/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 240 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 7 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 291 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), 39 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 13 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 454 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: gt(0, y) -> false gt(s(x), 0) -> true gt(s(x), s(y)) -> gt(x, y) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) double(0) -> 0 double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0) aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z 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(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (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: gt(0, y) -> false gt(s(x), 0) -> true gt(s(x), s(y)) -> gt(x, y) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) double(0) -> 0 double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0) aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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: gt(0, y) -> false gt(s(x), 0) -> true gt(s(x), s(y)) -> gt(x, y) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) double(0) -> 0 double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0) aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) 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: gt(0', y) -> false gt(s(x), 0') -> true gt(s(x), s(y)) -> gt(x, y) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) double(0') -> 0' double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0') aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: gt(0', y) -> false gt(s(x), 0') -> true gt(s(x), s(y)) -> gt(x, y) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) double(0') -> 0' double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0') aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if 0' :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encArg :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_0 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if hole_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if1_4 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4 :: Nat -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: gt, plus, double, aver, if, encArg They will be analysed ascendingly in the following order: gt < aver gt < encArg plus < encArg double < aver double < encArg aver = if aver < encArg if < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: gt(0', y) -> false gt(s(x), 0') -> true gt(s(x), s(y)) -> gt(x, y) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) double(0') -> 0' double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0') aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if 0' :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encArg :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_0 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if hole_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if1_4 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4 :: Nat -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if Generator Equations: gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0) <=> 0' gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(x, 1)) <=> s(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(x)) The following defined symbols remain to be analysed: gt, plus, double, aver, if, encArg They will be analysed ascendingly in the following order: gt < aver gt < encArg plus < encArg double < aver double < encArg aver = if aver < encArg if < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: gt(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4)) -> false, rt in Omega(1 + n4_4) Induction Base: gt(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0)) ->_R^Omega(1) false Induction Step: gt(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(n4_4, 1)), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(n4_4, 1))) ->_R^Omega(1) gt(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4)) ->_IH false 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: gt(0', y) -> false gt(s(x), 0') -> true gt(s(x), s(y)) -> gt(x, y) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) double(0') -> 0' double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0') aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if 0' :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encArg :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_0 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if hole_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if1_4 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4 :: Nat -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if Generator Equations: gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0) <=> 0' gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(x, 1)) <=> s(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(x)) The following defined symbols remain to be analysed: gt, plus, double, aver, if, encArg They will be analysed ascendingly in the following order: gt < aver gt < encArg plus < encArg double < aver double < encArg aver = if aver < encArg if < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: gt(0', y) -> false gt(s(x), 0') -> true gt(s(x), s(y)) -> gt(x, y) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) double(0') -> 0' double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0') aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if 0' :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encArg :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_0 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if hole_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if1_4 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4 :: Nat -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if Lemmas: gt(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4)) -> false, rt in Omega(1 + n4_4) Generator Equations: gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0) <=> 0' gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(x, 1)) <=> s(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(x)) The following defined symbols remain to be analysed: plus, double, aver, if, encArg They will be analysed ascendingly in the following order: plus < encArg double < aver double < encArg aver = if aver < encArg if < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n563_4), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(b)) -> gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(n563_4, b)), rt in Omega(1 + n563_4) Induction Base: plus(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(b)) ->_R^Omega(1) gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(b) Induction Step: plus(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(n563_4, 1)), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(b)) ->_R^Omega(1) s(plus(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n563_4), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(b))) ->_IH s(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(b, c564_4))) 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: gt(0', y) -> false gt(s(x), 0') -> true gt(s(x), s(y)) -> gt(x, y) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) double(0') -> 0' double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0') aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if 0' :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encArg :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_0 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if hole_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if1_4 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4 :: Nat -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if Lemmas: gt(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4)) -> false, rt in Omega(1 + n4_4) plus(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n563_4), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(b)) -> gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(n563_4, b)), rt in Omega(1 + n563_4) Generator Equations: gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0) <=> 0' gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(x, 1)) <=> s(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(x)) The following defined symbols remain to be analysed: double, aver, if, encArg They will be analysed ascendingly in the following order: double < aver double < encArg aver = if aver < encArg if < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: double(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n1714_4)) -> gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(*(2, n1714_4)), rt in Omega(1 + n1714_4) Induction Base: double(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0)) ->_R^Omega(1) 0' Induction Step: double(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(n1714_4, 1))) ->_R^Omega(1) s(s(double(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n1714_4)))) ->_IH s(s(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(*(2, c1715_4)))) 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: gt(0', y) -> false gt(s(x), 0') -> true gt(s(x), s(y)) -> gt(x, y) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) double(0') -> 0' double(s(x)) -> s(s(double(x))) average(x, y) -> aver(plus(x, y), 0') aver(sum, z) -> if(gt(sum, double(z)), sum, z) if(true, sum, z) -> aver(sum, s(z)) if(false, sum, z) -> z encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(true) -> true encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_aver(x_1, x_2)) -> aver(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3)) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_false -> false encode_s(x_1) -> s(encArg(x_1)) encode_true -> true encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_double(x_1) -> double(encArg(x_1)) encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_aver(x_1, x_2) -> aver(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) Types: gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if 0' :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encArg :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if cons_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_gt :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_0 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_false :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_s :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_true :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_plus :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_double :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_average :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_aver :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if encode_if :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if hole_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if1_4 :: 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4 :: Nat -> 0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if Lemmas: gt(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n4_4)) -> false, rt in Omega(1 + n4_4) plus(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n563_4), gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(b)) -> gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(n563_4, b)), rt in Omega(1 + n563_4) double(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n1714_4)) -> gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(*(2, n1714_4)), rt in Omega(1 + n1714_4) Generator Equations: gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0) <=> 0' gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(x, 1)) <=> s(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(x)) The following defined symbols remain to be analysed: if, aver, encArg They will be analysed ascendingly in the following order: aver = if aver < encArg if < encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n2242_4)) -> gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n2242_4), rt in Omega(0) Induction Base: encArg(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(+(n2242_4, 1))) ->_R^Omega(0) s(encArg(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(n2242_4))) ->_IH s(gen_0':false:s:true:cons_gt:cons_plus:cons_double:cons_average:cons_aver:cons_if2_4(c2243_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)