/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 384 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 4 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection (13) InfiniteLowerBoundProof [FINISHED, 83 ms] (14) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: nats -> cons(0, n__incr(n__nats)) pairs -> cons(0, n__incr(n__odds)) odds -> incr(pairs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) head(cons(X, XS)) -> X tail(cons(X, XS)) -> activate(XS) incr(X) -> n__incr(X) nats -> n__nats odds -> n__odds activate(n__incr(X)) -> incr(activate(X)) activate(n__nats) -> nats activate(n__odds) -> odds 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(0) -> 0 encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__nats) -> n__nats encArg(n__odds) -> n__odds encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_pairs) -> pairs encArg(cons_odds) -> odds encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__nats -> n__nats encode_pairs -> pairs encode_n__odds -> n__odds encode_odds -> odds encode_incr(x_1) -> incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: nats -> cons(0, n__incr(n__nats)) pairs -> cons(0, n__incr(n__odds)) odds -> incr(pairs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) head(cons(X, XS)) -> X tail(cons(X, XS)) -> activate(XS) incr(X) -> n__incr(X) nats -> n__nats odds -> n__odds activate(n__incr(X)) -> incr(activate(X)) activate(n__nats) -> nats activate(n__odds) -> odds 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(0) -> 0 encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__nats) -> n__nats encArg(n__odds) -> n__odds encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_pairs) -> pairs encArg(cons_odds) -> odds encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__nats -> n__nats encode_pairs -> pairs encode_n__odds -> n__odds encode_odds -> odds encode_incr(x_1) -> incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(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(INF, INF). The TRS R consists of the following rules: nats -> cons(0, n__incr(n__nats)) pairs -> cons(0, n__incr(n__odds)) odds -> incr(pairs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) head(cons(X, XS)) -> X tail(cons(X, XS)) -> activate(XS) incr(X) -> n__incr(X) nats -> n__nats odds -> n__odds activate(n__incr(X)) -> incr(activate(X)) activate(n__nats) -> nats activate(n__odds) -> odds 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(0) -> 0 encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__nats) -> n__nats encArg(n__odds) -> n__odds encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_pairs) -> pairs encArg(cons_odds) -> odds encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__nats -> n__nats encode_pairs -> pairs encode_n__odds -> n__odds encode_odds -> odds encode_incr(x_1) -> incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(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(INF, INF). The TRS R consists of the following rules: nats -> cons(0, n__incr(n__nats)) pairs -> cons(0, n__incr(n__odds)) odds -> incr(pairs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) head(cons(X, XS)) -> X tail(cons(X, XS)) -> activate(XS) incr(X) -> n__incr(X) nats -> n__nats odds -> n__odds activate(n__incr(X)) -> incr(activate(X)) activate(n__nats) -> nats activate(n__odds) -> odds 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(0) -> 0 encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__nats) -> n__nats encArg(n__odds) -> n__odds encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_pairs) -> pairs encArg(cons_odds) -> odds encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__nats -> n__nats encode_pairs -> pairs encode_n__odds -> n__odds encode_odds -> odds encode_incr(x_1) -> incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(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__incr(X)) ->^+ incr(activate(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / n__incr(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(INF, INF). The TRS R consists of the following rules: nats -> cons(0, n__incr(n__nats)) pairs -> cons(0, n__incr(n__odds)) odds -> incr(pairs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) head(cons(X, XS)) -> X tail(cons(X, XS)) -> activate(XS) incr(X) -> n__incr(X) nats -> n__nats odds -> n__odds activate(n__incr(X)) -> incr(activate(X)) activate(n__nats) -> nats activate(n__odds) -> odds 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(0) -> 0 encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__nats) -> n__nats encArg(n__odds) -> n__odds encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_pairs) -> pairs encArg(cons_odds) -> odds encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__nats -> n__nats encode_pairs -> pairs encode_n__odds -> n__odds encode_odds -> odds encode_incr(x_1) -> incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(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(INF, INF). The TRS R consists of the following rules: nats -> cons(0, n__incr(n__nats)) pairs -> cons(0, n__incr(n__odds)) odds -> incr(pairs) incr(cons(X, XS)) -> cons(s(X), n__incr(activate(XS))) head(cons(X, XS)) -> X tail(cons(X, XS)) -> activate(XS) incr(X) -> n__incr(X) nats -> n__nats odds -> n__odds activate(n__incr(X)) -> incr(activate(X)) activate(n__nats) -> nats activate(n__odds) -> odds 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(0) -> 0 encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__nats) -> n__nats encArg(n__odds) -> n__odds encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_pairs) -> pairs encArg(cons_odds) -> odds encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_head(x_1)) -> head(encArg(x_1)) encArg(cons_tail(x_1)) -> tail(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__nats -> n__nats encode_pairs -> pairs encode_n__odds -> n__odds encode_odds -> odds encode_incr(x_1) -> incr(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_head(x_1) -> head(encArg(x_1)) encode_tail(x_1) -> tail(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) InfiniteLowerBoundProof (FINISHED) The following loop proves infinite runtime complexity: The rewrite sequence incr(cons(X, n__incr(n__nats))) ->^+ cons(s(X), n__incr(incr(cons(0, n__incr(n__nats))))) gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0]. The pumping substitution is [ ]. The result substitution is [X / 0]. ---------------------------------------- (14) BOUNDS(INF, INF)