/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 494 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 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: app(nil, YS) -> YS app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS)) from(X) -> cons(X, n__from(n__s(X))) zWadr(nil, YS) -> nil zWadr(XS, nil) -> nil zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS))) prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L))) app(X1, X2) -> n__app(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) nil -> n__nil zWadr(X1, X2) -> n__zWadr(X1, X2) prefix(X) -> n__prefix(X) activate(n__app(X1, X2)) -> app(activate(X1), activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__nil) -> nil activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2)) activate(n__prefix(X)) -> prefix(activate(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__app(x_1, x_2)) -> n__app(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__zWadr(x_1, x_2)) -> n__zWadr(encArg(x_1), encArg(x_2)) encArg(n__prefix(x_1)) -> n__prefix(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2)) encArg(cons_prefix(x_1)) -> prefix(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__app(x_1, x_2) -> n__app(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__zWadr(x_1, x_2) -> n__zWadr(encArg(x_1), encArg(x_2)) encode_prefix(x_1) -> prefix(encArg(x_1)) encode_n__prefix(x_1) -> n__prefix(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) ---------------------------------------- (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: app(nil, YS) -> YS app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS)) from(X) -> cons(X, n__from(n__s(X))) zWadr(nil, YS) -> nil zWadr(XS, nil) -> nil zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS))) prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L))) app(X1, X2) -> n__app(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) nil -> n__nil zWadr(X1, X2) -> n__zWadr(X1, X2) prefix(X) -> n__prefix(X) activate(n__app(X1, X2)) -> app(activate(X1), activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__nil) -> nil activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2)) activate(n__prefix(X)) -> prefix(activate(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__app(x_1, x_2)) -> n__app(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__zWadr(x_1, x_2)) -> n__zWadr(encArg(x_1), encArg(x_2)) encArg(n__prefix(x_1)) -> n__prefix(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2)) encArg(cons_prefix(x_1)) -> prefix(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__app(x_1, x_2) -> n__app(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__zWadr(x_1, x_2) -> n__zWadr(encArg(x_1), encArg(x_2)) encode_prefix(x_1) -> prefix(encArg(x_1)) encode_n__prefix(x_1) -> n__prefix(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) 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: app(nil, YS) -> YS app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS)) from(X) -> cons(X, n__from(n__s(X))) zWadr(nil, YS) -> nil zWadr(XS, nil) -> nil zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS))) prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L))) app(X1, X2) -> n__app(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) nil -> n__nil zWadr(X1, X2) -> n__zWadr(X1, X2) prefix(X) -> n__prefix(X) activate(n__app(X1, X2)) -> app(activate(X1), activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__nil) -> nil activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2)) activate(n__prefix(X)) -> prefix(activate(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__app(x_1, x_2)) -> n__app(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__zWadr(x_1, x_2)) -> n__zWadr(encArg(x_1), encArg(x_2)) encArg(n__prefix(x_1)) -> n__prefix(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2)) encArg(cons_prefix(x_1)) -> prefix(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__app(x_1, x_2) -> n__app(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__zWadr(x_1, x_2) -> n__zWadr(encArg(x_1), encArg(x_2)) encode_prefix(x_1) -> prefix(encArg(x_1)) encode_n__prefix(x_1) -> n__prefix(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) 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: app(nil, YS) -> YS app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS)) from(X) -> cons(X, n__from(n__s(X))) zWadr(nil, YS) -> nil zWadr(XS, nil) -> nil zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS))) prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L))) app(X1, X2) -> n__app(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) nil -> n__nil zWadr(X1, X2) -> n__zWadr(X1, X2) prefix(X) -> n__prefix(X) activate(n__app(X1, X2)) -> app(activate(X1), activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__nil) -> nil activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2)) activate(n__prefix(X)) -> prefix(activate(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__app(x_1, x_2)) -> n__app(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__zWadr(x_1, x_2)) -> n__zWadr(encArg(x_1), encArg(x_2)) encArg(n__prefix(x_1)) -> n__prefix(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2)) encArg(cons_prefix(x_1)) -> prefix(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__app(x_1, x_2) -> n__app(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__zWadr(x_1, x_2) -> n__zWadr(encArg(x_1), encArg(x_2)) encode_prefix(x_1) -> prefix(encArg(x_1)) encode_n__prefix(x_1) -> n__prefix(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) 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 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 [ ]. ---------------------------------------- (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: app(nil, YS) -> YS app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS)) from(X) -> cons(X, n__from(n__s(X))) zWadr(nil, YS) -> nil zWadr(XS, nil) -> nil zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS))) prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L))) app(X1, X2) -> n__app(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) nil -> n__nil zWadr(X1, X2) -> n__zWadr(X1, X2) prefix(X) -> n__prefix(X) activate(n__app(X1, X2)) -> app(activate(X1), activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__nil) -> nil activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2)) activate(n__prefix(X)) -> prefix(activate(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__app(x_1, x_2)) -> n__app(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__zWadr(x_1, x_2)) -> n__zWadr(encArg(x_1), encArg(x_2)) encArg(n__prefix(x_1)) -> n__prefix(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2)) encArg(cons_prefix(x_1)) -> prefix(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__app(x_1, x_2) -> n__app(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__zWadr(x_1, x_2) -> n__zWadr(encArg(x_1), encArg(x_2)) encode_prefix(x_1) -> prefix(encArg(x_1)) encode_n__prefix(x_1) -> n__prefix(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) 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: app(nil, YS) -> YS app(cons(X, XS), YS) -> cons(X, n__app(activate(XS), YS)) from(X) -> cons(X, n__from(n__s(X))) zWadr(nil, YS) -> nil zWadr(XS, nil) -> nil zWadr(cons(X, XS), cons(Y, YS)) -> cons(app(Y, cons(X, n__nil)), n__zWadr(activate(XS), activate(YS))) prefix(L) -> cons(nil, n__zWadr(L, n__prefix(L))) app(X1, X2) -> n__app(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) nil -> n__nil zWadr(X1, X2) -> n__zWadr(X1, X2) prefix(X) -> n__prefix(X) activate(n__app(X1, X2)) -> app(activate(X1), activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__nil) -> nil activate(n__zWadr(X1, X2)) -> zWadr(activate(X1), activate(X2)) activate(n__prefix(X)) -> prefix(activate(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__app(x_1, x_2)) -> n__app(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__nil) -> n__nil encArg(n__zWadr(x_1, x_2)) -> n__zWadr(encArg(x_1), encArg(x_2)) encArg(n__prefix(x_1)) -> n__prefix(encArg(x_1)) encArg(cons_app(x_1, x_2)) -> app(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_zWadr(x_1, x_2)) -> zWadr(encArg(x_1), encArg(x_2)) encArg(cons_prefix(x_1)) -> prefix(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_nil) -> nil encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_app(x_1, x_2) -> app(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__app(x_1, x_2) -> n__app(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_zWadr(x_1, x_2) -> zWadr(encArg(x_1), encArg(x_2)) encode_n__nil -> n__nil encode_n__zWadr(x_1, x_2) -> n__zWadr(encArg(x_1), encArg(x_2)) encode_prefix(x_1) -> prefix(encArg(x_1)) encode_n__prefix(x_1) -> n__prefix(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST