1146.73/291.58 WORST_CASE(Omega(n^1), ?) 1157.99/294.45 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1157.99/294.45 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1157.99/294.45 1157.99/294.45 1157.99/294.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.99/294.45 1157.99/294.45 (0) CpxTRS 1157.99/294.45 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1157.99/294.45 (2) TRS for Loop Detection 1157.99/294.45 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1157.99/294.45 (4) BEST 1157.99/294.45 (5) proven lower bound 1157.99/294.45 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1157.99/294.45 (7) BOUNDS(n^1, INF) 1157.99/294.45 (8) TRS for Loop Detection 1157.99/294.45 1157.99/294.45 1157.99/294.45 ---------------------------------------- 1157.99/294.45 1157.99/294.45 (0) 1157.99/294.45 Obligation: 1157.99/294.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.99/294.45 1157.99/294.45 1157.99/294.45 The TRS R consists of the following rules: 1157.99/294.45 1157.99/294.45 active(zeros) -> mark(cons(0, zeros)) 1157.99/294.45 active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2)) 1157.99/294.45 active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2)) 1157.99/294.45 active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2)) 1157.99/294.45 active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2)) 1157.99/294.45 active(U105(tt, V2)) -> mark(U106(isNatIList(V2))) 1157.99/294.45 active(U106(tt)) -> mark(tt) 1157.99/294.45 active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) 1157.99/294.45 active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N)) 1157.99/294.45 active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N)) 1157.99/294.45 active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L)) 1157.99/294.45 active(U114(tt, L)) -> mark(s(length(L))) 1157.99/294.45 active(U12(tt, V1)) -> mark(U13(isNatList(V1))) 1157.99/294.45 active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL))) 1157.99/294.45 active(U122(tt)) -> mark(nil) 1157.99/294.45 active(U13(tt)) -> mark(tt) 1157.99/294.45 active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N)) 1157.99/294.45 active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N)) 1157.99/294.45 active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N)) 1157.99/294.45 active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N)) 1157.99/294.45 active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N)) 1157.99/294.45 active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) 1157.99/294.45 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 1157.99/294.45 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 1157.99/294.45 active(U23(tt)) -> mark(tt) 1157.99/294.45 active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) 1157.99/294.45 active(U32(tt, V)) -> mark(U33(isNatList(V))) 1157.99/294.45 active(U33(tt)) -> mark(tt) 1157.99/294.45 active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) 1157.99/294.45 active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) 1157.99/294.45 active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) 1157.99/294.45 active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) 1157.99/294.45 active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) 1157.99/294.45 active(U46(tt)) -> mark(tt) 1157.99/294.45 active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) 1157.99/294.45 active(U52(tt)) -> mark(tt) 1157.99/294.45 active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2))) 1157.99/294.45 active(U62(tt)) -> mark(tt) 1157.99/294.45 active(U71(tt)) -> mark(tt) 1157.99/294.45 active(U81(tt)) -> mark(tt) 1157.99/294.45 active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2)) 1157.99/294.45 active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2)) 1157.99/294.45 active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2)) 1157.99/294.45 active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2)) 1157.99/294.45 active(U95(tt, V2)) -> mark(U96(isNatList(V2))) 1157.99/294.45 active(U96(tt)) -> mark(tt) 1157.99/294.45 active(isNat(0)) -> mark(tt) 1157.99/294.45 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1157.99/294.45 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1157.99/294.45 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1157.99/294.45 active(isNatIList(zeros)) -> mark(tt) 1157.99/294.45 active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) 1157.99/294.45 active(isNatIListKind(nil)) -> mark(tt) 1157.99/294.45 active(isNatIListKind(zeros)) -> mark(tt) 1157.99/294.45 active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) 1157.99/294.45 active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2)) 1157.99/294.45 active(isNatKind(0)) -> mark(tt) 1157.99/294.45 active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1))) 1157.99/294.45 active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1))) 1157.99/294.45 active(isNatList(nil)) -> mark(tt) 1157.99/294.45 active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2)) 1157.99/294.45 active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2)) 1157.99/294.45 active(length(nil)) -> mark(0) 1157.99/294.45 active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N)) 1157.99/294.45 active(take(0, IL)) -> mark(U121(isNatIList(IL), IL)) 1157.99/294.45 active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N)) 1157.99/294.45 active(cons(X1, X2)) -> cons(active(X1), X2) 1157.99/294.45 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 1157.99/294.45 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 1157.99/294.45 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 1157.99/294.45 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 1157.99/294.45 active(U105(X1, X2)) -> U105(active(X1), X2) 1157.99/294.45 active(U106(X)) -> U106(active(X)) 1157.99/294.45 active(U11(X1, X2)) -> U11(active(X1), X2) 1157.99/294.45 active(U12(X1, X2)) -> U12(active(X1), X2) 1157.99/294.45 active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3) 1157.99/294.45 active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3) 1157.99/294.45 active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3) 1157.99/294.45 active(U114(X1, X2)) -> U114(active(X1), X2) 1157.99/294.45 active(s(X)) -> s(active(X)) 1157.99/294.45 active(length(X)) -> length(active(X)) 1157.99/294.45 active(U13(X)) -> U13(active(X)) 1157.99/294.45 active(U121(X1, X2)) -> U121(active(X1), X2) 1157.99/294.45 active(U122(X)) -> U122(active(X)) 1157.99/294.45 active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4) 1157.99/294.45 active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4) 1157.99/294.45 active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4) 1157.99/294.45 active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4) 1157.99/294.45 active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4) 1157.99/294.45 active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4) 1157.99/294.45 active(take(X1, X2)) -> take(active(X1), X2) 1157.99/294.45 active(take(X1, X2)) -> take(X1, active(X2)) 1157.99/294.45 active(U21(X1, X2)) -> U21(active(X1), X2) 1157.99/294.45 active(U22(X1, X2)) -> U22(active(X1), X2) 1157.99/294.45 active(U23(X)) -> U23(active(X)) 1157.99/294.45 active(U31(X1, X2)) -> U31(active(X1), X2) 1157.99/294.45 active(U32(X1, X2)) -> U32(active(X1), X2) 1157.99/294.45 active(U33(X)) -> U33(active(X)) 1157.99/294.45 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1157.99/294.45 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 1157.99/294.45 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 1157.99/294.45 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 1157.99/294.45 active(U45(X1, X2)) -> U45(active(X1), X2) 1157.99/294.45 active(U46(X)) -> U46(active(X)) 1157.99/294.45 active(U51(X1, X2)) -> U51(active(X1), X2) 1157.99/294.45 active(U52(X)) -> U52(active(X)) 1157.99/294.45 active(U61(X1, X2)) -> U61(active(X1), X2) 1157.99/294.45 active(U62(X)) -> U62(active(X)) 1157.99/294.45 active(U71(X)) -> U71(active(X)) 1157.99/294.45 active(U81(X)) -> U81(active(X)) 1157.99/294.45 active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) 1157.99/294.45 active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) 1157.99/294.45 active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) 1157.99/294.45 active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3) 1157.99/294.45 active(U95(X1, X2)) -> U95(active(X1), X2) 1157.99/294.45 active(U96(X)) -> U96(active(X)) 1157.99/294.45 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1157.99/294.45 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 1157.99/294.45 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 1157.99/294.45 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 1157.99/294.45 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 1157.99/294.45 U105(mark(X1), X2) -> mark(U105(X1, X2)) 1157.99/294.45 U106(mark(X)) -> mark(U106(X)) 1157.99/294.45 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1157.99/294.45 U12(mark(X1), X2) -> mark(U12(X1, X2)) 1157.99/294.45 U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3)) 1157.99/294.45 U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3)) 1157.99/294.45 U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3)) 1157.99/294.45 U114(mark(X1), X2) -> mark(U114(X1, X2)) 1157.99/294.45 s(mark(X)) -> mark(s(X)) 1157.99/294.45 length(mark(X)) -> mark(length(X)) 1157.99/294.45 U13(mark(X)) -> mark(U13(X)) 1157.99/294.45 U121(mark(X1), X2) -> mark(U121(X1, X2)) 1157.99/294.45 U122(mark(X)) -> mark(U122(X)) 1157.99/294.45 U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4)) 1157.99/294.45 U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4)) 1157.99/294.45 U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4)) 1157.99/294.45 U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4)) 1157.99/294.45 U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4)) 1157.99/294.45 U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4)) 1157.99/294.45 take(mark(X1), X2) -> mark(take(X1, X2)) 1157.99/294.45 take(X1, mark(X2)) -> mark(take(X1, X2)) 1157.99/294.45 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1157.99/294.45 U22(mark(X1), X2) -> mark(U22(X1, X2)) 1157.99/294.45 U23(mark(X)) -> mark(U23(X)) 1157.99/294.45 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1157.99/294.45 U32(mark(X1), X2) -> mark(U32(X1, X2)) 1157.99/294.45 U33(mark(X)) -> mark(U33(X)) 1157.99/294.45 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1157.99/294.45 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 1157.99/294.45 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 1157.99/294.45 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 1157.99/294.45 U45(mark(X1), X2) -> mark(U45(X1, X2)) 1157.99/294.45 U46(mark(X)) -> mark(U46(X)) 1157.99/294.45 U51(mark(X1), X2) -> mark(U51(X1, X2)) 1157.99/294.45 U52(mark(X)) -> mark(U52(X)) 1157.99/294.45 U61(mark(X1), X2) -> mark(U61(X1, X2)) 1157.99/294.45 U62(mark(X)) -> mark(U62(X)) 1157.99/294.45 U71(mark(X)) -> mark(U71(X)) 1157.99/294.45 U81(mark(X)) -> mark(U81(X)) 1157.99/294.45 U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) 1157.99/294.45 U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) 1157.99/294.45 U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) 1157.99/294.45 U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3)) 1157.99/294.45 U95(mark(X1), X2) -> mark(U95(X1, X2)) 1157.99/294.45 U96(mark(X)) -> mark(U96(X)) 1157.99/294.45 proper(zeros) -> ok(zeros) 1157.99/294.45 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1157.99/294.45 proper(0) -> ok(0) 1157.99/294.45 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(tt) -> ok(tt) 1157.99/294.45 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(isNatKind(X)) -> isNatKind(proper(X)) 1157.99/294.45 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1157.99/294.45 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U105(X1, X2)) -> U105(proper(X1), proper(X2)) 1157.99/294.45 proper(isNat(X)) -> isNat(proper(X)) 1157.99/294.45 proper(U106(X)) -> U106(proper(X)) 1157.99/294.45 proper(isNatIList(X)) -> isNatIList(proper(X)) 1157.99/294.45 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1157.99/294.45 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 1157.99/294.45 proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U114(X1, X2)) -> U114(proper(X1), proper(X2)) 1157.99/294.45 proper(s(X)) -> s(proper(X)) 1157.99/294.45 proper(length(X)) -> length(proper(X)) 1157.99/294.45 proper(U13(X)) -> U13(proper(X)) 1157.99/294.45 proper(isNatList(X)) -> isNatList(proper(X)) 1157.99/294.45 proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)) 1157.99/294.45 proper(U122(X)) -> U122(proper(X)) 1157.99/294.45 proper(nil) -> ok(nil) 1157.99/294.45 proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.45 proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.45 proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.45 proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.45 proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.45 proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.45 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1157.99/294.45 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1157.99/294.45 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 1157.99/294.45 proper(U23(X)) -> U23(proper(X)) 1157.99/294.45 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1157.99/294.45 proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) 1157.99/294.45 proper(U33(X)) -> U33(proper(X)) 1157.99/294.45 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 1157.99/294.45 proper(U46(X)) -> U46(proper(X)) 1157.99/294.45 proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) 1157.99/294.45 proper(U52(X)) -> U52(proper(X)) 1157.99/294.45 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 1157.99/294.45 proper(U62(X)) -> U62(proper(X)) 1157.99/294.45 proper(U71(X)) -> U71(proper(X)) 1157.99/294.45 proper(U81(X)) -> U81(proper(X)) 1157.99/294.45 proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3)) 1157.99/294.45 proper(U95(X1, X2)) -> U95(proper(X1), proper(X2)) 1157.99/294.45 proper(U96(X)) -> U96(proper(X)) 1157.99/294.45 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1157.99/294.45 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 1157.99/294.45 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 1157.99/294.45 isNatKind(ok(X)) -> ok(isNatKind(X)) 1157.99/294.45 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 1157.99/294.45 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1157.99/294.45 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 1157.99/294.45 U105(ok(X1), ok(X2)) -> ok(U105(X1, X2)) 1157.99/294.45 isNat(ok(X)) -> ok(isNat(X)) 1157.99/294.45 U106(ok(X)) -> ok(U106(X)) 1157.99/294.45 isNatIList(ok(X)) -> ok(isNatIList(X)) 1157.99/294.45 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1157.99/294.45 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 1157.99/294.45 U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3)) 1157.99/294.45 U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3)) 1157.99/294.45 U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3)) 1157.99/294.45 U114(ok(X1), ok(X2)) -> ok(U114(X1, X2)) 1157.99/294.45 s(ok(X)) -> ok(s(X)) 1157.99/294.45 length(ok(X)) -> ok(length(X)) 1157.99/294.45 U13(ok(X)) -> ok(U13(X)) 1157.99/294.45 isNatList(ok(X)) -> ok(isNatList(X)) 1157.99/294.45 U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)) 1157.99/294.45 U122(ok(X)) -> ok(U122(X)) 1157.99/294.45 U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4)) 1157.99/294.45 U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4)) 1157.99/294.45 U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4)) 1157.99/294.45 U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4)) 1157.99/294.45 U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4)) 1157.99/294.45 U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4)) 1157.99/294.45 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1157.99/294.45 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1157.99/294.45 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 1157.99/294.45 U23(ok(X)) -> ok(U23(X)) 1157.99/294.45 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1157.99/294.45 U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) 1157.99/294.45 U33(ok(X)) -> ok(U33(X)) 1157.99/294.45 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1157.99/294.45 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 1157.99/294.45 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 1157.99/294.45 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 1157.99/294.45 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 1157.99/294.45 U46(ok(X)) -> ok(U46(X)) 1157.99/294.45 U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) 1157.99/294.45 U52(ok(X)) -> ok(U52(X)) 1157.99/294.45 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 1157.99/294.45 U62(ok(X)) -> ok(U62(X)) 1157.99/294.45 U71(ok(X)) -> ok(U71(X)) 1157.99/294.45 U81(ok(X)) -> ok(U81(X)) 1157.99/294.45 U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) 1157.99/294.45 U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) 1157.99/294.45 U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) 1157.99/294.45 U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3)) 1157.99/294.45 U95(ok(X1), ok(X2)) -> ok(U95(X1, X2)) 1157.99/294.45 U96(ok(X)) -> ok(U96(X)) 1157.99/294.45 top(mark(X)) -> top(proper(X)) 1157.99/294.45 top(ok(X)) -> top(active(X)) 1157.99/294.45 1157.99/294.45 S is empty. 1157.99/294.45 Rewrite Strategy: FULL 1157.99/294.45 ---------------------------------------- 1157.99/294.45 1157.99/294.45 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1157.99/294.45 Transformed a relative TRS into a decreasing-loop problem. 1157.99/294.45 ---------------------------------------- 1157.99/294.45 1157.99/294.45 (2) 1157.99/294.45 Obligation: 1157.99/294.45 Analyzing the following TRS for decreasing loops: 1157.99/294.45 1157.99/294.45 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.99/294.45 1157.99/294.45 1157.99/294.45 The TRS R consists of the following rules: 1157.99/294.45 1157.99/294.45 active(zeros) -> mark(cons(0, zeros)) 1157.99/294.45 active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2)) 1157.99/294.45 active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2)) 1157.99/294.45 active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2)) 1157.99/294.46 active(U105(tt, V2)) -> mark(U106(isNatIList(V2))) 1157.99/294.46 active(U106(tt)) -> mark(tt) 1157.99/294.46 active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) 1157.99/294.46 active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N)) 1157.99/294.46 active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N)) 1157.99/294.46 active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L)) 1157.99/294.46 active(U114(tt, L)) -> mark(s(length(L))) 1157.99/294.46 active(U12(tt, V1)) -> mark(U13(isNatList(V1))) 1157.99/294.46 active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL))) 1157.99/294.46 active(U122(tt)) -> mark(nil) 1157.99/294.46 active(U13(tt)) -> mark(tt) 1157.99/294.46 active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N)) 1157.99/294.46 active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N)) 1157.99/294.46 active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N)) 1157.99/294.46 active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N)) 1157.99/294.46 active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N)) 1157.99/294.46 active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) 1157.99/294.46 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 1157.99/294.46 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 1157.99/294.46 active(U23(tt)) -> mark(tt) 1157.99/294.46 active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) 1157.99/294.46 active(U32(tt, V)) -> mark(U33(isNatList(V))) 1157.99/294.46 active(U33(tt)) -> mark(tt) 1157.99/294.46 active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) 1157.99/294.46 active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) 1157.99/294.46 active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) 1157.99/294.46 active(U46(tt)) -> mark(tt) 1157.99/294.46 active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) 1157.99/294.46 active(U52(tt)) -> mark(tt) 1157.99/294.46 active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2))) 1157.99/294.46 active(U62(tt)) -> mark(tt) 1157.99/294.46 active(U71(tt)) -> mark(tt) 1157.99/294.46 active(U81(tt)) -> mark(tt) 1157.99/294.46 active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2)) 1157.99/294.46 active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2)) 1157.99/294.46 active(U95(tt, V2)) -> mark(U96(isNatList(V2))) 1157.99/294.46 active(U96(tt)) -> mark(tt) 1157.99/294.46 active(isNat(0)) -> mark(tt) 1157.99/294.46 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1157.99/294.46 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1157.99/294.46 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1157.99/294.46 active(isNatIList(zeros)) -> mark(tt) 1157.99/294.46 active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) 1157.99/294.46 active(isNatIListKind(nil)) -> mark(tt) 1157.99/294.46 active(isNatIListKind(zeros)) -> mark(tt) 1157.99/294.46 active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) 1157.99/294.46 active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2)) 1157.99/294.46 active(isNatKind(0)) -> mark(tt) 1157.99/294.46 active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1))) 1157.99/294.46 active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1))) 1157.99/294.46 active(isNatList(nil)) -> mark(tt) 1157.99/294.46 active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2)) 1157.99/294.46 active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2)) 1157.99/294.46 active(length(nil)) -> mark(0) 1157.99/294.46 active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N)) 1157.99/294.46 active(take(0, IL)) -> mark(U121(isNatIList(IL), IL)) 1157.99/294.46 active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N)) 1157.99/294.46 active(cons(X1, X2)) -> cons(active(X1), X2) 1157.99/294.46 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 1157.99/294.46 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 1157.99/294.46 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 1157.99/294.46 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 1157.99/294.46 active(U105(X1, X2)) -> U105(active(X1), X2) 1157.99/294.46 active(U106(X)) -> U106(active(X)) 1157.99/294.46 active(U11(X1, X2)) -> U11(active(X1), X2) 1157.99/294.46 active(U12(X1, X2)) -> U12(active(X1), X2) 1157.99/294.46 active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3) 1157.99/294.46 active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3) 1157.99/294.46 active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3) 1157.99/294.46 active(U114(X1, X2)) -> U114(active(X1), X2) 1157.99/294.46 active(s(X)) -> s(active(X)) 1157.99/294.46 active(length(X)) -> length(active(X)) 1157.99/294.46 active(U13(X)) -> U13(active(X)) 1157.99/294.46 active(U121(X1, X2)) -> U121(active(X1), X2) 1157.99/294.46 active(U122(X)) -> U122(active(X)) 1157.99/294.46 active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4) 1157.99/294.46 active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4) 1157.99/294.46 active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4) 1157.99/294.46 active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4) 1157.99/294.46 active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4) 1157.99/294.46 active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4) 1157.99/294.46 active(take(X1, X2)) -> take(active(X1), X2) 1157.99/294.46 active(take(X1, X2)) -> take(X1, active(X2)) 1157.99/294.46 active(U21(X1, X2)) -> U21(active(X1), X2) 1157.99/294.46 active(U22(X1, X2)) -> U22(active(X1), X2) 1157.99/294.46 active(U23(X)) -> U23(active(X)) 1157.99/294.46 active(U31(X1, X2)) -> U31(active(X1), X2) 1157.99/294.46 active(U32(X1, X2)) -> U32(active(X1), X2) 1157.99/294.46 active(U33(X)) -> U33(active(X)) 1157.99/294.46 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1157.99/294.46 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 1157.99/294.46 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 1157.99/294.46 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 1157.99/294.46 active(U45(X1, X2)) -> U45(active(X1), X2) 1157.99/294.46 active(U46(X)) -> U46(active(X)) 1157.99/294.46 active(U51(X1, X2)) -> U51(active(X1), X2) 1157.99/294.46 active(U52(X)) -> U52(active(X)) 1157.99/294.46 active(U61(X1, X2)) -> U61(active(X1), X2) 1157.99/294.46 active(U62(X)) -> U62(active(X)) 1157.99/294.46 active(U71(X)) -> U71(active(X)) 1157.99/294.46 active(U81(X)) -> U81(active(X)) 1157.99/294.46 active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) 1157.99/294.46 active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) 1157.99/294.46 active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) 1157.99/294.46 active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3) 1157.99/294.46 active(U95(X1, X2)) -> U95(active(X1), X2) 1157.99/294.46 active(U96(X)) -> U96(active(X)) 1157.99/294.46 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1157.99/294.46 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 1157.99/294.46 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 1157.99/294.46 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 1157.99/294.46 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 1157.99/294.46 U105(mark(X1), X2) -> mark(U105(X1, X2)) 1157.99/294.46 U106(mark(X)) -> mark(U106(X)) 1157.99/294.46 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1157.99/294.46 U12(mark(X1), X2) -> mark(U12(X1, X2)) 1157.99/294.46 U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3)) 1157.99/294.46 U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3)) 1157.99/294.46 U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3)) 1157.99/294.46 U114(mark(X1), X2) -> mark(U114(X1, X2)) 1157.99/294.46 s(mark(X)) -> mark(s(X)) 1157.99/294.46 length(mark(X)) -> mark(length(X)) 1157.99/294.46 U13(mark(X)) -> mark(U13(X)) 1157.99/294.46 U121(mark(X1), X2) -> mark(U121(X1, X2)) 1157.99/294.46 U122(mark(X)) -> mark(U122(X)) 1157.99/294.46 U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4)) 1157.99/294.46 U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4)) 1157.99/294.46 U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4)) 1157.99/294.46 U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4)) 1157.99/294.46 U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4)) 1157.99/294.46 U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4)) 1157.99/294.46 take(mark(X1), X2) -> mark(take(X1, X2)) 1157.99/294.46 take(X1, mark(X2)) -> mark(take(X1, X2)) 1157.99/294.46 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1157.99/294.46 U22(mark(X1), X2) -> mark(U22(X1, X2)) 1157.99/294.46 U23(mark(X)) -> mark(U23(X)) 1157.99/294.46 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1157.99/294.46 U32(mark(X1), X2) -> mark(U32(X1, X2)) 1157.99/294.46 U33(mark(X)) -> mark(U33(X)) 1157.99/294.46 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1157.99/294.46 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 1157.99/294.46 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 1157.99/294.46 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 1157.99/294.46 U45(mark(X1), X2) -> mark(U45(X1, X2)) 1157.99/294.46 U46(mark(X)) -> mark(U46(X)) 1157.99/294.46 U51(mark(X1), X2) -> mark(U51(X1, X2)) 1157.99/294.46 U52(mark(X)) -> mark(U52(X)) 1157.99/294.46 U61(mark(X1), X2) -> mark(U61(X1, X2)) 1157.99/294.46 U62(mark(X)) -> mark(U62(X)) 1157.99/294.46 U71(mark(X)) -> mark(U71(X)) 1157.99/294.46 U81(mark(X)) -> mark(U81(X)) 1157.99/294.46 U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) 1157.99/294.46 U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) 1157.99/294.46 U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) 1157.99/294.46 U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3)) 1157.99/294.46 U95(mark(X1), X2) -> mark(U95(X1, X2)) 1157.99/294.46 U96(mark(X)) -> mark(U96(X)) 1157.99/294.46 proper(zeros) -> ok(zeros) 1157.99/294.46 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1157.99/294.46 proper(0) -> ok(0) 1157.99/294.46 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(tt) -> ok(tt) 1157.99/294.46 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(isNatKind(X)) -> isNatKind(proper(X)) 1157.99/294.46 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1157.99/294.46 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U105(X1, X2)) -> U105(proper(X1), proper(X2)) 1157.99/294.46 proper(isNat(X)) -> isNat(proper(X)) 1157.99/294.46 proper(U106(X)) -> U106(proper(X)) 1157.99/294.46 proper(isNatIList(X)) -> isNatIList(proper(X)) 1157.99/294.46 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1157.99/294.46 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 1157.99/294.46 proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U114(X1, X2)) -> U114(proper(X1), proper(X2)) 1157.99/294.46 proper(s(X)) -> s(proper(X)) 1157.99/294.46 proper(length(X)) -> length(proper(X)) 1157.99/294.46 proper(U13(X)) -> U13(proper(X)) 1157.99/294.46 proper(isNatList(X)) -> isNatList(proper(X)) 1157.99/294.46 proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)) 1157.99/294.46 proper(U122(X)) -> U122(proper(X)) 1157.99/294.46 proper(nil) -> ok(nil) 1157.99/294.46 proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1157.99/294.46 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1157.99/294.46 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 1157.99/294.46 proper(U23(X)) -> U23(proper(X)) 1157.99/294.46 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1157.99/294.46 proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) 1157.99/294.46 proper(U33(X)) -> U33(proper(X)) 1157.99/294.46 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 1157.99/294.46 proper(U46(X)) -> U46(proper(X)) 1157.99/294.46 proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) 1157.99/294.46 proper(U52(X)) -> U52(proper(X)) 1157.99/294.46 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 1157.99/294.46 proper(U62(X)) -> U62(proper(X)) 1157.99/294.46 proper(U71(X)) -> U71(proper(X)) 1157.99/294.46 proper(U81(X)) -> U81(proper(X)) 1157.99/294.46 proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U95(X1, X2)) -> U95(proper(X1), proper(X2)) 1157.99/294.46 proper(U96(X)) -> U96(proper(X)) 1157.99/294.46 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1157.99/294.46 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 1157.99/294.46 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 1157.99/294.46 isNatKind(ok(X)) -> ok(isNatKind(X)) 1157.99/294.46 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 1157.99/294.46 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1157.99/294.46 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 1157.99/294.46 U105(ok(X1), ok(X2)) -> ok(U105(X1, X2)) 1157.99/294.46 isNat(ok(X)) -> ok(isNat(X)) 1157.99/294.46 U106(ok(X)) -> ok(U106(X)) 1157.99/294.46 isNatIList(ok(X)) -> ok(isNatIList(X)) 1157.99/294.46 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1157.99/294.46 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 1157.99/294.46 U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3)) 1157.99/294.46 U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3)) 1157.99/294.46 U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3)) 1157.99/294.46 U114(ok(X1), ok(X2)) -> ok(U114(X1, X2)) 1157.99/294.46 s(ok(X)) -> ok(s(X)) 1157.99/294.46 length(ok(X)) -> ok(length(X)) 1157.99/294.46 U13(ok(X)) -> ok(U13(X)) 1157.99/294.46 isNatList(ok(X)) -> ok(isNatList(X)) 1157.99/294.46 U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)) 1157.99/294.46 U122(ok(X)) -> ok(U122(X)) 1157.99/294.46 U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4)) 1157.99/294.46 U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4)) 1157.99/294.46 U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4)) 1157.99/294.46 U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4)) 1157.99/294.46 U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4)) 1157.99/294.46 U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4)) 1157.99/294.46 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1157.99/294.46 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1157.99/294.46 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 1157.99/294.46 U23(ok(X)) -> ok(U23(X)) 1157.99/294.46 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1157.99/294.46 U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) 1157.99/294.46 U33(ok(X)) -> ok(U33(X)) 1157.99/294.46 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1157.99/294.46 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 1157.99/294.46 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 1157.99/294.46 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 1157.99/294.46 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 1157.99/294.46 U46(ok(X)) -> ok(U46(X)) 1157.99/294.46 U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) 1157.99/294.46 U52(ok(X)) -> ok(U52(X)) 1157.99/294.46 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 1157.99/294.46 U62(ok(X)) -> ok(U62(X)) 1157.99/294.46 U71(ok(X)) -> ok(U71(X)) 1157.99/294.46 U81(ok(X)) -> ok(U81(X)) 1157.99/294.46 U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) 1157.99/294.46 U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) 1157.99/294.46 U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) 1157.99/294.46 U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3)) 1157.99/294.46 U95(ok(X1), ok(X2)) -> ok(U95(X1, X2)) 1157.99/294.46 U96(ok(X)) -> ok(U96(X)) 1157.99/294.46 top(mark(X)) -> top(proper(X)) 1157.99/294.46 top(ok(X)) -> top(active(X)) 1157.99/294.46 1157.99/294.46 S is empty. 1157.99/294.46 Rewrite Strategy: FULL 1157.99/294.46 ---------------------------------------- 1157.99/294.46 1157.99/294.46 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1157.99/294.46 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1157.99/294.46 1157.99/294.46 The rewrite sequence 1157.99/294.46 1157.99/294.46 U22(mark(X1), X2) ->^+ mark(U22(X1, X2)) 1157.99/294.46 1157.99/294.46 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1157.99/294.46 1157.99/294.46 The pumping substitution is [X1 / mark(X1)]. 1157.99/294.46 1157.99/294.46 The result substitution is [ ]. 1157.99/294.46 1157.99/294.46 1157.99/294.46 1157.99/294.46 1157.99/294.46 ---------------------------------------- 1157.99/294.46 1157.99/294.46 (4) 1157.99/294.46 Complex Obligation (BEST) 1157.99/294.46 1157.99/294.46 ---------------------------------------- 1157.99/294.46 1157.99/294.46 (5) 1157.99/294.46 Obligation: 1157.99/294.46 Proved the lower bound n^1 for the following obligation: 1157.99/294.46 1157.99/294.46 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.99/294.46 1157.99/294.46 1157.99/294.46 The TRS R consists of the following rules: 1157.99/294.46 1157.99/294.46 active(zeros) -> mark(cons(0, zeros)) 1157.99/294.46 active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2)) 1157.99/294.46 active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2)) 1157.99/294.46 active(U105(tt, V2)) -> mark(U106(isNatIList(V2))) 1157.99/294.46 active(U106(tt)) -> mark(tt) 1157.99/294.46 active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) 1157.99/294.46 active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N)) 1157.99/294.46 active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N)) 1157.99/294.46 active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L)) 1157.99/294.46 active(U114(tt, L)) -> mark(s(length(L))) 1157.99/294.46 active(U12(tt, V1)) -> mark(U13(isNatList(V1))) 1157.99/294.46 active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL))) 1157.99/294.46 active(U122(tt)) -> mark(nil) 1157.99/294.46 active(U13(tt)) -> mark(tt) 1157.99/294.46 active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N)) 1157.99/294.46 active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N)) 1157.99/294.46 active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N)) 1157.99/294.46 active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N)) 1157.99/294.46 active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N)) 1157.99/294.46 active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) 1157.99/294.46 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 1157.99/294.46 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 1157.99/294.46 active(U23(tt)) -> mark(tt) 1157.99/294.46 active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) 1157.99/294.46 active(U32(tt, V)) -> mark(U33(isNatList(V))) 1157.99/294.46 active(U33(tt)) -> mark(tt) 1157.99/294.46 active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) 1157.99/294.46 active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) 1157.99/294.46 active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) 1157.99/294.46 active(U46(tt)) -> mark(tt) 1157.99/294.46 active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) 1157.99/294.46 active(U52(tt)) -> mark(tt) 1157.99/294.46 active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2))) 1157.99/294.46 active(U62(tt)) -> mark(tt) 1157.99/294.46 active(U71(tt)) -> mark(tt) 1157.99/294.46 active(U81(tt)) -> mark(tt) 1157.99/294.46 active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2)) 1157.99/294.46 active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2)) 1157.99/294.46 active(U95(tt, V2)) -> mark(U96(isNatList(V2))) 1157.99/294.46 active(U96(tt)) -> mark(tt) 1157.99/294.46 active(isNat(0)) -> mark(tt) 1157.99/294.46 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1157.99/294.46 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1157.99/294.46 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1157.99/294.46 active(isNatIList(zeros)) -> mark(tt) 1157.99/294.46 active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) 1157.99/294.46 active(isNatIListKind(nil)) -> mark(tt) 1157.99/294.46 active(isNatIListKind(zeros)) -> mark(tt) 1157.99/294.46 active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) 1157.99/294.46 active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2)) 1157.99/294.46 active(isNatKind(0)) -> mark(tt) 1157.99/294.46 active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1))) 1157.99/294.46 active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1))) 1157.99/294.46 active(isNatList(nil)) -> mark(tt) 1157.99/294.46 active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2)) 1157.99/294.46 active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2)) 1157.99/294.46 active(length(nil)) -> mark(0) 1157.99/294.46 active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N)) 1157.99/294.46 active(take(0, IL)) -> mark(U121(isNatIList(IL), IL)) 1157.99/294.46 active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N)) 1157.99/294.46 active(cons(X1, X2)) -> cons(active(X1), X2) 1157.99/294.46 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 1157.99/294.46 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 1157.99/294.46 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 1157.99/294.46 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 1157.99/294.46 active(U105(X1, X2)) -> U105(active(X1), X2) 1157.99/294.46 active(U106(X)) -> U106(active(X)) 1157.99/294.46 active(U11(X1, X2)) -> U11(active(X1), X2) 1157.99/294.46 active(U12(X1, X2)) -> U12(active(X1), X2) 1157.99/294.46 active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3) 1157.99/294.46 active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3) 1157.99/294.46 active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3) 1157.99/294.46 active(U114(X1, X2)) -> U114(active(X1), X2) 1157.99/294.46 active(s(X)) -> s(active(X)) 1157.99/294.46 active(length(X)) -> length(active(X)) 1157.99/294.46 active(U13(X)) -> U13(active(X)) 1157.99/294.46 active(U121(X1, X2)) -> U121(active(X1), X2) 1157.99/294.46 active(U122(X)) -> U122(active(X)) 1157.99/294.46 active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4) 1157.99/294.46 active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4) 1157.99/294.46 active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4) 1157.99/294.46 active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4) 1157.99/294.46 active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4) 1157.99/294.46 active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4) 1157.99/294.46 active(take(X1, X2)) -> take(active(X1), X2) 1157.99/294.46 active(take(X1, X2)) -> take(X1, active(X2)) 1157.99/294.46 active(U21(X1, X2)) -> U21(active(X1), X2) 1157.99/294.46 active(U22(X1, X2)) -> U22(active(X1), X2) 1157.99/294.46 active(U23(X)) -> U23(active(X)) 1157.99/294.46 active(U31(X1, X2)) -> U31(active(X1), X2) 1157.99/294.46 active(U32(X1, X2)) -> U32(active(X1), X2) 1157.99/294.46 active(U33(X)) -> U33(active(X)) 1157.99/294.46 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1157.99/294.46 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 1157.99/294.46 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 1157.99/294.46 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 1157.99/294.46 active(U45(X1, X2)) -> U45(active(X1), X2) 1157.99/294.46 active(U46(X)) -> U46(active(X)) 1157.99/294.46 active(U51(X1, X2)) -> U51(active(X1), X2) 1157.99/294.46 active(U52(X)) -> U52(active(X)) 1157.99/294.46 active(U61(X1, X2)) -> U61(active(X1), X2) 1157.99/294.46 active(U62(X)) -> U62(active(X)) 1157.99/294.46 active(U71(X)) -> U71(active(X)) 1157.99/294.46 active(U81(X)) -> U81(active(X)) 1157.99/294.46 active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) 1157.99/294.46 active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) 1157.99/294.46 active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) 1157.99/294.46 active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3) 1157.99/294.46 active(U95(X1, X2)) -> U95(active(X1), X2) 1157.99/294.46 active(U96(X)) -> U96(active(X)) 1157.99/294.46 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1157.99/294.46 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 1157.99/294.46 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 1157.99/294.46 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 1157.99/294.46 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 1157.99/294.46 U105(mark(X1), X2) -> mark(U105(X1, X2)) 1157.99/294.46 U106(mark(X)) -> mark(U106(X)) 1157.99/294.46 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1157.99/294.46 U12(mark(X1), X2) -> mark(U12(X1, X2)) 1157.99/294.46 U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3)) 1157.99/294.46 U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3)) 1157.99/294.46 U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3)) 1157.99/294.46 U114(mark(X1), X2) -> mark(U114(X1, X2)) 1157.99/294.46 s(mark(X)) -> mark(s(X)) 1157.99/294.46 length(mark(X)) -> mark(length(X)) 1157.99/294.46 U13(mark(X)) -> mark(U13(X)) 1157.99/294.46 U121(mark(X1), X2) -> mark(U121(X1, X2)) 1157.99/294.46 U122(mark(X)) -> mark(U122(X)) 1157.99/294.46 U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4)) 1157.99/294.46 U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4)) 1157.99/294.46 U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4)) 1157.99/294.46 U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4)) 1157.99/294.46 U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4)) 1157.99/294.46 U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4)) 1157.99/294.46 take(mark(X1), X2) -> mark(take(X1, X2)) 1157.99/294.46 take(X1, mark(X2)) -> mark(take(X1, X2)) 1157.99/294.46 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1157.99/294.46 U22(mark(X1), X2) -> mark(U22(X1, X2)) 1157.99/294.46 U23(mark(X)) -> mark(U23(X)) 1157.99/294.46 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1157.99/294.46 U32(mark(X1), X2) -> mark(U32(X1, X2)) 1157.99/294.46 U33(mark(X)) -> mark(U33(X)) 1157.99/294.46 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1157.99/294.46 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 1157.99/294.46 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 1157.99/294.46 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 1157.99/294.46 U45(mark(X1), X2) -> mark(U45(X1, X2)) 1157.99/294.46 U46(mark(X)) -> mark(U46(X)) 1157.99/294.46 U51(mark(X1), X2) -> mark(U51(X1, X2)) 1157.99/294.46 U52(mark(X)) -> mark(U52(X)) 1157.99/294.46 U61(mark(X1), X2) -> mark(U61(X1, X2)) 1157.99/294.46 U62(mark(X)) -> mark(U62(X)) 1157.99/294.46 U71(mark(X)) -> mark(U71(X)) 1157.99/294.46 U81(mark(X)) -> mark(U81(X)) 1157.99/294.46 U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) 1157.99/294.46 U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) 1157.99/294.46 U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) 1157.99/294.46 U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3)) 1157.99/294.46 U95(mark(X1), X2) -> mark(U95(X1, X2)) 1157.99/294.46 U96(mark(X)) -> mark(U96(X)) 1157.99/294.46 proper(zeros) -> ok(zeros) 1157.99/294.46 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1157.99/294.46 proper(0) -> ok(0) 1157.99/294.46 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(tt) -> ok(tt) 1157.99/294.46 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(isNatKind(X)) -> isNatKind(proper(X)) 1157.99/294.46 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1157.99/294.46 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U105(X1, X2)) -> U105(proper(X1), proper(X2)) 1157.99/294.46 proper(isNat(X)) -> isNat(proper(X)) 1157.99/294.46 proper(U106(X)) -> U106(proper(X)) 1157.99/294.46 proper(isNatIList(X)) -> isNatIList(proper(X)) 1157.99/294.46 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1157.99/294.46 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 1157.99/294.46 proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U114(X1, X2)) -> U114(proper(X1), proper(X2)) 1157.99/294.46 proper(s(X)) -> s(proper(X)) 1157.99/294.46 proper(length(X)) -> length(proper(X)) 1157.99/294.46 proper(U13(X)) -> U13(proper(X)) 1157.99/294.46 proper(isNatList(X)) -> isNatList(proper(X)) 1157.99/294.46 proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)) 1157.99/294.46 proper(U122(X)) -> U122(proper(X)) 1157.99/294.46 proper(nil) -> ok(nil) 1157.99/294.46 proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1157.99/294.46 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1157.99/294.46 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 1157.99/294.46 proper(U23(X)) -> U23(proper(X)) 1157.99/294.46 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1157.99/294.46 proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) 1157.99/294.46 proper(U33(X)) -> U33(proper(X)) 1157.99/294.46 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 1157.99/294.46 proper(U46(X)) -> U46(proper(X)) 1157.99/294.46 proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) 1157.99/294.46 proper(U52(X)) -> U52(proper(X)) 1157.99/294.46 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 1157.99/294.46 proper(U62(X)) -> U62(proper(X)) 1157.99/294.46 proper(U71(X)) -> U71(proper(X)) 1157.99/294.46 proper(U81(X)) -> U81(proper(X)) 1157.99/294.46 proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U95(X1, X2)) -> U95(proper(X1), proper(X2)) 1157.99/294.46 proper(U96(X)) -> U96(proper(X)) 1157.99/294.46 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1157.99/294.46 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 1157.99/294.46 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 1157.99/294.46 isNatKind(ok(X)) -> ok(isNatKind(X)) 1157.99/294.46 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 1157.99/294.46 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1157.99/294.46 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 1157.99/294.46 U105(ok(X1), ok(X2)) -> ok(U105(X1, X2)) 1157.99/294.46 isNat(ok(X)) -> ok(isNat(X)) 1157.99/294.46 U106(ok(X)) -> ok(U106(X)) 1157.99/294.46 isNatIList(ok(X)) -> ok(isNatIList(X)) 1157.99/294.46 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1157.99/294.46 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 1157.99/294.46 U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3)) 1157.99/294.46 U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3)) 1157.99/294.46 U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3)) 1157.99/294.46 U114(ok(X1), ok(X2)) -> ok(U114(X1, X2)) 1157.99/294.46 s(ok(X)) -> ok(s(X)) 1157.99/294.46 length(ok(X)) -> ok(length(X)) 1157.99/294.46 U13(ok(X)) -> ok(U13(X)) 1157.99/294.46 isNatList(ok(X)) -> ok(isNatList(X)) 1157.99/294.46 U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)) 1157.99/294.46 U122(ok(X)) -> ok(U122(X)) 1157.99/294.46 U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4)) 1157.99/294.46 U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4)) 1157.99/294.46 U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4)) 1157.99/294.46 U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4)) 1157.99/294.46 U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4)) 1157.99/294.46 U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4)) 1157.99/294.46 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1157.99/294.46 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1157.99/294.46 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 1157.99/294.46 U23(ok(X)) -> ok(U23(X)) 1157.99/294.46 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1157.99/294.46 U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) 1157.99/294.46 U33(ok(X)) -> ok(U33(X)) 1157.99/294.46 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1157.99/294.46 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 1157.99/294.46 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 1157.99/294.46 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 1157.99/294.46 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 1157.99/294.46 U46(ok(X)) -> ok(U46(X)) 1157.99/294.46 U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) 1157.99/294.46 U52(ok(X)) -> ok(U52(X)) 1157.99/294.46 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 1157.99/294.46 U62(ok(X)) -> ok(U62(X)) 1157.99/294.46 U71(ok(X)) -> ok(U71(X)) 1157.99/294.46 U81(ok(X)) -> ok(U81(X)) 1157.99/294.46 U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) 1157.99/294.46 U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) 1157.99/294.46 U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) 1157.99/294.46 U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3)) 1157.99/294.46 U95(ok(X1), ok(X2)) -> ok(U95(X1, X2)) 1157.99/294.46 U96(ok(X)) -> ok(U96(X)) 1157.99/294.46 top(mark(X)) -> top(proper(X)) 1157.99/294.46 top(ok(X)) -> top(active(X)) 1157.99/294.46 1157.99/294.46 S is empty. 1157.99/294.46 Rewrite Strategy: FULL 1157.99/294.46 ---------------------------------------- 1157.99/294.46 1157.99/294.46 (6) LowerBoundPropagationProof (FINISHED) 1157.99/294.46 Propagated lower bound. 1157.99/294.46 ---------------------------------------- 1157.99/294.46 1157.99/294.46 (7) 1157.99/294.46 BOUNDS(n^1, INF) 1157.99/294.46 1157.99/294.46 ---------------------------------------- 1157.99/294.46 1157.99/294.46 (8) 1157.99/294.46 Obligation: 1157.99/294.46 Analyzing the following TRS for decreasing loops: 1157.99/294.46 1157.99/294.46 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1157.99/294.46 1157.99/294.46 1157.99/294.46 The TRS R consists of the following rules: 1157.99/294.46 1157.99/294.46 active(zeros) -> mark(cons(0, zeros)) 1157.99/294.46 active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2)) 1157.99/294.46 active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2)) 1157.99/294.46 active(U105(tt, V2)) -> mark(U106(isNatIList(V2))) 1157.99/294.46 active(U106(tt)) -> mark(tt) 1157.99/294.46 active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1)) 1157.99/294.46 active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N)) 1157.99/294.46 active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N)) 1157.99/294.46 active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L)) 1157.99/294.46 active(U114(tt, L)) -> mark(s(length(L))) 1157.99/294.46 active(U12(tt, V1)) -> mark(U13(isNatList(V1))) 1157.99/294.46 active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL))) 1157.99/294.46 active(U122(tt)) -> mark(nil) 1157.99/294.46 active(U13(tt)) -> mark(tt) 1157.99/294.46 active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N)) 1157.99/294.46 active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N)) 1157.99/294.46 active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N)) 1157.99/294.46 active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N)) 1157.99/294.46 active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N)) 1157.99/294.46 active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) 1157.99/294.46 active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1)) 1157.99/294.46 active(U22(tt, V1)) -> mark(U23(isNat(V1))) 1157.99/294.46 active(U23(tt)) -> mark(tt) 1157.99/294.46 active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V)) 1157.99/294.46 active(U32(tt, V)) -> mark(U33(isNatList(V))) 1157.99/294.46 active(U33(tt)) -> mark(tt) 1157.99/294.46 active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2)) 1157.99/294.46 active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2)) 1157.99/294.46 active(U45(tt, V2)) -> mark(U46(isNatIList(V2))) 1157.99/294.46 active(U46(tt)) -> mark(tt) 1157.99/294.46 active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2))) 1157.99/294.46 active(U52(tt)) -> mark(tt) 1157.99/294.46 active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2))) 1157.99/294.46 active(U62(tt)) -> mark(tt) 1157.99/294.46 active(U71(tt)) -> mark(tt) 1157.99/294.46 active(U81(tt)) -> mark(tt) 1157.99/294.46 active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2)) 1157.99/294.46 active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2)) 1157.99/294.46 active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2)) 1157.99/294.46 active(U95(tt, V2)) -> mark(U96(isNatList(V2))) 1157.99/294.46 active(U96(tt)) -> mark(tt) 1157.99/294.46 active(isNat(0)) -> mark(tt) 1157.99/294.46 active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1)) 1157.99/294.46 active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1)) 1157.99/294.46 active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V)) 1157.99/294.46 active(isNatIList(zeros)) -> mark(tt) 1157.99/294.46 active(isNatIList(cons(V1, V2))) -> mark(U41(isNatKind(V1), V1, V2)) 1157.99/294.46 active(isNatIListKind(nil)) -> mark(tt) 1157.99/294.46 active(isNatIListKind(zeros)) -> mark(tt) 1157.99/294.46 active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2)) 1157.99/294.46 active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2)) 1157.99/294.46 active(isNatKind(0)) -> mark(tt) 1157.99/294.46 active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1))) 1157.99/294.46 active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1))) 1157.99/294.46 active(isNatList(nil)) -> mark(tt) 1157.99/294.46 active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2)) 1157.99/294.46 active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2)) 1157.99/294.46 active(length(nil)) -> mark(0) 1157.99/294.46 active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N)) 1157.99/294.46 active(take(0, IL)) -> mark(U121(isNatIList(IL), IL)) 1157.99/294.46 active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N)) 1157.99/294.46 active(cons(X1, X2)) -> cons(active(X1), X2) 1157.99/294.46 active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3) 1157.99/294.46 active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3) 1157.99/294.46 active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3) 1157.99/294.46 active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3) 1157.99/294.46 active(U105(X1, X2)) -> U105(active(X1), X2) 1157.99/294.46 active(U106(X)) -> U106(active(X)) 1157.99/294.46 active(U11(X1, X2)) -> U11(active(X1), X2) 1157.99/294.46 active(U12(X1, X2)) -> U12(active(X1), X2) 1157.99/294.46 active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3) 1157.99/294.46 active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3) 1157.99/294.46 active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3) 1157.99/294.46 active(U114(X1, X2)) -> U114(active(X1), X2) 1157.99/294.46 active(s(X)) -> s(active(X)) 1157.99/294.46 active(length(X)) -> length(active(X)) 1157.99/294.46 active(U13(X)) -> U13(active(X)) 1157.99/294.46 active(U121(X1, X2)) -> U121(active(X1), X2) 1157.99/294.46 active(U122(X)) -> U122(active(X)) 1157.99/294.46 active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4) 1157.99/294.46 active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4) 1157.99/294.46 active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4) 1157.99/294.46 active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4) 1157.99/294.46 active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4) 1157.99/294.46 active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4) 1157.99/294.46 active(take(X1, X2)) -> take(active(X1), X2) 1157.99/294.46 active(take(X1, X2)) -> take(X1, active(X2)) 1157.99/294.46 active(U21(X1, X2)) -> U21(active(X1), X2) 1157.99/294.46 active(U22(X1, X2)) -> U22(active(X1), X2) 1157.99/294.46 active(U23(X)) -> U23(active(X)) 1157.99/294.46 active(U31(X1, X2)) -> U31(active(X1), X2) 1157.99/294.46 active(U32(X1, X2)) -> U32(active(X1), X2) 1157.99/294.46 active(U33(X)) -> U33(active(X)) 1157.99/294.46 active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3) 1157.99/294.46 active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3) 1157.99/294.46 active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3) 1157.99/294.46 active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3) 1157.99/294.46 active(U45(X1, X2)) -> U45(active(X1), X2) 1157.99/294.46 active(U46(X)) -> U46(active(X)) 1157.99/294.46 active(U51(X1, X2)) -> U51(active(X1), X2) 1157.99/294.46 active(U52(X)) -> U52(active(X)) 1157.99/294.46 active(U61(X1, X2)) -> U61(active(X1), X2) 1157.99/294.46 active(U62(X)) -> U62(active(X)) 1157.99/294.46 active(U71(X)) -> U71(active(X)) 1157.99/294.46 active(U81(X)) -> U81(active(X)) 1157.99/294.46 active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3) 1157.99/294.46 active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3) 1157.99/294.46 active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3) 1157.99/294.46 active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3) 1157.99/294.46 active(U95(X1, X2)) -> U95(active(X1), X2) 1157.99/294.46 active(U96(X)) -> U96(active(X)) 1157.99/294.46 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1157.99/294.46 U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3)) 1157.99/294.46 U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3)) 1157.99/294.46 U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3)) 1157.99/294.46 U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3)) 1157.99/294.46 U105(mark(X1), X2) -> mark(U105(X1, X2)) 1157.99/294.46 U106(mark(X)) -> mark(U106(X)) 1157.99/294.46 U11(mark(X1), X2) -> mark(U11(X1, X2)) 1157.99/294.46 U12(mark(X1), X2) -> mark(U12(X1, X2)) 1157.99/294.46 U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3)) 1157.99/294.46 U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3)) 1157.99/294.46 U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3)) 1157.99/294.46 U114(mark(X1), X2) -> mark(U114(X1, X2)) 1157.99/294.46 s(mark(X)) -> mark(s(X)) 1157.99/294.46 length(mark(X)) -> mark(length(X)) 1157.99/294.46 U13(mark(X)) -> mark(U13(X)) 1157.99/294.46 U121(mark(X1), X2) -> mark(U121(X1, X2)) 1157.99/294.46 U122(mark(X)) -> mark(U122(X)) 1157.99/294.46 U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4)) 1157.99/294.46 U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4)) 1157.99/294.46 U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4)) 1157.99/294.46 U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4)) 1157.99/294.46 U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4)) 1157.99/294.46 U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4)) 1157.99/294.46 take(mark(X1), X2) -> mark(take(X1, X2)) 1157.99/294.46 take(X1, mark(X2)) -> mark(take(X1, X2)) 1157.99/294.46 U21(mark(X1), X2) -> mark(U21(X1, X2)) 1157.99/294.46 U22(mark(X1), X2) -> mark(U22(X1, X2)) 1157.99/294.46 U23(mark(X)) -> mark(U23(X)) 1157.99/294.46 U31(mark(X1), X2) -> mark(U31(X1, X2)) 1157.99/294.46 U32(mark(X1), X2) -> mark(U32(X1, X2)) 1157.99/294.46 U33(mark(X)) -> mark(U33(X)) 1157.99/294.46 U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3)) 1157.99/294.46 U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3)) 1157.99/294.46 U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3)) 1157.99/294.46 U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3)) 1157.99/294.46 U45(mark(X1), X2) -> mark(U45(X1, X2)) 1157.99/294.46 U46(mark(X)) -> mark(U46(X)) 1157.99/294.46 U51(mark(X1), X2) -> mark(U51(X1, X2)) 1157.99/294.46 U52(mark(X)) -> mark(U52(X)) 1157.99/294.46 U61(mark(X1), X2) -> mark(U61(X1, X2)) 1157.99/294.46 U62(mark(X)) -> mark(U62(X)) 1157.99/294.46 U71(mark(X)) -> mark(U71(X)) 1157.99/294.46 U81(mark(X)) -> mark(U81(X)) 1157.99/294.46 U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3)) 1157.99/294.46 U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3)) 1157.99/294.46 U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3)) 1157.99/294.46 U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3)) 1157.99/294.46 U95(mark(X1), X2) -> mark(U95(X1, X2)) 1157.99/294.46 U96(mark(X)) -> mark(U96(X)) 1157.99/294.46 proper(zeros) -> ok(zeros) 1157.99/294.46 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1157.99/294.46 proper(0) -> ok(0) 1157.99/294.46 proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(tt) -> ok(tt) 1157.99/294.46 proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(isNatKind(X)) -> isNatKind(proper(X)) 1157.99/294.46 proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(isNatIListKind(X)) -> isNatIListKind(proper(X)) 1157.99/294.46 proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U105(X1, X2)) -> U105(proper(X1), proper(X2)) 1157.99/294.46 proper(isNat(X)) -> isNat(proper(X)) 1157.99/294.46 proper(U106(X)) -> U106(proper(X)) 1157.99/294.46 proper(isNatIList(X)) -> isNatIList(proper(X)) 1157.99/294.46 proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) 1157.99/294.46 proper(U12(X1, X2)) -> U12(proper(X1), proper(X2)) 1157.99/294.46 proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U114(X1, X2)) -> U114(proper(X1), proper(X2)) 1157.99/294.46 proper(s(X)) -> s(proper(X)) 1157.99/294.46 proper(length(X)) -> length(proper(X)) 1157.99/294.46 proper(U13(X)) -> U13(proper(X)) 1157.99/294.46 proper(isNatList(X)) -> isNatList(proper(X)) 1157.99/294.46 proper(U121(X1, X2)) -> U121(proper(X1), proper(X2)) 1157.99/294.46 proper(U122(X)) -> U122(proper(X)) 1157.99/294.46 proper(nil) -> ok(nil) 1157.99/294.46 proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4)) 1157.99/294.46 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1157.99/294.46 proper(U21(X1, X2)) -> U21(proper(X1), proper(X2)) 1157.99/294.46 proper(U22(X1, X2)) -> U22(proper(X1), proper(X2)) 1157.99/294.46 proper(U23(X)) -> U23(proper(X)) 1157.99/294.46 proper(U31(X1, X2)) -> U31(proper(X1), proper(X2)) 1157.99/294.46 proper(U32(X1, X2)) -> U32(proper(X1), proper(X2)) 1157.99/294.46 proper(U33(X)) -> U33(proper(X)) 1157.99/294.46 proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U45(X1, X2)) -> U45(proper(X1), proper(X2)) 1157.99/294.46 proper(U46(X)) -> U46(proper(X)) 1157.99/294.46 proper(U51(X1, X2)) -> U51(proper(X1), proper(X2)) 1157.99/294.46 proper(U52(X)) -> U52(proper(X)) 1157.99/294.46 proper(U61(X1, X2)) -> U61(proper(X1), proper(X2)) 1157.99/294.46 proper(U62(X)) -> U62(proper(X)) 1157.99/294.46 proper(U71(X)) -> U71(proper(X)) 1157.99/294.46 proper(U81(X)) -> U81(proper(X)) 1157.99/294.46 proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3)) 1157.99/294.46 proper(U95(X1, X2)) -> U95(proper(X1), proper(X2)) 1157.99/294.46 proper(U96(X)) -> U96(proper(X)) 1157.99/294.46 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1157.99/294.46 U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3)) 1157.99/294.46 U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3)) 1157.99/294.46 isNatKind(ok(X)) -> ok(isNatKind(X)) 1157.99/294.46 U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3)) 1157.99/294.46 isNatIListKind(ok(X)) -> ok(isNatIListKind(X)) 1157.99/294.46 U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3)) 1157.99/294.46 U105(ok(X1), ok(X2)) -> ok(U105(X1, X2)) 1157.99/294.46 isNat(ok(X)) -> ok(isNat(X)) 1157.99/294.46 U106(ok(X)) -> ok(U106(X)) 1157.99/294.46 isNatIList(ok(X)) -> ok(isNatIList(X)) 1157.99/294.46 U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) 1157.99/294.46 U12(ok(X1), ok(X2)) -> ok(U12(X1, X2)) 1157.99/294.46 U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3)) 1158.32/294.46 U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3)) 1158.32/294.46 U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3)) 1158.32/294.46 U114(ok(X1), ok(X2)) -> ok(U114(X1, X2)) 1158.32/294.46 s(ok(X)) -> ok(s(X)) 1158.32/294.46 length(ok(X)) -> ok(length(X)) 1158.32/294.46 U13(ok(X)) -> ok(U13(X)) 1158.32/294.46 isNatList(ok(X)) -> ok(isNatList(X)) 1158.32/294.46 U121(ok(X1), ok(X2)) -> ok(U121(X1, X2)) 1158.32/294.46 U122(ok(X)) -> ok(U122(X)) 1158.32/294.46 U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4)) 1158.32/294.46 U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4)) 1158.32/294.46 U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4)) 1158.32/294.46 U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4)) 1158.32/294.46 U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4)) 1158.32/294.46 U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4)) 1158.32/294.46 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1158.32/294.46 U21(ok(X1), ok(X2)) -> ok(U21(X1, X2)) 1158.32/294.46 U22(ok(X1), ok(X2)) -> ok(U22(X1, X2)) 1158.32/294.46 U23(ok(X)) -> ok(U23(X)) 1158.32/294.46 U31(ok(X1), ok(X2)) -> ok(U31(X1, X2)) 1158.32/294.46 U32(ok(X1), ok(X2)) -> ok(U32(X1, X2)) 1158.32/294.46 U33(ok(X)) -> ok(U33(X)) 1158.32/294.46 U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3)) 1158.32/294.46 U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3)) 1158.32/294.46 U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3)) 1158.32/294.46 U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3)) 1158.32/294.46 U45(ok(X1), ok(X2)) -> ok(U45(X1, X2)) 1158.32/294.46 U46(ok(X)) -> ok(U46(X)) 1158.32/294.46 U51(ok(X1), ok(X2)) -> ok(U51(X1, X2)) 1158.32/294.46 U52(ok(X)) -> ok(U52(X)) 1158.32/294.46 U61(ok(X1), ok(X2)) -> ok(U61(X1, X2)) 1158.32/294.46 U62(ok(X)) -> ok(U62(X)) 1158.32/294.46 U71(ok(X)) -> ok(U71(X)) 1158.32/294.46 U81(ok(X)) -> ok(U81(X)) 1158.32/294.46 U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3)) 1158.32/294.46 U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3)) 1158.32/294.46 U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3)) 1158.32/294.46 U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3)) 1158.32/294.46 U95(ok(X1), ok(X2)) -> ok(U95(X1, X2)) 1158.32/294.46 U96(ok(X)) -> ok(U96(X)) 1158.32/294.46 top(mark(X)) -> top(proper(X)) 1158.32/294.46 top(ok(X)) -> top(active(X)) 1158.32/294.46 1158.32/294.46 S is empty. 1158.32/294.46 Rewrite Strategy: FULL 1158.43/294.54 EOF