/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 45 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 11 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 456 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), 97 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: c(a(a(x1))) -> a(c(x1)) a(c(b(x1))) -> a(d(b(x1))) a(d(x1)) -> d(a(a(a(x1)))) b(d(x1)) -> b(c(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(d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_d(x_1) -> d(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: c(a(a(x1))) -> a(c(x1)) a(c(b(x1))) -> a(d(b(x1))) a(d(x1)) -> d(a(a(a(x1)))) b(d(x1)) -> b(c(x1)) The (relative) TRS S consists of the following rules: encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_d(x_1) -> d(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: c(a(a(x1))) -> a(c(x1)) a(c(b(x1))) -> a(d(b(x1))) a(d(x1)) -> d(a(a(a(x1)))) b(d(x1)) -> b(c(x1)) The (relative) TRS S consists of the following rules: encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_d(x_1) -> d(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: c(a(a(x1))) -> a(c(x1)) a(c(b(x1))) -> a(d(b(x1))) a(d(x1)) -> d(a(a(a(x1)))) b(d(x1)) -> b(c(x1)) The (relative) TRS S consists of the following rules: encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: c(a(a(x1))) -> a(c(x1)) a(c(b(x1))) -> a(d(b(x1))) a(d(x1)) -> d(a(a(a(x1)))) b(d(x1)) -> b(c(x1)) encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Types: c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b d :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encArg :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_d :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b hole_d:cons_c:cons_a:cons_b1_0 :: d:cons_c:cons_a:cons_b gen_d:cons_c:cons_a:cons_b2_0 :: Nat -> d:cons_c:cons_a:cons_b ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: c, a, b, encArg They will be analysed ascendingly in the following order: c = a c = b c < encArg a = b a < encArg b < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: c(a(a(x1))) -> a(c(x1)) a(c(b(x1))) -> a(d(b(x1))) a(d(x1)) -> d(a(a(a(x1)))) b(d(x1)) -> b(c(x1)) encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Types: c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b d :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encArg :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_d :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b hole_d:cons_c:cons_a:cons_b1_0 :: d:cons_c:cons_a:cons_b gen_d:cons_c:cons_a:cons_b2_0 :: Nat -> d:cons_c:cons_a:cons_b Generator Equations: gen_d:cons_c:cons_a:cons_b2_0(0) <=> hole_d:cons_c:cons_a:cons_b1_0 gen_d:cons_c:cons_a:cons_b2_0(+(x, 1)) <=> d(gen_d:cons_c:cons_a:cons_b2_0(x)) The following defined symbols remain to be analysed: a, c, b, encArg They will be analysed ascendingly in the following order: c = a c = b c < encArg a = b a < encArg b < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: a(gen_d:cons_c:cons_a:cons_b2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Induction Base: a(gen_d:cons_c:cons_a:cons_b2_0(+(1, 0))) Induction Step: a(gen_d:cons_c:cons_a:cons_b2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) d(a(a(a(gen_d:cons_c:cons_a:cons_b2_0(+(1, n4_0)))))) ->_IH d(a(a(*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: c(a(a(x1))) -> a(c(x1)) a(c(b(x1))) -> a(d(b(x1))) a(d(x1)) -> d(a(a(a(x1)))) b(d(x1)) -> b(c(x1)) encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Types: c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b d :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encArg :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_d :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b hole_d:cons_c:cons_a:cons_b1_0 :: d:cons_c:cons_a:cons_b gen_d:cons_c:cons_a:cons_b2_0 :: Nat -> d:cons_c:cons_a:cons_b Generator Equations: gen_d:cons_c:cons_a:cons_b2_0(0) <=> hole_d:cons_c:cons_a:cons_b1_0 gen_d:cons_c:cons_a:cons_b2_0(+(x, 1)) <=> d(gen_d:cons_c:cons_a:cons_b2_0(x)) The following defined symbols remain to be analysed: a, c, b, encArg They will be analysed ascendingly in the following order: c = a c = b c < encArg a = b a < encArg b < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: c(a(a(x1))) -> a(c(x1)) a(c(b(x1))) -> a(d(b(x1))) a(d(x1)) -> d(a(a(a(x1)))) b(d(x1)) -> b(c(x1)) encArg(d(x_1)) -> d(encArg(x_1)) encArg(cons_c(x_1)) -> c(encArg(x_1)) encArg(cons_a(x_1)) -> a(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_c(x_1) -> c(encArg(x_1)) encode_a(x_1) -> a(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_d(x_1) -> d(encArg(x_1)) Types: c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b d :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encArg :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b cons_b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_c :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_a :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_b :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b encode_d :: d:cons_c:cons_a:cons_b -> d:cons_c:cons_a:cons_b hole_d:cons_c:cons_a:cons_b1_0 :: d:cons_c:cons_a:cons_b gen_d:cons_c:cons_a:cons_b2_0 :: Nat -> d:cons_c:cons_a:cons_b Lemmas: a(gen_d:cons_c:cons_a:cons_b2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_d:cons_c:cons_a:cons_b2_0(0) <=> hole_d:cons_c:cons_a:cons_b1_0 gen_d:cons_c:cons_a:cons_b2_0(+(x, 1)) <=> d(gen_d:cons_c:cons_a:cons_b2_0(x)) The following defined symbols remain to be analysed: b, c, encArg They will be analysed ascendingly in the following order: c = a c = b c < encArg a = b a < encArg b < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_d:cons_c:cons_a:cons_b2_0(+(1, n476_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_d:cons_c:cons_a:cons_b2_0(+(1, 0))) Induction Step: encArg(gen_d:cons_c:cons_a:cons_b2_0(+(1, +(n476_0, 1)))) ->_R^Omega(0) d(encArg(gen_d:cons_c:cons_a:cons_b2_0(+(1, n476_0)))) ->_IH d(*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)