/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 48 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), 16.8 s] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 14.3 s] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 14.0 s] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 15.0 s] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 14.3 s] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 108 ms] (26) 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(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> 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(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(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(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(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(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(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(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: a(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encArg :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p hole_s:cons_a:cons_b:cons_c:cons_p1_0 :: s:cons_a:cons_b:cons_c:cons_p gen_s:cons_a:cons_b:cons_c:cons_p2_0 :: Nat -> s:cons_a:cons_b:cons_c:cons_p ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a, p, b, c, encArg They will be analysed ascendingly in the following order: p < a a = b a = c a < encArg p < b p < c p < encArg b = c b < encArg c < encArg ---------------------------------------- (10) Obligation: TRS: Rules: a(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encArg :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p hole_s:cons_a:cons_b:cons_c:cons_p1_0 :: s:cons_a:cons_b:cons_c:cons_p gen_s:cons_a:cons_b:cons_c:cons_p2_0 :: Nat -> s:cons_a:cons_b:cons_c:cons_p Generator Equations: gen_s:cons_a:cons_b:cons_c:cons_p2_0(0) <=> hole_s:cons_a:cons_b:cons_c:cons_p1_0 gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(x)) The following defined symbols remain to be analysed: p, a, b, c, encArg They will be analysed ascendingly in the following order: p < a a = b a = c a < encArg p < b p < c p < encArg b = c b < encArg c < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n11_0)))) -> *3_0, rt in Omega(n11_0) Induction Base: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, 0)))) Induction Step: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, +(n11_0, 1))))) ->_R^Omega(1) s(s(s(p(p(s(s(c(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n11_0))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(c(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n11_0))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n11_0))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(a(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n11_0))))))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(a(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n11_0))))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n11_0))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(s(s(s(p(s(b(p(p(s(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n11_0)))))))))))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(s(s(s(p(s(b(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n11_0)))))))))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(s(s(s(p(s(b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n11_0)))))))))))))))))))) ->_IH s(s(s(p(p(s(s(p(s(p(s(s(s(s(p(s(*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(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encArg :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p hole_s:cons_a:cons_b:cons_c:cons_p1_0 :: s:cons_a:cons_b:cons_c:cons_p gen_s:cons_a:cons_b:cons_c:cons_p2_0 :: Nat -> s:cons_a:cons_b:cons_c:cons_p Generator Equations: gen_s:cons_a:cons_b:cons_c:cons_p2_0(0) <=> hole_s:cons_a:cons_b:cons_c:cons_p1_0 gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(x)) The following defined symbols remain to be analysed: b, a, c, encArg They will be analysed ascendingly in the following order: a = b a = c a < encArg b = c b < encArg c < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: a(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encArg :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p hole_s:cons_a:cons_b:cons_c:cons_p1_0 :: s:cons_a:cons_b:cons_c:cons_p gen_s:cons_a:cons_b:cons_c:cons_p2_0 :: Nat -> s:cons_a:cons_b:cons_c:cons_p Lemmas: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n11_0)))) -> *3_0, rt in Omega(n11_0) Generator Equations: gen_s:cons_a:cons_b:cons_c:cons_p2_0(0) <=> hole_s:cons_a:cons_b:cons_c:cons_p1_0 gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(x)) The following defined symbols remain to be analysed: c, a, encArg They will be analysed ascendingly in the following order: a = b a = c a < encArg b = c b < encArg c < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n735_0)))) -> *3_0, rt in Omega(n735_0) Induction Base: c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, 0)))) Induction Step: c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, +(n735_0, 1))))) ->_R^Omega(1) p(s(p(s(a(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n735_0)))))))))))) ->_R^Omega(1) p(s(p(s(a(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n735_0)))))))))) ->_R^Omega(1) p(s(p(s(a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n735_0)))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(b(p(p(s(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n735_0))))))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(b(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n735_0))))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n735_0))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(s(s(s(p(p(s(s(c(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n735_0)))))))))))))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(s(s(s(p(p(s(s(c(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n735_0)))))))))))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(s(s(s(p(p(s(s(c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n735_0)))))))))))))))))))) ->_IH p(s(p(s(s(s(s(p(s(s(s(s(p(p(s(s(*3_0)))))))))))))))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: TRS: Rules: a(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encArg :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p hole_s:cons_a:cons_b:cons_c:cons_p1_0 :: s:cons_a:cons_b:cons_c:cons_p gen_s:cons_a:cons_b:cons_c:cons_p2_0 :: Nat -> s:cons_a:cons_b:cons_c:cons_p Lemmas: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n11_0)))) -> *3_0, rt in Omega(n11_0) c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n735_0)))) -> *3_0, rt in Omega(n735_0) Generator Equations: gen_s:cons_a:cons_b:cons_c:cons_p2_0(0) <=> hole_s:cons_a:cons_b:cons_c:cons_p1_0 gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(x)) The following defined symbols remain to be analysed: a, b, encArg They will be analysed ascendingly in the following order: a = b a = c a < encArg b = c b < encArg c < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n1562_0)))) -> *3_0, rt in Omega(n1562_0) Induction Base: a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, 0)))) Induction Step: a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, +(n1562_0, 1))))) ->_R^Omega(1) s(s(s(p(s(b(p(p(s(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n1562_0))))))))))))) ->_R^Omega(1) s(s(s(p(s(b(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n1562_0))))))))))) ->_R^Omega(1) s(s(s(p(s(b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n1562_0))))))))) ->_R^Omega(1) s(s(s(p(s(s(s(s(p(p(s(s(c(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n1562_0)))))))))))))))))))) ->_R^Omega(1) s(s(s(p(s(s(s(s(p(p(s(s(c(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n1562_0)))))))))))))))))) ->_R^Omega(1) s(s(s(p(s(s(s(s(p(p(s(s(c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n1562_0)))))))))))))))) ->_R^Omega(1) s(s(s(p(s(s(s(s(p(p(s(s(p(s(p(s(a(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n1562_0)))))))))))))))))))))))) ->_R^Omega(1) s(s(s(p(s(s(s(s(p(p(s(s(p(s(p(s(a(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n1562_0)))))))))))))))))))))) ->_R^Omega(1) s(s(s(p(s(s(s(s(p(p(s(s(p(s(p(s(a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n1562_0)))))))))))))))))))) ->_IH s(s(s(p(s(s(s(s(p(p(s(s(p(s(p(s(*3_0)))))))))))))))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: a(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encArg :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p hole_s:cons_a:cons_b:cons_c:cons_p1_0 :: s:cons_a:cons_b:cons_c:cons_p gen_s:cons_a:cons_b:cons_c:cons_p2_0 :: Nat -> s:cons_a:cons_b:cons_c:cons_p Lemmas: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n11_0)))) -> *3_0, rt in Omega(n11_0) c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n735_0)))) -> *3_0, rt in Omega(n735_0) a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n1562_0)))) -> *3_0, rt in Omega(n1562_0) Generator Equations: gen_s:cons_a:cons_b:cons_c:cons_p2_0(0) <=> hole_s:cons_a:cons_b:cons_c:cons_p1_0 gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(x)) The following defined symbols remain to be analysed: b, c, encArg They will be analysed ascendingly in the following order: a = b a = c a < encArg b = c b < encArg c < encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n2545_0)))) -> *3_0, rt in Omega(n2545_0) Induction Base: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, 0)))) Induction Step: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, +(n2545_0, 1))))) ->_R^Omega(1) s(s(s(p(p(s(s(c(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n2545_0))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(c(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n2545_0))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n2545_0))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(a(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n2545_0))))))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(a(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n2545_0))))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n2545_0))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(s(s(s(p(s(b(p(p(s(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n2545_0)))))))))))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(s(s(s(p(s(b(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n2545_0)))))))))))))))))))))) ->_R^Omega(1) s(s(s(p(p(s(s(p(s(p(s(s(s(s(p(s(b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n2545_0)))))))))))))))))))) ->_IH s(s(s(p(p(s(s(p(s(p(s(s(s(s(p(s(*3_0)))))))))))))))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: TRS: Rules: a(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encArg :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p hole_s:cons_a:cons_b:cons_c:cons_p1_0 :: s:cons_a:cons_b:cons_c:cons_p gen_s:cons_a:cons_b:cons_c:cons_p2_0 :: Nat -> s:cons_a:cons_b:cons_c:cons_p Lemmas: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n2545_0)))) -> *3_0, rt in Omega(n2545_0) c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n735_0)))) -> *3_0, rt in Omega(n735_0) a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n1562_0)))) -> *3_0, rt in Omega(n1562_0) Generator Equations: gen_s:cons_a:cons_b:cons_c:cons_p2_0(0) <=> hole_s:cons_a:cons_b:cons_c:cons_p1_0 gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(x)) The following defined symbols remain to be analysed: c, encArg They will be analysed ascendingly in the following order: a = b a = c a < encArg b = c b < encArg c < encArg ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n3734_0)))) -> *3_0, rt in Omega(n3734_0) Induction Base: c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, 0)))) Induction Step: c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, +(n3734_0, 1))))) ->_R^Omega(1) p(s(p(s(a(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n3734_0)))))))))))) ->_R^Omega(1) p(s(p(s(a(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n3734_0)))))))))) ->_R^Omega(1) p(s(p(s(a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(3, *(3, n3734_0)))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(b(p(p(s(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n3734_0))))))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(b(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n3734_0))))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(2, *(3, n3734_0))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(s(s(s(p(p(s(s(c(p(s(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n3734_0)))))))))))))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(s(s(s(p(p(s(s(c(p(s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n3734_0)))))))))))))))))))))) ->_R^Omega(1) p(s(p(s(s(s(s(p(s(s(s(s(p(p(s(s(c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n3734_0)))))))))))))))))))) ->_IH p(s(p(s(s(s(s(p(s(s(s(s(p(p(s(s(*3_0)))))))))))))))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: TRS: Rules: a(s(x1)) -> s(s(s(p(s(b(p(p(s(s(x1)))))))))) b(s(x1)) -> s(s(s(p(p(s(s(c(p(s(p(s(x1)))))))))))) c(s(x1)) -> p(s(p(s(a(p(s(p(s(x1))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encArg :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p cons_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_a :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_s :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_p :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_b :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p encode_c :: s:cons_a:cons_b:cons_c:cons_p -> s:cons_a:cons_b:cons_c:cons_p hole_s:cons_a:cons_b:cons_c:cons_p1_0 :: s:cons_a:cons_b:cons_c:cons_p gen_s:cons_a:cons_b:cons_c:cons_p2_0 :: Nat -> s:cons_a:cons_b:cons_c:cons_p Lemmas: b(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n2545_0)))) -> *3_0, rt in Omega(n2545_0) c(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n3734_0)))) -> *3_0, rt in Omega(n3734_0) a(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, *(3, n1562_0)))) -> *3_0, rt in Omega(n1562_0) Generator Equations: gen_s:cons_a:cons_b:cons_c:cons_p2_0(0) <=> hole_s:cons_a:cons_b:cons_c:cons_p1_0 gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c:cons_p2_0(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, n4871_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, 0))) Induction Step: encArg(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, +(n4871_0, 1)))) ->_R^Omega(0) s(encArg(gen_s:cons_a:cons_b:cons_c:cons_p2_0(+(1, n4871_0)))) ->_IH s(*3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (26) BOUNDS(1, INF)