/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 671 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 23 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: natsFrom(N) -> cons(N, n__natsFrom(s(N))) fst(pair(XS, YS)) -> XS snd(pair(XS, YS)) -> YS splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS)) u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS) head(cons(N, XS)) -> N tail(cons(N, XS)) -> activate(XS) sel(N, XS) -> head(afterNth(N, XS)) take(N, XS) -> fst(splitAt(N, XS)) afterNth(N, XS) -> snd(splitAt(N, XS)) natsFrom(X) -> n__natsFrom(X) activate(n__natsFrom(X)) -> natsFrom(X) activate(X) -> X S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__natsFrom(x_1)) -> n__natsFrom(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(nil) -> nil encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__natsFrom(x_1) -> n__natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_snd(x_1) -> snd(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: natsFrom(N) -> cons(N, n__natsFrom(s(N))) fst(pair(XS, YS)) -> XS snd(pair(XS, YS)) -> YS splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS)) u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS) head(cons(N, XS)) -> N tail(cons(N, XS)) -> activate(XS) sel(N, XS) -> head(afterNth(N, XS)) take(N, XS) -> fst(splitAt(N, XS)) afterNth(N, XS) -> snd(splitAt(N, XS)) natsFrom(X) -> n__natsFrom(X) activate(n__natsFrom(X)) -> natsFrom(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__natsFrom(x_1)) -> n__natsFrom(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(nil) -> nil encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__natsFrom(x_1) -> n__natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_snd(x_1) -> snd(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: natsFrom(N) -> cons(N, n__natsFrom(s(N))) fst(pair(XS, YS)) -> XS snd(pair(XS, YS)) -> YS splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS)) u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS) head(cons(N, XS)) -> N tail(cons(N, XS)) -> activate(XS) sel(N, XS) -> head(afterNth(N, XS)) take(N, XS) -> fst(splitAt(N, XS)) afterNth(N, XS) -> snd(splitAt(N, XS)) natsFrom(X) -> n__natsFrom(X) activate(n__natsFrom(X)) -> natsFrom(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__natsFrom(x_1)) -> n__natsFrom(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(nil) -> nil encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__natsFrom(x_1) -> n__natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_snd(x_1) -> snd(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: natsFrom(N) -> cons(N, n__natsFrom(s(N))) fst(pair(XS, YS)) -> XS snd(pair(XS, YS)) -> YS splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS)) u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS) head(cons(N, XS)) -> N tail(cons(N, XS)) -> activate(XS) sel(N, XS) -> head(afterNth(N, XS)) take(N, XS) -> fst(splitAt(N, XS)) afterNth(N, XS) -> snd(splitAt(N, XS)) natsFrom(X) -> n__natsFrom(X) activate(n__natsFrom(X)) -> natsFrom(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__natsFrom(x_1)) -> n__natsFrom(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(nil) -> nil encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__natsFrom(x_1) -> n__natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_snd(x_1) -> snd(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence splitAt(s(N), cons(X, XS)) ->^+ u(splitAt(N, XS), N, X, activate(XS)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [N / s(N), XS / cons(X, XS)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: natsFrom(N) -> cons(N, n__natsFrom(s(N))) fst(pair(XS, YS)) -> XS snd(pair(XS, YS)) -> YS splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS)) u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS) head(cons(N, XS)) -> N tail(cons(N, XS)) -> activate(XS) sel(N, XS) -> head(afterNth(N, XS)) take(N, XS) -> fst(splitAt(N, XS)) afterNth(N, XS) -> snd(splitAt(N, XS)) natsFrom(X) -> n__natsFrom(X) activate(n__natsFrom(X)) -> natsFrom(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__natsFrom(x_1)) -> n__natsFrom(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(nil) -> nil encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__natsFrom(x_1) -> n__natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_snd(x_1) -> snd(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: natsFrom(N) -> cons(N, n__natsFrom(s(N))) fst(pair(XS, YS)) -> XS snd(pair(XS, YS)) -> YS splitAt(0, XS) -> pair(nil, XS) splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS)) u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS) head(cons(N, XS)) -> N tail(cons(N, XS)) -> activate(XS) sel(N, XS) -> head(afterNth(N, XS)) take(N, XS) -> fst(splitAt(N, XS)) afterNth(N, XS) -> snd(splitAt(N, XS)) natsFrom(X) -> n__natsFrom(X) activate(n__natsFrom(X)) -> natsFrom(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__natsFrom(x_1)) -> n__natsFrom(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(pair(x_1, x_2)) -> pair(encArg(x_1), encArg(x_2)) encArg(0) -> 0 encArg(nil) -> nil encArg(cons_natsFrom(x_1)) -> natsFrom(encArg(x_1)) encArg(cons_fst(x_1)) -> fst(encArg(x_1)) encArg(cons_snd(x_1)) -> snd(encArg(x_1)) encArg(cons_splitAt(x_1, x_2)) -> splitAt(encArg(x_1), encArg(x_2)) encArg(cons_u(x_1, x_2, x_3, x_4)) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_afterNth(x_1, x_2)) -> afterNth(encArg(x_1), encArg(x_2)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_natsFrom(x_1) -> natsFrom(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__natsFrom(x_1) -> n__natsFrom(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_fst(x_1) -> fst(encArg(x_1)) encode_pair(x_1, x_2) -> pair(encArg(x_1), encArg(x_2)) encode_snd(x_1) -> snd(encArg(x_1)) encode_splitAt(x_1, x_2) -> splitAt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_u(x_1, x_2, x_3, x_4) -> u(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) encode_afterNth(x_1, x_2) -> afterNth(encArg(x_1), encArg(x_2)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST