/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 183 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), 501 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), 94 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: f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) and(x, true) -> x and(x, false) -> false 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_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(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(and(gt(x, y), gt(x, z)), x, s(y), z) f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) and(x, true) -> x and(x, false) -> false 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_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(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(and(gt(x, y), gt(x, z)), x, s(y), z) f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) and(x, true) -> x and(x, false) -> false 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_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(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(and(gt(x, y), gt(x, z)), x, s(y), z) f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) and(x, true) -> x and(x, false) -> false 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_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(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(and(gt(x, y), gt(x, z)), x, s(y), z) f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) and(x, true) -> x and(x, false) -> false 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_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(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_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and true :: true:s:0':false:cons_f:cons_gt:cons_and and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and s :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and 0' :: true:s:0':false:cons_f:cons_gt:cons_and false :: true:s:0':false:cons_f:cons_gt:cons_and encArg :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_f :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_f :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_true :: true:s:0':false:cons_f:cons_gt:cons_and encode_and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_s :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_0 :: true:s:0':false:cons_f:cons_gt:cons_and encode_false :: true:s:0':false:cons_f:cons_gt:cons_and hole_true:s:0':false:cons_f:cons_gt:cons_and1_5 :: true:s:0':false:cons_f:cons_gt:cons_and gen_true:s:0':false:cons_f:cons_gt:cons_and2_5 :: Nat -> true:s:0':false:cons_f:cons_gt:cons_and ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: f, gt, encArg They will be analysed ascendingly in the following order: gt < f f < encArg gt < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) and(x, true) -> x and(x, false) -> false 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_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(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_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and true :: true:s:0':false:cons_f:cons_gt:cons_and and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and s :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and 0' :: true:s:0':false:cons_f:cons_gt:cons_and false :: true:s:0':false:cons_f:cons_gt:cons_and encArg :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_f :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_f :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_true :: true:s:0':false:cons_f:cons_gt:cons_and encode_and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_s :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_0 :: true:s:0':false:cons_f:cons_gt:cons_and encode_false :: true:s:0':false:cons_f:cons_gt:cons_and hole_true:s:0':false:cons_f:cons_gt:cons_and1_5 :: true:s:0':false:cons_f:cons_gt:cons_and gen_true:s:0':false:cons_f:cons_gt:cons_and2_5 :: Nat -> true:s:0':false:cons_f:cons_gt:cons_and Generator Equations: gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(0) <=> true gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(x, 1)) <=> s(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(x)) The following defined symbols remain to be analysed: gt, f, encArg They will be analysed ascendingly in the following order: gt < f f < encArg gt < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: gt(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(1, n4_5)), gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(1, n4_5))) -> *3_5, rt in Omega(n4_5) Induction Base: gt(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(1, 0)), gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(1, 0))) Induction Step: gt(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(1, +(n4_5, 1))), gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(1, +(n4_5, 1)))) ->_R^Omega(1) gt(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(1, n4_5)), gen_true:s:0':false:cons_f:cons_gt:cons_and2_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(and(gt(x, y), gt(x, z)), x, s(y), z) f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) and(x, true) -> x and(x, false) -> false 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_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(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_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and true :: true:s:0':false:cons_f:cons_gt:cons_and and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and s :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and 0' :: true:s:0':false:cons_f:cons_gt:cons_and false :: true:s:0':false:cons_f:cons_gt:cons_and encArg :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_f :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_f :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_true :: true:s:0':false:cons_f:cons_gt:cons_and encode_and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_s :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_0 :: true:s:0':false:cons_f:cons_gt:cons_and encode_false :: true:s:0':false:cons_f:cons_gt:cons_and hole_true:s:0':false:cons_f:cons_gt:cons_and1_5 :: true:s:0':false:cons_f:cons_gt:cons_and gen_true:s:0':false:cons_f:cons_gt:cons_and2_5 :: Nat -> true:s:0':false:cons_f:cons_gt:cons_and Generator Equations: gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(0) <=> true gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(x, 1)) <=> s(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(x)) The following defined symbols remain to be analysed: gt, f, encArg They will be analysed ascendingly in the following order: gt < f f < encArg gt < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, s(y), z) f(true, x, y, z) -> f(and(gt(x, y), gt(x, z)), x, y, s(z)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) and(x, true) -> x and(x, false) -> false 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_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_and(x_1, x_2)) -> and(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_and(x_1, x_2) -> and(encArg(x_1), encArg(x_2)) encode_gt(x_1, x_2) -> gt(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_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and true :: true:s:0':false:cons_f:cons_gt:cons_and and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and s :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and 0' :: true:s:0':false:cons_f:cons_gt:cons_and false :: true:s:0':false:cons_f:cons_gt:cons_and encArg :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_f :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and cons_and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_f :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_true :: true:s:0':false:cons_f:cons_gt:cons_and encode_and :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_gt :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_s :: true:s:0':false:cons_f:cons_gt:cons_and -> true:s:0':false:cons_f:cons_gt:cons_and encode_0 :: true:s:0':false:cons_f:cons_gt:cons_and encode_false :: true:s:0':false:cons_f:cons_gt:cons_and hole_true:s:0':false:cons_f:cons_gt:cons_and1_5 :: true:s:0':false:cons_f:cons_gt:cons_and gen_true:s:0':false:cons_f:cons_gt:cons_and2_5 :: Nat -> true:s:0':false:cons_f:cons_gt:cons_and Lemmas: gt(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(1, n4_5)), gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(1, n4_5))) -> *3_5, rt in Omega(n4_5) Generator Equations: gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(0) <=> true gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(x, 1)) <=> s(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(x)) The following defined symbols remain to be analysed: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(n2333_5)) -> gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(n2333_5), rt in Omega(0) Induction Base: encArg(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(0)) ->_R^Omega(0) true Induction Step: encArg(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(+(n2333_5, 1))) ->_R^Omega(0) s(encArg(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(n2333_5))) ->_IH s(gen_true:s:0':false:cons_f:cons_gt:cons_and2_5(c2334_5)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)