/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 191 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), 642 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), 41 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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) 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(t) -> t encArg(1) -> 1 encArg(constant) -> constant encArg(0) -> 0 encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1 encode_constant -> constant encode_0 -> 0 encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) ---------------------------------------- (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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(1) -> 1 encArg(constant) -> constant encArg(0) -> 0 encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1 encode_constant -> constant encode_0 -> 0 encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) 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: D(t) -> 1 D(constant) -> 0 D(+(x, y)) -> +(D(x), D(y)) D(*(x, y)) -> +(*(y, D(x)), *(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(1) -> 1 encArg(constant) -> constant encArg(0) -> 0 encArg(+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1 encode_constant -> constant encode_0 -> 0 encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) 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: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) The (relative) TRS S consists of the following rules: encArg(t) -> t encArg(1') -> 1' encArg(constant) -> constant encArg(0') -> 0' encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1' encode_constant -> constant encode_0 -> 0' encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) encArg(t) -> t encArg(1') -> 1' encArg(constant) -> constant encArg(0') -> 0' encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1' encode_constant -> constant encode_0 -> 0' encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) Types: D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D t :: t:1':constant:0':+':*':-:cons_D 1' :: t:1':constant:0':+':*':-:cons_D constant :: t:1':constant:0':+':*':-:cons_D 0' :: t:1':constant:0':+':*':-:cons_D +' :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D *' :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D - :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encArg :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D cons_D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_t :: t:1':constant:0':+':*':-:cons_D encode_1 :: t:1':constant:0':+':*':-:cons_D encode_constant :: t:1':constant:0':+':*':-:cons_D encode_0 :: t:1':constant:0':+':*':-:cons_D encode_+ :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_* :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_- :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D hole_t:1':constant:0':+':*':-:cons_D1_3 :: t:1':constant:0':+':*':-:cons_D gen_t:1':constant:0':+':*':-:cons_D2_3 :: Nat -> t:1':constant:0':+':*':-:cons_D ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: D, encArg They will be analysed ascendingly in the following order: D < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) encArg(t) -> t encArg(1') -> 1' encArg(constant) -> constant encArg(0') -> 0' encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1' encode_constant -> constant encode_0 -> 0' encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) Types: D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D t :: t:1':constant:0':+':*':-:cons_D 1' :: t:1':constant:0':+':*':-:cons_D constant :: t:1':constant:0':+':*':-:cons_D 0' :: t:1':constant:0':+':*':-:cons_D +' :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D *' :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D - :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encArg :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D cons_D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_t :: t:1':constant:0':+':*':-:cons_D encode_1 :: t:1':constant:0':+':*':-:cons_D encode_constant :: t:1':constant:0':+':*':-:cons_D encode_0 :: t:1':constant:0':+':*':-:cons_D encode_+ :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_* :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_- :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D hole_t:1':constant:0':+':*':-:cons_D1_3 :: t:1':constant:0':+':*':-:cons_D gen_t:1':constant:0':+':*':-:cons_D2_3 :: Nat -> t:1':constant:0':+':*':-:cons_D Generator Equations: gen_t:1':constant:0':+':*':-:cons_D2_3(0) <=> t gen_t:1':constant:0':+':*':-:cons_D2_3(+(x, 1)) <=> +'(t, gen_t:1':constant:0':+':*':-:cons_D2_3(x)) The following defined symbols remain to be analysed: D, encArg They will be analysed ascendingly in the following order: D < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: D(gen_t:1':constant:0':+':*':-:cons_D2_3(n4_3)) -> *3_3, rt in Omega(n4_3) Induction Base: D(gen_t:1':constant:0':+':*':-:cons_D2_3(0)) Induction Step: D(gen_t:1':constant:0':+':*':-:cons_D2_3(+(n4_3, 1))) ->_R^Omega(1) +'(D(t), D(gen_t:1':constant:0':+':*':-:cons_D2_3(n4_3))) ->_R^Omega(1) +'(1', D(gen_t:1':constant:0':+':*':-:cons_D2_3(n4_3))) ->_IH +'(1', *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: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) encArg(t) -> t encArg(1') -> 1' encArg(constant) -> constant encArg(0') -> 0' encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1' encode_constant -> constant encode_0 -> 0' encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) Types: D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D t :: t:1':constant:0':+':*':-:cons_D 1' :: t:1':constant:0':+':*':-:cons_D constant :: t:1':constant:0':+':*':-:cons_D 0' :: t:1':constant:0':+':*':-:cons_D +' :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D *' :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D - :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encArg :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D cons_D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_t :: t:1':constant:0':+':*':-:cons_D encode_1 :: t:1':constant:0':+':*':-:cons_D encode_constant :: t:1':constant:0':+':*':-:cons_D encode_0 :: t:1':constant:0':+':*':-:cons_D encode_+ :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_* :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_- :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D hole_t:1':constant:0':+':*':-:cons_D1_3 :: t:1':constant:0':+':*':-:cons_D gen_t:1':constant:0':+':*':-:cons_D2_3 :: Nat -> t:1':constant:0':+':*':-:cons_D Generator Equations: gen_t:1':constant:0':+':*':-:cons_D2_3(0) <=> t gen_t:1':constant:0':+':*':-:cons_D2_3(+(x, 1)) <=> +'(t, gen_t:1':constant:0':+':*':-:cons_D2_3(x)) The following defined symbols remain to be analysed: D, encArg They will be analysed ascendingly in the following order: D < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: D(t) -> 1' D(constant) -> 0' D(+'(x, y)) -> +'(D(x), D(y)) D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y))) D(-(x, y)) -> -(D(x), D(y)) encArg(t) -> t encArg(1') -> 1' encArg(constant) -> constant encArg(0') -> 0' encArg(+'(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(*'(x_1, x_2)) -> *'(encArg(x_1), encArg(x_2)) encArg(-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_D(x_1)) -> D(encArg(x_1)) encode_D(x_1) -> D(encArg(x_1)) encode_t -> t encode_1 -> 1' encode_constant -> constant encode_0 -> 0' encode_+(x_1, x_2) -> +'(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *'(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) Types: D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D t :: t:1':constant:0':+':*':-:cons_D 1' :: t:1':constant:0':+':*':-:cons_D constant :: t:1':constant:0':+':*':-:cons_D 0' :: t:1':constant:0':+':*':-:cons_D +' :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D *' :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D - :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encArg :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D cons_D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_D :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_t :: t:1':constant:0':+':*':-:cons_D encode_1 :: t:1':constant:0':+':*':-:cons_D encode_constant :: t:1':constant:0':+':*':-:cons_D encode_0 :: t:1':constant:0':+':*':-:cons_D encode_+ :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_* :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D encode_- :: t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D -> t:1':constant:0':+':*':-:cons_D hole_t:1':constant:0':+':*':-:cons_D1_3 :: t:1':constant:0':+':*':-:cons_D gen_t:1':constant:0':+':*':-:cons_D2_3 :: Nat -> t:1':constant:0':+':*':-:cons_D Lemmas: D(gen_t:1':constant:0':+':*':-:cons_D2_3(n4_3)) -> *3_3, rt in Omega(n4_3) Generator Equations: gen_t:1':constant:0':+':*':-:cons_D2_3(0) <=> t gen_t:1':constant:0':+':*':-:cons_D2_3(+(x, 1)) <=> +'(t, gen_t:1':constant:0':+':*':-:cons_D2_3(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_t:1':constant:0':+':*':-:cons_D2_3(n5258_3)) -> gen_t:1':constant:0':+':*':-:cons_D2_3(n5258_3), rt in Omega(0) Induction Base: encArg(gen_t:1':constant:0':+':*':-:cons_D2_3(0)) ->_R^Omega(0) t Induction Step: encArg(gen_t:1':constant:0':+':*':-:cons_D2_3(+(n5258_3, 1))) ->_R^Omega(0) +'(encArg(t), encArg(gen_t:1':constant:0':+':*':-:cons_D2_3(n5258_3))) ->_R^Omega(0) +'(t, encArg(gen_t:1':constant:0':+':*':-:cons_D2_3(n5258_3))) ->_IH +'(t, gen_t:1':constant:0':+':*':-:cons_D2_3(c5259_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)