/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 247 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 6 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 263 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), 916 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 532 ms] (20) 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) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) pred(s(x)) -> x minus(x, 0) -> x minus(x, s(y)) -> pred(minus(x, y)) gcd(0, y) -> y gcd(s(x), 0) -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, x, y) -> gcd(minus(x, y), y) if_gcd(false, x, y) -> gcd(minus(y, x), x) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_pred(x_1) -> pred(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) pred(s(x)) -> x minus(x, 0) -> x minus(x, s(y)) -> pred(minus(x, y)) gcd(0, y) -> y gcd(s(x), 0) -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, x, y) -> gcd(minus(x, y), y) if_gcd(false, x, y) -> gcd(minus(y, x), x) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_pred(x_1) -> pred(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) pred(s(x)) -> x minus(x, 0) -> x minus(x, s(y)) -> pred(minus(x, y)) gcd(0, y) -> y gcd(s(x), 0) -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, x, y) -> gcd(minus(x, y), y) if_gcd(false, x, y) -> gcd(minus(y, x), x) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_pred(x_1) -> pred(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(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: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) pred(s(x)) -> x minus(x, 0') -> x minus(x, s(y)) -> pred(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, x, y) -> gcd(minus(x, y), y) if_gcd(false, x, y) -> gcd(minus(y, x), x) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_pred(x_1) -> pred(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(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: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) pred(s(x)) -> x minus(x, 0') -> x minus(x, s(y)) -> pred(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, x, y) -> gcd(minus(x, y), y) if_gcd(false, x, y) -> gcd(minus(y, x), x) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_pred(x_1) -> pred(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Types: le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd 0' :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encArg :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_0 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd hole_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd1_4 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4 :: Nat -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: le, minus, gcd, if_gcd, encArg They will be analysed ascendingly in the following order: le < gcd le < encArg minus < if_gcd minus < encArg gcd = if_gcd gcd < encArg if_gcd < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) pred(s(x)) -> x minus(x, 0') -> x minus(x, s(y)) -> pred(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, x, y) -> gcd(minus(x, y), y) if_gcd(false, x, y) -> gcd(minus(y, x), x) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_pred(x_1) -> pred(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Types: le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd 0' :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encArg :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_0 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd hole_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd1_4 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4 :: Nat -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd Generator Equations: gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(0) <=> 0' gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(x)) The following defined symbols remain to be analysed: le, minus, gcd, if_gcd, encArg They will be analysed ascendingly in the following order: le < gcd le < encArg minus < if_gcd minus < encArg gcd = if_gcd gcd < encArg if_gcd < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: le(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n4_4), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n4_4)) -> true, rt in Omega(1 + n4_4) Induction Base: le(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(0), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(0)) ->_R^Omega(1) true Induction Step: le(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(n4_4, 1)), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(n4_4, 1))) ->_R^Omega(1) le(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n4_4), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n4_4)) ->_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: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) pred(s(x)) -> x minus(x, 0') -> x minus(x, s(y)) -> pred(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, x, y) -> gcd(minus(x, y), y) if_gcd(false, x, y) -> gcd(minus(y, x), x) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_pred(x_1) -> pred(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Types: le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd 0' :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encArg :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_0 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd hole_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd1_4 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4 :: Nat -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd Generator Equations: gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(0) <=> 0' gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(x)) The following defined symbols remain to be analysed: le, minus, gcd, if_gcd, encArg They will be analysed ascendingly in the following order: le < gcd le < encArg minus < if_gcd minus < encArg gcd = if_gcd gcd < encArg if_gcd < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) pred(s(x)) -> x minus(x, 0') -> x minus(x, s(y)) -> pred(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, x, y) -> gcd(minus(x, y), y) if_gcd(false, x, y) -> gcd(minus(y, x), x) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_pred(x_1) -> pred(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Types: le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd 0' :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encArg :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_0 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd hole_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd1_4 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4 :: Nat -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd Lemmas: le(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n4_4), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n4_4)) -> true, rt in Omega(1 + n4_4) Generator Equations: gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(0) <=> 0' gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(x)) The following defined symbols remain to be analysed: minus, gcd, if_gcd, encArg They will be analysed ascendingly in the following order: minus < if_gcd minus < encArg gcd = if_gcd gcd < encArg if_gcd < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: minus(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(a), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(1, n539_4))) -> *3_4, rt in Omega(n539_4) Induction Base: minus(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(a), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(1, 0))) Induction Step: minus(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(a), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(1, +(n539_4, 1)))) ->_R^Omega(1) pred(minus(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(a), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(1, n539_4)))) ->_IH pred(*3_4) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) pred(s(x)) -> x minus(x, 0') -> x minus(x, s(y)) -> pred(minus(x, y)) gcd(0', y) -> y gcd(s(x), 0') -> s(x) gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y)) if_gcd(true, x, y) -> gcd(minus(x, y), y) if_gcd(false, x, y) -> gcd(minus(y, x), x) encArg(0') -> 0' encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(false) -> false encArg(cons_le(x_1, x_2)) -> le(encArg(x_1), encArg(x_2)) encArg(cons_pred(x_1)) -> pred(encArg(x_1)) encArg(cons_minus(x_1, x_2)) -> minus(encArg(x_1), encArg(x_2)) encArg(cons_gcd(x_1, x_2)) -> gcd(encArg(x_1), encArg(x_2)) encArg(cons_if_gcd(x_1, x_2, x_3)) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) encode_le(x_1, x_2) -> le(encArg(x_1), encArg(x_2)) encode_0 -> 0' encode_true -> true encode_s(x_1) -> s(encArg(x_1)) encode_false -> false encode_pred(x_1) -> pred(encArg(x_1)) encode_minus(x_1, x_2) -> minus(encArg(x_1), encArg(x_2)) encode_gcd(x_1, x_2) -> gcd(encArg(x_1), encArg(x_2)) encode_if_gcd(x_1, x_2, x_3) -> if_gcd(encArg(x_1), encArg(x_2), encArg(x_3)) Types: le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd 0' :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encArg :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd cons_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_le :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_0 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_true :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_s :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_false :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_pred :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_minus :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd encode_if_gcd :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd hole_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd1_4 :: 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4 :: Nat -> 0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd Lemmas: le(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n4_4), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n4_4)) -> true, rt in Omega(1 + n4_4) minus(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(a), gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(1, n539_4))) -> *3_4, rt in Omega(n539_4) Generator Equations: gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(0) <=> 0' gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(x, 1)) <=> s(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(x)) The following defined symbols remain to be analysed: if_gcd, gcd, encArg They will be analysed ascendingly in the following order: gcd = if_gcd gcd < encArg if_gcd < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n11119_4)) -> gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n11119_4), rt in Omega(0) Induction Base: encArg(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(+(n11119_4, 1))) ->_R^Omega(0) s(encArg(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(n11119_4))) ->_IH s(gen_0':true:s:false:cons_le:cons_pred:cons_minus:cons_gcd:cons_if_gcd2_4(c11120_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (20) BOUNDS(1, INF)