/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 85 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), 727 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), 355 ms] (18) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: p(0(x1)) -> s(s(0(s(s(p(x1)))))) p(s(0(x1))) -> 0(x1) p(s(s(x1))) -> s(p(s(x1))) f(s(x1)) -> g(q(i(x1))) g(x1) -> f(p(p(x1))) q(i(x1)) -> q(s(x1)) q(s(x1)) -> s(s(x1)) i(x1) -> s(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(0(x_1)) -> 0(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_i(x_1) -> i(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: p(0(x1)) -> s(s(0(s(s(p(x1)))))) p(s(0(x1))) -> 0(x1) p(s(s(x1))) -> s(p(s(x1))) f(s(x1)) -> g(q(i(x1))) g(x1) -> f(p(p(x1))) q(i(x1)) -> q(s(x1)) q(s(x1)) -> s(s(x1)) i(x1) -> s(x1) The (relative) TRS S consists of the following rules: encArg(0(x_1)) -> 0(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_i(x_1) -> i(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: p(0(x1)) -> s(s(0(s(s(p(x1)))))) p(s(0(x1))) -> 0(x1) p(s(s(x1))) -> s(p(s(x1))) f(s(x1)) -> g(q(i(x1))) g(x1) -> f(p(p(x1))) q(i(x1)) -> q(s(x1)) q(s(x1)) -> s(s(x1)) i(x1) -> s(x1) The (relative) TRS S consists of the following rules: encArg(0(x_1)) -> 0(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_i(x_1) -> i(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: p(0(x1)) -> s(s(0(s(s(p(x1)))))) p(s(0(x1))) -> 0(x1) p(s(s(x1))) -> s(p(s(x1))) f(s(x1)) -> g(q(i(x1))) g(x1) -> f(p(p(x1))) q(i(x1)) -> q(s(x1)) q(s(x1)) -> s(s(x1)) i(x1) -> s(x1) The (relative) TRS S consists of the following rules: encArg(0(x_1)) -> 0(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: p(0(x1)) -> s(s(0(s(s(p(x1)))))) p(s(0(x1))) -> 0(x1) p(s(s(x1))) -> s(p(s(x1))) f(s(x1)) -> g(q(i(x1))) g(x1) -> f(p(p(x1))) q(i(x1)) -> q(s(x1)) q(s(x1)) -> s(s(x1)) i(x1) -> s(x1) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i 0 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i s :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encArg :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_s :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i hole_0:s:cons_p:cons_f:cons_g:cons_q:cons_i1_2 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: p, f, g, q, encArg They will be analysed ascendingly in the following order: p < g p < encArg f = g q < f f < encArg g < encArg q < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: p(0(x1)) -> s(s(0(s(s(p(x1)))))) p(s(0(x1))) -> 0(x1) p(s(s(x1))) -> s(p(s(x1))) f(s(x1)) -> g(q(i(x1))) g(x1) -> f(p(p(x1))) q(i(x1)) -> q(s(x1)) q(s(x1)) -> s(s(x1)) i(x1) -> s(x1) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i 0 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i s :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encArg :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_s :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i hole_0:s:cons_p:cons_f:cons_g:cons_q:cons_i1_2 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i Generator Equations: gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(0) <=> hole_0:s:cons_p:cons_f:cons_g:cons_q:cons_i1_2 gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(x, 1)) <=> 0(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(x)) The following defined symbols remain to be analysed: p, f, g, q, encArg They will be analysed ascendingly in the following order: p < g p < encArg f = g q < f f < encArg g < encArg q < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: p(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(1, n4_2))) -> *3_2, rt in Omega(n4_2) Induction Base: p(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(1, 0))) Induction Step: p(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(1, +(n4_2, 1)))) ->_R^Omega(1) s(s(0(s(s(p(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(1, n4_2)))))))) ->_IH s(s(0(s(s(*3_2))))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: p(0(x1)) -> s(s(0(s(s(p(x1)))))) p(s(0(x1))) -> 0(x1) p(s(s(x1))) -> s(p(s(x1))) f(s(x1)) -> g(q(i(x1))) g(x1) -> f(p(p(x1))) q(i(x1)) -> q(s(x1)) q(s(x1)) -> s(s(x1)) i(x1) -> s(x1) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i 0 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i s :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encArg :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_s :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i hole_0:s:cons_p:cons_f:cons_g:cons_q:cons_i1_2 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i Generator Equations: gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(0) <=> hole_0:s:cons_p:cons_f:cons_g:cons_q:cons_i1_2 gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(x, 1)) <=> 0(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(x)) The following defined symbols remain to be analysed: p, f, g, q, encArg They will be analysed ascendingly in the following order: p < g p < encArg f = g q < f f < encArg g < encArg q < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: p(0(x1)) -> s(s(0(s(s(p(x1)))))) p(s(0(x1))) -> 0(x1) p(s(s(x1))) -> s(p(s(x1))) f(s(x1)) -> g(q(i(x1))) g(x1) -> f(p(p(x1))) q(i(x1)) -> q(s(x1)) q(s(x1)) -> s(s(x1)) i(x1) -> s(x1) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encArg(cons_q(x_1)) -> q(encArg(x_1)) encArg(cons_i(x_1)) -> i(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_f(x_1) -> f(encArg(x_1)) encode_g(x_1) -> g(encArg(x_1)) encode_q(x_1) -> q(encArg(x_1)) encode_i(x_1) -> i(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i 0 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i s :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encArg :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i cons_i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_p :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_s :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_f :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_g :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_q :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i encode_i :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i hole_0:s:cons_p:cons_f:cons_g:cons_q:cons_i1_2 :: 0:s:cons_p:cons_f:cons_g:cons_q:cons_i gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_q:cons_i Lemmas: p(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(1, n4_2))) -> *3_2, rt in Omega(n4_2) Generator Equations: gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(0) <=> hole_0:s:cons_p:cons_f:cons_g:cons_q:cons_i1_2 gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(x, 1)) <=> 0(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(x)) The following defined symbols remain to be analysed: q, f, g, encArg They will be analysed ascendingly in the following order: f = g q < f f < encArg g < encArg q < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(1, n1369_2))) -> *3_2, rt in Omega(0) Induction Base: encArg(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(1, 0))) Induction Step: encArg(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(1, +(n1369_2, 1)))) ->_R^Omega(0) 0(encArg(gen_0:s:cons_p:cons_f:cons_g:cons_q:cons_i2_2(+(1, n1369_2)))) ->_IH 0(*3_2) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)