/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). (0) CpxTRS (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (2) TRS for Loop Detection (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (4) BEST (5) proven lower bound (6) LowerBoundPropagationProof [FINISHED, 0 ms] (7) BOUNDS(n^1, INF) (8) TRS for Loop Detection (9) DecreasingLoopProof [FINISHED, 318 ms] (10) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N, XS) -> a__U12(tt, N, XS) a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__U21(tt, X) -> a__U22(tt, X) a__U22(tt, X) -> mark(X) a__U31(tt, N) -> a__U32(tt, N) a__U32(tt, N) -> mark(N) a__U41(tt, N, XS) -> a__U42(tt, N, XS) a__U42(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__U51(tt, Y) -> a__U52(tt, Y) a__U52(tt, Y) -> mark(Y) a__U61(tt, N, X, XS) -> a__U62(tt, N, X, XS) a__U62(tt, N, X, XS) -> a__U63(tt, N, X, XS) a__U63(tt, N, X, XS) -> a__U64(a__splitAt(mark(N), mark(XS)), X) a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__U71(tt, XS) -> a__U72(tt, XS) a__U72(tt, XS) -> mark(XS) a__U81(tt, N, XS) -> a__U82(tt, N, XS) a__U82(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) a__afterNth(N, XS) -> a__U11(tt, N, XS) a__fst(pair(X, Y)) -> a__U21(tt, X) a__head(cons(N, XS)) -> a__U31(tt, N) a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) a__sel(N, XS) -> a__U41(tt, N, XS) a__snd(pair(X, Y)) -> a__U51(tt, Y) a__splitAt(0, XS) -> pair(nil, mark(XS)) a__splitAt(s(N), cons(X, XS)) -> a__U61(tt, N, X, XS) a__tail(cons(N, XS)) -> a__U71(tt, XS) a__take(N, XS) -> a__U81(tt, N, XS) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) mark(snd(X)) -> a__snd(mark(X)) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(head(X)) -> a__head(mark(X)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4) mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4) mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4) mark(U64(X1, X2)) -> a__U64(mark(X1), X2) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X1, X2)) -> a__U72(mark(X1), X2) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(fst(X)) -> a__fst(mark(X)) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(tt) -> tt mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__U12(X1, X2, X3) -> U12(X1, X2, X3) a__snd(X) -> snd(X) a__splitAt(X1, X2) -> splitAt(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__head(X) -> head(X) a__afterNth(X1, X2) -> afterNth(X1, X2) a__U51(X1, X2) -> U51(X1, X2) a__U52(X1, X2) -> U52(X1, X2) a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4) a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4) a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4) a__U64(X1, X2) -> U64(X1, X2) a__U71(X1, X2) -> U71(X1, X2) a__U72(X1, X2) -> U72(X1, X2) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__fst(X) -> fst(X) a__natsFrom(X) -> natsFrom(X) a__sel(X1, X2) -> sel(X1, X2) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (2) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N, XS) -> a__U12(tt, N, XS) a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__U21(tt, X) -> a__U22(tt, X) a__U22(tt, X) -> mark(X) a__U31(tt, N) -> a__U32(tt, N) a__U32(tt, N) -> mark(N) a__U41(tt, N, XS) -> a__U42(tt, N, XS) a__U42(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__U51(tt, Y) -> a__U52(tt, Y) a__U52(tt, Y) -> mark(Y) a__U61(tt, N, X, XS) -> a__U62(tt, N, X, XS) a__U62(tt, N, X, XS) -> a__U63(tt, N, X, XS) a__U63(tt, N, X, XS) -> a__U64(a__splitAt(mark(N), mark(XS)), X) a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__U71(tt, XS) -> a__U72(tt, XS) a__U72(tt, XS) -> mark(XS) a__U81(tt, N, XS) -> a__U82(tt, N, XS) a__U82(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) a__afterNth(N, XS) -> a__U11(tt, N, XS) a__fst(pair(X, Y)) -> a__U21(tt, X) a__head(cons(N, XS)) -> a__U31(tt, N) a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) a__sel(N, XS) -> a__U41(tt, N, XS) a__snd(pair(X, Y)) -> a__U51(tt, Y) a__splitAt(0, XS) -> pair(nil, mark(XS)) a__splitAt(s(N), cons(X, XS)) -> a__U61(tt, N, X, XS) a__tail(cons(N, XS)) -> a__U71(tt, XS) a__take(N, XS) -> a__U81(tt, N, XS) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) mark(snd(X)) -> a__snd(mark(X)) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(head(X)) -> a__head(mark(X)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4) mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4) mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4) mark(U64(X1, X2)) -> a__U64(mark(X1), X2) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X1, X2)) -> a__U72(mark(X1), X2) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(fst(X)) -> a__fst(mark(X)) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(tt) -> tt mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__U12(X1, X2, X3) -> U12(X1, X2, X3) a__snd(X) -> snd(X) a__splitAt(X1, X2) -> splitAt(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__head(X) -> head(X) a__afterNth(X1, X2) -> afterNth(X1, X2) a__U51(X1, X2) -> U51(X1, X2) a__U52(X1, X2) -> U52(X1, X2) a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4) a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4) a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4) a__U64(X1, X2) -> U64(X1, X2) a__U71(X1, X2) -> U71(X1, X2) a__U72(X1, X2) -> U72(X1, X2) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__fst(X) -> fst(X) a__natsFrom(X) -> natsFrom(X) a__sel(X1, X2) -> sel(X1, X2) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence mark(afterNth(X1, X2)) ->^+ a__afterNth(mark(X1), mark(X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / afterNth(X1, X2)]. The result substitution is [ ]. ---------------------------------------- (4) Complex Obligation (BEST) ---------------------------------------- (5) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N, XS) -> a__U12(tt, N, XS) a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__U21(tt, X) -> a__U22(tt, X) a__U22(tt, X) -> mark(X) a__U31(tt, N) -> a__U32(tt, N) a__U32(tt, N) -> mark(N) a__U41(tt, N, XS) -> a__U42(tt, N, XS) a__U42(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__U51(tt, Y) -> a__U52(tt, Y) a__U52(tt, Y) -> mark(Y) a__U61(tt, N, X, XS) -> a__U62(tt, N, X, XS) a__U62(tt, N, X, XS) -> a__U63(tt, N, X, XS) a__U63(tt, N, X, XS) -> a__U64(a__splitAt(mark(N), mark(XS)), X) a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__U71(tt, XS) -> a__U72(tt, XS) a__U72(tt, XS) -> mark(XS) a__U81(tt, N, XS) -> a__U82(tt, N, XS) a__U82(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) a__afterNth(N, XS) -> a__U11(tt, N, XS) a__fst(pair(X, Y)) -> a__U21(tt, X) a__head(cons(N, XS)) -> a__U31(tt, N) a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) a__sel(N, XS) -> a__U41(tt, N, XS) a__snd(pair(X, Y)) -> a__U51(tt, Y) a__splitAt(0, XS) -> pair(nil, mark(XS)) a__splitAt(s(N), cons(X, XS)) -> a__U61(tt, N, X, XS) a__tail(cons(N, XS)) -> a__U71(tt, XS) a__take(N, XS) -> a__U81(tt, N, XS) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) mark(snd(X)) -> a__snd(mark(X)) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(head(X)) -> a__head(mark(X)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4) mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4) mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4) mark(U64(X1, X2)) -> a__U64(mark(X1), X2) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X1, X2)) -> a__U72(mark(X1), X2) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(fst(X)) -> a__fst(mark(X)) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(tt) -> tt mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__U12(X1, X2, X3) -> U12(X1, X2, X3) a__snd(X) -> snd(X) a__splitAt(X1, X2) -> splitAt(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__head(X) -> head(X) a__afterNth(X1, X2) -> afterNth(X1, X2) a__U51(X1, X2) -> U51(X1, X2) a__U52(X1, X2) -> U52(X1, X2) a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4) a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4) a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4) a__U64(X1, X2) -> U64(X1, X2) a__U71(X1, X2) -> U71(X1, X2) a__U72(X1, X2) -> U72(X1, X2) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__fst(X) -> fst(X) a__natsFrom(X) -> natsFrom(X) a__sel(X1, X2) -> sel(X1, X2) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (6) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (7) BOUNDS(n^1, INF) ---------------------------------------- (8) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: a__U11(tt, N, XS) -> a__U12(tt, N, XS) a__U12(tt, N, XS) -> a__snd(a__splitAt(mark(N), mark(XS))) a__U21(tt, X) -> a__U22(tt, X) a__U22(tt, X) -> mark(X) a__U31(tt, N) -> a__U32(tt, N) a__U32(tt, N) -> mark(N) a__U41(tt, N, XS) -> a__U42(tt, N, XS) a__U42(tt, N, XS) -> a__head(a__afterNth(mark(N), mark(XS))) a__U51(tt, Y) -> a__U52(tt, Y) a__U52(tt, Y) -> mark(Y) a__U61(tt, N, X, XS) -> a__U62(tt, N, X, XS) a__U62(tt, N, X, XS) -> a__U63(tt, N, X, XS) a__U63(tt, N, X, XS) -> a__U64(a__splitAt(mark(N), mark(XS)), X) a__U64(pair(YS, ZS), X) -> pair(cons(mark(X), YS), mark(ZS)) a__U71(tt, XS) -> a__U72(tt, XS) a__U72(tt, XS) -> mark(XS) a__U81(tt, N, XS) -> a__U82(tt, N, XS) a__U82(tt, N, XS) -> a__fst(a__splitAt(mark(N), mark(XS))) a__afterNth(N, XS) -> a__U11(tt, N, XS) a__fst(pair(X, Y)) -> a__U21(tt, X) a__head(cons(N, XS)) -> a__U31(tt, N) a__natsFrom(N) -> cons(mark(N), natsFrom(s(N))) a__sel(N, XS) -> a__U41(tt, N, XS) a__snd(pair(X, Y)) -> a__U51(tt, Y) a__splitAt(0, XS) -> pair(nil, mark(XS)) a__splitAt(s(N), cons(X, XS)) -> a__U61(tt, N, X, XS) a__tail(cons(N, XS)) -> a__U71(tt, XS) a__take(N, XS) -> a__U81(tt, N, XS) mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) mark(U12(X1, X2, X3)) -> a__U12(mark(X1), X2, X3) mark(snd(X)) -> a__snd(mark(X)) mark(splitAt(X1, X2)) -> a__splitAt(mark(X1), mark(X2)) mark(U21(X1, X2)) -> a__U21(mark(X1), X2) mark(U22(X1, X2)) -> a__U22(mark(X1), X2) mark(U31(X1, X2)) -> a__U31(mark(X1), X2) mark(U32(X1, X2)) -> a__U32(mark(X1), X2) mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) mark(U42(X1, X2, X3)) -> a__U42(mark(X1), X2, X3) mark(head(X)) -> a__head(mark(X)) mark(afterNth(X1, X2)) -> a__afterNth(mark(X1), mark(X2)) mark(U51(X1, X2)) -> a__U51(mark(X1), X2) mark(U52(X1, X2)) -> a__U52(mark(X1), X2) mark(U61(X1, X2, X3, X4)) -> a__U61(mark(X1), X2, X3, X4) mark(U62(X1, X2, X3, X4)) -> a__U62(mark(X1), X2, X3, X4) mark(U63(X1, X2, X3, X4)) -> a__U63(mark(X1), X2, X3, X4) mark(U64(X1, X2)) -> a__U64(mark(X1), X2) mark(U71(X1, X2)) -> a__U71(mark(X1), X2) mark(U72(X1, X2)) -> a__U72(mark(X1), X2) mark(U81(X1, X2, X3)) -> a__U81(mark(X1), X2, X3) mark(U82(X1, X2, X3)) -> a__U82(mark(X1), X2, X3) mark(fst(X)) -> a__fst(mark(X)) mark(natsFrom(X)) -> a__natsFrom(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) mark(tt) -> tt mark(pair(X1, X2)) -> pair(mark(X1), mark(X2)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__U11(X1, X2, X3) -> U11(X1, X2, X3) a__U12(X1, X2, X3) -> U12(X1, X2, X3) a__snd(X) -> snd(X) a__splitAt(X1, X2) -> splitAt(X1, X2) a__U21(X1, X2) -> U21(X1, X2) a__U22(X1, X2) -> U22(X1, X2) a__U31(X1, X2) -> U31(X1, X2) a__U32(X1, X2) -> U32(X1, X2) a__U41(X1, X2, X3) -> U41(X1, X2, X3) a__U42(X1, X2, X3) -> U42(X1, X2, X3) a__head(X) -> head(X) a__afterNth(X1, X2) -> afterNth(X1, X2) a__U51(X1, X2) -> U51(X1, X2) a__U52(X1, X2) -> U52(X1, X2) a__U61(X1, X2, X3, X4) -> U61(X1, X2, X3, X4) a__U62(X1, X2, X3, X4) -> U62(X1, X2, X3, X4) a__U63(X1, X2, X3, X4) -> U63(X1, X2, X3, X4) a__U64(X1, X2) -> U64(X1, X2) a__U71(X1, X2) -> U71(X1, X2) a__U72(X1, X2) -> U72(X1, X2) a__U81(X1, X2, X3) -> U81(X1, X2, X3) a__U82(X1, X2, X3) -> U82(X1, X2, X3) a__fst(X) -> fst(X) a__natsFrom(X) -> natsFrom(X) a__sel(X1, X2) -> sel(X1, X2) a__tail(X) -> tail(X) a__take(X1, X2) -> take(X1, X2) S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence mark(natsFrom(X)) ->^+ cons(mark(mark(X)), natsFrom(s(mark(X)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [X / natsFrom(X)]. The result substitution is [ ]. The rewrite sequence mark(natsFrom(X)) ->^+ cons(mark(mark(X)), natsFrom(s(mark(X)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0]. The pumping substitution is [X / natsFrom(X)]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)