/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 208 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), 247 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), 57 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 80 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 54 ms] (22) 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: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, 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(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(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: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(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: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) min(x, 0) -> 0 min(0, y) -> 0 min(s(x), s(y)) -> s(min(x, y)) twice(0) -> 0 twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(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: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) Types: - :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f 0' :: 0':s:cons_-:cons_min:cons_twice:cons_f s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encArg :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_0 :: 0':s:cons_-:cons_min:cons_twice:cons_f encode_s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f hole_0':s:cons_-:cons_min:cons_twice:cons_f1_3 :: 0':s:cons_-:cons_min:cons_twice:cons_f gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3 :: Nat -> 0':s:cons_-:cons_min:cons_twice:cons_f ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: -, min, twice, f, encArg They will be analysed ascendingly in the following order: - < f - < encArg min < f min < encArg twice < f twice < encArg f < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) Types: - :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f 0' :: 0':s:cons_-:cons_min:cons_twice:cons_f s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encArg :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_0 :: 0':s:cons_-:cons_min:cons_twice:cons_f encode_s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f hole_0':s:cons_-:cons_min:cons_twice:cons_f1_3 :: 0':s:cons_-:cons_min:cons_twice:cons_f gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3 :: Nat -> 0':s:cons_-:cons_min:cons_twice:cons_f Generator Equations: gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0) <=> 0' gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(x)) The following defined symbols remain to be analysed: -, min, twice, f, encArg They will be analysed ascendingly in the following order: - < f - < encArg min < f min < encArg twice < f twice < encArg f < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: -(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0), rt in Omega(1 + n4_3) Induction Base: -(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0)) ->_R^Omega(1) gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0) Induction Step: -(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(n4_3, 1)), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(n4_3, 1))) ->_R^Omega(1) -(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3)) ->_IH gen_0':s:cons_-:cons_min:cons_twice:cons_f2_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: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) Types: - :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f 0' :: 0':s:cons_-:cons_min:cons_twice:cons_f s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encArg :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_0 :: 0':s:cons_-:cons_min:cons_twice:cons_f encode_s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f hole_0':s:cons_-:cons_min:cons_twice:cons_f1_3 :: 0':s:cons_-:cons_min:cons_twice:cons_f gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3 :: Nat -> 0':s:cons_-:cons_min:cons_twice:cons_f Generator Equations: gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0) <=> 0' gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(x)) The following defined symbols remain to be analysed: -, min, twice, f, encArg They will be analysed ascendingly in the following order: - < f - < encArg min < f min < encArg twice < f twice < encArg f < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) Types: - :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f 0' :: 0':s:cons_-:cons_min:cons_twice:cons_f s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encArg :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_0 :: 0':s:cons_-:cons_min:cons_twice:cons_f encode_s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f hole_0':s:cons_-:cons_min:cons_twice:cons_f1_3 :: 0':s:cons_-:cons_min:cons_twice:cons_f gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3 :: Nat -> 0':s:cons_-:cons_min:cons_twice:cons_f Lemmas: -(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0), rt in Omega(1 + n4_3) Generator Equations: gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0) <=> 0' gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(x)) The following defined symbols remain to be analysed: min, twice, f, encArg They will be analysed ascendingly in the following order: min < f min < encArg twice < f twice < encArg f < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: min(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3), rt in Omega(1 + n504_3) Induction Base: min(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0)) ->_R^Omega(1) 0' Induction Step: min(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(n504_3, 1)), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(n504_3, 1))) ->_R^Omega(1) s(min(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3))) ->_IH s(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(c505_3)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) Types: - :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f 0' :: 0':s:cons_-:cons_min:cons_twice:cons_f s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encArg :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_0 :: 0':s:cons_-:cons_min:cons_twice:cons_f encode_s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f hole_0':s:cons_-:cons_min:cons_twice:cons_f1_3 :: 0':s:cons_-:cons_min:cons_twice:cons_f gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3 :: Nat -> 0':s:cons_-:cons_min:cons_twice:cons_f Lemmas: -(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0), rt in Omega(1 + n4_3) min(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3), rt in Omega(1 + n504_3) Generator Equations: gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0) <=> 0' gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(x)) The following defined symbols remain to be analysed: twice, f, encArg They will be analysed ascendingly in the following order: twice < f twice < encArg f < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: twice(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n1024_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(*(2, n1024_3)), rt in Omega(1 + n1024_3) Induction Base: twice(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0)) ->_R^Omega(1) 0' Induction Step: twice(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(n1024_3, 1))) ->_R^Omega(1) s(s(twice(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n1024_3)))) ->_IH s(s(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(*(2, c1025_3)))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: Innermost TRS: Rules: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) min(x, 0') -> 0' min(0', y) -> 0' min(s(x), s(y)) -> s(min(x, y)) twice(0') -> 0' twice(s(x)) -> s(s(twice(x))) f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y)))) f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y)))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_-(x_1, x_2)) -> -(encArg(x_1), encArg(x_2)) encArg(cons_min(x_1, x_2)) -> min(encArg(x_1), encArg(x_2)) encArg(cons_twice(x_1)) -> twice(encArg(x_1)) encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encode_-(x_1, x_2) -> -(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_min(x_1, x_2) -> min(encArg(x_1), encArg(x_2)) encode_twice(x_1) -> twice(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) Types: - :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f 0' :: 0':s:cons_-:cons_min:cons_twice:cons_f s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encArg :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f cons_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_- :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_0 :: 0':s:cons_-:cons_min:cons_twice:cons_f encode_s :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_min :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_twice :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f encode_f :: 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f -> 0':s:cons_-:cons_min:cons_twice:cons_f hole_0':s:cons_-:cons_min:cons_twice:cons_f1_3 :: 0':s:cons_-:cons_min:cons_twice:cons_f gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3 :: Nat -> 0':s:cons_-:cons_min:cons_twice:cons_f Lemmas: -(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n4_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0), rt in Omega(1 + n4_3) min(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3), gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n504_3), rt in Omega(1 + n504_3) twice(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n1024_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(*(2, n1024_3)), rt in Omega(1 + n1024_3) Generator Equations: gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0) <=> 0' gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(x, 1)) <=> s(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(x)) The following defined symbols remain to be analysed: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n1515_3)) -> gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n1515_3), rt in Omega(0) Induction Base: encArg(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(+(n1515_3, 1))) ->_R^Omega(0) s(encArg(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(n1515_3))) ->_IH s(gen_0':s:cons_-:cons_min:cons_twice:cons_f2_3(c1516_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)