/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(INF, INF).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), 429 ms] (4) CpxRelTRS (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 75 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, 2595 ms] (14) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: nats -> adx(zeros) zeros -> cons(n__0, n__zeros) incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) hd(cons(X, Y)) -> activate(X) tl(cons(X, Y)) -> activate(Y) 0 -> n__0 zeros -> n__zeros s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0) -> 0 activate(n__zeros) -> zeros activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(X) activate(n__adx(X)) -> adx(X) activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (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__0) -> n__0 encArg(n__zeros) -> n__zeros encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__adx(x_1)) -> n__adx(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_zeros) -> zeros encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_adx(x_1)) -> adx(encArg(x_1)) encArg(cons_hd(x_1)) -> hd(encArg(x_1)) encArg(cons_tl(x_1)) -> tl(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_adx(x_1) -> adx(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__zeros -> n__zeros encode_incr(x_1) -> incr(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__adx(x_1) -> n__adx(encArg(x_1)) encode_hd(x_1) -> hd(encArg(x_1)) encode_tl(x_1) -> tl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: nats -> adx(zeros) zeros -> cons(n__0, n__zeros) incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) hd(cons(X, Y)) -> activate(X) tl(cons(X, Y)) -> activate(Y) 0 -> n__0 zeros -> n__zeros s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0) -> 0 activate(n__zeros) -> zeros activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(X) activate(n__adx(X)) -> adx(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__0) -> n__0 encArg(n__zeros) -> n__zeros encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__adx(x_1)) -> n__adx(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_zeros) -> zeros encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_adx(x_1)) -> adx(encArg(x_1)) encArg(cons_hd(x_1)) -> hd(encArg(x_1)) encArg(cons_tl(x_1)) -> tl(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_adx(x_1) -> adx(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__zeros -> n__zeros encode_incr(x_1) -> incr(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__adx(x_1) -> n__adx(encArg(x_1)) encode_hd(x_1) -> hd(encArg(x_1)) encode_tl(x_1) -> tl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: nats -> adx(zeros) zeros -> cons(n__0, n__zeros) incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) hd(cons(X, Y)) -> activate(X) tl(cons(X, Y)) -> activate(Y) 0 -> n__0 zeros -> n__zeros s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0) -> 0 activate(n__zeros) -> zeros activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(X) activate(n__adx(X)) -> adx(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__0) -> n__0 encArg(n__zeros) -> n__zeros encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__adx(x_1)) -> n__adx(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_zeros) -> zeros encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_adx(x_1)) -> adx(encArg(x_1)) encArg(cons_hd(x_1)) -> hd(encArg(x_1)) encArg(cons_tl(x_1)) -> tl(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_adx(x_1) -> adx(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__zeros -> n__zeros encode_incr(x_1) -> incr(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__adx(x_1) -> n__adx(encArg(x_1)) encode_hd(x_1) -> hd(encArg(x_1)) encode_tl(x_1) -> tl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (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 (full) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: nats -> adx(zeros) zeros -> cons(n__0, n__zeros) incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) hd(cons(X, Y)) -> activate(X) tl(cons(X, Y)) -> activate(Y) 0 -> n__0 zeros -> n__zeros s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0) -> 0 activate(n__zeros) -> zeros activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(X) activate(n__adx(X)) -> adx(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__0) -> n__0 encArg(n__zeros) -> n__zeros encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__adx(x_1)) -> n__adx(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_zeros) -> zeros encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_adx(x_1)) -> adx(encArg(x_1)) encArg(cons_hd(x_1)) -> hd(encArg(x_1)) encArg(cons_tl(x_1)) -> tl(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_adx(x_1) -> adx(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__zeros -> n__zeros encode_incr(x_1) -> incr(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__adx(x_1) -> n__adx(encArg(x_1)) encode_hd(x_1) -> hd(encArg(x_1)) encode_tl(x_1) -> tl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (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(cons(X1_0, Y2_0))) ->^+ cons(n__s(activate(X1_0)), n__incr(activate(Y2_0))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [X1_0 / n__incr(cons(X1_0, Y2_0))]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: nats -> adx(zeros) zeros -> cons(n__0, n__zeros) incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) hd(cons(X, Y)) -> activate(X) tl(cons(X, Y)) -> activate(Y) 0 -> n__0 zeros -> n__zeros s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0) -> 0 activate(n__zeros) -> zeros activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(X) activate(n__adx(X)) -> adx(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__0) -> n__0 encArg(n__zeros) -> n__zeros encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__adx(x_1)) -> n__adx(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_zeros) -> zeros encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_adx(x_1)) -> adx(encArg(x_1)) encArg(cons_hd(x_1)) -> hd(encArg(x_1)) encArg(cons_tl(x_1)) -> tl(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_adx(x_1) -> adx(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__zeros -> n__zeros encode_incr(x_1) -> incr(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__adx(x_1) -> n__adx(encArg(x_1)) encode_hd(x_1) -> hd(encArg(x_1)) encode_tl(x_1) -> tl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(INF, INF). The TRS R consists of the following rules: nats -> adx(zeros) zeros -> cons(n__0, n__zeros) incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) hd(cons(X, Y)) -> activate(X) tl(cons(X, Y)) -> activate(Y) 0 -> n__0 zeros -> n__zeros s(X) -> n__s(X) incr(X) -> n__incr(X) adx(X) -> n__adx(X) activate(n__0) -> 0 activate(n__zeros) -> zeros activate(n__s(X)) -> s(X) activate(n__incr(X)) -> incr(X) activate(n__adx(X)) -> adx(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__0) -> n__0 encArg(n__zeros) -> n__zeros encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(n__incr(x_1)) -> n__incr(encArg(x_1)) encArg(n__adx(x_1)) -> n__adx(encArg(x_1)) encArg(cons_nats) -> nats encArg(cons_zeros) -> zeros encArg(cons_incr(x_1)) -> incr(encArg(x_1)) encArg(cons_adx(x_1)) -> adx(encArg(x_1)) encArg(cons_hd(x_1)) -> hd(encArg(x_1)) encArg(cons_tl(x_1)) -> tl(encArg(x_1)) encArg(cons_0) -> 0 encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_nats -> nats encode_adx(x_1) -> adx(encArg(x_1)) encode_zeros -> zeros encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__0 -> n__0 encode_n__zeros -> n__zeros encode_incr(x_1) -> incr(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_activate(x_1) -> activate(encArg(x_1)) encode_n__incr(x_1) -> n__incr(encArg(x_1)) encode_n__adx(x_1) -> n__adx(encArg(x_1)) encode_hd(x_1) -> hd(encArg(x_1)) encode_tl(x_1) -> tl(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (13) InfiniteLowerBoundProof (FINISHED) The following loop proves infinite runtime complexity: The rewrite sequence incr(cons(X, n__adx(cons(X1_1, n__zeros)))) ->^+ cons(n__s(activate(X)), n__incr(incr(cons(activate(X1_1), n__adx(cons(n__0, n__zeros)))))) 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 / activate(X1_1), X1_1 / n__0]. ---------------------------------------- (14) BOUNDS(INF, INF)