) = 2 42.27/12.91 POL( U43_3(x_1, ..., x_3) ) = 2 42.27/12.91 POL( a__U43_3(x_1, ..., x_3) ) = 2 42.27/12.91 POL( U44_3(x_1, ..., x_3) ) = 2 42.27/12.91 POL( a__U44_3(x_1, ..., x_3) ) = 2 42.27/12.91 POL( U45_2(x_1, x_2) ) = 2 42.27/12.91 POL( a__U45_2(x_1, x_2) ) = 2 42.27/12.91 POL( U46_1(x_1) ) = 2 42.27/12.91 POL( a__U46_1(x_1) ) = 2 42.27/12.91 POL( U51_2(x_1, x_2) ) = 1 42.27/12.91 POL( a__U51_2(x_1, x_2) ) = 1 42.27/12.91 POL( U52_1(x_1) ) = 1 42.27/12.91 POL( a__U52_1(x_1) ) = 1 42.27/12.91 POL( U61_2(x_1, x_2) ) = 1 42.27/12.91 POL( a__U61_2(x_1, x_2) ) = 1 42.27/12.91 POL( U62_1(x_1) ) = 1 42.27/12.91 POL( a__U62_1(x_1) ) = 1 42.27/12.91 POL( U71_1(x_1) ) = 1 42.27/12.91 POL( a__U71_1(x_1) ) = 1 42.27/12.91 POL( U81_1(x_1) ) = 1 42.27/12.91 POL( a__U81_1(x_1) ) = 1 42.27/12.91 POL( U91_3(x_1, ..., x_3) ) = 2x_3 42.27/12.91 POL( a__U91_3(x_1, ..., x_3) ) = 2x_3 42.27/12.91 POL( U92_3(x_1, ..., x_3) ) = 2x_3 42.27/12.91 POL( a__U92_3(x_1, ..., x_3) ) = 2x_3 42.27/12.91 POL( U93_3(x_1, ..., x_3) ) = 2x_3 42.27/12.91 POL( a__U93_3(x_1, ..., x_3) ) = 2x_3 42.27/12.91 POL( U94_3(x_1, ..., x_3) ) = 2x_3 42.27/12.91 POL( a__U94_3(x_1, ..., x_3) ) = 2x_3 42.27/12.91 POL( U95_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( a__U95_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( U96_1(x_1) ) = x_1 42.27/12.91 POL( a__U96_1(x_1) ) = x_1 42.27/12.91 POL( cons_2(x_1, x_2) ) = 2x_2 42.27/12.91 POL( 0 ) = 2 42.27/12.91 POL( tt ) = 1 42.27/12.91 POL( s_1(x_1) ) = 2x_1 42.27/12.91 POL( nil ) = 2 42.27/12.91 POL( A__U111_3(x_1, ..., x_3) ) = x_1 + 2x_2 42.27/12.91 POL( A__U113_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( A__U112_3(x_1, ..., x_3) ) = 2x_2 42.27/12.91 POL( A__U114_2(x_1, x_2) ) = 2x_2 42.27/12.91 42.27/12.91 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 42.27/12.91 42.27/12.91 mark(zeros) -> a__zeros 42.27/12.91 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.27/12.91 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.27/12.91 mark(isNatKind(X)) -> a__isNatKind(X) 42.27/12.91 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.27/12.91 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.27/12.91 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.27/12.91 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.27/12.91 mark(isNat(X)) -> a__isNat(X) 42.27/12.91 mark(U106(X)) -> a__U106(mark(X)) 42.27/12.91 mark(isNatIList(X)) -> a__isNatIList(X) 42.27/12.91 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.27/12.91 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.27/12.91 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.27/12.91 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.27/12.91 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.27/12.91 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.27/12.91 mark(length(X)) -> a__length(mark(X)) 42.27/12.91 mark(U13(X)) -> a__U13(mark(X)) 42.27/12.91 mark(isNatList(X)) -> a__isNatList(X) 42.27/12.91 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.27/12.91 mark(U122(X)) -> a__U122(mark(X)) 42.27/12.91 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.27/12.91 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.27/12.91 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.27/12.91 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.27/12.91 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.27/12.91 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.27/12.91 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.27/12.91 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.27/12.91 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.27/12.91 mark(U23(X)) -> a__U23(mark(X)) 42.27/12.91 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.27/12.91 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.27/12.91 mark(U33(X)) -> a__U33(mark(X)) 42.27/12.91 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.27/12.91 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.27/12.91 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.27/12.91 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.27/12.91 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.27/12.91 mark(U46(X)) -> a__U46(mark(X)) 42.27/12.91 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.27/12.91 mark(U52(X)) -> a__U52(mark(X)) 42.27/12.91 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.27/12.91 mark(U62(X)) -> a__U62(mark(X)) 42.27/12.91 mark(U71(X)) -> a__U71(mark(X)) 42.27/12.91 mark(U81(X)) -> a__U81(mark(X)) 42.27/12.91 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.27/12.91 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.27/12.91 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.27/12.91 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.27/12.91 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.27/12.91 mark(U96(X)) -> a__U96(mark(X)) 42.27/12.91 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.27/12.91 mark(0) -> 0 42.27/12.91 mark(tt) -> tt 42.27/12.91 mark(s(X)) -> s(mark(X)) 42.27/12.91 mark(nil) -> nil 42.27/12.91 a__isNatList(nil) -> tt 42.27/12.91 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatList(X) -> isNatList(X) 42.27/12.91 a__isNatIListKind(nil) -> tt 42.27/12.91 a__isNatIListKind(zeros) -> tt 42.27/12.91 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.27/12.91 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.27/12.91 a__isNatIListKind(X) -> isNatIListKind(X) 42.27/12.91 a__isNat(0) -> tt 42.27/12.91 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.27/12.91 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.27/12.91 a__isNat(X) -> isNat(X) 42.27/12.91 a__isNatKind(0) -> tt 42.27/12.91 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.27/12.91 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.27/12.91 a__isNatKind(X) -> isNatKind(X) 42.27/12.91 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.27/12.91 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.27/12.91 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.27/12.91 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.27/12.91 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.27/12.91 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.27/12.91 a__U105(X1, X2) -> U105(X1, X2) 42.27/12.91 a__U106(tt) -> tt 42.27/12.91 a__U106(X) -> U106(X) 42.27/12.91 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.27/12.91 a__U11(X1, X2) -> U11(X1, X2) 42.27/12.91 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.27/12.91 a__U12(X1, X2) -> U12(X1, X2) 42.27/12.91 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.27/12.91 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.27/12.91 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.27/12.91 a__U114(X1, X2) -> U114(X1, X2) 42.27/12.91 a__length(nil) -> 0 42.27/12.91 a__length(X) -> length(X) 42.27/12.91 a__U13(tt) -> tt 42.27/12.91 a__U13(X) -> U13(X) 42.27/12.91 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.27/12.91 a__U121(X1, X2) -> U121(X1, X2) 42.27/12.91 a__U122(tt) -> nil 42.27/12.91 a__U122(X) -> U122(X) 42.27/12.91 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.27/12.91 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.27/12.91 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.27/12.91 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.27/12.91 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.27/12.91 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.27/12.91 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.27/12.91 a__take(X1, X2) -> take(X1, X2) 42.27/12.91 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.27/12.91 a__U21(X1, X2) -> U21(X1, X2) 42.27/12.91 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.27/12.91 a__U22(X1, X2) -> U22(X1, X2) 42.27/12.91 a__U23(tt) -> tt 42.27/12.91 a__U23(X) -> U23(X) 42.27/12.91 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.27/12.91 a__U31(X1, X2) -> U31(X1, X2) 42.27/12.91 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.27/12.91 a__U32(X1, X2) -> U32(X1, X2) 42.27/12.91 a__U33(tt) -> tt 42.27/12.91 a__U33(X) -> U33(X) 42.27/12.91 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.27/12.91 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.27/12.91 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.27/12.91 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.27/12.91 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.27/12.91 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.27/12.91 a__U45(X1, X2) -> U45(X1, X2) 42.27/12.91 a__U46(tt) -> tt 42.27/12.91 a__U46(X) -> U46(X) 42.27/12.91 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.27/12.91 a__U51(X1, X2) -> U51(X1, X2) 42.27/12.91 a__U52(tt) -> tt 42.27/12.91 a__U52(X) -> U52(X) 42.27/12.91 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.27/12.91 a__U61(X1, X2) -> U61(X1, X2) 42.27/12.91 a__U62(tt) -> tt 42.27/12.91 a__U62(X) -> U62(X) 42.27/12.91 a__U71(tt) -> tt 42.27/12.91 a__U71(X) -> U71(X) 42.27/12.91 a__U81(tt) -> tt 42.27/12.91 a__U81(X) -> U81(X) 42.27/12.91 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.27/12.91 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.27/12.91 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.27/12.91 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.27/12.91 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.27/12.91 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.27/12.91 a__U95(X1, X2) -> U95(X1, X2) 42.27/12.91 a__U96(tt) -> tt 42.27/12.91 a__U96(X) -> U96(X) 42.27/12.91 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.27/12.91 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.27/12.91 a__isNatIList(zeros) -> tt 42.27/12.91 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatIList(X) -> isNatIList(X) 42.27/12.91 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.27/12.91 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.27/12.91 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.27/12.91 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.27/12.91 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.27/12.91 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.27/12.91 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.27/12.91 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.27/12.91 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.27/12.91 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.27/12.91 a__U114(tt, L) -> s(a__length(mark(L))) 42.27/12.91 a__zeros -> cons(0, zeros) 42.27/12.91 a__zeros -> zeros 42.27/12.91 42.27/12.91 42.27/12.91 ---------------------------------------- 42.27/12.91 42.27/12.91 (37) 42.27/12.91 Obligation: 42.27/12.91 Q DP problem: 42.27/12.91 The TRS P consists of the following rules: 42.27/12.91 42.27/12.91 A__U114(tt, L) -> A__LENGTH(mark(L)) 42.27/12.91 A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) 42.27/12.91 A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) 42.27/12.91 A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) 42.27/12.91 42.27/12.91 The TRS R consists of the following rules: 42.27/12.91 42.27/12.91 a__zeros -> cons(0, zeros) 42.27/12.91 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 42.27/12.91 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 42.27/12.91 a__U106(tt) -> tt 42.27/12.91 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 42.27/12.91 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 42.27/12.91 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 42.27/12.91 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 42.27/12.91 a__U114(tt, L) -> s(a__length(mark(L))) 42.27/12.91 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 42.27/12.91 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 42.27/12.91 a__U122(tt) -> nil 42.27/12.91 a__U13(tt) -> tt 42.27/12.91 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 42.27/12.91 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 42.27/12.91 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 42.27/12.91 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 42.27/12.91 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 42.27/12.91 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 42.27/12.91 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 42.27/12.91 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 42.27/12.91 a__U23(tt) -> tt 42.27/12.91 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 42.27/12.91 a__U32(tt, V) -> a__U33(a__isNatList(V)) 42.27/12.91 a__U33(tt) -> tt 42.27/12.91 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 42.27/12.91 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 42.27/12.91 a__U46(tt) -> tt 42.27/12.91 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 42.27/12.91 a__U52(tt) -> tt 42.27/12.91 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 42.27/12.91 a__U62(tt) -> tt 42.27/12.91 a__U71(tt) -> tt 42.27/12.91 a__U81(tt) -> tt 42.27/12.91 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 42.27/12.91 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 42.27/12.91 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 42.27/12.91 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 42.27/12.91 a__U96(tt) -> tt 42.27/12.91 a__isNat(0) -> tt 42.27/12.91 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 42.27/12.91 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 42.27/12.91 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 42.27/12.91 a__isNatIList(zeros) -> tt 42.27/12.91 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatIListKind(nil) -> tt 42.27/12.91 a__isNatIListKind(zeros) -> tt 42.27/12.91 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 42.27/12.91 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 42.27/12.91 a__isNatKind(0) -> tt 42.27/12.91 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 42.27/12.91 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 42.27/12.91 a__isNatList(nil) -> tt 42.27/12.91 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 42.27/12.91 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 42.27/12.91 a__length(nil) -> 0 42.27/12.91 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 42.27/12.91 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 42.27/12.91 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 42.27/12.91 mark(zeros) -> a__zeros 42.27/12.91 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 42.27/12.91 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 42.27/12.91 mark(isNatKind(X)) -> a__isNatKind(X) 42.27/12.91 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 42.27/12.91 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 42.27/12.91 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 42.27/12.91 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 42.27/12.91 mark(isNat(X)) -> a__isNat(X) 42.27/12.91 mark(U106(X)) -> a__U106(mark(X)) 42.27/12.91 mark(isNatIList(X)) -> a__isNatIList(X) 42.27/12.91 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 42.27/12.91 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 42.27/12.91 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 42.27/12.91 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 42.27/12.91 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 42.27/12.91 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 42.27/12.91 mark(length(X)) -> a__length(mark(X)) 42.27/12.91 mark(U13(X)) -> a__U13(mark(X)) 42.27/12.91 mark(isNatList(X)) -> a__isNatList(X) 42.27/12.91 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 42.27/12.91 mark(U122(X)) -> a__U122(mark(X)) 42.27/12.91 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 42.27/12.91 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 42.27/12.91 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 42.27/12.91 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 42.27/12.91 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 42.27/12.91 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 42.27/12.91 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 42.27/12.91 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 42.27/12.91 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 42.27/12.91 mark(U23(X)) -> a__U23(mark(X)) 42.27/12.91 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 42.27/12.91 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 42.27/12.92 mark(U33(X)) -> a__U33(mark(X)) 42.27/12.92 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 42.27/12.92 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 42.27/12.92 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 42.27/12.92 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 42.27/12.92 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 42.27/12.92 mark(U46(X)) -> a__U46(mark(X)) 42.27/12.92 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 42.27/12.92 mark(U52(X)) -> a__U52(mark(X)) 42.27/12.92 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 42.27/12.92 mark(U62(X)) -> a__U62(mark(X)) 42.27/12.92 mark(U71(X)) -> a__U71(mark(X)) 42.27/12.92 mark(U81(X)) -> a__U81(mark(X)) 42.27/12.92 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 42.27/12.92 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 42.27/12.92 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 42.27/12.92 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 42.27/12.92 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 42.27/12.92 mark(U96(X)) -> a__U96(mark(X)) 42.27/12.92 mark(cons(X1, X2)) -> cons(mark(X1), X2) 42.27/12.92 mark(0) -> 0 42.27/12.92 mark(tt) -> tt 42.27/12.92 mark(s(X)) -> s(mark(X)) 42.27/12.92 mark(nil) -> nil 42.27/12.92 a__zeros -> zeros 42.27/12.92 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 42.27/12.92 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 42.27/12.92 a__isNatKind(X) -> isNatKind(X) 42.27/12.92 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 42.27/12.92 a__isNatIListKind(X) -> isNatIListKind(X) 42.27/12.92 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 42.27/12.92 a__U105(X1, X2) -> U105(X1, X2) 42.27/12.92 a__isNat(X) -> isNat(X) 42.27/12.92 a__U106(X) -> U106(X) 42.27/12.92 a__isNatIList(X) -> isNatIList(X) 42.27/12.92 a__U11(X1, X2) -> U11(X1, X2) 42.27/12.92 a__U12(X1, X2) -> U12(X1, X2) 42.27/12.92 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 42.27/12.92 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 42.27/12.92 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 42.27/12.92 a__U114(X1, X2) -> U114(X1, X2) 42.27/12.92 a__length(X) -> length(X) 42.27/12.92 a__U13(X) -> U13(X) 42.27/12.92 a__isNatList(X) -> isNatList(X) 42.27/12.92 a__U121(X1, X2) -> U121(X1, X2) 42.27/12.92 a__U122(X) -> U122(X) 42.27/12.92 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 42.27/12.92 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 42.27/12.92 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 42.27/12.92 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 42.27/12.92 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 42.27/12.92 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 42.27/12.92 a__take(X1, X2) -> take(X1, X2) 42.27/12.92 a__U21(X1, X2) -> U21(X1, X2) 42.27/12.92 a__U22(X1, X2) -> U22(X1, X2) 42.27/12.92 a__U23(X) -> U23(X) 42.27/12.92 a__U31(X1, X2) -> U31(X1, X2) 42.27/12.92 a__U32(X1, X2) -> U32(X1, X2) 42.27/12.92 a__U33(X) -> U33(X) 42.27/12.92 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 42.27/12.92 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 42.27/12.92 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 42.27/12.92 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 42.27/12.92 a__U45(X1, X2) -> U45(X1, X2) 42.27/12.92 a__U46(X) -> U46(X) 42.27/12.92 a__U51(X1, X2) -> U51(X1, X2) 42.27/12.92 a__U52(X) -> U52(X) 42.27/12.92 a__U61(X1, X2) -> U61(X1, X2) 42.27/12.92 a__U62(X) -> U62(X) 42.27/12.92 a__U71(X) -> U71(X) 42.27/12.92 a__U81(X) -> U81(X) 42.27/12.92 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 42.27/12.92 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 42.27/12.92 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 42.27/12.92 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 42.27/12.92 a__U95(X1, X2) -> U95(X1, X2) 42.27/12.92 a__U96(X) -> U96(X) 42.27/12.92 42.27/12.92 The set Q consists of the following terms: 42.27/12.92 42.27/12.92 a__zeros 42.27/12.92 a__isNatIList(x0) 42.27/12.92 mark(zeros) 42.27/12.92 mark(U101(x0, x1, x2)) 42.27/12.92 mark(U102(x0, x1, x2)) 42.27/12.92 mark(isNatKind(x0)) 42.27/12.92 mark(U103(x0, x1, x2)) 42.27/12.92 mark(isNatIListKind(x0)) 42.27/12.92 mark(U104(x0, x1, x2)) 42.27/12.92 mark(U105(x0, x1)) 42.27/12.92 mark(isNat(x0)) 42.27/12.92 mark(U106(x0)) 42.27/12.92 mark(isNatIList(x0)) 42.27/12.92 mark(U11(x0, x1)) 42.27/12.92 mark(U12(x0, x1)) 42.27/12.92 mark(U111(x0, x1, x2)) 42.27/12.92 mark(U112(x0, x1, x2)) 42.27/12.92 mark(U113(x0, x1, x2)) 42.27/12.92 mark(U114(x0, x1)) 42.27/12.92 mark(length(x0)) 42.27/12.92 mark(U13(x0)) 42.27/12.92 mark(isNatList(x0)) 42.27/12.92 mark(U121(x0, x1)) 42.27/12.92 mark(U122(x0)) 42.27/12.92 mark(U131(x0, x1, x2, x3)) 42.27/12.92 mark(U132(x0, x1, x2, x3)) 42.27/12.92 mark(U133(x0, x1, x2, x3)) 42.27/12.92 mark(U134(x0, x1, x2, x3)) 42.27/12.92 mark(U135(x0, x1, x2, x3)) 42.27/12.92 mark(U136(x0, x1, x2, x3)) 42.27/12.92 mark(take(x0, x1)) 42.27/12.92 mark(U21(x0, x1)) 42.27/12.92 mark(U22(x0, x1)) 42.27/12.92 mark(U23(x0)) 42.27/12.92 mark(U31(x0, x1)) 42.27/12.92 mark(U32(x0, x1)) 42.27/12.92 mark(U33(x0)) 42.27/12.92 mark(U41(x0, x1, x2)) 42.27/12.92 mark(U42(x0, x1, x2)) 42.27/12.92 mark(U43(x0, x1, x2)) 42.27/12.92 mark(U44(x0, x1, x2)) 42.27/12.92 mark(U45(x0, x1)) 42.27/12.92 mark(U46(x0)) 42.27/12.92 mark(U51(x0, x1)) 42.27/12.92 mark(U52(x0)) 42.27/12.92 mark(U61(x0, x1)) 42.27/12.92 mark(U62(x0)) 42.27/12.92 mark(U71(x0)) 42.27/12.92 mark(U81(x0)) 42.27/12.92 mark(U91(x0, x1, x2)) 42.27/12.92 mark(U92(x0, x1, x2)) 42.27/12.92 mark(U93(x0, x1, x2)) 42.27/12.92 mark(U94(x0, x1, x2)) 42.27/12.92 mark(U95(x0, x1)) 42.27/12.92 mark(U96(x0)) 42.27/12.92 mark(cons(x0, x1)) 42.27/12.92 mark(0) 42.27/12.92 mark(tt) 42.27/12.92 mark(s(x0)) 42.27/12.92 mark(nil) 42.27/12.92 a__U101(x0, x1, x2) 42.27/12.92 a__U102(x0, x1, x2) 42.27/12.92 a__isNatKind(x0) 42.27/12.92 a__U103(x0, x1, x2) 42.27/12.92 a__isNatIListKind(x0) 42.27/12.92 a__U104(x0, x1, x2) 42.27/12.92 a__U105(x0, x1) 42.27/12.92 a__isNat(x0) 42.27/12.92 a__U106(x0) 42.27/12.92 a__U11(x0, x1) 42.27/12.92 a__U12(x0, x1) 42.27/12.92 a__U111(x0, x1, x2) 42.27/12.92 a__U112(x0, x1, x2) 42.27/12.92 a__U113(x0, x1, x2) 42.27/12.92 a__U114(x0, x1) 42.27/12.92 a__length(x0) 42.27/12.92 a__U13(x0) 42.27/12.92 a__isNatList(x0) 42.27/12.92 a__U121(x0, x1) 42.27/12.92 a__U122(x0) 42.27/12.92 a__U131(x0, x1, x2, x3) 42.27/12.92 a__U132(x0, x1, x2, x3) 42.27/12.92 a__U133(x0, x1, x2, x3) 42.27/12.92 a__U134(x0, x1, x2, x3) 42.27/12.92 a__U135(x0, x1, x2, x3) 42.27/12.92 a__U136(x0, x1, x2, x3) 42.27/12.92 a__take(x0, x1) 42.27/12.92 a__U21(x0, x1) 42.27/12.92 a__U22(x0, x1) 42.27/12.92 a__U23(x0) 42.27/12.92 a__U31(x0, x1) 42.27/12.92 a__U32(x0, x1) 42.27/12.92 a__U33(x0) 42.27/12.92 a__U41(x0, x1, x2) 42.27/12.92 a__U42(x0, x1, x2) 42.27/12.92 a__U43(x0, x1, x2) 42.27/12.92 a__U44(x0, x1, x2) 42.27/12.92 a__U45(x0, x1) 42.27/12.92 a__U46(x0) 42.27/12.92 a__U51(x0, x1) 42.27/12.92 a__U52(x0) 42.27/12.92 a__U61(x0, x1) 42.27/12.92 a__U62(x0) 42.27/12.92 a__U71(x0) 42.27/12.92 a__U81(x0) 42.27/12.92 a__U91(x0, x1, x2) 42.27/12.92 a__U92(x0, x1, x2) 42.27/12.92 a__U93(x0, x1, x2) 42.27/12.92 a__U94(x0, x1, x2) 42.27/12.92 a__U95(x0, x1) 42.27/12.92 a__U96(x0) 42.27/12.92 42.27/12.92 We have to consider all minimal (P,Q,R)-chains. 42.27/12.92 ---------------------------------------- 42.27/12.92 42.27/12.92 (38) DependencyGraphProof (EQUIVALENT) 42.27/12.92 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. 42.27/12.92 ---------------------------------------- 42.27/12.92 42.27/12.92 (39) 42.27/12.92 TRUE 42.33/13.01 EOF