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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 44 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), 21.0 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), 2207 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 110 ms] (20) 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: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(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_thrice(x_1)) -> thrice(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_sixtimes(x_1)) -> sixtimes(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_thrice(x_1) -> thrice(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_sixtimes(x_1) -> sixtimes(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: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_thrice(x_1)) -> thrice(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_sixtimes(x_1)) -> sixtimes(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_thrice(x_1) -> thrice(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_sixtimes(x_1) -> sixtimes(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: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_thrice(x_1)) -> thrice(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_sixtimes(x_1)) -> sixtimes(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_thrice(x_1) -> thrice(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_sixtimes(x_1) -> sixtimes(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: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(x1) -> x1 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_thrice(x_1)) -> thrice(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_sixtimes(x_1)) -> sixtimes(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_thrice(x_1) -> thrice(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_sixtimes(x_1) -> sixtimes(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(x1) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_thrice(x_1)) -> thrice(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_sixtimes(x_1)) -> sixtimes(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_thrice(x_1) -> thrice(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_sixtimes(x_1) -> sixtimes(encArg(x_1)) Types: thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encArg :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 hole_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_01_2 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2 :: Nat -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: p, half, sixtimes, encArg They will be analysed ascendingly in the following order: p < half p < sixtimes p < encArg half < encArg sixtimes < encArg ---------------------------------------- (10) Obligation: TRS: Rules: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(x1) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_thrice(x_1)) -> thrice(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_sixtimes(x_1)) -> sixtimes(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_thrice(x_1) -> thrice(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_sixtimes(x_1) -> sixtimes(encArg(x_1)) Types: thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encArg :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 hole_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_01_2 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2 :: Nat -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 Generator Equations: gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(0) <=> hole_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_01_2 gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(x, 1)) <=> s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(x)) The following defined symbols remain to be analysed: p, half, sixtimes, encArg They will be analysed ascendingly in the following order: p < half p < sixtimes p < encArg half < encArg sixtimes < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: half(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n17_2))) -> *3_2, rt in Omega(n17_2) Induction Base: half(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, 0))) Induction Step: half(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, +(n17_2, 1)))) ->_R^Omega(1) p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n17_2))))))))))))))))))) ->_R^Omega(1) p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n17_2))))))))))))))))) ->_R^Omega(1) p(s(p(p(s(s(p(p(s(s(half(p(s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n17_2))))))))))))))) ->_R^Omega(1) p(s(p(p(s(s(p(p(s(s(half(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n17_2))))))))))))) ->_IH p(s(p(p(s(s(p(p(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: TRS: Rules: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(x1) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_thrice(x_1)) -> thrice(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_sixtimes(x_1)) -> sixtimes(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_thrice(x_1) -> thrice(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_sixtimes(x_1) -> sixtimes(encArg(x_1)) Types: thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encArg :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 hole_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_01_2 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2 :: Nat -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 Generator Equations: gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(0) <=> hole_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_01_2 gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(x, 1)) <=> s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(x)) The following defined symbols remain to be analysed: half, sixtimes, encArg They will be analysed ascendingly in the following order: half < encArg sixtimes < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(x1) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_thrice(x_1)) -> thrice(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_sixtimes(x_1)) -> sixtimes(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_thrice(x_1) -> thrice(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_sixtimes(x_1) -> sixtimes(encArg(x_1)) Types: thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encArg :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 hole_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_01_2 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2 :: Nat -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 Lemmas: half(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n17_2))) -> *3_2, rt in Omega(n17_2) Generator Equations: gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(0) <=> hole_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_01_2 gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(x, 1)) <=> s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(x)) The following defined symbols remain to be analysed: sixtimes, encArg They will be analysed ascendingly in the following order: sixtimes < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: sixtimes(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n1386_2))) -> *3_2, rt in Omega(n1386_2) Induction Base: sixtimes(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, 0))) Induction Step: sixtimes(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, +(n1386_2, 1)))) ->_R^Omega(1) p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n1386_2))))))))))))))))))))))))))) ->_R^Omega(1) p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(s(s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n1386_2))))))))))))))))))))))))) ->_R^Omega(1) p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n1386_2))))))))))))))))))))))) ->_R^Omega(1) p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n1386_2))))))))))))))))))))) ->_R^Omega(1) p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n1386_2))))))))))))))))))) ->_IH p(p(s(s(s(s(s(s(s(p(p(s(p(s(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: TRS: Rules: thrice(0(x1)) -> p(s(p(p(p(s(s(s(0(p(s(p(s(x1))))))))))))) thrice(s(x1)) -> p(p(s(s(half(p(p(s(s(p(s(sixtimes(p(s(p(p(s(s(x1)))))))))))))))))) half(0(x1)) -> p(p(s(s(p(s(0(p(s(s(s(s(x1)))))))))))) half(s(x1)) -> p(s(p(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1))))))))))))))))) half(s(s(x1))) -> p(s(p(s(s(p(p(s(s(half(p(p(s(s(p(s(x1)))))))))))))))) sixtimes(0(x1)) -> p(s(p(s(0(s(s(s(s(s(p(s(p(s(x1)))))))))))))) sixtimes(s(x1)) -> p(p(s(s(s(s(s(s(s(p(p(s(p(s(s(s(sixtimes(p(s(p(p(p(s(s(s(x1))))))))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(x1))))) 0(x1) -> x1 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_thrice(x_1)) -> thrice(encArg(x_1)) encArg(cons_half(x_1)) -> half(encArg(x_1)) encArg(cons_sixtimes(x_1)) -> sixtimes(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_0(x_1)) -> 0(encArg(x_1)) encode_thrice(x_1) -> thrice(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_half(x_1) -> half(encArg(x_1)) encode_sixtimes(x_1) -> sixtimes(encArg(x_1)) Types: thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encArg :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 cons_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_thrice :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_0 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_p :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_s :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_half :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 encode_sixtimes :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 hole_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_01_2 :: s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2 :: Nat -> s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_0 Lemmas: half(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n17_2))) -> *3_2, rt in Omega(n17_2) sixtimes(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n1386_2))) -> *3_2, rt in Omega(n1386_2) Generator Equations: gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(0) <=> hole_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_01_2 gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(x, 1)) <=> s(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n2946_2))) -> *3_2, rt in Omega(0) Induction Base: encArg(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, 0))) Induction Step: encArg(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, +(n2946_2, 1)))) ->_R^Omega(0) s(encArg(gen_s:cons_thrice:cons_half:cons_sixtimes:cons_p:cons_02_2(+(1, n2946_2)))) ->_IH s(*3_2) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (20) BOUNDS(1, INF)