/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 330 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: from(X) -> cons(X, n__from(n__s(X))) head(cons(X, XS)) -> X 2nd(cons(X, XS)) -> head(activate(XS)) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1, X2) -> n__take(X1, X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 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__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(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: from(X) -> cons(X, n__from(n__s(X))) head(cons(X, XS)) -> X 2nd(cons(X, XS)) -> head(activate(XS)) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1, X2) -> n__take(X1, X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 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__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(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: from(X) -> cons(X, n__from(n__s(X))) head(cons(X, XS)) -> X 2nd(cons(X, XS)) -> head(activate(XS)) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1, X2) -> n__take(X1, X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 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__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(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: from(X) -> cons(X, n__from(n__s(X))) head(cons(X, XS)) -> X 2nd(cons(X, XS)) -> head(activate(XS)) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1, X2) -> n__take(X1, X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 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__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(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 activate(n__take(X1, X2)) ->^+ take(activate(X1), activate(X2)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X1 / n__take(X1, X2)]. 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: from(X) -> cons(X, n__from(n__s(X))) head(cons(X, XS)) -> X 2nd(cons(X, XS)) -> head(activate(XS)) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1, X2) -> n__take(X1, X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 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__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(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: from(X) -> cons(X, n__from(n__s(X))) head(cons(X, XS)) -> X 2nd(cons(X, XS)) -> head(activate(XS)) take(0, XS) -> nil take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS))) sel(0, cons(X, XS)) -> X sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) from(X) -> n__from(X) s(X) -> n__s(X) take(X1, X2) -> n__take(X1, X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__take(X1, X2)) -> take(activate(X1), activate(X2)) 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__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(0) -> 0 encArg(nil) -> nil encArg(n__take(x_1, x_2)) -> n__take(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_take(x_1, x_2)) -> take(encArg(x_1), encArg(x_2)) encArg(cons_sel(x_1, x_2)) -> sel(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_take(x_1, x_2) -> take(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_nil -> nil encode_s(x_1) -> s(encArg(x_1)) encode_n__take(x_1, x_2) -> n__take(encArg(x_1), encArg(x_2)) encode_sel(x_1, x_2) -> sel(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST