/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 46 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), 930 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), 200 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: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) ---------------------------------------- (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: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(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^1, INF). The TRS R consists of the following rules: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(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^1, INF). The TRS R consists of the following rules: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a s :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encArg :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_s :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a hole_s:cons_g:cons_f:cons_h:cons_b:cons_a1_0 :: s:cons_g:cons_f:cons_h:cons_b:cons_a gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0 :: Nat -> s:cons_g:cons_f:cons_h:cons_b:cons_a ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: g, f, b, a, encArg They will be analysed ascendingly in the following order: f < g g < encArg f < encArg b = a b < encArg a < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a s :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encArg :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_s :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a hole_s:cons_g:cons_f:cons_h:cons_b:cons_a1_0 :: s:cons_g:cons_f:cons_h:cons_b:cons_a gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0 :: Nat -> s:cons_g:cons_f:cons_h:cons_b:cons_a Generator Equations: gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(0) <=> hole_s:cons_g:cons_f:cons_h:cons_b:cons_a1_0 gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(x, 1)) <=> s(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(x)) The following defined symbols remain to be analysed: f, g, b, a, encArg They will be analysed ascendingly in the following order: f < g g < encArg f < encArg b = a b < encArg a < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: f(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(3, *(2, n4_0)))) -> *3_0, rt in Omega(n4_0) Induction Base: f(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(3, *(2, 0)))) Induction Step: f(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(3, *(2, +(n4_0, 1))))) ->_R^Omega(1) h(f(s(h(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(2, *(2, n4_0))))))) ->_R^Omega(1) h(f(s(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(2, *(2, n4_0)))))) ->_IH h(*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: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a s :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encArg :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_s :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a hole_s:cons_g:cons_f:cons_h:cons_b:cons_a1_0 :: s:cons_g:cons_f:cons_h:cons_b:cons_a gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0 :: Nat -> s:cons_g:cons_f:cons_h:cons_b:cons_a Generator Equations: gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(0) <=> hole_s:cons_g:cons_f:cons_h:cons_b:cons_a1_0 gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(x, 1)) <=> s(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(x)) The following defined symbols remain to be analysed: f, g, b, a, encArg They will be analysed ascendingly in the following order: f < g g < encArg f < encArg b = a b < encArg a < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: g(h(x1)) -> g(f(s(x1))) f(s(s(s(x1)))) -> h(f(s(h(x1)))) f(h(x1)) -> h(f(s(h(x1)))) h(x1) -> x1 f(f(s(s(x1)))) -> s(s(s(f(f(x1))))) b(a(x1)) -> a(b(x1)) a(a(a(x1))) -> b(a(a(b(x1)))) b(b(b(b(x1)))) -> a(x1) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) Types: g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a s :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encArg :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a cons_a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_g :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_h :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_f :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_s :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_b :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a encode_a :: s:cons_g:cons_f:cons_h:cons_b:cons_a -> s:cons_g:cons_f:cons_h:cons_b:cons_a hole_s:cons_g:cons_f:cons_h:cons_b:cons_a1_0 :: s:cons_g:cons_f:cons_h:cons_b:cons_a gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0 :: Nat -> s:cons_g:cons_f:cons_h:cons_b:cons_a Lemmas: f(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(3, *(2, n4_0)))) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(0) <=> hole_s:cons_g:cons_f:cons_h:cons_b:cons_a1_0 gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(x, 1)) <=> s(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(x)) The following defined symbols remain to be analysed: g, b, a, encArg They will be analysed ascendingly in the following order: g < encArg b = a b < encArg a < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(1, n649_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(1, 0))) Induction Step: encArg(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(1, +(n649_0, 1)))) ->_R^Omega(0) s(encArg(gen_s:cons_g:cons_f:cons_h:cons_b:cons_a2_0(+(1, n649_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). ---------------------------------------- (18) BOUNDS(1, INF)