WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 147 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), 505 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), 45.0 s] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 216 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 235 ms] (22) 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)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(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)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(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)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(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)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: p(0(x1)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd 0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encArg :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: p, f, g, j, half, rd, encArg They will be analysed ascendingly in the following order: p < f p < g p < j p < half p < encArg f = g f = j f < encArg g = j g < encArg j < encArg half < encArg rd < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: p(0(x1)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd 0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encArg :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd Generator Equations: gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(0) <=> hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(x, 1)) <=> 0(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(x)) The following defined symbols remain to be analysed: p, f, g, j, half, rd, encArg They will be analysed ascendingly in the following order: p < f p < g p < j p < half p < encArg f = g f = j f < encArg g = j g < encArg j < encArg half < encArg rd < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: p(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n4_2))) -> *3_2, rt in Omega(n4_2) Induction Base: p(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, 0))) Induction Step: p(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, +(n4_2, 1)))) ->_R^Omega(1) 0(s(s(p(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n4_2)))))) ->_IH 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)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd 0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encArg :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd Generator Equations: gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(0) <=> hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(x, 1)) <=> 0(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(x)) The following defined symbols remain to be analysed: p, f, g, j, half, rd, encArg They will be analysed ascendingly in the following order: p < f p < g p < j p < half p < encArg f = g f = j f < encArg g = j g < encArg j < encArg half < encArg rd < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: p(0(x1)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd 0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encArg :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd Lemmas: p(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n4_2))) -> *3_2, rt in Omega(n4_2) Generator Equations: gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(0) <=> hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(x, 1)) <=> 0(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(x)) The following defined symbols remain to be analysed: half, f, g, j, rd, encArg They will be analysed ascendingly in the following order: f = g f = j f < encArg g = j g < encArg j < encArg half < encArg rd < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: half(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n725_2))) -> *3_2, rt in Omega(n725_2) Induction Base: half(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, 0))) Induction Step: half(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, +(n725_2, 1)))) ->_R^Omega(1) 0(s(s(half(p(s(p(s(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n725_2)))))))))) ->_R^Omega(1) 0(s(s(half(p(s(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n725_2)))))))) ->_R^Omega(1) 0(s(s(half(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n725_2)))))) ->_IH 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). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: p(0(x1)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd 0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encArg :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd Lemmas: p(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n4_2))) -> *3_2, rt in Omega(n4_2) half(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n725_2))) -> *3_2, rt in Omega(n725_2) Generator Equations: gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(0) <=> hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(x, 1)) <=> 0(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(x)) The following defined symbols remain to be analysed: rd, f, g, j, encArg They will be analysed ascendingly in the following order: f = g f = j f < encArg g = j g < encArg j < encArg rd < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: rd(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n1623_2))) -> *3_2, rt in Omega(n1623_2) Induction Base: rd(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, 0))) Induction Step: rd(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, +(n1623_2, 1)))) ->_R^Omega(1) 0(s(0(0(0(0(s(0(rd(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n1623_2))))))))))) ->_IH 0(s(0(0(0(0(s(0(*3_2)))))))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: Innermost TRS: Rules: p(0(x1)) -> 0(s(s(p(x1)))) p(s(x1)) -> x1 p(p(s(x1))) -> p(x1) f(s(x1)) -> p(s(g(p(s(s(x1)))))) g(s(x1)) -> p(p(s(s(s(j(s(p(s(p(s(x1))))))))))) j(s(x1)) -> p(s(s(p(s(f(p(s(p(p(s(x1))))))))))) half(0(x1)) -> 0(s(s(half(p(s(p(s(x1)))))))) half(s(s(x1))) -> s(half(p(p(s(s(x1)))))) rd(0(x1)) -> 0(s(0(0(0(0(s(0(rd(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_j(x_1)) -> j(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_rd(x_1)) -> rd(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_j(x_1) -> j(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_rd(x_1) -> rd(encArg(x_1)) Types: p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd 0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encArg :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd cons_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_p :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_0 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_s :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_f :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_g :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_j :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_half :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd encode_rd :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 :: 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2 :: Nat -> 0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd Lemmas: p(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n4_2))) -> *3_2, rt in Omega(n4_2) half(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n725_2))) -> *3_2, rt in Omega(n725_2) rd(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n1623_2))) -> *3_2, rt in Omega(n1623_2) Generator Equations: gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(0) <=> hole_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd1_2 gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(x, 1)) <=> 0(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(x)) The following defined symbols remain to be analysed: g, f, j, encArg They will be analysed ascendingly in the following order: f = g f = j f < encArg g = j g < encArg j < encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n2157_2))) -> *3_2, rt in Omega(0) Induction Base: encArg(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, 0))) Induction Step: encArg(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, +(n2157_2, 1)))) ->_R^Omega(0) 0(encArg(gen_0:s:cons_p:cons_f:cons_g:cons_j:cons_half:cons_rd2_2(+(1, n2157_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). ---------------------------------------- (22) BOUNDS(1, INF)