/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 57 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 11 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 450 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), 247 ms] (18) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) e(x1) -> a(x1) b(d(x1)) -> d(d(x1)) 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(c(x_1)) -> c(encArg(x_1)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_e(x_1)) -> e(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) e(x1) -> a(x1) b(d(x1)) -> d(d(x1)) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_e(x_1)) -> e(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_f(x_1) -> f(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(n^1, INF). The TRS R consists of the following rules: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) e(x1) -> a(x1) b(d(x1)) -> d(d(x1)) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_e(x_1)) -> e(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) e(x1) -> a(x1) b(d(x1)) -> d(d(x1)) The (relative) TRS S consists of the following rules: encArg(c(x_1)) -> c(encArg(x_1)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_e(x_1)) -> e(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) e(x1) -> a(x1) b(d(x1)) -> d(d(x1)) encArg(c(x_1)) -> c(encArg(x_1)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_e(x_1)) -> e(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) Types: a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e c :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e f :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encArg :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_c :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_f :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e hole_c:f:cons_a:cons_b:cons_d:cons_e1_0 :: c:f:cons_a:cons_b:cons_d:cons_e gen_c:f:cons_a:cons_b:cons_d:cons_e2_0 :: Nat -> c:f:cons_a:cons_b:cons_d:cons_e ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a, b, d, e, encArg They will be analysed ascendingly in the following order: b < a d < a a < e a < encArg d < b b < encArg d < encArg e < encArg ---------------------------------------- (10) Obligation: TRS: Rules: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) e(x1) -> a(x1) b(d(x1)) -> d(d(x1)) encArg(c(x_1)) -> c(encArg(x_1)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_e(x_1)) -> e(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) Types: a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e c :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e f :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encArg :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_c :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_f :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e hole_c:f:cons_a:cons_b:cons_d:cons_e1_0 :: c:f:cons_a:cons_b:cons_d:cons_e gen_c:f:cons_a:cons_b:cons_d:cons_e2_0 :: Nat -> c:f:cons_a:cons_b:cons_d:cons_e Generator Equations: gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(0) <=> hole_c:f:cons_a:cons_b:cons_d:cons_e1_0 gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(x, 1)) <=> c(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(x)) The following defined symbols remain to be analysed: d, a, b, e, encArg They will be analysed ascendingly in the following order: b < a d < a a < e a < encArg d < b b < encArg d < encArg e < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: b(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(1, n8_0))) -> *3_0, rt in Omega(n8_0) Induction Base: b(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(1, 0))) Induction Step: b(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(1, +(n8_0, 1)))) ->_R^Omega(1) c(b(b(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(1, n8_0))))) ->_IH c(b(*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: TRS: Rules: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) e(x1) -> a(x1) b(d(x1)) -> d(d(x1)) encArg(c(x_1)) -> c(encArg(x_1)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_e(x_1)) -> e(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) Types: a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e c :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e f :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encArg :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_c :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_f :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e hole_c:f:cons_a:cons_b:cons_d:cons_e1_0 :: c:f:cons_a:cons_b:cons_d:cons_e gen_c:f:cons_a:cons_b:cons_d:cons_e2_0 :: Nat -> c:f:cons_a:cons_b:cons_d:cons_e Generator Equations: gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(0) <=> hole_c:f:cons_a:cons_b:cons_d:cons_e1_0 gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(x, 1)) <=> c(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(x)) The following defined symbols remain to be analysed: b, a, e, encArg They will be analysed ascendingly in the following order: b < a a < e a < encArg b < encArg e < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: a(b(x1)) -> b(c(a(x1))) b(c(x1)) -> c(b(b(x1))) a(c(x1)) -> c(a(b(x1))) a(a(x1)) -> a(d(d(d(x1)))) d(a(x1)) -> d(d(c(x1))) a(d(d(c(x1)))) -> a(a(a(d(x1)))) e(e(f(f(x1)))) -> f(f(f(e(e(x1))))) e(x1) -> a(x1) b(d(x1)) -> d(d(x1)) encArg(c(x_1)) -> c(encArg(x_1)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_d(x_1)) -> d(encArg(x_1)) encArg(cons_e(x_1)) -> e(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) encode_e(x_1) -> e(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) Types: a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e c :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e f :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encArg :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e cons_e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_a :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_b :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_c :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_d :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_e :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e encode_f :: c:f:cons_a:cons_b:cons_d:cons_e -> c:f:cons_a:cons_b:cons_d:cons_e hole_c:f:cons_a:cons_b:cons_d:cons_e1_0 :: c:f:cons_a:cons_b:cons_d:cons_e gen_c:f:cons_a:cons_b:cons_d:cons_e2_0 :: Nat -> c:f:cons_a:cons_b:cons_d:cons_e Lemmas: b(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(1, n8_0))) -> *3_0, rt in Omega(n8_0) Generator Equations: gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(0) <=> hole_c:f:cons_a:cons_b:cons_d:cons_e1_0 gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(x, 1)) <=> c(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(x)) The following defined symbols remain to be analysed: a, e, encArg They will be analysed ascendingly in the following order: a < e a < encArg e < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(1, n1112_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(1, 0))) Induction Step: encArg(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(1, +(n1112_0, 1)))) ->_R^Omega(0) c(encArg(gen_c:f:cons_a:cons_b:cons_d:cons_e2_0(+(1, n1112_0)))) ->_IH c(*3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)