/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 196 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), 482 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), 127 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 100 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: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) 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(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(false) -> false encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_false -> false ---------------------------------------- (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: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(false) -> false encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_false -> false 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: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(false) -> false encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 encode_false -> false 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: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0') -> n plus(n, s(m)) -> s(plus(n, m)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(false) -> false encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_false -> false Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0') -> n plus(n, s(m)) -> s(plus(n, m)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(false) -> false encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_false -> false Types: f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt true :: true:s:0':false:cons_f:cons_plus:cons_gt gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt 0' :: true:s:0':false:cons_f:cons_plus:cons_gt false :: true:s:0':false:cons_f:cons_plus:cons_gt encArg :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_true :: true:s:0':false:cons_f:cons_plus:cons_gt encode_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_0 :: true:s:0':false:cons_f:cons_plus:cons_gt encode_false :: true:s:0':false:cons_f:cons_plus:cons_gt hole_true:s:0':false:cons_f:cons_plus:cons_gt1_5 :: true:s:0':false:cons_f:cons_plus:cons_gt gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5 :: Nat -> true:s:0':false:cons_f:cons_plus:cons_gt ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: f, gt, plus, encArg They will be analysed ascendingly in the following order: gt < f plus < f f < encArg gt < encArg plus < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0') -> n plus(n, s(m)) -> s(plus(n, m)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(false) -> false encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_false -> false Types: f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt true :: true:s:0':false:cons_f:cons_plus:cons_gt gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt 0' :: true:s:0':false:cons_f:cons_plus:cons_gt false :: true:s:0':false:cons_f:cons_plus:cons_gt encArg :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_true :: true:s:0':false:cons_f:cons_plus:cons_gt encode_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_0 :: true:s:0':false:cons_f:cons_plus:cons_gt encode_false :: true:s:0':false:cons_f:cons_plus:cons_gt hole_true:s:0':false:cons_f:cons_plus:cons_gt1_5 :: true:s:0':false:cons_f:cons_plus:cons_gt gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5 :: Nat -> true:s:0':false:cons_f:cons_plus:cons_gt Generator Equations: gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(0) <=> true gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(x, 1)) <=> s(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(x)) The following defined symbols remain to be analysed: gt, f, plus, encArg They will be analysed ascendingly in the following order: gt < f plus < f f < encArg gt < encArg plus < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: gt(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n4_5)), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n4_5))) -> *3_5, rt in Omega(n4_5) Induction Base: gt(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, 0)), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, 0))) Induction Step: gt(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, +(n4_5, 1))), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, +(n4_5, 1)))) ->_R^Omega(1) gt(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n4_5)), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n4_5))) ->_IH *3_5 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: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0') -> n plus(n, s(m)) -> s(plus(n, m)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(false) -> false encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_false -> false Types: f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt true :: true:s:0':false:cons_f:cons_plus:cons_gt gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt 0' :: true:s:0':false:cons_f:cons_plus:cons_gt false :: true:s:0':false:cons_f:cons_plus:cons_gt encArg :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_true :: true:s:0':false:cons_f:cons_plus:cons_gt encode_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_0 :: true:s:0':false:cons_f:cons_plus:cons_gt encode_false :: true:s:0':false:cons_f:cons_plus:cons_gt hole_true:s:0':false:cons_f:cons_plus:cons_gt1_5 :: true:s:0':false:cons_f:cons_plus:cons_gt gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5 :: Nat -> true:s:0':false:cons_f:cons_plus:cons_gt Generator Equations: gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(0) <=> true gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(x, 1)) <=> s(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(x)) The following defined symbols remain to be analysed: gt, f, plus, encArg They will be analysed ascendingly in the following order: gt < f plus < f f < encArg gt < encArg plus < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0') -> n plus(n, s(m)) -> s(plus(n, m)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(false) -> false encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_false -> false Types: f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt true :: true:s:0':false:cons_f:cons_plus:cons_gt gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt 0' :: true:s:0':false:cons_f:cons_plus:cons_gt false :: true:s:0':false:cons_f:cons_plus:cons_gt encArg :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_true :: true:s:0':false:cons_f:cons_plus:cons_gt encode_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_0 :: true:s:0':false:cons_f:cons_plus:cons_gt encode_false :: true:s:0':false:cons_f:cons_plus:cons_gt hole_true:s:0':false:cons_f:cons_plus:cons_gt1_5 :: true:s:0':false:cons_f:cons_plus:cons_gt gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5 :: Nat -> true:s:0':false:cons_f:cons_plus:cons_gt Lemmas: gt(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n4_5)), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n4_5))) -> *3_5, rt in Omega(n4_5) Generator Equations: gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(0) <=> true gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(x, 1)) <=> s(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(x)) The following defined symbols remain to be analysed: plus, f, encArg They will be analysed ascendingly in the following order: plus < f f < encArg plus < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(a), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n1182_5))) -> *3_5, rt in Omega(n1182_5) Induction Base: plus(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(a), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, 0))) Induction Step: plus(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(a), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, +(n1182_5, 1)))) ->_R^Omega(1) s(plus(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(a), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n1182_5)))) ->_IH s(*3_5) 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: f(true, x, y, z) -> f(gt(x, plus(y, z)), x, s(y), z) f(true, x, y, z) -> f(gt(x, plus(y, z)), x, y, s(z)) plus(n, 0') -> n plus(n, s(m)) -> s(plus(n, m)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) encArg(true) -> true encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(false) -> false encArg(cons_f(x_1, x_2, x_3, x_4)) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2, x_3, x_4) -> f(encArg(x_1), encArg(x_2), encArg(x_3), encArg(x_4)) encode_true -> true encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' encode_false -> false Types: f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt true :: true:s:0':false:cons_f:cons_plus:cons_gt gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt 0' :: true:s:0':false:cons_f:cons_plus:cons_gt false :: true:s:0':false:cons_f:cons_plus:cons_gt encArg :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt cons_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_f :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_true :: true:s:0':false:cons_f:cons_plus:cons_gt encode_gt :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_plus :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_s :: true:s:0':false:cons_f:cons_plus:cons_gt -> true:s:0':false:cons_f:cons_plus:cons_gt encode_0 :: true:s:0':false:cons_f:cons_plus:cons_gt encode_false :: true:s:0':false:cons_f:cons_plus:cons_gt hole_true:s:0':false:cons_f:cons_plus:cons_gt1_5 :: true:s:0':false:cons_f:cons_plus:cons_gt gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5 :: Nat -> true:s:0':false:cons_f:cons_plus:cons_gt Lemmas: gt(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n4_5)), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n4_5))) -> *3_5, rt in Omega(n4_5) plus(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(a), gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(1, n1182_5))) -> *3_5, rt in Omega(n1182_5) Generator Equations: gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(0) <=> true gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(x, 1)) <=> s(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(x)) The following defined symbols remain to be analysed: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(n3351_5)) -> gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(n3351_5), rt in Omega(0) Induction Base: encArg(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(0)) ->_R^Omega(0) true Induction Step: encArg(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(+(n3351_5, 1))) ->_R^Omega(0) s(encArg(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(n3351_5))) ->_IH s(gen_true:s:0':false:cons_f:cons_plus:cons_gt2_5(c3352_5)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (20) BOUNDS(1, INF)