/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 181 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 4 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 487 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), 455 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: 2nd(cons(X, n__cons(Y, Z))) -> activate(Y) from(X) -> cons(X, n__from(n__s(X))) cons(X1, X2) -> n__cons(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X 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(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s(x_1) -> s(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: 2nd(cons(X, n__cons(Y, Z))) -> activate(Y) from(X) -> cons(X, n__from(n__s(X))) cons(X1, X2) -> n__cons(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s(x_1) -> s(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: 2nd(cons(X, n__cons(Y, Z))) -> activate(Y) from(X) -> cons(X, n__from(n__s(X))) cons(X1, X2) -> n__cons(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s(x_1) -> s(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: 2nd(cons(X, n__cons(Y, Z))) -> activate(Y) from(X) -> cons(X, n__from(n__s(X))) cons(X1, X2) -> n__cons(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X The (relative) TRS S consists of the following rules: encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: 2nd(cons(X, n__cons(Y, Z))) -> activate(Y) from(X) -> cons(X, n__from(n__s(X))) cons(X1, X2) -> n__cons(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) Types: 2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encArg :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0 :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0 :: Nat -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: activate, encArg They will be analysed ascendingly in the following order: activate < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: 2nd(cons(X, n__cons(Y, Z))) -> activate(Y) from(X) -> cons(X, n__from(n__s(X))) cons(X1, X2) -> n__cons(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) Types: 2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encArg :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0 :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0 :: Nat -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate Generator Equations: gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(0) <=> hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0 gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(x, 1)) <=> n__cons(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(x), hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0) The following defined symbols remain to be analysed: activate, encArg They will be analysed ascendingly in the following order: activate < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: activate(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Induction Base: activate(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(1, 0))) Induction Step: activate(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) cons(activate(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(1, n4_0))), hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0) ->_IH cons(*3_0, hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_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: 2nd(cons(X, n__cons(Y, Z))) -> activate(Y) from(X) -> cons(X, n__from(n__s(X))) cons(X1, X2) -> n__cons(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) Types: 2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encArg :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0 :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0 :: Nat -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate Generator Equations: gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(0) <=> hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0 gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(x, 1)) <=> n__cons(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(x), hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0) The following defined symbols remain to be analysed: activate, encArg They will be analysed ascendingly in the following order: activate < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: 2nd(cons(X, n__cons(Y, Z))) -> activate(Y) from(X) -> cons(X, n__from(n__s(X))) cons(X1, X2) -> n__cons(X1, X2) from(X) -> n__from(X) s(X) -> n__s(X) activate(n__cons(X1, X2)) -> cons(activate(X1), X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(X) -> X encArg(n__cons(x_1, x_2)) -> n__cons(encArg(x_1), encArg(x_2)) encArg(n__from(x_1)) -> n__from(encArg(x_1)) encArg(n__s(x_1)) -> n__s(encArg(x_1)) encArg(cons_2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_from(x_1)) -> from(encArg(x_1)) encArg(cons_cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_s(x_1)) -> s(encArg(x_1)) encArg(cons_activate(x_1)) -> activate(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_n__cons(x_1, x_2) -> n__cons(encArg(x_1), encArg(x_2)) encode_activate(x_1) -> activate(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_n__from(x_1) -> n__from(encArg(x_1)) encode_n__s(x_1) -> n__s(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) Types: 2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate n__s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encArg :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate cons_activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_2nd :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__cons :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_activate :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__from :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_n__s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate encode_s :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0 :: n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0 :: Nat -> n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate Lemmas: activate(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(0) <=> hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0 gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(x, 1)) <=> n__cons(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(x), hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0) The following defined symbols remain to be analysed: encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(1, n1419_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(1, 0))) Induction Step: encArg(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(1, +(n1419_0, 1)))) ->_R^Omega(0) n__cons(encArg(gen_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate2_0(+(1, n1419_0))), encArg(hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_0)) ->_IH n__cons(*3_0, encArg(hole_n__cons:n__s:n__from:cons_2nd:cons_from:cons_cons:cons_s:cons_activate1_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)