/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 74 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), 422 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), 95 ms] (18) 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: h(1(1(x1))) -> 1(h(x1)) 1(1(h(b(x1)))) -> 1(1(s(b(x1)))) 1(s(x1)) -> s(1(x1)) b(s(x1)) -> b(h(x1)) h(1(b(x1))) -> t(1(1(b(x1)))) 1(t(x1)) -> t(1(1(1(x1)))) b(t(x1)) -> b(h(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(t(x_1)) -> t(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_t(x_1) -> t(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: h(1(1(x1))) -> 1(h(x1)) 1(1(h(b(x1)))) -> 1(1(s(b(x1)))) 1(s(x1)) -> s(1(x1)) b(s(x1)) -> b(h(x1)) h(1(b(x1))) -> t(1(1(b(x1)))) 1(t(x1)) -> t(1(1(1(x1)))) b(t(x1)) -> b(h(x1)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(t(x_1)) -> t(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_t(x_1) -> t(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: h(1(1(x1))) -> 1(h(x1)) 1(1(h(b(x1)))) -> 1(1(s(b(x1)))) 1(s(x1)) -> s(1(x1)) b(s(x1)) -> b(h(x1)) h(1(b(x1))) -> t(1(1(b(x1)))) 1(t(x1)) -> t(1(1(1(x1)))) b(t(x1)) -> b(h(x1)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(t(x_1)) -> t(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_t(x_1) -> t(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: h(1(1(x1))) -> 1(h(x1)) 1(1(h(b(x1)))) -> 1(1(s(b(x1)))) 1(s(x1)) -> s(1(x1)) b(s(x1)) -> b(h(x1)) h(1(b(x1))) -> t(1(1(b(x1)))) 1(t(x1)) -> t(1(1(1(x1)))) b(t(x1)) -> b(h(x1)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(t(x_1)) -> t(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: h(1(1(x1))) -> 1(h(x1)) 1(1(h(b(x1)))) -> 1(1(s(b(x1)))) 1(s(x1)) -> s(1(x1)) b(s(x1)) -> b(h(x1)) h(1(b(x1))) -> t(1(1(b(x1)))) 1(t(x1)) -> t(1(1(1(x1)))) b(t(x1)) -> b(h(x1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(t(x_1)) -> t(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) Types: h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b 1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b s :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b t :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encArg :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_s :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_t :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b hole_s:t:cons_h:cons_1:cons_b1_0 :: s:t:cons_h:cons_1:cons_b gen_s:t:cons_h:cons_1:cons_b2_0 :: Nat -> s:t:cons_h:cons_1:cons_b ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: h, 1, b, encArg They will be analysed ascendingly in the following order: h = 1 h = b h < encArg 1 = b 1 < encArg b < encArg ---------------------------------------- (10) Obligation: TRS: Rules: h(1(1(x1))) -> 1(h(x1)) 1(1(h(b(x1)))) -> 1(1(s(b(x1)))) 1(s(x1)) -> s(1(x1)) b(s(x1)) -> b(h(x1)) h(1(b(x1))) -> t(1(1(b(x1)))) 1(t(x1)) -> t(1(1(1(x1)))) b(t(x1)) -> b(h(x1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(t(x_1)) -> t(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) Types: h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b 1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b s :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b t :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encArg :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_s :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_t :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b hole_s:t:cons_h:cons_1:cons_b1_0 :: s:t:cons_h:cons_1:cons_b gen_s:t:cons_h:cons_1:cons_b2_0 :: Nat -> s:t:cons_h:cons_1:cons_b Generator Equations: gen_s:t:cons_h:cons_1:cons_b2_0(0) <=> hole_s:t:cons_h:cons_1:cons_b1_0 gen_s:t:cons_h:cons_1:cons_b2_0(+(x, 1)) <=> s(gen_s:t:cons_h:cons_1:cons_b2_0(x)) The following defined symbols remain to be analysed: 1, h, b, encArg They will be analysed ascendingly in the following order: h = 1 h = b h < encArg 1 = b 1 < encArg b < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: 1(gen_s:t:cons_h:cons_1:cons_b2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Induction Base: 1(gen_s:t:cons_h:cons_1:cons_b2_0(+(1, 0))) Induction Step: 1(gen_s:t:cons_h:cons_1:cons_b2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) s(1(gen_s:t:cons_h:cons_1:cons_b2_0(+(1, n4_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: TRS: Rules: h(1(1(x1))) -> 1(h(x1)) 1(1(h(b(x1)))) -> 1(1(s(b(x1)))) 1(s(x1)) -> s(1(x1)) b(s(x1)) -> b(h(x1)) h(1(b(x1))) -> t(1(1(b(x1)))) 1(t(x1)) -> t(1(1(1(x1)))) b(t(x1)) -> b(h(x1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(t(x_1)) -> t(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) Types: h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b 1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b s :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b t :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encArg :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_s :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_t :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b hole_s:t:cons_h:cons_1:cons_b1_0 :: s:t:cons_h:cons_1:cons_b gen_s:t:cons_h:cons_1:cons_b2_0 :: Nat -> s:t:cons_h:cons_1:cons_b Generator Equations: gen_s:t:cons_h:cons_1:cons_b2_0(0) <=> hole_s:t:cons_h:cons_1:cons_b1_0 gen_s:t:cons_h:cons_1:cons_b2_0(+(x, 1)) <=> s(gen_s:t:cons_h:cons_1:cons_b2_0(x)) The following defined symbols remain to be analysed: 1, h, b, encArg They will be analysed ascendingly in the following order: h = 1 h = b h < encArg 1 = b 1 < encArg b < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: h(1(1(x1))) -> 1(h(x1)) 1(1(h(b(x1)))) -> 1(1(s(b(x1)))) 1(s(x1)) -> s(1(x1)) b(s(x1)) -> b(h(x1)) h(1(b(x1))) -> t(1(1(b(x1)))) 1(t(x1)) -> t(1(1(1(x1)))) b(t(x1)) -> b(h(x1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(t(x_1)) -> t(encArg(x_1)) encArg(cons_h(x_1)) -> h(encArg(x_1)) encArg(cons_1(x_1)) -> 1(encArg(x_1)) encArg(cons_b(x_1)) -> b(encArg(x_1)) encode_h(x_1) -> h(encArg(x_1)) encode_1(x_1) -> 1(encArg(x_1)) encode_b(x_1) -> b(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_t(x_1) -> t(encArg(x_1)) Types: h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b 1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b s :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b t :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encArg :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b cons_b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_h :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_1 :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_b :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_s :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b encode_t :: s:t:cons_h:cons_1:cons_b -> s:t:cons_h:cons_1:cons_b hole_s:t:cons_h:cons_1:cons_b1_0 :: s:t:cons_h:cons_1:cons_b gen_s:t:cons_h:cons_1:cons_b2_0 :: Nat -> s:t:cons_h:cons_1:cons_b Lemmas: 1(gen_s:t:cons_h:cons_1:cons_b2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_s:t:cons_h:cons_1:cons_b2_0(0) <=> hole_s:t:cons_h:cons_1:cons_b1_0 gen_s:t:cons_h:cons_1:cons_b2_0(+(x, 1)) <=> s(gen_s:t:cons_h:cons_1:cons_b2_0(x)) The following defined symbols remain to be analysed: b, h, encArg They will be analysed ascendingly in the following order: h = 1 h = b h < encArg 1 = b 1 < encArg b < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:t:cons_h:cons_1:cons_b2_0(+(1, n680_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_s:t:cons_h:cons_1:cons_b2_0(+(1, 0))) Induction Step: encArg(gen_s:t:cons_h:cons_1:cons_b2_0(+(1, +(n680_0, 1)))) ->_R^Omega(0) s(encArg(gen_s:t:cons_h:cons_1:cons_b2_0(+(1, n680_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)