/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^2), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 224 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), 254 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), 86 ms] (18) BEST (19) proven lower bound (20) LowerBoundPropagationProof [FINISHED, 0 ms] (21) BOUNDS(n^2, INF) (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 40 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 2041 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 31 ms] (28) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF). The TRS R consists of the following rules: plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) times(0, y) -> 0 times(x, 0) -> 0 times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 fac(s(x)) -> times(fac(p(s(x))), s(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(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF). The TRS R consists of the following rules: plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) times(0, y) -> 0 times(x, 0) -> 0 times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 fac(s(x)) -> times(fac(p(s(x))), s(x)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(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(n^2, INF). The TRS R consists of the following rules: plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) times(0, y) -> 0 times(x, 0) -> 0 times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 fac(s(x)) -> times(fac(p(s(x))), s(x)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) 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^2, INF). The TRS R consists of the following rules: plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(x)) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(x)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) Types: plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac 0' :: 0':s:cons_plus:cons_times:cons_p:cons_fac s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encArg :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_0 :: 0':s:cons_plus:cons_times:cons_p:cons_fac encode_s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac hole_0':s:cons_plus:cons_times:cons_p:cons_fac1_3 :: 0':s:cons_plus:cons_times:cons_p:cons_fac gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3 :: Nat -> 0':s:cons_plus:cons_times:cons_p:cons_fac ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: plus, times, p, fac, encArg They will be analysed ascendingly in the following order: plus < times plus < encArg times < fac times < encArg p < fac p < encArg fac < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(x)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) Types: plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac 0' :: 0':s:cons_plus:cons_times:cons_p:cons_fac s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encArg :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_0 :: 0':s:cons_plus:cons_times:cons_p:cons_fac encode_s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac hole_0':s:cons_plus:cons_times:cons_p:cons_fac1_3 :: 0':s:cons_plus:cons_times:cons_p:cons_fac gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3 :: Nat -> 0':s:cons_plus:cons_times:cons_p:cons_fac Generator Equations: gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(x)) The following defined symbols remain to be analysed: plus, times, p, fac, encArg They will be analysed ascendingly in the following order: plus < times plus < encArg times < fac times < encArg p < fac p < encArg fac < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n4_3)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(n4_3, a)), rt in Omega(1 + n4_3) Induction Base: plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0)) ->_R^Omega(1) gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a) Induction Step: plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(n4_3, 1))) ->_R^Omega(1) s(plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n4_3))) ->_IH s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(a, 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(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(x)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) Types: plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac 0' :: 0':s:cons_plus:cons_times:cons_p:cons_fac s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encArg :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_0 :: 0':s:cons_plus:cons_times:cons_p:cons_fac encode_s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac hole_0':s:cons_plus:cons_times:cons_p:cons_fac1_3 :: 0':s:cons_plus:cons_times:cons_p:cons_fac gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3 :: Nat -> 0':s:cons_plus:cons_times:cons_p:cons_fac Generator Equations: gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(x)) The following defined symbols remain to be analysed: plus, times, p, fac, encArg They will be analysed ascendingly in the following order: plus < times plus < encArg times < fac times < encArg p < fac p < encArg fac < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(x)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) Types: plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac 0' :: 0':s:cons_plus:cons_times:cons_p:cons_fac s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encArg :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_0 :: 0':s:cons_plus:cons_times:cons_p:cons_fac encode_s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac hole_0':s:cons_plus:cons_times:cons_p:cons_fac1_3 :: 0':s:cons_plus:cons_times:cons_p:cons_fac gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3 :: Nat -> 0':s:cons_plus:cons_times:cons_p:cons_fac Lemmas: plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n4_3)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(n4_3, a)), rt in Omega(1 + n4_3) Generator Equations: gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(x)) The following defined symbols remain to be analysed: times, p, fac, encArg They will be analysed ascendingly in the following order: times < fac times < encArg p < fac p < encArg fac < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: times(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n851_3), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(b)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(*(n851_3, b)), rt in Omega(1 + b*n851_3 + n851_3) Induction Base: times(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(b)) ->_R^Omega(1) 0' Induction Step: times(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(n851_3, 1)), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(b)) ->_R^Omega(1) plus(times(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n851_3), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(b)), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(b)) ->_IH plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(*(c852_3, b)), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(b)) ->_L^Omega(1 + b) gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(b, *(n851_3, b))) We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). ---------------------------------------- (18) Complex Obligation (BEST) ---------------------------------------- (19) Obligation: Proved the lower bound n^2 for the following obligation: Innermost TRS: Rules: plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(x)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) Types: plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac 0' :: 0':s:cons_plus:cons_times:cons_p:cons_fac s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encArg :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_0 :: 0':s:cons_plus:cons_times:cons_p:cons_fac encode_s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac hole_0':s:cons_plus:cons_times:cons_p:cons_fac1_3 :: 0':s:cons_plus:cons_times:cons_p:cons_fac gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3 :: Nat -> 0':s:cons_plus:cons_times:cons_p:cons_fac Lemmas: plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n4_3)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(n4_3, a)), rt in Omega(1 + n4_3) Generator Equations: gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(x)) The following defined symbols remain to be analysed: times, p, fac, encArg They will be analysed ascendingly in the following order: times < fac times < encArg p < fac p < encArg fac < encArg ---------------------------------------- (20) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (21) BOUNDS(n^2, INF) ---------------------------------------- (22) Obligation: Innermost TRS: Rules: plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(x)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) Types: plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac 0' :: 0':s:cons_plus:cons_times:cons_p:cons_fac s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encArg :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_0 :: 0':s:cons_plus:cons_times:cons_p:cons_fac encode_s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac hole_0':s:cons_plus:cons_times:cons_p:cons_fac1_3 :: 0':s:cons_plus:cons_times:cons_p:cons_fac gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3 :: Nat -> 0':s:cons_plus:cons_times:cons_p:cons_fac Lemmas: plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n4_3)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(n4_3, a)), rt in Omega(1 + n4_3) times(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n851_3), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(b)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(*(n851_3, b)), rt in Omega(1 + b*n851_3 + n851_3) Generator Equations: gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(x)) The following defined symbols remain to be analysed: p, fac, encArg They will be analysed ascendingly in the following order: p < fac p < encArg fac < encArg ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: p(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n1912_3))) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n1912_3), rt in Omega(1 + n1912_3) Induction Base: p(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, 0))) ->_R^Omega(1) 0' Induction Step: p(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, +(n1912_3, 1)))) ->_R^Omega(1) s(p(s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n1912_3)))) ->_IH s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(c1913_3)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: Innermost TRS: Rules: plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(x)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) Types: plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac 0' :: 0':s:cons_plus:cons_times:cons_p:cons_fac s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encArg :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_0 :: 0':s:cons_plus:cons_times:cons_p:cons_fac encode_s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac hole_0':s:cons_plus:cons_times:cons_p:cons_fac1_3 :: 0':s:cons_plus:cons_times:cons_p:cons_fac gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3 :: Nat -> 0':s:cons_plus:cons_times:cons_p:cons_fac Lemmas: plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n4_3)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(n4_3, a)), rt in Omega(1 + n4_3) times(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n851_3), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(b)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(*(n851_3, b)), rt in Omega(1 + b*n851_3 + n851_3) p(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n1912_3))) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n1912_3), rt in Omega(1 + n1912_3) Generator Equations: gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(x)) The following defined symbols remain to be analysed: fac, encArg They will be analysed ascendingly in the following order: fac < encArg ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: fac(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n2224_3))) -> *3_3, rt in Omega(n2224_3 + n2224_3^2) Induction Base: fac(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, 0))) Induction Step: fac(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, +(n2224_3, 1)))) ->_R^Omega(1) times(fac(p(s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n2224_3))))), s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n2224_3)))) ->_L^Omega(2 + n2224_3) times(fac(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n2224_3))), s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n2224_3)))) ->_IH times(*3_3, s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n2224_3)))) We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). ---------------------------------------- (26) Obligation: Innermost TRS: Rules: plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(x)) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_fac(x_1)) -> fac(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_p(x_1) -> p(encArg(x_1)) encode_fac(x_1) -> fac(encArg(x_1)) Types: plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac 0' :: 0':s:cons_plus:cons_times:cons_p:cons_fac s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encArg :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac cons_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_plus :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_0 :: 0':s:cons_plus:cons_times:cons_p:cons_fac encode_s :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_times :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_p :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac encode_fac :: 0':s:cons_plus:cons_times:cons_p:cons_fac -> 0':s:cons_plus:cons_times:cons_p:cons_fac hole_0':s:cons_plus:cons_times:cons_p:cons_fac1_3 :: 0':s:cons_plus:cons_times:cons_p:cons_fac gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3 :: Nat -> 0':s:cons_plus:cons_times:cons_p:cons_fac Lemmas: plus(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(a), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n4_3)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(n4_3, a)), rt in Omega(1 + n4_3) times(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n851_3), gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(b)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(*(n851_3, b)), rt in Omega(1 + b*n851_3 + n851_3) p(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n1912_3))) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n1912_3), rt in Omega(1 + n1912_3) fac(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(1, n2224_3))) -> *3_3, rt in Omega(n2224_3 + n2224_3^2) Generator Equations: gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0) <=> 0' gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(x, 1)) <=> s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n3125_3)) -> gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n3125_3), rt in Omega(0) Induction Base: encArg(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(+(n3125_3, 1))) ->_R^Omega(0) s(encArg(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(n3125_3))) ->_IH s(gen_0':s:cons_plus:cons_times:cons_p:cons_fac2_3(c3126_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (28) BOUNDS(1, INF)