/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 642 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 (13) DecreasingLoopProof [FINISHED, 25 ms] (14) BOUNDS(EXP, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: and(true, X) -> activate(X) and(false, Y) -> false if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) add(0, X) -> activate(X) add(s(X), Y) -> s(n__add(activate(X), activate(Y))) first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) from(X) -> cons(activate(X), n__from(n__s(activate(X)))) add(X1, X2) -> n__add(X1, X2) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1, X2)) -> add(activate(X1), X2) activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(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(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(n__add(x_1, x_2)) -> n__add(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__first(x_1, x_2)) -> n__first(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(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_true -> true encode_activate(x_1) -> activate(encArg(x_1)) encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_n__add(x_1, x_2) -> n__add(encArg(x_1), encArg(x_2)) encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) 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)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(EXP, INF). The TRS R consists of the following rules: and(true, X) -> activate(X) and(false, Y) -> false if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) add(0, X) -> activate(X) add(s(X), Y) -> s(n__add(activate(X), activate(Y))) first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) from(X) -> cons(activate(X), n__from(n__s(activate(X)))) add(X1, X2) -> n__add(X1, X2) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1, X2)) -> add(activate(X1), X2) activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(n__add(x_1, x_2)) -> n__add(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__first(x_1, x_2)) -> n__first(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(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_true -> true encode_activate(x_1) -> activate(encArg(x_1)) encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_n__add(x_1, x_2) -> n__add(encArg(x_1), encArg(x_2)) encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) 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)) 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(EXP, INF). The TRS R consists of the following rules: and(true, X) -> activate(X) and(false, Y) -> false if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) add(0, X) -> activate(X) add(s(X), Y) -> s(n__add(activate(X), activate(Y))) first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) from(X) -> cons(activate(X), n__from(n__s(activate(X)))) add(X1, X2) -> n__add(X1, X2) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1, X2)) -> add(activate(X1), X2) activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(n__add(x_1, x_2)) -> n__add(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__first(x_1, x_2)) -> n__first(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(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_true -> true encode_activate(x_1) -> activate(encArg(x_1)) encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_n__add(x_1, x_2) -> n__add(encArg(x_1), encArg(x_2)) encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) 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)) 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(EXP, INF). The TRS R consists of the following rules: and(true, X) -> activate(X) and(false, Y) -> false if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) add(0, X) -> activate(X) add(s(X), Y) -> s(n__add(activate(X), activate(Y))) first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) from(X) -> cons(activate(X), n__from(n__s(activate(X)))) add(X1, X2) -> n__add(X1, X2) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1, X2)) -> add(activate(X1), X2) activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(n__add(x_1, x_2)) -> n__add(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__first(x_1, x_2)) -> n__first(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(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_true -> true encode_activate(x_1) -> activate(encArg(x_1)) encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_n__add(x_1, x_2) -> n__add(encArg(x_1), encArg(x_2)) encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) 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)) 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__first(X1, X2)) ->^+ first(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__first(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(EXP, INF). The TRS R consists of the following rules: and(true, X) -> activate(X) and(false, Y) -> false if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) add(0, X) -> activate(X) add(s(X), Y) -> s(n__add(activate(X), activate(Y))) first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) from(X) -> cons(activate(X), n__from(n__s(activate(X)))) add(X1, X2) -> n__add(X1, X2) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1, X2)) -> add(activate(X1), X2) activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(n__add(x_1, x_2)) -> n__add(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__first(x_1, x_2)) -> n__first(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(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_true -> true encode_activate(x_1) -> activate(encArg(x_1)) encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_n__add(x_1, x_2) -> n__add(encArg(x_1), encArg(x_2)) encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) 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)) 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(EXP, INF). The TRS R consists of the following rules: and(true, X) -> activate(X) and(false, Y) -> false if(true, X, Y) -> activate(X) if(false, X, Y) -> activate(Y) add(0, X) -> activate(X) add(s(X), Y) -> s(n__add(activate(X), activate(Y))) first(0, X) -> nil first(s(X), cons(Y, Z)) -> cons(activate(Y), n__first(activate(X), activate(Z))) from(X) -> cons(activate(X), n__from(n__s(activate(X)))) add(X1, X2) -> n__add(X1, X2) first(X1, X2) -> n__first(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__add(X1, X2)) -> add(activate(X1), X2) activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) activate(n__from(X)) -> from(X) activate(n__s(X)) -> s(X) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(false) -> false encArg(0) -> 0 encArg(n__add(x_1, x_2)) -> n__add(encArg(x_1), encArg(x_2)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(n__first(x_1, x_2)) -> n__first(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(cons_and(x_1, x_2)) -> and(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_add(x_1, x_2)) -> add(encArg(x_1), encArg(x_2)) encArg(cons_first(x_1, x_2)) -> first(encArg(x_1), encArg(x_2)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_true -> true encode_activate(x_1) -> activate(encArg(x_1)) encode_false -> false encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_add(x_1, x_2) -> add(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_n__add(x_1, x_2) -> n__add(encArg(x_1), encArg(x_2)) encode_first(x_1, x_2) -> first(encArg(x_1), encArg(x_2)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__first(x_1, x_2) -> n__first(encArg(x_1), encArg(x_2)) 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)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) DecreasingLoopProof (FINISHED) The following loop(s) give(s) rise to the lower bound EXP: The rewrite sequence activate(n__from(X)) ->^+ cons(activate(X), n__from(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__from(X)]. The result substitution is [ ]. The rewrite sequence activate(n__from(X)) ->^+ cons(activate(X), n__from(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__from(X)]. The result substitution is [ ]. ---------------------------------------- (14) BOUNDS(EXP, INF)