/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 193 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) QDP (6) UsableRulesProof [EQUIVALENT, 0 ms] (7) QDP (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] (9) YES (10) QDP (11) QDPSizeChangeProof [EQUIVALENT, 15 ms] (12) YES (13) QDP (14) QDPOrderProof [EQUIVALENT, 923 ms] (15) QDP (16) DependencyGraphProof [EQUIVALENT, 0 ms] (17) QDP (18) QDPOrderProof [EQUIVALENT, 602 ms] (19) QDP (20) DependencyGraphProof [EQUIVALENT, 0 ms] (21) AND (22) QDP (23) UsableRulesProof [EQUIVALENT, 27 ms] (24) QDP (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] (26) YES (27) QDP (28) QDPOrderProof [EQUIVALENT, 794 ms] (29) QDP (30) DependencyGraphProof [EQUIVALENT, 0 ms] (31) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: A__U101(tt, V1, V2) -> A__U102(a__isNatKind(V1), V1, V2) A__U101(tt, V1, V2) -> A__ISNATKIND(V1) A__U102(tt, V1, V2) -> A__U103(a__isNatIListKind(V2), V1, V2) A__U102(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U103(tt, V1, V2) -> A__U104(a__isNatIListKind(V2), V1, V2) A__U103(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U104(tt, V1, V2) -> A__U105(a__isNat(V1), V2) A__U104(tt, V1, V2) -> A__ISNAT(V1) A__U105(tt, V2) -> A__U106(a__isNatIList(V2)) A__U105(tt, V2) -> A__ISNATILIST(V2) A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) A__U11(tt, V1) -> A__ISNATILISTKIND(V1) A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) A__U111(tt, L, N) -> A__ISNATILISTKIND(L) A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) A__U112(tt, L, N) -> A__ISNAT(N) A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) A__U113(tt, L, N) -> A__ISNATKIND(N) A__U114(tt, L) -> A__LENGTH(mark(L)) A__U114(tt, L) -> MARK(L) A__U12(tt, V1) -> A__U13(a__isNatList(V1)) A__U12(tt, V1) -> A__ISNATLIST(V1) A__U121(tt, IL) -> A__U122(a__isNatIListKind(IL)) A__U121(tt, IL) -> A__ISNATILISTKIND(IL) A__U131(tt, IL, M, N) -> A__U132(a__isNatIListKind(IL), IL, M, N) A__U131(tt, IL, M, N) -> A__ISNATILISTKIND(IL) A__U132(tt, IL, M, N) -> A__U133(a__isNat(M), IL, M, N) A__U132(tt, IL, M, N) -> A__ISNAT(M) A__U133(tt, IL, M, N) -> A__U134(a__isNatKind(M), IL, M, N) A__U133(tt, IL, M, N) -> A__ISNATKIND(M) A__U134(tt, IL, M, N) -> A__U135(a__isNat(N), IL, M, N) A__U134(tt, IL, M, N) -> A__ISNAT(N) A__U135(tt, IL, M, N) -> A__U136(a__isNatKind(N), IL, M, N) A__U135(tt, IL, M, N) -> A__ISNATKIND(N) A__U136(tt, IL, M, N) -> MARK(N) A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) A__U21(tt, V1) -> A__ISNATKIND(V1) A__U22(tt, V1) -> A__U23(a__isNat(V1)) A__U22(tt, V1) -> A__ISNAT(V1) A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) A__U31(tt, V) -> A__ISNATILISTKIND(V) A__U32(tt, V) -> A__U33(a__isNatList(V)) A__U32(tt, V) -> A__ISNATLIST(V) A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) A__U41(tt, V1, V2) -> A__ISNATKIND(V1) A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) A__U42(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) A__U43(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) A__U44(tt, V1, V2) -> A__ISNAT(V1) A__U45(tt, V2) -> A__U46(a__isNatIList(V2)) A__U45(tt, V2) -> A__ISNATILIST(V2) A__U51(tt, V2) -> A__U52(a__isNatIListKind(V2)) A__U51(tt, V2) -> A__ISNATILISTKIND(V2) A__U61(tt, V2) -> A__U62(a__isNatIListKind(V2)) A__U61(tt, V2) -> A__ISNATILISTKIND(V2) A__U91(tt, V1, V2) -> A__U92(a__isNatKind(V1), V1, V2) A__U91(tt, V1, V2) -> A__ISNATKIND(V1) A__U92(tt, V1, V2) -> A__U93(a__isNatIListKind(V2), V1, V2) A__U92(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U93(tt, V1, V2) -> A__U94(a__isNatIListKind(V2), V1, V2) A__U93(tt, V1, V2) -> A__ISNATILISTKIND(V2) A__U94(tt, V1, V2) -> A__U95(a__isNat(V1), V2) A__U94(tt, V1, V2) -> A__ISNAT(V1) A__U95(tt, V2) -> A__U96(a__isNatList(V2)) A__U95(tt, V2) -> A__ISNATLIST(V2) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) A__ISNAT(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__ISNAT(s(V1)) -> A__ISNATKIND(V1) A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) A__ISNATILIST(V) -> A__ISNATILISTKIND(V) A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) A__ISNATILIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATILISTKIND(take(V1, V2)) -> A__U61(a__isNatKind(V1), V2) A__ISNATILISTKIND(take(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__U71(a__isNatIListKind(V1)) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATKIND(s(V1)) -> A__U81(a__isNatKind(V1)) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) A__ISNATLIST(cons(V1, V2)) -> A__U91(a__isNatKind(V1), V1, V2) A__ISNATLIST(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATLIST(take(V1, V2)) -> A__U101(a__isNatKind(V1), V1, V2) A__ISNATLIST(take(V1, V2)) -> A__ISNATKIND(V1) A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) A__LENGTH(cons(N, L)) -> A__ISNATLIST(L) A__TAKE(0, IL) -> A__U121(a__isNatIList(IL), IL) A__TAKE(0, IL) -> A__ISNATILIST(IL) A__TAKE(s(M), cons(N, IL)) -> A__U131(a__isNatIList(IL), IL, M, N) A__TAKE(s(M), cons(N, IL)) -> A__ISNATILIST(IL) MARK(zeros) -> A__ZEROS MARK(U101(X1, X2, X3)) -> A__U101(mark(X1), X2, X3) MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U102(X1, X2, X3)) -> A__U102(mark(X1), X2, X3) MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(isNatKind(X)) -> A__ISNATKIND(X) MARK(U103(X1, X2, X3)) -> A__U103(mark(X1), X2, X3) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(isNatIListKind(X)) -> A__ISNATILISTKIND(X) MARK(U104(X1, X2, X3)) -> A__U104(mark(X1), X2, X3) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(U105(X1, X2)) -> A__U105(mark(X1), X2) MARK(U105(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> A__ISNAT(X) MARK(U106(X)) -> A__U106(mark(X)) MARK(U106(X)) -> MARK(X) MARK(isNatIList(X)) -> A__ISNATILIST(X) MARK(U11(X1, X2)) -> A__U11(mark(X1), X2) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> A__U12(mark(X1), X2) MARK(U12(X1, X2)) -> MARK(X1) MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) MARK(U111(X1, X2, X3)) -> MARK(X1) MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) MARK(U112(X1, X2, X3)) -> MARK(X1) MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) MARK(U113(X1, X2, X3)) -> MARK(X1) MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) MARK(U114(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(U13(X)) -> A__U13(mark(X)) MARK(U13(X)) -> MARK(X) MARK(isNatList(X)) -> A__ISNATLIST(X) MARK(U121(X1, X2)) -> A__U121(mark(X1), X2) MARK(U121(X1, X2)) -> MARK(X1) MARK(U122(X)) -> A__U122(mark(X)) MARK(U122(X)) -> MARK(X) MARK(U131(X1, X2, X3, X4)) -> A__U131(mark(X1), X2, X3, X4) MARK(U131(X1, X2, X3, X4)) -> MARK(X1) MARK(U132(X1, X2, X3, X4)) -> A__U132(mark(X1), X2, X3, X4) MARK(U132(X1, X2, X3, X4)) -> MARK(X1) MARK(U133(X1, X2, X3, X4)) -> A__U133(mark(X1), X2, X3, X4) MARK(U133(X1, X2, X3, X4)) -> MARK(X1) MARK(U134(X1, X2, X3, X4)) -> A__U134(mark(X1), X2, X3, X4) MARK(U134(X1, X2, X3, X4)) -> MARK(X1) MARK(U135(X1, X2, X3, X4)) -> A__U135(mark(X1), X2, X3, X4) MARK(U135(X1, X2, X3, X4)) -> MARK(X1) MARK(U136(X1, X2, X3, X4)) -> A__U136(mark(X1), X2, X3, X4) MARK(U136(X1, X2, X3, X4)) -> MARK(X1) MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) MARK(U21(X1, X2)) -> A__U21(mark(X1), X2) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> A__U22(mark(X1), X2) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> A__U23(mark(X)) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> A__U31(mark(X1), X2) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> A__U32(mark(X1), X2) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> A__U33(mark(X)) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> A__U41(mark(X1), X2, X3) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> A__U42(mark(X1), X2, X3) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> A__U43(mark(X1), X2, X3) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> A__U44(mark(X1), X2, X3) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> A__U45(mark(X1), X2) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> A__U46(mark(X)) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> A__U51(mark(X1), X2) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> A__U52(mark(X)) MARK(U52(X)) -> MARK(X) MARK(U61(X1, X2)) -> A__U61(mark(X1), X2) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> A__U62(mark(X)) MARK(U62(X)) -> MARK(X) MARK(U71(X)) -> A__U71(mark(X)) MARK(U71(X)) -> MARK(X) MARK(U81(X)) -> A__U81(mark(X)) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3)) -> A__U91(mark(X1), X2, X3) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> A__U92(mark(X1), X2, X3) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> A__U93(mark(X1), X2, X3) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2, X3)) -> A__U94(mark(X1), X2, X3) MARK(U94(X1, X2, X3)) -> MARK(X1) MARK(U95(X1, X2)) -> A__U95(mark(X1), X2) MARK(U95(X1, X2)) -> MARK(X1) MARK(U96(X)) -> A__U96(mark(X)) MARK(U96(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 83 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Q DP problem: The TRS P consists of the following rules: A__U51(tt, V2) -> A__ISNATILISTKIND(V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATILISTKIND(take(V1, V2)) -> A__U61(a__isNatKind(V1), V2) A__U61(tt, V2) -> A__ISNATILISTKIND(V2) A__ISNATILISTKIND(take(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (6) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: A__U51(tt, V2) -> A__ISNATILISTKIND(V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) A__ISNATILISTKIND(take(V1, V2)) -> A__U61(a__isNatKind(V1), V2) A__U61(tt, V2) -> A__ISNATILISTKIND(V2) A__ISNATILISTKIND(take(V1, V2)) -> A__ISNATKIND(V1) A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) The TRS R consists of the following rules: a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__U81(tt) -> tt a__U81(X) -> U81(X) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatIListKind(X) -> isNatIListKind(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U61(X1, X2) -> U61(X1, X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1, X2) -> U51(X1, X2) a__U52(tt) -> tt a__U52(X) -> U52(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *A__ISNATILISTKIND(cons(V1, V2)) -> A__U51(a__isNatKind(V1), V2) The graph contains the following edges 1 > 2 *A__U51(tt, V2) -> A__ISNATILISTKIND(V2) The graph contains the following edges 2 >= 1 *A__ISNATKIND(s(V1)) -> A__ISNATKIND(V1) The graph contains the following edges 1 > 1 *A__ISNATKIND(length(V1)) -> A__ISNATILISTKIND(V1) The graph contains the following edges 1 > 1 *A__U61(tt, V2) -> A__ISNATILISTKIND(V2) The graph contains the following edges 2 >= 1 *A__ISNATILISTKIND(take(V1, V2)) -> A__U61(a__isNatKind(V1), V2) The graph contains the following edges 1 > 2 *A__ISNATILISTKIND(cons(V1, V2)) -> A__ISNATKIND(V1) The graph contains the following edges 1 > 1 *A__ISNATILISTKIND(take(V1, V2)) -> A__ISNATKIND(V1) The graph contains the following edges 1 > 1 ---------------------------------------- (9) YES ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: A__U102(tt, V1, V2) -> A__U103(a__isNatIListKind(V2), V1, V2) A__U103(tt, V1, V2) -> A__U104(a__isNatIListKind(V2), V1, V2) A__U104(tt, V1, V2) -> A__U105(a__isNat(V1), V2) A__U105(tt, V2) -> A__ISNATILIST(V2) A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) A__U32(tt, V) -> A__ISNATLIST(V) A__ISNATLIST(cons(V1, V2)) -> A__U91(a__isNatKind(V1), V1, V2) A__U91(tt, V1, V2) -> A__U92(a__isNatKind(V1), V1, V2) A__U92(tt, V1, V2) -> A__U93(a__isNatIListKind(V2), V1, V2) A__U93(tt, V1, V2) -> A__U94(a__isNatIListKind(V2), V1, V2) A__U94(tt, V1, V2) -> A__U95(a__isNat(V1), V2) A__U95(tt, V2) -> A__ISNATLIST(V2) A__ISNATLIST(take(V1, V2)) -> A__U101(a__isNatKind(V1), V1, V2) A__U101(tt, V1, V2) -> A__U102(a__isNatKind(V1), V1, V2) A__U94(tt, V1, V2) -> A__ISNAT(V1) A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) A__U12(tt, V1) -> A__ISNATLIST(V1) A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) A__U22(tt, V1) -> A__ISNAT(V1) A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) A__U45(tt, V2) -> A__ISNATILIST(V2) A__U44(tt, V1, V2) -> A__ISNAT(V1) A__U104(tt, V1, V2) -> A__ISNAT(V1) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *A__U103(tt, V1, V2) -> A__U104(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U101(tt, V1, V2) -> A__U102(a__isNatKind(V1), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U102(tt, V1, V2) -> A__U103(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U105(tt, V2) -> A__ISNATILIST(V2) The graph contains the following edges 2 >= 1 *A__U104(tt, V1, V2) -> A__U105(a__isNat(V1), V2) The graph contains the following edges 3 >= 2 *A__U104(tt, V1, V2) -> A__ISNAT(V1) The graph contains the following edges 2 >= 1 *A__U31(tt, V) -> A__U32(a__isNatIListKind(V), V) The graph contains the following edges 2 >= 2 *A__U45(tt, V2) -> A__ISNATILIST(V2) The graph contains the following edges 2 >= 1 *A__ISNATILIST(V) -> A__U31(a__isNatIListKind(V), V) The graph contains the following edges 1 >= 2 *A__ISNATILIST(cons(V1, V2)) -> A__U41(a__isNatKind(V1), V1, V2) The graph contains the following edges 1 > 2, 1 > 3 *A__U32(tt, V) -> A__ISNATLIST(V) The graph contains the following edges 2 >= 1 *A__U91(tt, V1, V2) -> A__U92(a__isNatKind(V1), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__ISNATLIST(cons(V1, V2)) -> A__U91(a__isNatKind(V1), V1, V2) The graph contains the following edges 1 > 2, 1 > 3 *A__ISNATLIST(take(V1, V2)) -> A__U101(a__isNatKind(V1), V1, V2) The graph contains the following edges 1 > 2, 1 > 3 *A__U92(tt, V1, V2) -> A__U93(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U93(tt, V1, V2) -> A__U94(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U95(tt, V2) -> A__ISNATLIST(V2) The graph contains the following edges 2 >= 1 *A__U12(tt, V1) -> A__ISNATLIST(V1) The graph contains the following edges 2 >= 1 *A__U94(tt, V1, V2) -> A__U95(a__isNat(V1), V2) The graph contains the following edges 3 >= 2 *A__U94(tt, V1, V2) -> A__ISNAT(V1) The graph contains the following edges 2 >= 1 *A__U11(tt, V1) -> A__U12(a__isNatIListKind(V1), V1) The graph contains the following edges 2 >= 2 *A__ISNAT(length(V1)) -> A__U11(a__isNatIListKind(V1), V1) The graph contains the following edges 1 > 2 *A__ISNAT(s(V1)) -> A__U21(a__isNatKind(V1), V1) The graph contains the following edges 1 > 2 *A__U21(tt, V1) -> A__U22(a__isNatKind(V1), V1) The graph contains the following edges 2 >= 2 *A__U22(tt, V1) -> A__ISNAT(V1) The graph contains the following edges 2 >= 1 *A__U44(tt, V1, V2) -> A__ISNAT(V1) The graph contains the following edges 2 >= 1 *A__U41(tt, V1, V2) -> A__U42(a__isNatKind(V1), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U42(tt, V1, V2) -> A__U43(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U43(tt, V1, V2) -> A__U44(a__isNatIListKind(V2), V1, V2) The graph contains the following edges 2 >= 2, 3 >= 3 *A__U44(tt, V1, V2) -> A__U45(a__isNat(V1), V2) The graph contains the following edges 3 >= 2 ---------------------------------------- (12) YES ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(U105(X1, X2)) -> MARK(X1) MARK(U106(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> MARK(X1) MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) A__U114(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) A__U114(tt, L) -> MARK(L) MARK(U111(X1, X2, X3)) -> MARK(X1) MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) MARK(U112(X1, X2, X3)) -> MARK(X1) MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) MARK(U113(X1, X2, X3)) -> MARK(X1) MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) MARK(U114(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(U13(X)) -> MARK(X) MARK(U121(X1, X2)) -> MARK(X1) MARK(U122(X)) -> MARK(X) MARK(U131(X1, X2, X3, X4)) -> A__U131(mark(X1), X2, X3, X4) A__U131(tt, IL, M, N) -> A__U132(a__isNatIListKind(IL), IL, M, N) A__U132(tt, IL, M, N) -> A__U133(a__isNat(M), IL, M, N) A__U133(tt, IL, M, N) -> A__U134(a__isNatKind(M), IL, M, N) A__U134(tt, IL, M, N) -> A__U135(a__isNat(N), IL, M, N) A__U135(tt, IL, M, N) -> A__U136(a__isNatKind(N), IL, M, N) A__U136(tt, IL, M, N) -> MARK(N) MARK(U131(X1, X2, X3, X4)) -> MARK(X1) MARK(U132(X1, X2, X3, X4)) -> A__U132(mark(X1), X2, X3, X4) MARK(U132(X1, X2, X3, X4)) -> MARK(X1) MARK(U133(X1, X2, X3, X4)) -> A__U133(mark(X1), X2, X3, X4) MARK(U133(X1, X2, X3, X4)) -> MARK(X1) MARK(U134(X1, X2, X3, X4)) -> A__U134(mark(X1), X2, X3, X4) MARK(U134(X1, X2, X3, X4)) -> MARK(X1) MARK(U135(X1, X2, X3, X4)) -> A__U135(mark(X1), X2, X3, X4) MARK(U135(X1, X2, X3, X4)) -> MARK(X1) MARK(U136(X1, X2, X3, X4)) -> A__U136(mark(X1), X2, X3, X4) MARK(U136(X1, X2, X3, X4)) -> MARK(X1) MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) A__TAKE(s(M), cons(N, IL)) -> A__U131(a__isNatIList(IL), IL, M, N) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2, X3)) -> MARK(X1) MARK(U95(X1, X2)) -> MARK(X1) MARK(U96(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U121(X1, X2)) -> MARK(X1) MARK(U122(X)) -> MARK(X) A__U133(tt, IL, M, N) -> A__U134(a__isNatKind(M), IL, M, N) MARK(U131(X1, X2, X3, X4)) -> MARK(X1) MARK(U132(X1, X2, X3, X4)) -> MARK(X1) MARK(U133(X1, X2, X3, X4)) -> MARK(X1) MARK(U134(X1, X2, X3, X4)) -> A__U134(mark(X1), X2, X3, X4) MARK(U134(X1, X2, X3, X4)) -> MARK(X1) MARK(U135(X1, X2, X3, X4)) -> A__U135(mark(X1), X2, X3, X4) MARK(U135(X1, X2, X3, X4)) -> MARK(X1) MARK(U136(X1, X2, X3, X4)) -> A__U136(mark(X1), X2, X3, X4) MARK(U136(X1, X2, X3, X4)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__LENGTH_1(x_1) ) = x_1 POL( A__TAKE_2(x_1, x_2) ) = x_2 + 1 POL( A__U111_3(x_1, ..., x_3) ) = x_2 + x_3 POL( A__U112_3(x_1, ..., x_3) ) = x_2 POL( A__U113_3(x_1, ..., x_3) ) = x_2 POL( A__U114_2(x_1, x_2) ) = x_2 POL( A__U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 POL( A__U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 1 POL( A__U133_4(x_1, ..., x_4) ) = 2x_4 + 1 POL( A__U134_4(x_1, ..., x_4) ) = x_4 POL( A__U135_4(x_1, ..., x_4) ) = x_4 POL( A__U136_4(x_1, ..., x_4) ) = x_4 POL( mark_1(x_1) ) = x_1 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( U101_3(x_1, ..., x_3) ) = 2x_1 POL( a__U101_3(x_1, ..., x_3) ) = 2x_1 POL( U102_3(x_1, ..., x_3) ) = x_1 POL( a__U102_3(x_1, ..., x_3) ) = x_1 POL( isNatKind_1(x_1) ) = 0 POL( a__isNatKind_1(x_1) ) = 0 POL( U103_3(x_1, ..., x_3) ) = x_1 POL( a__U103_3(x_1, ..., x_3) ) = x_1 POL( isNatIListKind_1(x_1) ) = 0 POL( a__isNatIListKind_1(x_1) ) = 0 POL( U104_3(x_1, ..., x_3) ) = 2x_1 POL( a__U104_3(x_1, ..., x_3) ) = 2x_1 POL( U105_2(x_1, x_2) ) = x_1 POL( a__U105_2(x_1, x_2) ) = x_1 POL( isNat_1(x_1) ) = 0 POL( a__isNat_1(x_1) ) = 0 POL( U106_1(x_1) ) = x_1 POL( a__U106_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 0 POL( a__isNatIList_1(x_1) ) = 0 POL( U11_2(x_1, x_2) ) = 2x_1 POL( a__U11_2(x_1, x_2) ) = 2x_1 POL( U12_2(x_1, x_2) ) = 2x_1 POL( a__U12_2(x_1, x_2) ) = 2x_1 POL( U111_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 POL( a__U111_3(x_1, ..., x_3) ) = 2x_1 + x_2 + x_3 POL( U112_3(x_1, ..., x_3) ) = 2x_1 + x_2 POL( a__U112_3(x_1, ..., x_3) ) = 2x_1 + x_2 POL( U113_3(x_1, ..., x_3) ) = x_1 + x_2 POL( a__U113_3(x_1, ..., x_3) ) = x_1 + x_2 POL( U114_2(x_1, x_2) ) = x_1 + x_2 POL( a__U114_2(x_1, x_2) ) = x_1 + x_2 POL( length_1(x_1) ) = x_1 POL( a__length_1(x_1) ) = x_1 POL( U13_1(x_1) ) = x_1 POL( a__U13_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 0 POL( a__isNatList_1(x_1) ) = 0 POL( U121_2(x_1, x_2) ) = x_1 + 1 POL( a__U121_2(x_1, x_2) ) = x_1 + 1 POL( U122_1(x_1) ) = 2x_1 + 1 POL( a__U122_1(x_1) ) = 2x_1 + 1 POL( U131_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 POL( a__U131_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 POL( U132_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 POL( a__U132_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 POL( U133_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 POL( a__U133_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 POL( U134_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 POL( a__U134_4(x_1, ..., x_4) ) = 2x_1 + x_2 + x_3 + 2x_4 + 1 POL( U135_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 1 POL( a__U135_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 1 POL( U136_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 1 POL( a__U136_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + 2x_4 + 1 POL( take_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( a__take_2(x_1, x_2) ) = x_1 + x_2 + 1 POL( U21_2(x_1, x_2) ) = x_1 POL( a__U21_2(x_1, x_2) ) = x_1 POL( U22_2(x_1, x_2) ) = 2x_1 POL( a__U22_2(x_1, x_2) ) = 2x_1 POL( U23_1(x_1) ) = x_1 POL( a__U23_1(x_1) ) = x_1 POL( U31_2(x_1, x_2) ) = x_1 POL( a__U31_2(x_1, x_2) ) = x_1 POL( U32_2(x_1, x_2) ) = x_1 POL( a__U32_2(x_1, x_2) ) = x_1 POL( U33_1(x_1) ) = x_1 POL( a__U33_1(x_1) ) = x_1 POL( U41_3(x_1, ..., x_3) ) = 2x_1 POL( a__U41_3(x_1, ..., x_3) ) = 2x_1 POL( U42_3(x_1, ..., x_3) ) = x_1 POL( a__U42_3(x_1, ..., x_3) ) = x_1 POL( U43_3(x_1, ..., x_3) ) = x_1 POL( a__U43_3(x_1, ..., x_3) ) = x_1 POL( U44_3(x_1, ..., x_3) ) = x_1 POL( a__U44_3(x_1, ..., x_3) ) = x_1 POL( U45_2(x_1, x_2) ) = 2x_1 POL( a__U45_2(x_1, x_2) ) = 2x_1 POL( U46_1(x_1) ) = 2x_1 POL( a__U46_1(x_1) ) = 2x_1 POL( U51_2(x_1, x_2) ) = x_1 POL( a__U51_2(x_1, x_2) ) = x_1 POL( U52_1(x_1) ) = 2x_1 POL( a__U52_1(x_1) ) = 2x_1 POL( U61_2(x_1, x_2) ) = x_1 POL( a__U61_2(x_1, x_2) ) = x_1 POL( U62_1(x_1) ) = x_1 POL( a__U62_1(x_1) ) = x_1 POL( U71_1(x_1) ) = 2x_1 POL( a__U71_1(x_1) ) = 2x_1 POL( U81_1(x_1) ) = 2x_1 POL( a__U81_1(x_1) ) = 2x_1 POL( U91_3(x_1, ..., x_3) ) = x_1 POL( a__U91_3(x_1, ..., x_3) ) = x_1 POL( U92_3(x_1, ..., x_3) ) = x_1 POL( a__U92_3(x_1, ..., x_3) ) = x_1 POL( U93_3(x_1, ..., x_3) ) = 2x_1 POL( a__U93_3(x_1, ..., x_3) ) = 2x_1 POL( U94_3(x_1, ..., x_3) ) = 2x_1 POL( a__U94_3(x_1, ..., x_3) ) = 2x_1 POL( U95_2(x_1, x_2) ) = 2x_1 POL( a__U95_2(x_1, x_2) ) = 2x_1 POL( U96_1(x_1) ) = x_1 POL( a__U96_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( 0 ) = 0 POL( tt ) = 0 POL( s_1(x_1) ) = x_1 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(X) -> isNat(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__isNatList(X) -> isNatList(X) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIList(X) -> isNatIList(X) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U105(X1, X2) -> U105(X1, X2) a__U106(tt) -> tt a__U106(X) -> U106(X) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U11(X1, X2) -> U11(X1, X2) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(nil) -> 0 a__length(X) -> length(X) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U121(X1, X2) -> U121(X1, X2) a__U122(tt) -> nil a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(X1, X2) -> take(X1, X2) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U22(X1, X2) -> U22(X1, X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U31(X1, X2) -> U31(X1, X2) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U32(X1, X2) -> U32(X1, X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U45(X1, X2) -> U45(X1, X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1, X2) -> U51(X1, X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U61(X1, X2) -> U61(X1, X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt) -> tt a__U81(X) -> U81(X) a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U95(X1, X2) -> U95(X1, X2) a__U96(tt) -> tt a__U96(X) -> U96(X) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(U105(X1, X2)) -> MARK(X1) MARK(U106(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> MARK(X1) MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) A__U114(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) A__U114(tt, L) -> MARK(L) MARK(U111(X1, X2, X3)) -> MARK(X1) MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) MARK(U112(X1, X2, X3)) -> MARK(X1) MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) MARK(U113(X1, X2, X3)) -> MARK(X1) MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) MARK(U114(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(U13(X)) -> MARK(X) MARK(U131(X1, X2, X3, X4)) -> A__U131(mark(X1), X2, X3, X4) A__U131(tt, IL, M, N) -> A__U132(a__isNatIListKind(IL), IL, M, N) A__U132(tt, IL, M, N) -> A__U133(a__isNat(M), IL, M, N) A__U134(tt, IL, M, N) -> A__U135(a__isNat(N), IL, M, N) A__U135(tt, IL, M, N) -> A__U136(a__isNatKind(N), IL, M, N) A__U136(tt, IL, M, N) -> MARK(N) MARK(U132(X1, X2, X3, X4)) -> A__U132(mark(X1), X2, X3, X4) MARK(U133(X1, X2, X3, X4)) -> A__U133(mark(X1), X2, X3, X4) MARK(take(X1, X2)) -> A__TAKE(mark(X1), mark(X2)) A__TAKE(s(M), cons(N, IL)) -> A__U131(a__isNatIList(IL), IL, M, N) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2, X3)) -> MARK(X1) MARK(U95(X1, X2)) -> MARK(X1) MARK(U96(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes. ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(U105(X1, X2)) -> MARK(X1) MARK(U106(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> MARK(X1) MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) A__U114(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) A__U114(tt, L) -> MARK(L) MARK(U111(X1, X2, X3)) -> MARK(X1) MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) MARK(U112(X1, X2, X3)) -> MARK(X1) MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) MARK(U113(X1, X2, X3)) -> MARK(X1) MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) MARK(U114(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(U13(X)) -> MARK(X) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2, X3)) -> MARK(X1) MARK(U95(X1, X2)) -> MARK(X1) MARK(U96(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U111(X1, X2, X3)) -> A__U111(mark(X1), X2, X3) MARK(U111(X1, X2, X3)) -> MARK(X1) MARK(U112(X1, X2, X3)) -> A__U112(mark(X1), X2, X3) MARK(U112(X1, X2, X3)) -> MARK(X1) MARK(U113(X1, X2, X3)) -> A__U113(mark(X1), X2, X3) MARK(U113(X1, X2, X3)) -> MARK(X1) MARK(U114(X1, X2)) -> A__U114(mark(X1), X2) MARK(U114(X1, X2)) -> MARK(X1) MARK(length(X)) -> A__LENGTH(mark(X)) MARK(length(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__LENGTH_1(x_1) ) = x_1 POL( A__U111_3(x_1, ..., x_3) ) = x_2 + 2x_3 POL( A__U112_3(x_1, ..., x_3) ) = x_2 POL( A__U113_3(x_1, ..., x_3) ) = x_2 POL( A__U114_2(x_1, x_2) ) = x_2 POL( mark_1(x_1) ) = x_1 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( U101_3(x_1, ..., x_3) ) = x_1 POL( a__U101_3(x_1, ..., x_3) ) = x_1 POL( U102_3(x_1, ..., x_3) ) = x_1 POL( a__U102_3(x_1, ..., x_3) ) = x_1 POL( isNatKind_1(x_1) ) = 0 POL( a__isNatKind_1(x_1) ) = 0 POL( U103_3(x_1, ..., x_3) ) = 2x_1 POL( a__U103_3(x_1, ..., x_3) ) = 2x_1 POL( isNatIListKind_1(x_1) ) = 0 POL( a__isNatIListKind_1(x_1) ) = 0 POL( U104_3(x_1, ..., x_3) ) = 2x_1 POL( a__U104_3(x_1, ..., x_3) ) = 2x_1 POL( U105_2(x_1, x_2) ) = 2x_1 POL( a__U105_2(x_1, x_2) ) = 2x_1 POL( isNat_1(x_1) ) = 0 POL( a__isNat_1(x_1) ) = 0 POL( U106_1(x_1) ) = 2x_1 POL( a__U106_1(x_1) ) = 2x_1 POL( isNatIList_1(x_1) ) = 0 POL( a__isNatIList_1(x_1) ) = 0 POL( U11_2(x_1, x_2) ) = x_1 POL( a__U11_2(x_1, x_2) ) = x_1 POL( U12_2(x_1, x_2) ) = x_1 POL( a__U12_2(x_1, x_2) ) = x_1 POL( U111_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2 POL( a__U111_3(x_1, ..., x_3) ) = 2x_1 + x_2 + 2x_3 + 2 POL( U112_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 POL( a__U112_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 POL( U113_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 POL( a__U113_3(x_1, ..., x_3) ) = x_1 + x_2 + x_3 + 2 POL( U114_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( a__U114_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( length_1(x_1) ) = x_1 + 2 POL( a__length_1(x_1) ) = x_1 + 2 POL( U13_1(x_1) ) = 2x_1 POL( a__U13_1(x_1) ) = 2x_1 POL( isNatList_1(x_1) ) = 0 POL( a__isNatList_1(x_1) ) = 0 POL( U121_2(x_1, x_2) ) = 0 POL( a__U121_2(x_1, x_2) ) = max{0, -2} POL( U122_1(x_1) ) = 0 POL( a__U122_1(x_1) ) = max{0, -2} POL( U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( a__U131_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( a__U132_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( U133_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( a__U133_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( U134_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( a__U134_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( U135_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( a__U135_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( U136_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( a__U136_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( take_2(x_1, x_2) ) = x_2 + 2 POL( a__take_2(x_1, x_2) ) = x_2 + 2 POL( U21_2(x_1, x_2) ) = x_1 POL( a__U21_2(x_1, x_2) ) = x_1 POL( U22_2(x_1, x_2) ) = 2x_1 POL( a__U22_2(x_1, x_2) ) = 2x_1 POL( U23_1(x_1) ) = 2x_1 POL( a__U23_1(x_1) ) = 2x_1 POL( U31_2(x_1, x_2) ) = x_1 POL( a__U31_2(x_1, x_2) ) = x_1 POL( U32_2(x_1, x_2) ) = 2x_1 POL( a__U32_2(x_1, x_2) ) = 2x_1 POL( U33_1(x_1) ) = 2x_1 POL( a__U33_1(x_1) ) = 2x_1 POL( U41_3(x_1, ..., x_3) ) = x_1 POL( a__U41_3(x_1, ..., x_3) ) = x_1 POL( U42_3(x_1, ..., x_3) ) = x_1 POL( a__U42_3(x_1, ..., x_3) ) = x_1 POL( U43_3(x_1, ..., x_3) ) = 2x_1 POL( a__U43_3(x_1, ..., x_3) ) = 2x_1 POL( U44_3(x_1, ..., x_3) ) = 2x_1 POL( a__U44_3(x_1, ..., x_3) ) = 2x_1 POL( U45_2(x_1, x_2) ) = x_1 POL( a__U45_2(x_1, x_2) ) = x_1 POL( U46_1(x_1) ) = 2x_1 POL( a__U46_1(x_1) ) = 2x_1 POL( U51_2(x_1, x_2) ) = x_1 POL( a__U51_2(x_1, x_2) ) = x_1 POL( U52_1(x_1) ) = 2x_1 POL( a__U52_1(x_1) ) = 2x_1 POL( U61_2(x_1, x_2) ) = 2x_1 POL( a__U61_2(x_1, x_2) ) = 2x_1 POL( U62_1(x_1) ) = x_1 POL( a__U62_1(x_1) ) = x_1 POL( U71_1(x_1) ) = x_1 POL( a__U71_1(x_1) ) = x_1 POL( U81_1(x_1) ) = 2x_1 POL( a__U81_1(x_1) ) = 2x_1 POL( U91_3(x_1, ..., x_3) ) = 2x_1 POL( a__U91_3(x_1, ..., x_3) ) = 2x_1 POL( U92_3(x_1, ..., x_3) ) = 2x_1 POL( a__U92_3(x_1, ..., x_3) ) = 2x_1 POL( U93_3(x_1, ..., x_3) ) = x_1 POL( a__U93_3(x_1, ..., x_3) ) = x_1 POL( U94_3(x_1, ..., x_3) ) = x_1 POL( a__U94_3(x_1, ..., x_3) ) = x_1 POL( U95_2(x_1, x_2) ) = 2x_1 POL( a__U95_2(x_1, x_2) ) = 2x_1 POL( U96_1(x_1) ) = 2x_1 POL( a__U96_1(x_1) ) = 2x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( 0 ) = 0 POL( tt ) = 0 POL( s_1(x_1) ) = x_1 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(X) -> isNat(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__isNatList(X) -> isNatList(X) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U105(X1, X2) -> U105(X1, X2) a__U106(tt) -> tt a__U106(X) -> U106(X) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U11(X1, X2) -> U11(X1, X2) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(nil) -> 0 a__length(X) -> length(X) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U121(X1, X2) -> U121(X1, X2) a__U122(tt) -> nil a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(X1, X2) -> take(X1, X2) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U22(X1, X2) -> U22(X1, X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U31(X1, X2) -> U31(X1, X2) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U32(X1, X2) -> U32(X1, X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U45(X1, X2) -> U45(X1, X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1, X2) -> U51(X1, X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U61(X1, X2) -> U61(X1, X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt) -> tt a__U81(X) -> U81(X) a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U95(X1, X2) -> U95(X1, X2) a__U96(tt) -> tt a__U96(X) -> U96(X) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIList(X) -> isNatIList(X) a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(U105(X1, X2)) -> MARK(X1) MARK(U106(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> MARK(X1) A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) A__U114(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) A__U114(tt, L) -> MARK(L) MARK(U13(X)) -> MARK(X) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2, X3)) -> MARK(X1) MARK(U95(X1, X2)) -> MARK(X1) MARK(U96(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (21) Complex Obligation (AND) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(U105(X1, X2)) -> MARK(X1) MARK(U106(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2, X3)) -> MARK(X1) MARK(U95(X1, X2)) -> MARK(X1) MARK(U96(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U101(X1, X2, X3)) -> MARK(X1) MARK(U102(X1, X2, X3)) -> MARK(X1) MARK(U103(X1, X2, X3)) -> MARK(X1) MARK(U104(X1, X2, X3)) -> MARK(X1) MARK(U105(X1, X2)) -> MARK(X1) MARK(U106(X)) -> MARK(X) MARK(U11(X1, X2)) -> MARK(X1) MARK(U12(X1, X2)) -> MARK(X1) MARK(U13(X)) -> MARK(X) MARK(U21(X1, X2)) -> MARK(X1) MARK(U22(X1, X2)) -> MARK(X1) MARK(U23(X)) -> MARK(X) MARK(U31(X1, X2)) -> MARK(X1) MARK(U32(X1, X2)) -> MARK(X1) MARK(U33(X)) -> MARK(X) MARK(U41(X1, X2, X3)) -> MARK(X1) MARK(U42(X1, X2, X3)) -> MARK(X1) MARK(U43(X1, X2, X3)) -> MARK(X1) MARK(U44(X1, X2, X3)) -> MARK(X1) MARK(U45(X1, X2)) -> MARK(X1) MARK(U46(X)) -> MARK(X) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X)) -> MARK(X) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3)) -> MARK(X1) MARK(U92(X1, X2, X3)) -> MARK(X1) MARK(U93(X1, X2, X3)) -> MARK(X1) MARK(U94(X1, X2, X3)) -> MARK(X1) MARK(U95(X1, X2)) -> MARK(X1) MARK(U96(X)) -> MARK(X) MARK(cons(X1, X2)) -> MARK(X1) MARK(s(X)) -> MARK(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *MARK(U101(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U102(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U103(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U104(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U105(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U106(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U11(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U12(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U13(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U21(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U22(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U23(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U31(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U32(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U33(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U41(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U42(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U43(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U44(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U45(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U46(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U51(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U52(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U61(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U62(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U71(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U81(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(U91(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U92(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U93(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U94(X1, X2, X3)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U95(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(U96(X)) -> MARK(X) The graph contains the following edges 1 > 1 *MARK(cons(X1, X2)) -> MARK(X1) The graph contains the following edges 1 > 1 *MARK(s(X)) -> MARK(X) The graph contains the following edges 1 > 1 ---------------------------------------- (26) YES ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: A__U114(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A__U111(tt, L, N) -> A__U112(a__isNatIListKind(L), L, N) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( A__LENGTH_1(x_1) ) = 2x_1 POL( mark_1(x_1) ) = x_1 POL( zeros ) = 0 POL( a__zeros ) = 0 POL( U101_3(x_1, ..., x_3) ) = 2x_2 POL( a__U101_3(x_1, ..., x_3) ) = 2x_2 POL( U102_3(x_1, ..., x_3) ) = 2x_2 POL( a__U102_3(x_1, ..., x_3) ) = 2x_2 POL( isNatKind_1(x_1) ) = 2 POL( a__isNatKind_1(x_1) ) = 2 POL( U103_3(x_1, ..., x_3) ) = 2x_2 POL( a__U103_3(x_1, ..., x_3) ) = 2x_2 POL( isNatIListKind_1(x_1) ) = 1 POL( a__isNatIListKind_1(x_1) ) = 1 POL( U104_3(x_1, ..., x_3) ) = 2x_2 POL( a__U104_3(x_1, ..., x_3) ) = 2x_2 POL( U105_2(x_1, x_2) ) = x_1 POL( a__U105_2(x_1, x_2) ) = x_1 POL( isNat_1(x_1) ) = 2x_1 POL( a__isNat_1(x_1) ) = 2x_1 POL( U106_1(x_1) ) = 1 POL( a__U106_1(x_1) ) = 1 POL( isNatIList_1(x_1) ) = 2 POL( a__isNatIList_1(x_1) ) = 2 POL( U11_2(x_1, x_2) ) = 2x_2 POL( a__U11_2(x_1, x_2) ) = 2x_2 POL( U12_2(x_1, x_2) ) = 2x_2 POL( a__U12_2(x_1, x_2) ) = 2x_2 POL( U111_3(x_1, ..., x_3) ) = 2x_2 POL( a__U111_3(x_1, ..., x_3) ) = 2x_2 POL( U112_3(x_1, ..., x_3) ) = 2x_2 POL( a__U112_3(x_1, ..., x_3) ) = 2x_2 POL( U113_3(x_1, ..., x_3) ) = 2x_2 POL( a__U113_3(x_1, ..., x_3) ) = 2x_2 POL( U114_2(x_1, x_2) ) = 2x_2 POL( a__U114_2(x_1, x_2) ) = 2x_2 POL( length_1(x_1) ) = x_1 POL( a__length_1(x_1) ) = x_1 POL( U13_1(x_1) ) = x_1 POL( a__U13_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 2x_1 POL( a__isNatList_1(x_1) ) = 2x_1 POL( U121_2(x_1, x_2) ) = 2 POL( a__U121_2(x_1, x_2) ) = 2 POL( U122_1(x_1) ) = x_1 + 1 POL( a__U122_1(x_1) ) = x_1 + 1 POL( U131_4(x_1, ..., x_4) ) = 2x_3 POL( a__U131_4(x_1, ..., x_4) ) = 2x_3 POL( U132_4(x_1, ..., x_4) ) = 2x_3 POL( a__U132_4(x_1, ..., x_4) ) = 2x_3 POL( U133_4(x_1, ..., x_4) ) = 2x_3 POL( a__U133_4(x_1, ..., x_4) ) = 2x_3 POL( U134_4(x_1, ..., x_4) ) = 2x_3 POL( a__U134_4(x_1, ..., x_4) ) = 2x_3 POL( U135_4(x_1, ..., x_4) ) = 2x_3 POL( a__U135_4(x_1, ..., x_4) ) = 2x_3 POL( U136_4(x_1, ..., x_4) ) = 2x_3 POL( a__U136_4(x_1, ..., x_4) ) = 2x_3 POL( take_2(x_1, x_2) ) = x_1 POL( a__take_2(x_1, x_2) ) = x_1 POL( U21_2(x_1, x_2) ) = 2x_2 POL( a__U21_2(x_1, x_2) ) = 2x_2 POL( U22_2(x_1, x_2) ) = 2x_2 POL( a__U22_2(x_1, x_2) ) = 2x_2 POL( U23_1(x_1) ) = x_1 POL( a__U23_1(x_1) ) = x_1 POL( U31_2(x_1, x_2) ) = 1 POL( a__U31_2(x_1, x_2) ) = 1 POL( U32_2(x_1, x_2) ) = 1 POL( a__U32_2(x_1, x_2) ) = 1 POL( U33_1(x_1) ) = 1 POL( a__U33_1(x_1) ) = 1 POL( U41_3(x_1, ..., x_3) ) = 2 POL( a__U41_3(x_1, ..., x_3) ) = 2 POL( U42_3(x_1, ..., x_3) ) = 2 POL( a__U42_3(x_1, ..., x_3) ) = 2 POL( U43_3(x_1, ..., x_3) ) = 2 POL( a__U43_3(x_1, ..., x_3) ) = 2 POL( U44_3(x_1, ..., x_3) ) = 2 POL( a__U44_3(x_1, ..., x_3) ) = 2 POL( U45_2(x_1, x_2) ) = 2 POL( a__U45_2(x_1, x_2) ) = 2 POL( U46_1(x_1) ) = 2 POL( a__U46_1(x_1) ) = 2 POL( U51_2(x_1, x_2) ) = 1 POL( a__U51_2(x_1, x_2) ) = 1 POL( U52_1(x_1) ) = 1 POL( a__U52_1(x_1) ) = 1 POL( U61_2(x_1, x_2) ) = 1 POL( a__U61_2(x_1, x_2) ) = 1 POL( U62_1(x_1) ) = 1 POL( a__U62_1(x_1) ) = 1 POL( U71_1(x_1) ) = 1 POL( a__U71_1(x_1) ) = 1 POL( U81_1(x_1) ) = 1 POL( a__U81_1(x_1) ) = 1 POL( U91_3(x_1, ..., x_3) ) = 2x_3 POL( a__U91_3(x_1, ..., x_3) ) = 2x_3 POL( U92_3(x_1, ..., x_3) ) = 2x_3 POL( a__U92_3(x_1, ..., x_3) ) = 2x_3 POL( U93_3(x_1, ..., x_3) ) = 2x_3 POL( a__U93_3(x_1, ..., x_3) ) = 2x_3 POL( U94_3(x_1, ..., x_3) ) = 2x_3 POL( a__U94_3(x_1, ..., x_3) ) = 2x_3 POL( U95_2(x_1, x_2) ) = 2x_2 POL( a__U95_2(x_1, x_2) ) = 2x_2 POL( U96_1(x_1) ) = x_1 POL( a__U96_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_2 POL( 0 ) = 2 POL( tt ) = 1 POL( s_1(x_1) ) = 2x_1 POL( nil ) = 2 POL( A__U111_3(x_1, ..., x_3) ) = x_1 + 2x_2 POL( A__U113_3(x_1, ..., x_3) ) = 2x_2 POL( A__U112_3(x_1, ..., x_3) ) = 2x_2 POL( A__U114_2(x_1, x_2) ) = 2x_2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__isNatList(X) -> isNatList(X) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatIListKind(X) -> isNatIListKind(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNat(X) -> isNat(X) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatKind(X) -> isNatKind(X) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U105(X1, X2) -> U105(X1, X2) a__U106(tt) -> tt a__U106(X) -> U106(X) a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U11(X1, X2) -> U11(X1, X2) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(nil) -> 0 a__length(X) -> length(X) a__U13(tt) -> tt a__U13(X) -> U13(X) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U121(X1, X2) -> U121(X1, X2) a__U122(tt) -> nil a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(X1, X2) -> take(X1, X2) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U21(X1, X2) -> U21(X1, X2) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U22(X1, X2) -> U22(X1, X2) a__U23(tt) -> tt a__U23(X) -> U23(X) a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U31(X1, X2) -> U31(X1, X2) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U32(X1, X2) -> U32(X1, X2) a__U33(tt) -> tt a__U33(X) -> U33(X) a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U45(X1, X2) -> U45(X1, X2) a__U46(tt) -> tt a__U46(X) -> U46(X) a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U51(X1, X2) -> U51(X1, X2) a__U52(tt) -> tt a__U52(X) -> U52(X) a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U61(X1, X2) -> U61(X1, X2) a__U62(tt) -> tt a__U62(X) -> U62(X) a__U71(tt) -> tt a__U71(X) -> U71(X) a__U81(tt) -> tt a__U81(X) -> U81(X) a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U95(X1, X2) -> U95(X1, X2) a__U96(tt) -> tt a__U96(X) -> U96(X) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIList(X) -> isNatIList(X) a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__zeros -> cons(0, zeros) a__zeros -> zeros ---------------------------------------- (29) Obligation: Q DP problem: The TRS P consists of the following rules: A__U114(tt, L) -> A__LENGTH(mark(L)) A__LENGTH(cons(N, L)) -> A__U111(a__isNatList(L), L, N) A__U112(tt, L, N) -> A__U113(a__isNat(N), L, N) A__U113(tt, L, N) -> A__U114(a__isNatKind(N), L) The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) a__U106(tt) -> tt a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) a__U114(tt, L) -> s(a__length(mark(L))) a__U12(tt, V1) -> a__U13(a__isNatList(V1)) a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) a__U122(tt) -> nil a__U13(tt) -> tt a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) a__U22(tt, V1) -> a__U23(a__isNat(V1)) a__U23(tt) -> tt a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) a__U32(tt, V) -> a__U33(a__isNatList(V)) a__U33(tt) -> tt a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) a__U46(tt) -> tt a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) a__U52(tt) -> tt a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) a__U62(tt) -> tt a__U71(tt) -> tt a__U81(tt) -> tt a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) a__U95(tt, V2) -> a__U96(a__isNatList(V2)) a__U96(tt) -> tt a__isNat(0) -> tt a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) a__isNatIListKind(nil) -> tt a__isNatIListKind(zeros) -> tt a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) a__isNatKind(0) -> tt a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) mark(zeros) -> a__zeros mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) mark(isNatKind(X)) -> a__isNatKind(X) mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) mark(isNatIListKind(X)) -> a__isNatIListKind(X) mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) mark(U105(X1, X2)) -> a__U105(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(U106(X)) -> a__U106(mark(X)) mark(isNatIList(X)) -> a__isNatIList(X) mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) mark(U114(X1, X2)) -> a__U114(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(U13(X)) -> a__U13(mark(X)) mark(isNatList(X)) -> a__isNatList(X) mark(U121(X1, X2)) -> a__U121(mark(X1), X2) mark(U122(X)) -> a__U122(mark(X)) mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U23(X)) -> a__U23(mark(X)) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U33(X)) -> a__U33(mark(X)) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) mark(U45(X1, X2)) -> a__U45(mark(X1), X2) mark(U46(X)) -> a__U46(mark(X)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X)) -> a__U52(mark(X)) mark(U61(X1, X2)) -> a__U61(mark(X1), X2) mark(U62(X)) -> a__U62(mark(X)) mark(U71(X)) -> a__U71(mark(X)) mark(U81(X)) -> a__U81(mark(X)) mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) mark(U95(X1, X2)) -> a__U95(mark(X1), X2) mark(U96(X)) -> a__U96(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U101(X1, X2, X3) -> U101(X1, X2, X3) a__U102(X1, X2, X3) -> U102(X1, X2, X3) a__isNatKind(X) -> isNatKind(X) a__U103(X1, X2, X3) -> U103(X1, X2, X3) a__isNatIListKind(X) -> isNatIListKind(X) a__U104(X1, X2, X3) -> U104(X1, X2, X3) a__U105(X1, X2) -> U105(X1, X2) a__isNat(X) -> isNat(X) a__U106(X) -> U106(X) a__isNatIList(X) -> isNatIList(X) a__U11(X1, X2) -> U11(X1, X2) a__U12(X1, X2) -> U12(X1, X2) a__U111(X1, X2, X3) -> U111(X1, X2, X3) a__U112(X1, X2, X3) -> U112(X1, X2, X3) a__U113(X1, X2, X3) -> U113(X1, X2, X3) a__U114(X1, X2) -> U114(X1, X2) a__length(X) -> length(X) a__U13(X) -> U13(X) a__isNatList(X) -> isNatList(X) a__U121(X1, X2) -> U121(X1, X2) a__U122(X) -> U122(X) a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) a__take(X1, X2) -> take(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U23(X) -> U23(X) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U33(X) -> U33(X) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__U43(X1, X2, X3) -> U43(X1, X2, X3) a__U44(X1, X2, X3) -> U44(X1, X2, X3) a__U45(X1, X2) -> U45(X1, X2) a__U46(X) -> U46(X) a__U51(X1, X2) -> U51(X1, X2) a__U52(X) -> U52(X) a__U61(X1, X2) -> U61(X1, X2) a__U62(X) -> U62(X) a__U71(X) -> U71(X) a__U81(X) -> U81(X) a__U91(X1, X2, X3) -> U91(X1, X2, X3) a__U92(X1, X2, X3) -> U92(X1, X2, X3) a__U93(X1, X2, X3) -> U93(X1, X2, X3) a__U94(X1, X2, X3) -> U94(X1, X2, X3) a__U95(X1, X2) -> U95(X1, X2) a__U96(X) -> U96(X) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (30) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. ---------------------------------------- (31) TRUE