/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).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, 121 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: U11(tt, N, XS) -> U12(tt, activate(N), activate(XS)) U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) U21(tt, X) -> U22(tt, activate(X)) U22(tt, X) -> activate(X) U31(tt, N) -> U32(tt, activate(N)) U32(tt, N) -> activate(N) U41(tt, N, XS) -> U42(tt, activate(N), activate(XS)) U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) U51(tt, Y) -> U52(tt, activate(Y)) U52(tt, Y) -> activate(Y) U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS)) U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS)) U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X)) U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) U71(tt, XS) -> U72(tt, activate(XS)) U72(tt, XS) -> activate(XS) U81(tt, N, XS) -> U82(tt, activate(N), activate(XS)) U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) afterNth(N, XS) -> U11(tt, N, XS) fst(pair(X, Y)) -> U21(tt, X) head(cons(N, XS)) -> U31(tt, N) natsFrom(N) -> cons(N, n__natsFrom(n__s(N))) sel(N, XS) -> U41(tt, N, XS) snd(pair(X, Y)) -> U51(tt, Y) splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS)) tail(cons(N, XS)) -> U71(tt, activate(XS)) take(N, XS) -> U81(tt, N, XS) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X 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: U11(tt, N, XS) -> U12(tt, activate(N), activate(XS)) U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) U21(tt, X) -> U22(tt, activate(X)) U22(tt, X) -> activate(X) U31(tt, N) -> U32(tt, activate(N)) U32(tt, N) -> activate(N) U41(tt, N, XS) -> U42(tt, activate(N), activate(XS)) U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) U51(tt, Y) -> U52(tt, activate(Y)) U52(tt, Y) -> activate(Y) U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS)) U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS)) U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X)) U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) U71(tt, XS) -> U72(tt, activate(XS)) U72(tt, XS) -> activate(XS) U81(tt, N, XS) -> U82(tt, activate(N), activate(XS)) U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) afterNth(N, XS) -> U11(tt, N, XS) fst(pair(X, Y)) -> U21(tt, X) head(cons(N, XS)) -> U31(tt, N) natsFrom(N) -> cons(N, n__natsFrom(n__s(N))) sel(N, XS) -> U41(tt, N, XS) snd(pair(X, Y)) -> U51(tt, Y) splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS)) tail(cons(N, XS)) -> U71(tt, activate(XS)) take(N, XS) -> U81(tt, N, XS) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X 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 activate(n__s(X)) ->^+ s(activate(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / n__s(X)]. 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: U11(tt, N, XS) -> U12(tt, activate(N), activate(XS)) U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) U21(tt, X) -> U22(tt, activate(X)) U22(tt, X) -> activate(X) U31(tt, N) -> U32(tt, activate(N)) U32(tt, N) -> activate(N) U41(tt, N, XS) -> U42(tt, activate(N), activate(XS)) U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) U51(tt, Y) -> U52(tt, activate(Y)) U52(tt, Y) -> activate(Y) U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS)) U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS)) U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X)) U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) U71(tt, XS) -> U72(tt, activate(XS)) U72(tt, XS) -> activate(XS) U81(tt, N, XS) -> U82(tt, activate(N), activate(XS)) U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) afterNth(N, XS) -> U11(tt, N, XS) fst(pair(X, Y)) -> U21(tt, X) head(cons(N, XS)) -> U31(tt, N) natsFrom(N) -> cons(N, n__natsFrom(n__s(N))) sel(N, XS) -> U41(tt, N, XS) snd(pair(X, Y)) -> U51(tt, Y) splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS)) tail(cons(N, XS)) -> U71(tt, activate(XS)) take(N, XS) -> U81(tt, N, XS) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X 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: U11(tt, N, XS) -> U12(tt, activate(N), activate(XS)) U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS))) U21(tt, X) -> U22(tt, activate(X)) U22(tt, X) -> activate(X) U31(tt, N) -> U32(tt, activate(N)) U32(tt, N) -> activate(N) U41(tt, N, XS) -> U42(tt, activate(N), activate(XS)) U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS))) U51(tt, Y) -> U52(tt, activate(Y)) U52(tt, Y) -> activate(Y) U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS)) U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS)) U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X)) U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS) U71(tt, XS) -> U72(tt, activate(XS)) U72(tt, XS) -> activate(XS) U81(tt, N, XS) -> U82(tt, activate(N), activate(XS)) U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS))) afterNth(N, XS) -> U11(tt, N, XS) fst(pair(X, Y)) -> U21(tt, X) head(cons(N, XS)) -> U31(tt, N) natsFrom(N) -> cons(N, n__natsFrom(n__s(N))) sel(N, XS) -> U41(tt, N, XS) snd(pair(X, Y)) -> U51(tt, Y) splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS)) tail(cons(N, XS)) -> U71(tt, activate(XS)) take(N, XS) -> U81(tt, N, XS) natsFrom(X) -> n__natsFrom(X) s(X) -> n__s(X) activate(n__natsFrom(X)) -> natsFrom(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (9) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence activate(n__natsFrom(X)) ->^+ cons(activate(X), n__natsFrom(n__s(activate(X)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / n__natsFrom(X)]. The result substitution is [ ]. The rewrite sequence activate(n__natsFrom(X)) ->^+ cons(activate(X), n__natsFrom(n__s(activate(X)))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0]. The pumping substitution is [X / n__natsFrom(X)]. The result substitution is [ ]. ---------------------------------------- (10) BOUNDS(EXP, INF)