/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 161 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 332 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 180 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 25 ms] (20) BOUNDS(1, INF) ---------------------------------------- (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: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0), y) -> s(y) plus(0, y) -> y ack(0, y) -> s(y) ack(s(x), 0) -> ack(x, s(0)) ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) 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(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_ack(x_1, x_2) -> ack(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: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0), y) -> s(y) plus(0, y) -> y ack(0, y) -> s(y) ack(s(x), 0) -> ack(x, s(0)) ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_ack(x_1, x_2) -> ack(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: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0), y) -> s(y) plus(0, y) -> y ack(0, y) -> s(y) ack(s(x), 0) -> ack(x, s(0)) ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) 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: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0'), y) -> s(y) plus(0', y) -> y ack(0', y) -> s(y) ack(s(x), 0') -> ack(x, s(0')) ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0'), y) -> s(y) plus(0', y) -> y ack(0', y) -> s(y) ack(s(x), 0') -> ack(x, s(0')) ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2)) Types: plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack 0' :: s:0':cons_plus:cons_ack ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encArg :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_0 :: s:0':cons_plus:cons_ack encode_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack hole_s:0':cons_plus:cons_ack1_3 :: s:0':cons_plus:cons_ack gen_s:0':cons_plus:cons_ack2_3 :: Nat -> s:0':cons_plus:cons_ack ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: plus, ack, encArg They will be analysed ascendingly in the following order: plus < ack plus < encArg ack < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0'), y) -> s(y) plus(0', y) -> y ack(0', y) -> s(y) ack(s(x), 0') -> ack(x, s(0')) ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2)) Types: plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack 0' :: s:0':cons_plus:cons_ack ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encArg :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_0 :: s:0':cons_plus:cons_ack encode_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack hole_s:0':cons_plus:cons_ack1_3 :: s:0':cons_plus:cons_ack gen_s:0':cons_plus:cons_ack2_3 :: Nat -> s:0':cons_plus:cons_ack Generator Equations: gen_s:0':cons_plus:cons_ack2_3(0) <=> 0' gen_s:0':cons_plus:cons_ack2_3(+(x, 1)) <=> s(gen_s:0':cons_plus:cons_ack2_3(x)) The following defined symbols remain to be analysed: plus, ack, encArg They will be analysed ascendingly in the following order: plus < ack plus < encArg ack < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_s:0':cons_plus:cons_ack2_3(+(1, *(2, n4_3))), gen_s:0':cons_plus:cons_ack2_3(b)) -> gen_s:0':cons_plus:cons_ack2_3(+(+(1, *(2, n4_3)), b)), rt in Omega(1 + n4_3) Induction Base: plus(gen_s:0':cons_plus:cons_ack2_3(+(1, *(2, 0))), gen_s:0':cons_plus:cons_ack2_3(b)) ->_R^Omega(1) s(gen_s:0':cons_plus:cons_ack2_3(b)) Induction Step: plus(gen_s:0':cons_plus:cons_ack2_3(+(1, *(2, +(n4_3, 1)))), gen_s:0':cons_plus:cons_ack2_3(b)) ->_R^Omega(1) s(plus(gen_s:0':cons_plus:cons_ack2_3(+(1, *(2, n4_3))), s(gen_s:0':cons_plus:cons_ack2_3(b)))) ->_IH s(gen_s:0':cons_plus:cons_ack2_3(+(+(1, +(b, 1)), *(2, c5_3)))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0'), y) -> s(y) plus(0', y) -> y ack(0', y) -> s(y) ack(s(x), 0') -> ack(x, s(0')) ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2)) Types: plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack 0' :: s:0':cons_plus:cons_ack ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encArg :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_0 :: s:0':cons_plus:cons_ack encode_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack hole_s:0':cons_plus:cons_ack1_3 :: s:0':cons_plus:cons_ack gen_s:0':cons_plus:cons_ack2_3 :: Nat -> s:0':cons_plus:cons_ack Generator Equations: gen_s:0':cons_plus:cons_ack2_3(0) <=> 0' gen_s:0':cons_plus:cons_ack2_3(+(x, 1)) <=> s(gen_s:0':cons_plus:cons_ack2_3(x)) The following defined symbols remain to be analysed: plus, ack, encArg They will be analysed ascendingly in the following order: plus < ack plus < encArg ack < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0'), y) -> s(y) plus(0', y) -> y ack(0', y) -> s(y) ack(s(x), 0') -> ack(x, s(0')) ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2)) Types: plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack 0' :: s:0':cons_plus:cons_ack ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encArg :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_0 :: s:0':cons_plus:cons_ack encode_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack hole_s:0':cons_plus:cons_ack1_3 :: s:0':cons_plus:cons_ack gen_s:0':cons_plus:cons_ack2_3 :: Nat -> s:0':cons_plus:cons_ack Lemmas: plus(gen_s:0':cons_plus:cons_ack2_3(+(1, *(2, n4_3))), gen_s:0':cons_plus:cons_ack2_3(b)) -> gen_s:0':cons_plus:cons_ack2_3(+(+(1, *(2, n4_3)), b)), rt in Omega(1 + n4_3) Generator Equations: gen_s:0':cons_plus:cons_ack2_3(0) <=> 0' gen_s:0':cons_plus:cons_ack2_3(+(x, 1)) <=> s(gen_s:0':cons_plus:cons_ack2_3(x)) The following defined symbols remain to be analysed: ack, encArg They will be analysed ascendingly in the following order: ack < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: ack(gen_s:0':cons_plus:cons_ack2_3(1), gen_s:0':cons_plus:cons_ack2_3(+(1, n1092_3))) -> *3_3, rt in Omega(n1092_3) Induction Base: ack(gen_s:0':cons_plus:cons_ack2_3(1), gen_s:0':cons_plus:cons_ack2_3(+(1, 0))) Induction Step: ack(gen_s:0':cons_plus:cons_ack2_3(1), gen_s:0':cons_plus:cons_ack2_3(+(1, +(n1092_3, 1)))) ->_R^Omega(1) ack(gen_s:0':cons_plus:cons_ack2_3(0), plus(gen_s:0':cons_plus:cons_ack2_3(+(1, n1092_3)), ack(s(gen_s:0':cons_plus:cons_ack2_3(0)), gen_s:0':cons_plus:cons_ack2_3(+(1, n1092_3))))) ->_IH ack(gen_s:0':cons_plus:cons_ack2_3(0), plus(gen_s:0':cons_plus:cons_ack2_3(+(1, n1092_3)), *3_3)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: plus(s(s(x)), y) -> s(plus(x, s(y))) plus(x, s(s(y))) -> s(plus(s(x), y)) plus(s(0'), y) -> s(y) plus(0', y) -> y ack(0', y) -> s(y) ack(s(x), 0') -> ack(x, s(0')) ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_ack(x_1, x_2)) -> ack(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_ack(x_1, x_2) -> ack(encArg(x_1), encArg(x_2)) Types: plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack 0' :: s:0':cons_plus:cons_ack ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encArg :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack cons_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_plus :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_s :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack encode_0 :: s:0':cons_plus:cons_ack encode_ack :: s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack -> s:0':cons_plus:cons_ack hole_s:0':cons_plus:cons_ack1_3 :: s:0':cons_plus:cons_ack gen_s:0':cons_plus:cons_ack2_3 :: Nat -> s:0':cons_plus:cons_ack Lemmas: plus(gen_s:0':cons_plus:cons_ack2_3(+(1, *(2, n4_3))), gen_s:0':cons_plus:cons_ack2_3(b)) -> gen_s:0':cons_plus:cons_ack2_3(+(+(1, *(2, n4_3)), b)), rt in Omega(1 + n4_3) ack(gen_s:0':cons_plus:cons_ack2_3(1), gen_s:0':cons_plus:cons_ack2_3(+(1, n1092_3))) -> *3_3, rt in Omega(n1092_3) Generator Equations: gen_s:0':cons_plus:cons_ack2_3(0) <=> 0' gen_s:0':cons_plus:cons_ack2_3(+(x, 1)) <=> s(gen_s:0':cons_plus:cons_ack2_3(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:0':cons_plus:cons_ack2_3(n3823_3)) -> gen_s:0':cons_plus:cons_ack2_3(n3823_3), rt in Omega(0) Induction Base: encArg(gen_s:0':cons_plus:cons_ack2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_s:0':cons_plus:cons_ack2_3(+(n3823_3, 1))) ->_R^Omega(0) s(encArg(gen_s:0':cons_plus:cons_ack2_3(n3823_3))) ->_IH s(gen_s:0':cons_plus:cons_ack2_3(c3824_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (20) BOUNDS(1, INF)