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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 47 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), 1261 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), 632 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 436 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 158 ms] (22) 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: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(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(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(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: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(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: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(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: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) Types: v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p 0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encArg :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p hole_s:0:cons_v:cons_w:cons_p1_2 :: s:0:cons_v:cons_w:cons_p gen_s:0:cons_v:cons_w:cons_p2_2 :: Nat -> s:0:cons_v:cons_w:cons_p ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: v, p, w, encArg They will be analysed ascendingly in the following order: p < v v = w v < encArg p < w p < encArg w < encArg ---------------------------------------- (10) Obligation: TRS: Rules: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) Types: v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p 0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encArg :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p hole_s:0:cons_v:cons_w:cons_p1_2 :: s:0:cons_v:cons_w:cons_p gen_s:0:cons_v:cons_w:cons_p2_2 :: Nat -> s:0:cons_v:cons_w:cons_p Generator Equations: gen_s:0:cons_v:cons_w:cons_p2_2(0) <=> hole_s:0:cons_v:cons_w:cons_p1_2 gen_s:0:cons_v:cons_w:cons_p2_2(+(x, 1)) <=> s(gen_s:0:cons_v:cons_w:cons_p2_2(x)) The following defined symbols remain to be analysed: p, v, w, encArg They will be analysed ascendingly in the following order: p < v v = w v < encArg p < w p < encArg w < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2))) -> *3_2, rt in Omega(n17_2) Induction Base: w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, 0))) Induction Step: w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, +(n17_2, 1)))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(p(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2)))))))))))))))))))))))) ->_IH s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(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). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) Types: v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p 0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encArg :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p hole_s:0:cons_v:cons_w:cons_p1_2 :: s:0:cons_v:cons_w:cons_p gen_s:0:cons_v:cons_w:cons_p2_2 :: Nat -> s:0:cons_v:cons_w:cons_p Generator Equations: gen_s:0:cons_v:cons_w:cons_p2_2(0) <=> hole_s:0:cons_v:cons_w:cons_p1_2 gen_s:0:cons_v:cons_w:cons_p2_2(+(x, 1)) <=> s(gen_s:0:cons_v:cons_w:cons_p2_2(x)) The following defined symbols remain to be analysed: w, v, encArg They will be analysed ascendingly in the following order: v = w v < encArg w < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) Types: v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p 0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encArg :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p hole_s:0:cons_v:cons_w:cons_p1_2 :: s:0:cons_v:cons_w:cons_p gen_s:0:cons_v:cons_w:cons_p2_2 :: Nat -> s:0:cons_v:cons_w:cons_p Lemmas: w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2))) -> *3_2, rt in Omega(n17_2) Generator Equations: gen_s:0:cons_v:cons_w:cons_p2_2(0) <=> hole_s:0:cons_v:cons_w:cons_p1_2 gen_s:0:cons_v:cons_w:cons_p2_2(+(x, 1)) <=> s(gen_s:0:cons_v:cons_w:cons_p2_2(x)) The following defined symbols remain to be analysed: v, encArg They will be analysed ascendingly in the following order: v = w v < encArg w < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: v(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n1507_2))) -> *3_2, rt in Omega(n1507_2) Induction Base: v(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, 0))) Induction Step: v(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, +(n1507_2, 1)))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n1507_2)))))))))))))))))))))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n1507_2)))))))))))))))))))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n1507_2)))))))))))))))))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(w(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n1507_2)))))))))))))))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n1507_2)))))))))))))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(n1507_2)))))))))))))))))))))))))))))))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(n1507_2)))))))))))))))))))))))))))))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(n1507_2)))))))))))))))))))))))))))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(p(s(s(v(p(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(n1507_2)))))))))))))))))))))))))) ->_R^Omega(1) s(p(p(s(s(s(s(s(s(s(s(s(s(s(s(s(s(p(p(s(s(v(s(gen_s:0:cons_v:cons_w:cons_p2_2(n1507_2)))))))))))))))))))))))) ->_IH s(p(p(s(s(s(s(s(s(s(s(s(s(s(s(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). ---------------------------------------- (18) Obligation: TRS: Rules: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) Types: v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p 0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encArg :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p hole_s:0:cons_v:cons_w:cons_p1_2 :: s:0:cons_v:cons_w:cons_p gen_s:0:cons_v:cons_w:cons_p2_2 :: Nat -> s:0:cons_v:cons_w:cons_p Lemmas: w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n17_2))) -> *3_2, rt in Omega(n17_2) v(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n1507_2))) -> *3_2, rt in Omega(n1507_2) Generator Equations: gen_s:0:cons_v:cons_w:cons_p2_2(0) <=> hole_s:0:cons_v:cons_w:cons_p1_2 gen_s:0:cons_v:cons_w:cons_p2_2(+(x, 1)) <=> s(gen_s:0:cons_v:cons_w:cons_p2_2(x)) The following defined symbols remain to be analysed: w, encArg They will be analysed ascendingly in the following order: v = w v < encArg w < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2))) -> *3_2, rt in Omega(n3107_2) Induction Base: w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, 0))) Induction Step: w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, +(n3107_2, 1)))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(p(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(v(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(p(s(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))))))))))))))) ->_R^Omega(1) s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(s(s(s(w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2)))))))))))))))))))))))) ->_IH s(s(s(s(s(s(p(p(s(s(s(p(p(s(s(s(s(s(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). ---------------------------------------- (20) Obligation: TRS: Rules: v(s(x1)) -> s(p(p(s(s(s(s(s(s(s(s(w(p(p(s(s(p(s(p(s(x1)))))))))))))))))))) v(0(x1)) -> p(p(s(s(0(p(p(s(s(s(s(s(x1)))))))))))) w(s(x1)) -> s(s(s(s(s(s(p(p(s(s(v(p(p(s(s(s(p(p(s(s(x1)))))))))))))))))))) w(0(x1)) -> p(s(p(p(p(p(p(p(p(p(s(s(0(s(s(s(s(s(s(x1))))))))))))))))))) p(p(s(x1))) -> p(x1) p(s(x1)) -> x1 p(0(x1)) -> 0(s(s(s(s(s(s(s(p(s(x1)))))))))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0(x_1)) -> 0(encArg(x_1)) encArg(cons_v(x_1)) -> v(encArg(x_1)) encArg(cons_w(x_1)) -> w(encArg(x_1)) encArg(cons_p(x_1)) -> p(encArg(x_1)) encode_v(x_1) -> v(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) encode_w(x_1) -> w(encArg(x_1)) encode_0(x_1) -> 0(encArg(x_1)) Types: v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p 0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encArg :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p cons_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_v :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_s :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_p :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_w :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p encode_0 :: s:0:cons_v:cons_w:cons_p -> s:0:cons_v:cons_w:cons_p hole_s:0:cons_v:cons_w:cons_p1_2 :: s:0:cons_v:cons_w:cons_p gen_s:0:cons_v:cons_w:cons_p2_2 :: Nat -> s:0:cons_v:cons_w:cons_p Lemmas: w(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n3107_2))) -> *3_2, rt in Omega(n3107_2) v(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n1507_2))) -> *3_2, rt in Omega(n1507_2) Generator Equations: gen_s:0:cons_v:cons_w:cons_p2_2(0) <=> hole_s:0:cons_v:cons_w:cons_p1_2 gen_s:0:cons_v:cons_w:cons_p2_2(+(x, 1)) <=> s(gen_s:0:cons_v:cons_w:cons_p2_2(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n4989_2))) -> *3_2, rt in Omega(0) Induction Base: encArg(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, 0))) Induction Step: encArg(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, +(n4989_2, 1)))) ->_R^Omega(0) s(encArg(gen_s:0:cons_v:cons_w:cons_p2_2(+(1, n4989_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). ---------------------------------------- (22) BOUNDS(1, INF)