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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 388 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), 277 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), 128 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 63 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 890 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: le(0, y, z) -> greater(y, z) le(s(x), 0, z) -> false le(s(x), s(y), 0) -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0) -> first greater(0, s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0, 0) if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), 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(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true ---------------------------------------- (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: le(0, y, z) -> greater(y, z) le(s(x), 0, z) -> false le(s(x), s(y), 0) -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0) -> first greater(0, s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0, 0) if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true 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: le(0, y, z) -> greater(y, z) le(s(x), 0, z) -> false le(s(x), s(y), 0) -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0) -> first greater(0, s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0) -> 0 double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0, 0) if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true 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: le(0', y, z) -> greater(y, z) le(s(x), 0', z) -> false le(s(x), s(y), 0') -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0') -> first greater(0', s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0') -> 0' double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0', 0') if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: le(0', y, z) -> greater(y, z) le(s(x), 0', z) -> false le(s(x), s(y), 0') -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0') -> first greater(0', s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0') -> 0' double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0', 0') if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true Types: le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if 0' :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encArg :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_0 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if hole_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if1_5 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5 :: Nat -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: le, greater, double, if, encArg They will be analysed ascendingly in the following order: greater < le le < if le < encArg greater < encArg double < encArg if < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: le(0', y, z) -> greater(y, z) le(s(x), 0', z) -> false le(s(x), s(y), 0') -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0') -> first greater(0', s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0') -> 0' double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0', 0') if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true Types: le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if 0' :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encArg :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_0 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if hole_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if1_5 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5 :: Nat -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if Generator Equations: gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0) <=> 0' gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(x, 1)) <=> s(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(x)) The following defined symbols remain to be analysed: greater, le, double, if, encArg They will be analysed ascendingly in the following order: greater < le le < if le < encArg greater < encArg double < encArg if < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: greater(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5)) -> first, rt in Omega(1 + n4_5) Induction Base: greater(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0)) ->_R^Omega(1) first Induction Step: greater(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(n4_5, 1)), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(n4_5, 1))) ->_R^Omega(1) greater(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5)) ->_IH first 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: le(0', y, z) -> greater(y, z) le(s(x), 0', z) -> false le(s(x), s(y), 0') -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0') -> first greater(0', s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0') -> 0' double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0', 0') if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true Types: le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if 0' :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encArg :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_0 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if hole_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if1_5 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5 :: Nat -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if Generator Equations: gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0) <=> 0' gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(x, 1)) <=> s(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(x)) The following defined symbols remain to be analysed: greater, le, double, if, encArg They will be analysed ascendingly in the following order: greater < le le < if le < encArg greater < encArg double < encArg if < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: le(0', y, z) -> greater(y, z) le(s(x), 0', z) -> false le(s(x), s(y), 0') -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0') -> first greater(0', s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0') -> 0' double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0', 0') if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true Types: le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if 0' :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encArg :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_0 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if hole_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if1_5 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5 :: Nat -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if Lemmas: greater(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5)) -> first, rt in Omega(1 + n4_5) Generator Equations: gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0) <=> 0' gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(x, 1)) <=> s(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(x)) The following defined symbols remain to be analysed: le, double, if, encArg They will be analysed ascendingly in the following order: le < if le < encArg double < encArg if < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: le(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(1, n610_5)), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n610_5), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n610_5)) -> false, rt in Omega(1 + n610_5) Induction Base: le(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(1, 0)), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0)) ->_R^Omega(1) false Induction Step: le(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(1, +(n610_5, 1))), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(n610_5, 1)), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(n610_5, 1))) ->_R^Omega(1) le(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(1, n610_5)), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n610_5), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n610_5)) ->_IH false 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: le(0', y, z) -> greater(y, z) le(s(x), 0', z) -> false le(s(x), s(y), 0') -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0') -> first greater(0', s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0') -> 0' double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0', 0') if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true Types: le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if 0' :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encArg :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_0 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if hole_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if1_5 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5 :: Nat -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if Lemmas: greater(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5)) -> first, rt in Omega(1 + n4_5) le(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(1, n610_5)), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n610_5), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n610_5)) -> false, rt in Omega(1 + n610_5) Generator Equations: gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0) <=> 0' gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(x, 1)) <=> s(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(x)) The following defined symbols remain to be analysed: double, if, encArg They will be analysed ascendingly in the following order: double < encArg if < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: double(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n1874_5)) -> gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(*(2, n1874_5)), rt in Omega(1 + n1874_5) Induction Base: double(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0)) ->_R^Omega(1) 0' Induction Step: double(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(n1874_5, 1))) ->_R^Omega(1) s(s(double(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n1874_5)))) ->_IH s(s(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(*(2, c1875_5)))) 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: le(0', y, z) -> greater(y, z) le(s(x), 0', z) -> false le(s(x), s(y), 0') -> false le(s(x), s(y), s(z)) -> le(x, y, z) greater(x, 0') -> first greater(0', s(y)) -> second greater(s(x), s(y)) -> greater(x, y) double(0') -> 0' double(s(x)) -> s(s(double(x))) triple(x) -> if(le(x, x, double(x)), x, 0', 0') if(false, x, y, z) -> true if(first, x, y, z) -> if(le(s(x), y, s(z)), s(x), y, s(z)) if(second, x, y, z) -> if(le(s(x), s(y), z), s(x), s(y), z) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(first) -> first encArg(second) -> second encArg(true) -> true encArg(cons_le(x_1, x_2, x_3)) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_greater(x_1, x_2)) -> greater(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_triple(x_1)) -> triple(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3, x_4)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_le(x_1, x_2, x_3) -> le(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_greater(x_1, x_2) -> greater(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_first -> first encode_second -> second encode_double(x_1) -> double(encArg(x_1)) encode_triple(x_1) -> triple(encArg(x_1)) encode_if(x_1, x_2, x_3, x_4) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true Types: le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if 0' :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encArg :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if cons_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_le :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_0 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_greater :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_s :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_false :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_first :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_second :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_double :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_triple :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_if :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if encode_true :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if hole_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if1_5 :: 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5 :: Nat -> 0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if Lemmas: greater(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n4_5)) -> first, rt in Omega(1 + n4_5) le(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(1, n610_5)), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n610_5), gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n610_5)) -> false, rt in Omega(1 + n610_5) double(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n1874_5)) -> gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(*(2, n1874_5)), rt in Omega(1 + n1874_5) Generator Equations: gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0) <=> 0' gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(x, 1)) <=> s(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(x)) The following defined symbols remain to be analysed: if, encArg They will be analysed ascendingly in the following order: if < encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n2395_5)) -> gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n2395_5), rt in Omega(0) Induction Base: encArg(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(+(n2395_5, 1))) ->_R^Omega(0) s(encArg(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(n2395_5))) ->_IH s(gen_0':s:false:first:second:true:cons_le:cons_greater:cons_double:cons_triple:cons_if2_5(c2396_5)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)