/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 61 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), 3 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 394 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), 148 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: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(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: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(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: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(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: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c s :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encArg :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_s :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c hole_s:cons_a:cons_b:cons_c1_0 :: s:cons_a:cons_b:cons_c gen_s:cons_a:cons_b:cons_c2_0 :: Nat -> s:cons_a:cons_b:cons_c ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a, b, c, encArg They will be analysed ascendingly in the following order: a = b a = c a < encArg b = c b < encArg c < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c s :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encArg :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_s :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c hole_s:cons_a:cons_b:cons_c1_0 :: s:cons_a:cons_b:cons_c gen_s:cons_a:cons_b:cons_c2_0 :: Nat -> s:cons_a:cons_b:cons_c Generator Equations: gen_s:cons_a:cons_b:cons_c2_0(0) <=> hole_s:cons_a:cons_b:cons_c1_0 gen_s:cons_a:cons_b:cons_c2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c2_0(x)) The following defined symbols remain to be analysed: b, a, c, encArg They will be analysed ascendingly in the following order: a = b a = c a < encArg b = c b < encArg c < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: a(gen_s:cons_a:cons_b:cons_c2_0(+(1, n11_0))) -> *3_0, rt in Omega(n11_0) Induction Base: a(gen_s:cons_a:cons_b:cons_c2_0(+(1, 0))) Induction Step: a(gen_s:cons_a:cons_b:cons_c2_0(+(1, +(n11_0, 1)))) ->_R^Omega(1) s(a(gen_s:cons_a:cons_b:cons_c2_0(+(1, n11_0)))) ->_IH s(*3_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c s :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encArg :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_s :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c hole_s:cons_a:cons_b:cons_c1_0 :: s:cons_a:cons_b:cons_c gen_s:cons_a:cons_b:cons_c2_0 :: Nat -> s:cons_a:cons_b:cons_c Generator Equations: gen_s:cons_a:cons_b:cons_c2_0(0) <=> hole_s:cons_a:cons_b:cons_c1_0 gen_s:cons_a:cons_b:cons_c2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c2_0(x)) The following defined symbols remain to be analysed: a, c, encArg They will be analysed ascendingly in the following order: a = b a = c a < encArg b = c b < encArg c < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: a(s(x1)) -> s(a(x1)) b(a(b(s(x1)))) -> a(b(s(a(x1)))) b(a(b(b(x1)))) -> c(s(x1)) c(s(x1)) -> a(b(a(b(x1)))) a(b(a(a(x1)))) -> b(a(b(a(x1)))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) Types: a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c s :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encArg :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c cons_c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_a :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_s :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_b :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c encode_c :: s:cons_a:cons_b:cons_c -> s:cons_a:cons_b:cons_c hole_s:cons_a:cons_b:cons_c1_0 :: s:cons_a:cons_b:cons_c gen_s:cons_a:cons_b:cons_c2_0 :: Nat -> s:cons_a:cons_b:cons_c Lemmas: a(gen_s:cons_a:cons_b:cons_c2_0(+(1, n11_0))) -> *3_0, rt in Omega(n11_0) Generator Equations: gen_s:cons_a:cons_b:cons_c2_0(0) <=> hole_s:cons_a:cons_b:cons_c1_0 gen_s:cons_a:cons_b:cons_c2_0(+(x, 1)) <=> s(gen_s:cons_a:cons_b:cons_c2_0(x)) The following defined symbols remain to be analysed: c, b, encArg They will be analysed ascendingly in the following order: a = b a = c a < encArg b = c b < encArg c < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:cons_a:cons_b:cons_c2_0(+(1, n736_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_s:cons_a:cons_b:cons_c2_0(+(1, 0))) Induction Step: encArg(gen_s:cons_a:cons_b:cons_c2_0(+(1, +(n736_0, 1)))) ->_R^Omega(0) s(encArg(gen_s:cons_a:cons_b:cons_c2_0(+(1, n736_0)))) ->_IH s(*3_0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)