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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 372 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), 306 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), 963 ms] (18) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: p(s(x)) -> x p(0) -> 0 le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) average(x, y) -> if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) if2(true, b2, b3, x, y) -> 0 if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) if3(true, b3, x, y) -> 0 if3(false, b3, x, y) -> if4(b3, x, y) if4(true, x, y) -> s(0) if4(false, x, y) -> average(s(x), p(p(y))) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if3(x_1, x_2, x_3, x_4)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if4(x_1, x_2, x_3)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if3(x_1, x_2, x_3, x_4) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if4(x_1, x_2, x_3) -> if4(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: p(s(x)) -> x p(0) -> 0 le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) average(x, y) -> if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) if2(true, b2, b3, x, y) -> 0 if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) if3(true, b3, x, y) -> 0 if3(false, b3, x, y) -> if4(b3, x, y) if4(true, x, y) -> s(0) if4(false, x, y) -> average(s(x), p(p(y))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if3(x_1, x_2, x_3, x_4)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if4(x_1, x_2, x_3)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if3(x_1, x_2, x_3, x_4) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if4(x_1, x_2, x_3) -> if4(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: p(s(x)) -> x p(0) -> 0 le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) average(x, y) -> if(le(x, 0), le(y, 0), le(y, s(0)), le(y, s(s(0))), x, y) if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) if2(true, b2, b3, x, y) -> 0 if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) if3(true, b3, x, y) -> 0 if3(false, b3, x, y) -> if4(b3, x, y) if4(true, x, y) -> s(0) if4(false, x, y) -> average(s(x), p(p(y))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if3(x_1, x_2, x_3, x_4)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if4(x_1, x_2, x_3)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if3(x_1, x_2, x_3, x_4) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if4(x_1, x_2, x_3) -> if4(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: p(s(x)) -> x p(0') -> 0' le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y) if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) if2(true, b2, b3, x, y) -> 0' if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) if3(true, b3, x, y) -> 0' if3(false, b3, x, y) -> if4(b3, x, y) if4(true, x, y) -> s(0') if4(false, x, y) -> average(s(x), p(p(y))) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if3(x_1, x_2, x_3, x_4)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if4(x_1, x_2, x_3)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if3(x_1, x_2, x_3, x_4) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if4(x_1, x_2, x_3) -> if4(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: p(s(x)) -> x p(0') -> 0' le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y) if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) if2(true, b2, b3, x, y) -> 0' if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) if3(true, b3, x, y) -> 0' if3(false, b3, x, y) -> if4(b3, x, y) if4(true, x, y) -> s(0') if4(false, x, y) -> average(s(x), p(p(y))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if3(x_1, x_2, x_3, x_4)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if4(x_1, x_2, x_3)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if3(x_1, x_2, x_3, x_4) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if4(x_1, x_2, x_3) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) Types: p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 s :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 0' :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 true :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 false :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encArg :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_s :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_0 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_true :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_false :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 hole_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if41_7 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7 :: Nat -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: le, average, if, if2, if3, if4, encArg They will be analysed ascendingly in the following order: le < average le < encArg average = if average = if2 average = if3 average = if4 average < encArg if = if2 if = if3 if = if4 if < encArg if2 = if3 if2 = if4 if2 < encArg if3 = if4 if3 < encArg if4 < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: p(s(x)) -> x p(0') -> 0' le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y) if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) if2(true, b2, b3, x, y) -> 0' if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) if3(true, b3, x, y) -> 0' if3(false, b3, x, y) -> if4(b3, x, y) if4(true, x, y) -> s(0') if4(false, x, y) -> average(s(x), p(p(y))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if3(x_1, x_2, x_3, x_4)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if4(x_1, x_2, x_3)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if3(x_1, x_2, x_3, x_4) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if4(x_1, x_2, x_3) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) Types: p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 s :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 0' :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 true :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 false :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encArg :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_s :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_0 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_true :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_false :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 hole_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if41_7 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7 :: Nat -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 Generator Equations: gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(0) <=> 0' gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(+(x, 1)) <=> s(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(x)) The following defined symbols remain to be analysed: le, average, if, if2, if3, if4, encArg They will be analysed ascendingly in the following order: le < average le < encArg average = if average = if2 average = if3 average = if4 average < encArg if = if2 if = if3 if = if4 if < encArg if2 = if3 if2 = if4 if2 < encArg if3 = if4 if3 < encArg if4 < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: le(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(n4_7), gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(n4_7)) -> true, rt in Omega(1 + n4_7) Induction Base: le(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(0), gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(0)) ->_R^Omega(1) true Induction Step: le(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(+(n4_7, 1)), gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(+(n4_7, 1))) ->_R^Omega(1) le(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(n4_7), gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(n4_7)) ->_IH true 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: p(s(x)) -> x p(0') -> 0' le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y) if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) if2(true, b2, b3, x, y) -> 0' if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) if3(true, b3, x, y) -> 0' if3(false, b3, x, y) -> if4(b3, x, y) if4(true, x, y) -> s(0') if4(false, x, y) -> average(s(x), p(p(y))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if3(x_1, x_2, x_3, x_4)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if4(x_1, x_2, x_3)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if3(x_1, x_2, x_3, x_4) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if4(x_1, x_2, x_3) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) Types: p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 s :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 0' :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 true :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 false :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encArg :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_s :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_0 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_true :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_false :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 hole_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if41_7 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7 :: Nat -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 Generator Equations: gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(0) <=> 0' gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(+(x, 1)) <=> s(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(x)) The following defined symbols remain to be analysed: le, average, if, if2, if3, if4, encArg They will be analysed ascendingly in the following order: le < average le < encArg average = if average = if2 average = if3 average = if4 average < encArg if = if2 if = if3 if = if4 if < encArg if2 = if3 if2 = if4 if2 < encArg if3 = if4 if3 < encArg if4 < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: p(s(x)) -> x p(0') -> 0' le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) average(x, y) -> if(le(x, 0'), le(y, 0'), le(y, s(0')), le(y, s(s(0'))), x, y) if(true, b1, b2, b3, x, y) -> if2(b1, b2, b3, x, y) if(false, b1, b2, b3, x, y) -> average(p(x), s(y)) if2(true, b2, b3, x, y) -> 0' if2(false, b2, b3, x, y) -> if3(b2, b3, x, y) if3(true, b3, x, y) -> 0' if3(false, b3, x, y) -> if4(b3, x, y) if4(true, x, y) -> s(0') if4(false, x, y) -> average(s(x), p(p(y))) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(true) -> true encArg(false) -> false encArg(cons_p(x_1)) -> p(encArg(x_1)) encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_average(x_1, x_2)) -> average(encArg(x_1), encArg(x_2)) encArg(cons_if(x_1, x_2, x_3, x_4, x_5, x_6)) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encArg(cons_if2(x_1, x_2, x_3, x_4, x_5)) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encArg(cons_if3(x_1, x_2, x_3, x_4)) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_if4(x_1, x_2, x_3)) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) encode_p(x_1) -> p(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_true -> true encode_false -> false encode_average(x_1, x_2) -> average(encArg(x_1), encArg(x_2)) encode_if(x_1, x_2, x_3, x_4, x_5, x_6) -> if(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5), encArg(x_6)) encode_if2(x_1, x_2, x_3, x_4, x_5) -> if2(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4), encArg(x_5)) encode_if3(x_1, x_2, x_3, x_4) -> if3(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_if4(x_1, x_2, x_3) -> if4(encArg(x_1), encArg(x_2), encArg(x_3)) Types: p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 s :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 0' :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 true :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 false :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encArg :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 cons_if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_p :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_s :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_0 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_le :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_true :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_false :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_average :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if2 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if3 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 encode_if4 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 hole_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if41_7 :: s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7 :: Nat -> s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if4 Lemmas: le(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(n4_7), gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(n4_7)) -> true, rt in Omega(1 + n4_7) Generator Equations: gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(0) <=> 0' gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(+(x, 1)) <=> s(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(x)) The following defined symbols remain to be analysed: if, average, if2, if3, if4, encArg They will be analysed ascendingly in the following order: average = if average = if2 average = if3 average = if4 average < encArg if = if2 if = if3 if = if4 if < encArg if2 = if3 if2 = if4 if2 < encArg if3 = if4 if3 < encArg if4 < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(n930_7)) -> gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(n930_7), rt in Omega(0) Induction Base: encArg(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(+(n930_7, 1))) ->_R^Omega(0) s(encArg(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(n930_7))) ->_IH s(gen_s:0':true:false:cons_p:cons_le:cons_average:cons_if:cons_if2:cons_if3:cons_if42_7(c931_7)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)