/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 219 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), 7 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 301 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), 43 ms] (18) 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: p(0) -> 0 p(s(x)) -> x minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) minus(x, s(y)) -> p(minus(x, y)) div(0, s(y)) -> 0 div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y))) log(s(0), s(s(y))) -> 0 log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(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(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_log(x_1, x_2)) -> log(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_log(x_1, x_2) -> log(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: p(0) -> 0 p(s(x)) -> x minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) minus(x, s(y)) -> p(minus(x, y)) div(0, s(y)) -> 0 div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y))) log(s(0), s(s(y))) -> 0 log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y)))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_log(x_1, x_2)) -> log(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_log(x_1, x_2) -> log(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: p(0) -> 0 p(s(x)) -> x minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) minus(x, s(y)) -> p(minus(x, y)) div(0, s(y)) -> 0 div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y))) log(s(0), s(s(y))) -> 0 log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y)))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_log(x_1, x_2)) -> log(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_log(x_1, x_2) -> log(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: p(0') -> 0' p(s(x)) -> x minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) minus(x, s(y)) -> p(minus(x, y)) div(0', s(y)) -> 0' div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y))) log(s(0'), s(s(y))) -> 0' log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y)))) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_log(x_1, x_2)) -> log(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_log(x_1, x_2) -> log(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: p(0') -> 0' p(s(x)) -> x minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) minus(x, s(y)) -> p(minus(x, y)) div(0', s(y)) -> 0' div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y))) log(s(0'), s(s(y))) -> 0' log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_log(x_1, x_2)) -> log(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_log(x_1, x_2) -> log(encArg(x_1), encArg(x_2)) Types: p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log 0' :: 0':s:cons_p:cons_minus:cons_div:cons_log s :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encArg :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_0 :: 0':s:cons_p:cons_minus:cons_div:cons_log encode_s :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log hole_0':s:cons_p:cons_minus:cons_div:cons_log1_3 :: 0':s:cons_p:cons_minus:cons_div:cons_log gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3 :: Nat -> 0':s:cons_p:cons_minus:cons_div:cons_log ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: minus, div, log, encArg They will be analysed ascendingly in the following order: minus < div minus < log minus < encArg div < log div < encArg log < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: p(0') -> 0' p(s(x)) -> x minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) minus(x, s(y)) -> p(minus(x, y)) div(0', s(y)) -> 0' div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y))) log(s(0'), s(s(y))) -> 0' log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_log(x_1, x_2)) -> log(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_log(x_1, x_2) -> log(encArg(x_1), encArg(x_2)) Types: p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log 0' :: 0':s:cons_p:cons_minus:cons_div:cons_log s :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encArg :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_0 :: 0':s:cons_p:cons_minus:cons_div:cons_log encode_s :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log hole_0':s:cons_p:cons_minus:cons_div:cons_log1_3 :: 0':s:cons_p:cons_minus:cons_div:cons_log gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3 :: Nat -> 0':s:cons_p:cons_minus:cons_div:cons_log Generator Equations: gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0) <=> 0' gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(+(x, 1)) <=> s(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(x)) The following defined symbols remain to be analysed: minus, div, log, encArg They will be analysed ascendingly in the following order: minus < div minus < log minus < encArg div < log div < encArg log < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: minus(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(n4_3), gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(n4_3)) -> gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0), rt in Omega(1 + n4_3) Induction Base: minus(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0), gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0)) ->_R^Omega(1) gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0) Induction Step: minus(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(+(n4_3, 1)), gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(+(n4_3, 1))) ->_R^Omega(1) minus(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(n4_3), gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(n4_3)) ->_IH gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0) 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: p(0') -> 0' p(s(x)) -> x minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) minus(x, s(y)) -> p(minus(x, y)) div(0', s(y)) -> 0' div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y))) log(s(0'), s(s(y))) -> 0' log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_log(x_1, x_2)) -> log(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_log(x_1, x_2) -> log(encArg(x_1), encArg(x_2)) Types: p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log 0' :: 0':s:cons_p:cons_minus:cons_div:cons_log s :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encArg :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_0 :: 0':s:cons_p:cons_minus:cons_div:cons_log encode_s :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log hole_0':s:cons_p:cons_minus:cons_div:cons_log1_3 :: 0':s:cons_p:cons_minus:cons_div:cons_log gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3 :: Nat -> 0':s:cons_p:cons_minus:cons_div:cons_log Generator Equations: gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0) <=> 0' gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(+(x, 1)) <=> s(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(x)) The following defined symbols remain to be analysed: minus, div, log, encArg They will be analysed ascendingly in the following order: minus < div minus < log minus < encArg div < log div < encArg log < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: p(0') -> 0' p(s(x)) -> x minus(x, 0') -> x minus(s(x), s(y)) -> minus(x, y) minus(x, s(y)) -> p(minus(x, y)) div(0', s(y)) -> 0' div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y))) log(s(0'), s(s(y))) -> 0' log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_div(x_1, x_2)) -> div(encArg(x_1), encArg(x_2)) encArg(cons_log(x_1, x_2)) -> log(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_div(x_1, x_2) -> div(encArg(x_1), encArg(x_2)) encode_log(x_1, x_2) -> log(encArg(x_1), encArg(x_2)) Types: p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log 0' :: 0':s:cons_p:cons_minus:cons_div:cons_log s :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encArg :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log cons_log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_p :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_0 :: 0':s:cons_p:cons_minus:cons_div:cons_log encode_s :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_minus :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_div :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log encode_log :: 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log -> 0':s:cons_p:cons_minus:cons_div:cons_log hole_0':s:cons_p:cons_minus:cons_div:cons_log1_3 :: 0':s:cons_p:cons_minus:cons_div:cons_log gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3 :: Nat -> 0':s:cons_p:cons_minus:cons_div:cons_log Lemmas: minus(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(n4_3), gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(n4_3)) -> gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0), rt in Omega(1 + n4_3) Generator Equations: gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0) <=> 0' gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(+(x, 1)) <=> s(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(x)) The following defined symbols remain to be analysed: div, log, encArg They will be analysed ascendingly in the following order: div < log div < encArg log < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(n992_3)) -> gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(n992_3), rt in Omega(0) Induction Base: encArg(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(+(n992_3, 1))) ->_R^Omega(0) s(encArg(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(n992_3))) ->_IH s(gen_0':s:cons_p:cons_minus:cons_div:cons_log2_3(c993_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)