1145.63/291.63 WORST_CASE(Omega(n^1), ?) 1149.85/292.66 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1149.85/292.66 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1149.85/292.66 1149.85/292.66 1149.85/292.66 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1149.85/292.66 1149.85/292.66 (0) CpxTRS 1149.85/292.66 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1149.85/292.66 (2) TRS for Loop Detection 1149.85/292.66 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1149.85/292.66 (4) BEST 1149.85/292.66 (5) proven lower bound 1149.85/292.66 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1149.85/292.66 (7) BOUNDS(n^1, INF) 1149.85/292.66 (8) TRS for Loop Detection 1149.85/292.66 1149.85/292.66 1149.85/292.66 ---------------------------------------- 1149.85/292.66 1149.85/292.66 (0) 1149.85/292.66 Obligation: 1149.85/292.66 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1149.85/292.66 1149.85/292.66 1149.85/292.66 The TRS R consists of the following rules: 1149.85/292.66 1149.85/292.66 a__zeros -> cons(0, zeros) 1149.85/292.66 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 1149.85/292.66 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 1149.85/292.66 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 1149.85/292.66 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 1149.85/292.66 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 1149.85/292.66 a__U106(tt) -> tt 1149.85/292.66 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 1149.85/292.66 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 1149.85/292.66 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 1149.85/292.66 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 1149.85/292.66 a__U114(tt, L) -> s(a__length(mark(L))) 1149.85/292.66 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 1149.85/292.66 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 1149.85/292.66 a__U122(tt) -> nil 1149.85/292.66 a__U13(tt) -> tt 1149.85/292.66 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 1149.85/292.66 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 1149.85/292.66 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 1149.85/292.66 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 1149.85/292.66 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 1149.85/292.66 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 1149.85/292.66 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 1149.85/292.66 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 1149.85/292.66 a__U23(tt) -> tt 1149.85/292.66 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 1149.85/292.66 a__U32(tt, V) -> a__U33(a__isNatList(V)) 1149.85/292.66 a__U33(tt) -> tt 1149.85/292.66 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 1149.85/292.66 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 1149.85/292.66 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 1149.85/292.66 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 1149.85/292.66 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 1149.85/292.66 a__U46(tt) -> tt 1149.85/292.66 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 1149.85/292.66 a__U52(tt) -> tt 1149.85/292.66 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 1149.85/292.66 a__U62(tt) -> tt 1149.85/292.66 a__U71(tt) -> tt 1149.85/292.66 a__U81(tt) -> tt 1149.85/292.66 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 1149.85/292.66 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 1149.85/292.66 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 1149.85/292.66 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 1149.85/292.66 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 1149.85/292.66 a__U96(tt) -> tt 1149.85/292.66 a__isNat(0) -> tt 1149.85/292.66 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 1149.85/292.66 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1149.85/292.66 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 1149.85/292.66 a__isNatIList(zeros) -> tt 1149.85/292.66 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 1149.85/292.66 a__isNatIListKind(nil) -> tt 1149.85/292.66 a__isNatIListKind(zeros) -> tt 1149.85/292.66 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 1149.85/292.66 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 1149.85/292.66 a__isNatKind(0) -> tt 1149.85/292.66 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 1149.85/292.66 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 1149.85/292.66 a__isNatList(nil) -> tt 1149.85/292.66 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 1149.85/292.66 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 1149.85/292.66 a__length(nil) -> 0 1149.85/292.66 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 1149.85/292.66 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 1149.85/292.66 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 1149.85/292.66 mark(zeros) -> a__zeros 1149.85/292.66 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 1149.85/292.66 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 1149.85/292.66 mark(isNatKind(X)) -> a__isNatKind(X) 1149.85/292.66 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 1149.85/292.66 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 1149.85/292.66 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 1149.85/292.66 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 1149.85/292.66 mark(isNat(X)) -> a__isNat(X) 1149.85/292.66 mark(U106(X)) -> a__U106(mark(X)) 1149.85/292.66 mark(isNatIList(X)) -> a__isNatIList(X) 1149.85/292.66 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1149.85/292.66 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1149.85/292.66 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 1149.85/292.66 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 1149.85/292.66 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 1149.85/292.66 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 1149.85/292.66 mark(length(X)) -> a__length(mark(X)) 1149.85/292.66 mark(U13(X)) -> a__U13(mark(X)) 1149.85/292.66 mark(isNatList(X)) -> a__isNatList(X) 1149.85/292.66 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 1149.85/292.66 mark(U122(X)) -> a__U122(mark(X)) 1149.85/292.66 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 1149.85/292.66 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 1149.85/292.66 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 1149.85/292.66 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 1149.85/292.66 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 1149.85/292.66 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 1149.85/292.66 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1149.85/292.66 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1149.85/292.66 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1149.85/292.66 mark(U23(X)) -> a__U23(mark(X)) 1149.85/292.66 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1149.85/292.66 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1149.85/292.66 mark(U33(X)) -> a__U33(mark(X)) 1149.85/292.66 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1149.85/292.66 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 1149.85/292.66 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 1149.85/292.66 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 1149.85/292.66 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 1149.85/292.66 mark(U46(X)) -> a__U46(mark(X)) 1149.85/292.66 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1149.85/292.66 mark(U52(X)) -> a__U52(mark(X)) 1149.85/292.66 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 1149.85/292.66 mark(U62(X)) -> a__U62(mark(X)) 1149.85/292.66 mark(U71(X)) -> a__U71(mark(X)) 1149.85/292.66 mark(U81(X)) -> a__U81(mark(X)) 1149.85/292.66 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 1149.85/292.66 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 1149.85/292.66 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 1149.85/292.66 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 1149.85/292.66 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 1149.85/292.66 mark(U96(X)) -> a__U96(mark(X)) 1149.85/292.66 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1149.85/292.66 mark(0) -> 0 1149.85/292.66 mark(tt) -> tt 1149.85/292.66 mark(s(X)) -> s(mark(X)) 1149.85/292.66 mark(nil) -> nil 1149.85/292.66 a__zeros -> zeros 1149.85/292.66 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 1149.85/292.66 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 1149.85/292.66 a__isNatKind(X) -> isNatKind(X) 1149.85/292.66 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 1149.85/292.66 a__isNatIListKind(X) -> isNatIListKind(X) 1149.85/292.66 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 1149.85/292.66 a__U105(X1, X2) -> U105(X1, X2) 1149.85/292.66 a__isNat(X) -> isNat(X) 1149.85/292.66 a__U106(X) -> U106(X) 1149.85/292.66 a__isNatIList(X) -> isNatIList(X) 1149.85/292.66 a__U11(X1, X2) -> U11(X1, X2) 1149.85/292.66 a__U12(X1, X2) -> U12(X1, X2) 1149.85/292.66 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 1149.85/292.66 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 1149.85/292.66 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 1149.85/292.66 a__U114(X1, X2) -> U114(X1, X2) 1149.85/292.66 a__length(X) -> length(X) 1149.85/292.66 a__U13(X) -> U13(X) 1149.85/292.66 a__isNatList(X) -> isNatList(X) 1149.85/292.66 a__U121(X1, X2) -> U121(X1, X2) 1149.85/292.66 a__U122(X) -> U122(X) 1149.85/292.66 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 1149.85/292.66 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 1149.85/292.66 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 1149.85/292.66 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 1149.85/292.66 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 1149.85/292.66 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 1149.85/292.66 a__take(X1, X2) -> take(X1, X2) 1149.85/292.66 a__U21(X1, X2) -> U21(X1, X2) 1149.85/292.66 a__U22(X1, X2) -> U22(X1, X2) 1149.85/292.66 a__U23(X) -> U23(X) 1149.85/292.66 a__U31(X1, X2) -> U31(X1, X2) 1149.85/292.66 a__U32(X1, X2) -> U32(X1, X2) 1149.85/292.66 a__U33(X) -> U33(X) 1149.85/292.66 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1149.85/292.66 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 1149.85/292.66 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 1149.85/292.66 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 1149.85/292.66 a__U45(X1, X2) -> U45(X1, X2) 1149.85/292.66 a__U46(X) -> U46(X) 1149.85/292.66 a__U51(X1, X2) -> U51(X1, X2) 1149.85/292.66 a__U52(X) -> U52(X) 1149.85/292.66 a__U61(X1, X2) -> U61(X1, X2) 1149.85/292.66 a__U62(X) -> U62(X) 1149.85/292.66 a__U71(X) -> U71(X) 1149.85/292.66 a__U81(X) -> U81(X) 1149.85/292.66 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 1149.85/292.67 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 1149.85/292.67 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 1149.85/292.67 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 1149.85/292.67 a__U95(X1, X2) -> U95(X1, X2) 1149.85/292.67 a__U96(X) -> U96(X) 1149.85/292.67 1149.85/292.67 S is empty. 1149.85/292.67 Rewrite Strategy: INNERMOST 1149.85/292.67 ---------------------------------------- 1149.85/292.67 1149.85/292.67 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1149.85/292.67 Transformed a relative TRS into a decreasing-loop problem. 1149.85/292.67 ---------------------------------------- 1149.85/292.67 1149.85/292.67 (2) 1149.85/292.67 Obligation: 1149.85/292.67 Analyzing the following TRS for decreasing loops: 1149.85/292.67 1149.85/292.67 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1149.85/292.67 1149.85/292.67 1149.85/292.67 The TRS R consists of the following rules: 1149.85/292.67 1149.85/292.67 a__zeros -> cons(0, zeros) 1149.85/292.67 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 1149.85/292.67 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 1149.85/292.67 a__U106(tt) -> tt 1149.85/292.67 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 1149.85/292.67 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 1149.85/292.67 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 1149.85/292.67 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 1149.85/292.67 a__U114(tt, L) -> s(a__length(mark(L))) 1149.85/292.67 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 1149.85/292.67 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 1149.85/292.67 a__U122(tt) -> nil 1149.85/292.67 a__U13(tt) -> tt 1149.85/292.67 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 1149.85/292.67 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 1149.85/292.67 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 1149.85/292.67 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 1149.85/292.67 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 1149.85/292.67 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 1149.85/292.67 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 1149.85/292.67 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 1149.85/292.67 a__U23(tt) -> tt 1149.85/292.67 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 1149.85/292.67 a__U32(tt, V) -> a__U33(a__isNatList(V)) 1149.85/292.67 a__U33(tt) -> tt 1149.85/292.67 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 1149.85/292.67 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 1149.85/292.67 a__U46(tt) -> tt 1149.85/292.67 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 1149.85/292.67 a__U52(tt) -> tt 1149.85/292.67 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 1149.85/292.67 a__U62(tt) -> tt 1149.85/292.67 a__U71(tt) -> tt 1149.85/292.67 a__U81(tt) -> tt 1149.85/292.67 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 1149.85/292.67 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 1149.85/292.67 a__U96(tt) -> tt 1149.85/292.67 a__isNat(0) -> tt 1149.85/292.67 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 1149.85/292.67 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1149.85/292.67 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 1149.85/292.67 a__isNatIList(zeros) -> tt 1149.85/292.67 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__isNatIListKind(nil) -> tt 1149.85/292.67 a__isNatIListKind(zeros) -> tt 1149.85/292.67 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 1149.85/292.67 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 1149.85/292.67 a__isNatKind(0) -> tt 1149.85/292.67 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 1149.85/292.67 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 1149.85/292.67 a__isNatList(nil) -> tt 1149.85/292.67 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__length(nil) -> 0 1149.85/292.67 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 1149.85/292.67 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 1149.85/292.67 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 1149.85/292.67 mark(zeros) -> a__zeros 1149.85/292.67 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 1149.85/292.67 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 1149.85/292.67 mark(isNatKind(X)) -> a__isNatKind(X) 1149.85/292.67 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 1149.85/292.67 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 1149.85/292.67 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 1149.85/292.67 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 1149.85/292.67 mark(isNat(X)) -> a__isNat(X) 1149.85/292.67 mark(U106(X)) -> a__U106(mark(X)) 1149.85/292.67 mark(isNatIList(X)) -> a__isNatIList(X) 1149.85/292.67 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1149.85/292.67 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1149.85/292.67 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 1149.85/292.67 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 1149.85/292.67 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 1149.85/292.67 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 1149.85/292.67 mark(length(X)) -> a__length(mark(X)) 1149.85/292.67 mark(U13(X)) -> a__U13(mark(X)) 1149.85/292.67 mark(isNatList(X)) -> a__isNatList(X) 1149.85/292.67 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 1149.85/292.67 mark(U122(X)) -> a__U122(mark(X)) 1149.85/292.67 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 1149.85/292.67 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1149.85/292.67 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1149.85/292.67 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1149.85/292.67 mark(U23(X)) -> a__U23(mark(X)) 1149.85/292.67 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1149.85/292.67 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1149.85/292.67 mark(U33(X)) -> a__U33(mark(X)) 1149.85/292.67 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1149.85/292.67 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 1149.85/292.67 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 1149.85/292.67 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 1149.85/292.67 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 1149.85/292.67 mark(U46(X)) -> a__U46(mark(X)) 1149.85/292.67 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1149.85/292.67 mark(U52(X)) -> a__U52(mark(X)) 1149.85/292.67 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 1149.85/292.67 mark(U62(X)) -> a__U62(mark(X)) 1149.85/292.67 mark(U71(X)) -> a__U71(mark(X)) 1149.85/292.67 mark(U81(X)) -> a__U81(mark(X)) 1149.85/292.67 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 1149.85/292.67 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 1149.85/292.67 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 1149.85/292.67 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 1149.85/292.67 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 1149.85/292.67 mark(U96(X)) -> a__U96(mark(X)) 1149.85/292.67 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1149.85/292.67 mark(0) -> 0 1149.85/292.67 mark(tt) -> tt 1149.85/292.67 mark(s(X)) -> s(mark(X)) 1149.85/292.67 mark(nil) -> nil 1149.85/292.67 a__zeros -> zeros 1149.85/292.67 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 1149.85/292.67 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 1149.85/292.67 a__isNatKind(X) -> isNatKind(X) 1149.85/292.67 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 1149.85/292.67 a__isNatIListKind(X) -> isNatIListKind(X) 1149.85/292.67 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 1149.85/292.67 a__U105(X1, X2) -> U105(X1, X2) 1149.85/292.67 a__isNat(X) -> isNat(X) 1149.85/292.67 a__U106(X) -> U106(X) 1149.85/292.67 a__isNatIList(X) -> isNatIList(X) 1149.85/292.67 a__U11(X1, X2) -> U11(X1, X2) 1149.85/292.67 a__U12(X1, X2) -> U12(X1, X2) 1149.85/292.67 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 1149.85/292.67 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 1149.85/292.67 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 1149.85/292.67 a__U114(X1, X2) -> U114(X1, X2) 1149.85/292.67 a__length(X) -> length(X) 1149.85/292.67 a__U13(X) -> U13(X) 1149.85/292.67 a__isNatList(X) -> isNatList(X) 1149.85/292.67 a__U121(X1, X2) -> U121(X1, X2) 1149.85/292.67 a__U122(X) -> U122(X) 1149.85/292.67 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 1149.85/292.67 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 1149.85/292.67 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 1149.85/292.67 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 1149.85/292.67 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 1149.85/292.67 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 1149.85/292.67 a__take(X1, X2) -> take(X1, X2) 1149.85/292.67 a__U21(X1, X2) -> U21(X1, X2) 1149.85/292.67 a__U22(X1, X2) -> U22(X1, X2) 1149.85/292.67 a__U23(X) -> U23(X) 1149.85/292.67 a__U31(X1, X2) -> U31(X1, X2) 1149.85/292.67 a__U32(X1, X2) -> U32(X1, X2) 1149.85/292.67 a__U33(X) -> U33(X) 1149.85/292.67 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1149.85/292.67 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 1149.85/292.67 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 1149.85/292.67 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 1149.85/292.67 a__U45(X1, X2) -> U45(X1, X2) 1149.85/292.67 a__U46(X) -> U46(X) 1149.85/292.67 a__U51(X1, X2) -> U51(X1, X2) 1149.85/292.67 a__U52(X) -> U52(X) 1149.85/292.67 a__U61(X1, X2) -> U61(X1, X2) 1149.85/292.67 a__U62(X) -> U62(X) 1149.85/292.67 a__U71(X) -> U71(X) 1149.85/292.67 a__U81(X) -> U81(X) 1149.85/292.67 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 1149.85/292.67 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 1149.85/292.67 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 1149.85/292.67 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 1149.85/292.67 a__U95(X1, X2) -> U95(X1, X2) 1149.85/292.67 a__U96(X) -> U96(X) 1149.85/292.67 1149.85/292.67 S is empty. 1149.85/292.67 Rewrite Strategy: INNERMOST 1149.85/292.67 ---------------------------------------- 1149.85/292.67 1149.85/292.67 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1149.85/292.67 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1149.85/292.67 1149.85/292.67 The rewrite sequence 1149.85/292.67 1149.85/292.67 mark(U71(X)) ->^+ a__U71(mark(X)) 1149.85/292.67 1149.85/292.67 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1149.85/292.67 1149.85/292.67 The pumping substitution is [X / U71(X)]. 1149.85/292.67 1149.85/292.67 The result substitution is [ ]. 1149.85/292.67 1149.85/292.67 1149.85/292.67 1149.85/292.67 1149.85/292.67 ---------------------------------------- 1149.85/292.67 1149.85/292.67 (4) 1149.85/292.67 Complex Obligation (BEST) 1149.85/292.67 1149.85/292.67 ---------------------------------------- 1149.85/292.67 1149.85/292.67 (5) 1149.85/292.67 Obligation: 1149.85/292.67 Proved the lower bound n^1 for the following obligation: 1149.85/292.67 1149.85/292.67 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1149.85/292.67 1149.85/292.67 1149.85/292.67 The TRS R consists of the following rules: 1149.85/292.67 1149.85/292.67 a__zeros -> cons(0, zeros) 1149.85/292.67 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 1149.85/292.67 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 1149.85/292.67 a__U106(tt) -> tt 1149.85/292.67 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 1149.85/292.67 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 1149.85/292.67 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 1149.85/292.67 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 1149.85/292.67 a__U114(tt, L) -> s(a__length(mark(L))) 1149.85/292.67 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 1149.85/292.67 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 1149.85/292.67 a__U122(tt) -> nil 1149.85/292.67 a__U13(tt) -> tt 1149.85/292.67 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 1149.85/292.67 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 1149.85/292.67 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 1149.85/292.67 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 1149.85/292.67 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 1149.85/292.67 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 1149.85/292.67 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 1149.85/292.67 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 1149.85/292.67 a__U23(tt) -> tt 1149.85/292.67 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 1149.85/292.67 a__U32(tt, V) -> a__U33(a__isNatList(V)) 1149.85/292.67 a__U33(tt) -> tt 1149.85/292.67 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 1149.85/292.67 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 1149.85/292.67 a__U46(tt) -> tt 1149.85/292.67 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 1149.85/292.67 a__U52(tt) -> tt 1149.85/292.67 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 1149.85/292.67 a__U62(tt) -> tt 1149.85/292.67 a__U71(tt) -> tt 1149.85/292.67 a__U81(tt) -> tt 1149.85/292.67 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 1149.85/292.67 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 1149.85/292.67 a__U96(tt) -> tt 1149.85/292.67 a__isNat(0) -> tt 1149.85/292.67 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 1149.85/292.67 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1149.85/292.67 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 1149.85/292.67 a__isNatIList(zeros) -> tt 1149.85/292.67 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__isNatIListKind(nil) -> tt 1149.85/292.67 a__isNatIListKind(zeros) -> tt 1149.85/292.67 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 1149.85/292.67 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 1149.85/292.67 a__isNatKind(0) -> tt 1149.85/292.67 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 1149.85/292.67 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 1149.85/292.67 a__isNatList(nil) -> tt 1149.85/292.67 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__length(nil) -> 0 1149.85/292.67 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 1149.85/292.67 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 1149.85/292.67 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 1149.85/292.67 mark(zeros) -> a__zeros 1149.85/292.67 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 1149.85/292.67 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 1149.85/292.67 mark(isNatKind(X)) -> a__isNatKind(X) 1149.85/292.67 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 1149.85/292.67 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 1149.85/292.67 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 1149.85/292.67 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 1149.85/292.67 mark(isNat(X)) -> a__isNat(X) 1149.85/292.67 mark(U106(X)) -> a__U106(mark(X)) 1149.85/292.67 mark(isNatIList(X)) -> a__isNatIList(X) 1149.85/292.67 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1149.85/292.67 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1149.85/292.67 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 1149.85/292.67 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 1149.85/292.67 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 1149.85/292.67 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 1149.85/292.67 mark(length(X)) -> a__length(mark(X)) 1149.85/292.67 mark(U13(X)) -> a__U13(mark(X)) 1149.85/292.67 mark(isNatList(X)) -> a__isNatList(X) 1149.85/292.67 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 1149.85/292.67 mark(U122(X)) -> a__U122(mark(X)) 1149.85/292.67 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 1149.85/292.67 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1149.85/292.67 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1149.85/292.67 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1149.85/292.67 mark(U23(X)) -> a__U23(mark(X)) 1149.85/292.67 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1149.85/292.67 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1149.85/292.67 mark(U33(X)) -> a__U33(mark(X)) 1149.85/292.67 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1149.85/292.67 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 1149.85/292.67 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 1149.85/292.67 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 1149.85/292.67 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 1149.85/292.67 mark(U46(X)) -> a__U46(mark(X)) 1149.85/292.67 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1149.85/292.67 mark(U52(X)) -> a__U52(mark(X)) 1149.85/292.67 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 1149.85/292.67 mark(U62(X)) -> a__U62(mark(X)) 1149.85/292.67 mark(U71(X)) -> a__U71(mark(X)) 1149.85/292.67 mark(U81(X)) -> a__U81(mark(X)) 1149.85/292.67 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 1149.85/292.67 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 1149.85/292.67 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 1149.85/292.67 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 1149.85/292.67 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 1149.85/292.67 mark(U96(X)) -> a__U96(mark(X)) 1149.85/292.67 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1149.85/292.67 mark(0) -> 0 1149.85/292.67 mark(tt) -> tt 1149.85/292.67 mark(s(X)) -> s(mark(X)) 1149.85/292.67 mark(nil) -> nil 1149.85/292.67 a__zeros -> zeros 1149.85/292.67 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 1149.85/292.67 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 1149.85/292.67 a__isNatKind(X) -> isNatKind(X) 1149.85/292.67 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 1149.85/292.67 a__isNatIListKind(X) -> isNatIListKind(X) 1149.85/292.67 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 1149.85/292.67 a__U105(X1, X2) -> U105(X1, X2) 1149.85/292.67 a__isNat(X) -> isNat(X) 1149.85/292.67 a__U106(X) -> U106(X) 1149.85/292.67 a__isNatIList(X) -> isNatIList(X) 1149.85/292.67 a__U11(X1, X2) -> U11(X1, X2) 1149.85/292.67 a__U12(X1, X2) -> U12(X1, X2) 1149.85/292.67 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 1149.85/292.67 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 1149.85/292.67 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 1149.85/292.67 a__U114(X1, X2) -> U114(X1, X2) 1149.85/292.67 a__length(X) -> length(X) 1149.85/292.67 a__U13(X) -> U13(X) 1149.85/292.67 a__isNatList(X) -> isNatList(X) 1149.85/292.67 a__U121(X1, X2) -> U121(X1, X2) 1149.85/292.67 a__U122(X) -> U122(X) 1149.85/292.67 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 1149.85/292.67 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 1149.85/292.67 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 1149.85/292.67 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 1149.85/292.67 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 1149.85/292.67 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 1149.85/292.67 a__take(X1, X2) -> take(X1, X2) 1149.85/292.67 a__U21(X1, X2) -> U21(X1, X2) 1149.85/292.67 a__U22(X1, X2) -> U22(X1, X2) 1149.85/292.67 a__U23(X) -> U23(X) 1149.85/292.67 a__U31(X1, X2) -> U31(X1, X2) 1149.85/292.67 a__U32(X1, X2) -> U32(X1, X2) 1149.85/292.67 a__U33(X) -> U33(X) 1149.85/292.67 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1149.85/292.67 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 1149.85/292.67 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 1149.85/292.67 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 1149.85/292.67 a__U45(X1, X2) -> U45(X1, X2) 1149.85/292.67 a__U46(X) -> U46(X) 1149.85/292.67 a__U51(X1, X2) -> U51(X1, X2) 1149.85/292.67 a__U52(X) -> U52(X) 1149.85/292.67 a__U61(X1, X2) -> U61(X1, X2) 1149.85/292.67 a__U62(X) -> U62(X) 1149.85/292.67 a__U71(X) -> U71(X) 1149.85/292.67 a__U81(X) -> U81(X) 1149.85/292.67 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 1149.85/292.67 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 1149.85/292.67 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 1149.85/292.67 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 1149.85/292.67 a__U95(X1, X2) -> U95(X1, X2) 1149.85/292.67 a__U96(X) -> U96(X) 1149.85/292.67 1149.85/292.67 S is empty. 1149.85/292.67 Rewrite Strategy: INNERMOST 1149.85/292.67 ---------------------------------------- 1149.85/292.67 1149.85/292.67 (6) LowerBoundPropagationProof (FINISHED) 1149.85/292.67 Propagated lower bound. 1149.85/292.67 ---------------------------------------- 1149.85/292.67 1149.85/292.67 (7) 1149.85/292.67 BOUNDS(n^1, INF) 1149.85/292.67 1149.85/292.67 ---------------------------------------- 1149.85/292.67 1149.85/292.67 (8) 1149.85/292.67 Obligation: 1149.85/292.67 Analyzing the following TRS for decreasing loops: 1149.85/292.67 1149.85/292.67 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1149.85/292.67 1149.85/292.67 1149.85/292.67 The TRS R consists of the following rules: 1149.85/292.67 1149.85/292.67 a__zeros -> cons(0, zeros) 1149.85/292.67 a__U101(tt, V1, V2) -> a__U102(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__U102(tt, V1, V2) -> a__U103(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U103(tt, V1, V2) -> a__U104(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U104(tt, V1, V2) -> a__U105(a__isNat(V1), V2) 1149.85/292.67 a__U105(tt, V2) -> a__U106(a__isNatIList(V2)) 1149.85/292.67 a__U106(tt) -> tt 1149.85/292.67 a__U11(tt, V1) -> a__U12(a__isNatIListKind(V1), V1) 1149.85/292.67 a__U111(tt, L, N) -> a__U112(a__isNatIListKind(L), L, N) 1149.85/292.67 a__U112(tt, L, N) -> a__U113(a__isNat(N), L, N) 1149.85/292.67 a__U113(tt, L, N) -> a__U114(a__isNatKind(N), L) 1149.85/292.67 a__U114(tt, L) -> s(a__length(mark(L))) 1149.85/292.67 a__U12(tt, V1) -> a__U13(a__isNatList(V1)) 1149.85/292.67 a__U121(tt, IL) -> a__U122(a__isNatIListKind(IL)) 1149.85/292.67 a__U122(tt) -> nil 1149.85/292.67 a__U13(tt) -> tt 1149.85/292.67 a__U131(tt, IL, M, N) -> a__U132(a__isNatIListKind(IL), IL, M, N) 1149.85/292.67 a__U132(tt, IL, M, N) -> a__U133(a__isNat(M), IL, M, N) 1149.85/292.67 a__U133(tt, IL, M, N) -> a__U134(a__isNatKind(M), IL, M, N) 1149.85/292.67 a__U134(tt, IL, M, N) -> a__U135(a__isNat(N), IL, M, N) 1149.85/292.67 a__U135(tt, IL, M, N) -> a__U136(a__isNatKind(N), IL, M, N) 1149.85/292.67 a__U136(tt, IL, M, N) -> cons(mark(N), take(M, IL)) 1149.85/292.67 a__U21(tt, V1) -> a__U22(a__isNatKind(V1), V1) 1149.85/292.67 a__U22(tt, V1) -> a__U23(a__isNat(V1)) 1149.85/292.67 a__U23(tt) -> tt 1149.85/292.67 a__U31(tt, V) -> a__U32(a__isNatIListKind(V), V) 1149.85/292.67 a__U32(tt, V) -> a__U33(a__isNatList(V)) 1149.85/292.67 a__U33(tt) -> tt 1149.85/292.67 a__U41(tt, V1, V2) -> a__U42(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__U42(tt, V1, V2) -> a__U43(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U43(tt, V1, V2) -> a__U44(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U44(tt, V1, V2) -> a__U45(a__isNat(V1), V2) 1149.85/292.67 a__U45(tt, V2) -> a__U46(a__isNatIList(V2)) 1149.85/292.67 a__U46(tt) -> tt 1149.85/292.67 a__U51(tt, V2) -> a__U52(a__isNatIListKind(V2)) 1149.85/292.67 a__U52(tt) -> tt 1149.85/292.67 a__U61(tt, V2) -> a__U62(a__isNatIListKind(V2)) 1149.85/292.67 a__U62(tt) -> tt 1149.85/292.67 a__U71(tt) -> tt 1149.85/292.67 a__U81(tt) -> tt 1149.85/292.67 a__U91(tt, V1, V2) -> a__U92(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__U92(tt, V1, V2) -> a__U93(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U93(tt, V1, V2) -> a__U94(a__isNatIListKind(V2), V1, V2) 1149.85/292.67 a__U94(tt, V1, V2) -> a__U95(a__isNat(V1), V2) 1149.85/292.67 a__U95(tt, V2) -> a__U96(a__isNatList(V2)) 1149.85/292.67 a__U96(tt) -> tt 1149.85/292.67 a__isNat(0) -> tt 1149.85/292.67 a__isNat(length(V1)) -> a__U11(a__isNatIListKind(V1), V1) 1149.85/292.67 a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1149.85/292.67 a__isNatIList(V) -> a__U31(a__isNatIListKind(V), V) 1149.85/292.67 a__isNatIList(zeros) -> tt 1149.85/292.67 a__isNatIList(cons(V1, V2)) -> a__U41(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__isNatIListKind(nil) -> tt 1149.85/292.67 a__isNatIListKind(zeros) -> tt 1149.85/292.67 a__isNatIListKind(cons(V1, V2)) -> a__U51(a__isNatKind(V1), V2) 1149.85/292.67 a__isNatIListKind(take(V1, V2)) -> a__U61(a__isNatKind(V1), V2) 1149.85/292.67 a__isNatKind(0) -> tt 1149.85/292.67 a__isNatKind(length(V1)) -> a__U71(a__isNatIListKind(V1)) 1149.85/292.67 a__isNatKind(s(V1)) -> a__U81(a__isNatKind(V1)) 1149.85/292.67 a__isNatList(nil) -> tt 1149.85/292.67 a__isNatList(cons(V1, V2)) -> a__U91(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__isNatList(take(V1, V2)) -> a__U101(a__isNatKind(V1), V1, V2) 1149.85/292.67 a__length(nil) -> 0 1149.85/292.67 a__length(cons(N, L)) -> a__U111(a__isNatList(L), L, N) 1149.85/292.67 a__take(0, IL) -> a__U121(a__isNatIList(IL), IL) 1149.85/292.67 a__take(s(M), cons(N, IL)) -> a__U131(a__isNatIList(IL), IL, M, N) 1149.85/292.67 mark(zeros) -> a__zeros 1149.85/292.67 mark(U101(X1, X2, X3)) -> a__U101(mark(X1), X2, X3) 1149.85/292.67 mark(U102(X1, X2, X3)) -> a__U102(mark(X1), X2, X3) 1149.85/292.67 mark(isNatKind(X)) -> a__isNatKind(X) 1149.85/292.67 mark(U103(X1, X2, X3)) -> a__U103(mark(X1), X2, X3) 1149.85/292.67 mark(isNatIListKind(X)) -> a__isNatIListKind(X) 1149.85/292.67 mark(U104(X1, X2, X3)) -> a__U104(mark(X1), X2, X3) 1149.85/292.67 mark(U105(X1, X2)) -> a__U105(mark(X1), X2) 1149.85/292.67 mark(isNat(X)) -> a__isNat(X) 1149.85/292.67 mark(U106(X)) -> a__U106(mark(X)) 1149.85/292.67 mark(isNatIList(X)) -> a__isNatIList(X) 1149.85/292.67 mark(U11(X1, X2)) -> a__U11(mark(X1), X2) 1149.85/292.67 mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1149.85/292.67 mark(U111(X1, X2, X3)) -> a__U111(mark(X1), X2, X3) 1149.85/292.67 mark(U112(X1, X2, X3)) -> a__U112(mark(X1), X2, X3) 1149.85/292.67 mark(U113(X1, X2, X3)) -> a__U113(mark(X1), X2, X3) 1149.85/292.67 mark(U114(X1, X2)) -> a__U114(mark(X1), X2) 1149.85/292.67 mark(length(X)) -> a__length(mark(X)) 1149.85/292.67 mark(U13(X)) -> a__U13(mark(X)) 1149.85/292.67 mark(isNatList(X)) -> a__isNatList(X) 1149.85/292.67 mark(U121(X1, X2)) -> a__U121(mark(X1), X2) 1149.85/292.67 mark(U122(X)) -> a__U122(mark(X)) 1149.85/292.67 mark(U131(X1, X2, X3, X4)) -> a__U131(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U132(X1, X2, X3, X4)) -> a__U132(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U133(X1, X2, X3, X4)) -> a__U133(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U134(X1, X2, X3, X4)) -> a__U134(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U135(X1, X2, X3, X4)) -> a__U135(mark(X1), X2, X3, X4) 1149.85/292.67 mark(U136(X1, X2, X3, X4)) -> a__U136(mark(X1), X2, X3, X4) 1149.85/292.67 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1149.85/292.67 mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1149.85/292.67 mark(U22(X1, X2)) -> a__U22(mark(X1), X2) 1149.85/292.67 mark(U23(X)) -> a__U23(mark(X)) 1149.85/292.67 mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1149.85/292.67 mark(U32(X1, X2)) -> a__U32(mark(X1), X2) 1149.85/292.67 mark(U33(X)) -> a__U33(mark(X)) 1149.85/292.67 mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1149.85/292.67 mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) 1149.85/292.67 mark(U43(X1, X2, X3)) -> a__U43(mark(X1), X2, X3) 1149.85/292.67 mark(U44(X1, X2, X3)) -> a__U44(mark(X1), X2, X3) 1149.85/292.67 mark(U45(X1, X2)) -> a__U45(mark(X1), X2) 1149.85/292.67 mark(U46(X)) -> a__U46(mark(X)) 1149.85/292.67 mark(U51(X1, X2)) -> a__U51(mark(X1), X2) 1149.85/292.67 mark(U52(X)) -> a__U52(mark(X)) 1149.85/292.67 mark(U61(X1, X2)) -> a__U61(mark(X1), X2) 1149.85/292.67 mark(U62(X)) -> a__U62(mark(X)) 1149.85/292.67 mark(U71(X)) -> a__U71(mark(X)) 1149.85/292.67 mark(U81(X)) -> a__U81(mark(X)) 1149.85/292.67 mark(U91(X1, X2, X3)) -> a__U91(mark(X1), X2, X3) 1149.85/292.67 mark(U92(X1, X2, X3)) -> a__U92(mark(X1), X2, X3) 1149.85/292.67 mark(U93(X1, X2, X3)) -> a__U93(mark(X1), X2, X3) 1149.85/292.67 mark(U94(X1, X2, X3)) -> a__U94(mark(X1), X2, X3) 1149.85/292.67 mark(U95(X1, X2)) -> a__U95(mark(X1), X2) 1149.85/292.67 mark(U96(X)) -> a__U96(mark(X)) 1149.85/292.67 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1149.85/292.67 mark(0) -> 0 1149.85/292.67 mark(tt) -> tt 1149.85/292.67 mark(s(X)) -> s(mark(X)) 1149.85/292.67 mark(nil) -> nil 1149.85/292.67 a__zeros -> zeros 1149.85/292.67 a__U101(X1, X2, X3) -> U101(X1, X2, X3) 1149.85/292.67 a__U102(X1, X2, X3) -> U102(X1, X2, X3) 1149.85/292.67 a__isNatKind(X) -> isNatKind(X) 1149.85/292.67 a__U103(X1, X2, X3) -> U103(X1, X2, X3) 1149.85/292.67 a__isNatIListKind(X) -> isNatIListKind(X) 1149.85/292.67 a__U104(X1, X2, X3) -> U104(X1, X2, X3) 1149.85/292.67 a__U105(X1, X2) -> U105(X1, X2) 1149.85/292.67 a__isNat(X) -> isNat(X) 1149.85/292.67 a__U106(X) -> U106(X) 1149.85/292.67 a__isNatIList(X) -> isNatIList(X) 1149.85/292.67 a__U11(X1, X2) -> U11(X1, X2) 1149.85/292.67 a__U12(X1, X2) -> U12(X1, X2) 1149.85/292.67 a__U111(X1, X2, X3) -> U111(X1, X2, X3) 1149.85/292.67 a__U112(X1, X2, X3) -> U112(X1, X2, X3) 1149.85/292.67 a__U113(X1, X2, X3) -> U113(X1, X2, X3) 1149.85/292.67 a__U114(X1, X2) -> U114(X1, X2) 1149.85/292.67 a__length(X) -> length(X) 1149.85/292.67 a__U13(X) -> U13(X) 1149.85/292.67 a__isNatList(X) -> isNatList(X) 1149.85/292.67 a__U121(X1, X2) -> U121(X1, X2) 1149.85/292.67 a__U122(X) -> U122(X) 1149.85/292.67 a__U131(X1, X2, X3, X4) -> U131(X1, X2, X3, X4) 1149.85/292.67 a__U132(X1, X2, X3, X4) -> U132(X1, X2, X3, X4) 1149.85/292.67 a__U133(X1, X2, X3, X4) -> U133(X1, X2, X3, X4) 1149.85/292.67 a__U134(X1, X2, X3, X4) -> U134(X1, X2, X3, X4) 1149.85/292.67 a__U135(X1, X2, X3, X4) -> U135(X1, X2, X3, X4) 1149.85/292.67 a__U136(X1, X2, X3, X4) -> U136(X1, X2, X3, X4) 1149.85/292.67 a__take(X1, X2) -> take(X1, X2) 1149.85/292.67 a__U21(X1, X2) -> U21(X1, X2) 1149.85/292.67 a__U22(X1, X2) -> U22(X1, X2) 1149.85/292.67 a__U23(X) -> U23(X) 1149.85/292.67 a__U31(X1, X2) -> U31(X1, X2) 1149.85/292.67 a__U32(X1, X2) -> U32(X1, X2) 1149.85/292.67 a__U33(X) -> U33(X) 1149.85/292.67 a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1149.85/292.67 a__U42(X1, X2, X3) -> U42(X1, X2, X3) 1149.85/292.67 a__U43(X1, X2, X3) -> U43(X1, X2, X3) 1149.85/292.67 a__U44(X1, X2, X3) -> U44(X1, X2, X3) 1149.85/292.67 a__U45(X1, X2) -> U45(X1, X2) 1149.85/292.67 a__U46(X) -> U46(X) 1149.85/292.67 a__U51(X1, X2) -> U51(X1, X2) 1149.85/292.67 a__U52(X) -> U52(X) 1149.85/292.67 a__U61(X1, X2) -> U61(X1, X2) 1149.85/292.67 a__U62(X) -> U62(X) 1149.85/292.67 a__U71(X) -> U71(X) 1149.85/292.67 a__U81(X) -> U81(X) 1149.85/292.67 a__U91(X1, X2, X3) -> U91(X1, X2, X3) 1149.85/292.67 a__U92(X1, X2, X3) -> U92(X1, X2, X3) 1149.85/292.67 a__U93(X1, X2, X3) -> U93(X1, X2, X3) 1149.85/292.67 a__U94(X1, X2, X3) -> U94(X1, X2, X3) 1149.85/292.67 a__U95(X1, X2) -> U95(X1, X2) 1149.85/292.67 a__U96(X) -> U96(X) 1149.85/292.67 1149.85/292.67 S is empty. 1149.85/292.67 Rewrite Strategy: INNERMOST 1149.97/292.75 EOF