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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 155 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), 388 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), 34 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: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0) -> tt 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(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 ---------------------------------------- (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: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0) -> tt The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0) -> tt The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0') -> tt The (relative) TRS S consists of the following rules: encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0') -> tt encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat tt :: tt:s:0':cons_f:cons_isNat isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat s :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat 0' :: tt:s:0':cons_f:cons_isNat encArg :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat cons_f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat cons_isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_tt :: tt:s:0':cons_f:cons_isNat encode_isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_s :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_0 :: tt:s:0':cons_f:cons_isNat hole_tt:s:0':cons_f:cons_isNat1_3 :: tt:s:0':cons_f:cons_isNat gen_tt:s:0':cons_f:cons_isNat2_3 :: Nat -> tt:s:0':cons_f:cons_isNat ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: f, isNat, encArg They will be analysed ascendingly in the following order: isNat < f f < encArg isNat < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0') -> tt encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat tt :: tt:s:0':cons_f:cons_isNat isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat s :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat 0' :: tt:s:0':cons_f:cons_isNat encArg :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat cons_f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat cons_isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_tt :: tt:s:0':cons_f:cons_isNat encode_isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_s :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_0 :: tt:s:0':cons_f:cons_isNat hole_tt:s:0':cons_f:cons_isNat1_3 :: tt:s:0':cons_f:cons_isNat gen_tt:s:0':cons_f:cons_isNat2_3 :: Nat -> tt:s:0':cons_f:cons_isNat Generator Equations: gen_tt:s:0':cons_f:cons_isNat2_3(0) <=> tt gen_tt:s:0':cons_f:cons_isNat2_3(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_isNat2_3(x)) The following defined symbols remain to be analysed: isNat, f, encArg They will be analysed ascendingly in the following order: isNat < f f < encArg isNat < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: isNat(gen_tt:s:0':cons_f:cons_isNat2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) Induction Base: isNat(gen_tt:s:0':cons_f:cons_isNat2_3(+(1, 0))) Induction Step: isNat(gen_tt:s:0':cons_f:cons_isNat2_3(+(1, +(n4_3, 1)))) ->_R^Omega(1) isNat(gen_tt:s:0':cons_f:cons_isNat2_3(+(1, n4_3))) ->_IH *3_3 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: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0') -> tt encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat tt :: tt:s:0':cons_f:cons_isNat isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat s :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat 0' :: tt:s:0':cons_f:cons_isNat encArg :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat cons_f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat cons_isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_tt :: tt:s:0':cons_f:cons_isNat encode_isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_s :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_0 :: tt:s:0':cons_f:cons_isNat hole_tt:s:0':cons_f:cons_isNat1_3 :: tt:s:0':cons_f:cons_isNat gen_tt:s:0':cons_f:cons_isNat2_3 :: Nat -> tt:s:0':cons_f:cons_isNat Generator Equations: gen_tt:s:0':cons_f:cons_isNat2_3(0) <=> tt gen_tt:s:0':cons_f:cons_isNat2_3(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_isNat2_3(x)) The following defined symbols remain to be analysed: isNat, f, encArg They will be analysed ascendingly in the following order: isNat < f f < encArg isNat < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: f(tt, x) -> f(isNat(x), s(x)) isNat(s(x)) -> isNat(x) isNat(0') -> tt encArg(tt) -> tt encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_isNat(x_1)) -> isNat(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_isNat(x_1) -> isNat(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat tt :: tt:s:0':cons_f:cons_isNat isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat s :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat 0' :: tt:s:0':cons_f:cons_isNat encArg :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat cons_f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat cons_isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_f :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_tt :: tt:s:0':cons_f:cons_isNat encode_isNat :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_s :: tt:s:0':cons_f:cons_isNat -> tt:s:0':cons_f:cons_isNat encode_0 :: tt:s:0':cons_f:cons_isNat hole_tt:s:0':cons_f:cons_isNat1_3 :: tt:s:0':cons_f:cons_isNat gen_tt:s:0':cons_f:cons_isNat2_3 :: Nat -> tt:s:0':cons_f:cons_isNat Lemmas: isNat(gen_tt:s:0':cons_f:cons_isNat2_3(+(1, n4_3))) -> *3_3, rt in Omega(n4_3) Generator Equations: gen_tt:s:0':cons_f:cons_isNat2_3(0) <=> tt gen_tt:s:0':cons_f:cons_isNat2_3(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_isNat2_3(x)) The following defined symbols remain to be analysed: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_tt:s:0':cons_f:cons_isNat2_3(n520_3)) -> gen_tt:s:0':cons_f:cons_isNat2_3(n520_3), rt in Omega(0) Induction Base: encArg(gen_tt:s:0':cons_f:cons_isNat2_3(0)) ->_R^Omega(0) tt Induction Step: encArg(gen_tt:s:0':cons_f:cons_isNat2_3(+(n520_3, 1))) ->_R^Omega(0) s(encArg(gen_tt:s:0':cons_f:cons_isNat2_3(n520_3))) ->_IH s(gen_tt:s:0':cons_f:cons_isNat2_3(c521_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)