/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 338 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), 527 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), 116 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 305 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: diff(x, y) -> cond1(equal(x, y), x, y) cond1(true, x, y) -> 0 cond1(false, x, y) -> cond2(gt(x, y), x, y) cond2(true, x, y) -> s(diff(x, s(y))) cond2(false, x, y) -> s(diff(s(x), y)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, 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(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_true -> true encode_0 -> 0 encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) ---------------------------------------- (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: diff(x, y) -> cond1(equal(x, y), x, y) cond1(true, x, y) -> 0 cond1(false, x, y) -> cond2(gt(x, y), x, y) cond2(true, x, y) -> s(diff(x, s(y))) cond2(false, x, y) -> s(diff(s(x), y)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_true -> true encode_0 -> 0 encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) 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: diff(x, y) -> cond1(equal(x, y), x, y) cond1(true, x, y) -> 0 cond1(false, x, y) -> cond2(gt(x, y), x, y) cond2(true, x, y) -> s(diff(x, s(y))) cond2(false, x, y) -> s(diff(s(x), y)) gt(0, v) -> false gt(s(u), 0) -> true gt(s(u), s(v)) -> gt(u, v) equal(0, 0) -> true equal(s(x), 0) -> false equal(0, s(y)) -> false equal(s(x), s(y)) -> equal(x, y) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0) -> 0 encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_true -> true encode_0 -> 0 encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) 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: diff(x, y) -> cond1(equal(x, y), x, y) cond1(true, x, y) -> 0' cond1(false, x, y) -> cond2(gt(x, y), x, y) cond2(true, x, y) -> s(diff(x, s(y))) cond2(false, x, y) -> s(diff(s(x), y)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) equal(0', 0') -> true equal(s(x), 0') -> false equal(0', s(y)) -> false equal(s(x), s(y)) -> equal(x, y) The (relative) TRS S consists of the following rules: encArg(true) -> true encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_true -> true encode_0 -> 0' encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: diff(x, y) -> cond1(equal(x, y), x, y) cond1(true, x, y) -> 0' cond1(false, x, y) -> cond2(gt(x, y), x, y) cond2(true, x, y) -> s(diff(x, s(y))) cond2(false, x, y) -> s(diff(s(x), y)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) equal(0', 0') -> true equal(s(x), 0') -> false equal(0', s(y)) -> false equal(s(x), s(y)) -> equal(x, y) encArg(true) -> true encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_true -> true encode_0 -> 0' encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) Types: diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal 0' :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encArg :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_0 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal hole_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal1_4 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4 :: Nat -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: diff, cond1, equal, cond2, gt, encArg They will be analysed ascendingly in the following order: diff = cond1 equal < diff diff = cond2 diff < encArg cond1 = cond2 gt < cond1 cond1 < encArg equal < encArg cond2 < encArg gt < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: diff(x, y) -> cond1(equal(x, y), x, y) cond1(true, x, y) -> 0' cond1(false, x, y) -> cond2(gt(x, y), x, y) cond2(true, x, y) -> s(diff(x, s(y))) cond2(false, x, y) -> s(diff(s(x), y)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) equal(0', 0') -> true equal(s(x), 0') -> false equal(0', s(y)) -> false equal(s(x), s(y)) -> equal(x, y) encArg(true) -> true encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_true -> true encode_0 -> 0' encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) Types: diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal 0' :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encArg :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_0 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal hole_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal1_4 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4 :: Nat -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal Generator Equations: gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(0) <=> true gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(x, 1)) <=> s(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(x)) The following defined symbols remain to be analysed: equal, diff, cond1, cond2, gt, encArg They will be analysed ascendingly in the following order: diff = cond1 equal < diff diff = cond2 diff < encArg cond1 = cond2 gt < cond1 cond1 < encArg equal < encArg cond2 < encArg gt < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: equal(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n4_4)), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4) Induction Base: equal(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, 0)), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, 0))) Induction Step: equal(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, +(n4_4, 1))), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, +(n4_4, 1)))) ->_R^Omega(1) equal(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n4_4)), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n4_4))) ->_IH *3_4 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: diff(x, y) -> cond1(equal(x, y), x, y) cond1(true, x, y) -> 0' cond1(false, x, y) -> cond2(gt(x, y), x, y) cond2(true, x, y) -> s(diff(x, s(y))) cond2(false, x, y) -> s(diff(s(x), y)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) equal(0', 0') -> true equal(s(x), 0') -> false equal(0', s(y)) -> false equal(s(x), s(y)) -> equal(x, y) encArg(true) -> true encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_true -> true encode_0 -> 0' encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) Types: diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal 0' :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encArg :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_0 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal hole_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal1_4 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4 :: Nat -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal Generator Equations: gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(0) <=> true gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(x, 1)) <=> s(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(x)) The following defined symbols remain to be analysed: equal, diff, cond1, cond2, gt, encArg They will be analysed ascendingly in the following order: diff = cond1 equal < diff diff = cond2 diff < encArg cond1 = cond2 gt < cond1 cond1 < encArg equal < encArg cond2 < encArg gt < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: diff(x, y) -> cond1(equal(x, y), x, y) cond1(true, x, y) -> 0' cond1(false, x, y) -> cond2(gt(x, y), x, y) cond2(true, x, y) -> s(diff(x, s(y))) cond2(false, x, y) -> s(diff(s(x), y)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) equal(0', 0') -> true equal(s(x), 0') -> false equal(0', s(y)) -> false equal(s(x), s(y)) -> equal(x, y) encArg(true) -> true encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_true -> true encode_0 -> 0' encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) Types: diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal 0' :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encArg :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_0 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal hole_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal1_4 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4 :: Nat -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal Lemmas: equal(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n4_4)), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4) Generator Equations: gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(0) <=> true gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(x, 1)) <=> s(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(x)) The following defined symbols remain to be analysed: gt, diff, cond1, cond2, encArg They will be analysed ascendingly in the following order: diff = cond1 diff = cond2 diff < encArg cond1 = cond2 gt < cond1 cond1 < encArg cond2 < encArg gt < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: gt(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n2323_4)), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n2323_4))) -> *3_4, rt in Omega(n2323_4) Induction Base: gt(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, 0)), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, 0))) Induction Step: gt(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, +(n2323_4, 1))), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, +(n2323_4, 1)))) ->_R^Omega(1) gt(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n2323_4)), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n2323_4))) ->_IH *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: diff(x, y) -> cond1(equal(x, y), x, y) cond1(true, x, y) -> 0' cond1(false, x, y) -> cond2(gt(x, y), x, y) cond2(true, x, y) -> s(diff(x, s(y))) cond2(false, x, y) -> s(diff(s(x), y)) gt(0', v) -> false gt(s(u), 0') -> true gt(s(u), s(v)) -> gt(u, v) equal(0', 0') -> true equal(s(x), 0') -> false equal(0', s(y)) -> false equal(s(x), s(y)) -> equal(x, y) encArg(true) -> true encArg(0') -> 0' encArg(false) -> false encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_diff(x_1, x_2)) -> diff(encArg(x_1), encArg(x_2)) encArg(cons_cond1(x_1, x_2, x_3)) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_cond2(x_1, x_2, x_3)) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_gt(x_1, x_2)) -> gt(encArg(x_1), encArg(x_2)) encArg(cons_equal(x_1, x_2)) -> equal(encArg(x_1), encArg(x_2)) encode_diff(x_1, x_2) -> diff(encArg(x_1), encArg(x_2)) encode_cond1(x_1, x_2, x_3) -> cond1(encArg(x_1), encArg(x_2), encArg(x_3)) encode_equal(x_1, x_2) -> equal(encArg(x_1), encArg(x_2)) encode_true -> true encode_0 -> 0' encode_false -> false encode_cond2(x_1, x_2, x_3) -> cond2(encArg(x_1), encArg(x_2), encArg(x_3)) encode_gt(x_1, x_2) -> gt(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) Types: diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal 0' :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encArg :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal cons_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_diff :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond1 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_equal :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_true :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_0 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_false :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_cond2 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_gt :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal encode_s :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal hole_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal1_4 :: true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4 :: Nat -> true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal Lemmas: equal(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n4_4)), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4) gt(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n2323_4)), gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(1, n2323_4))) -> *3_4, rt in Omega(n2323_4) Generator Equations: gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(0) <=> true gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(x, 1)) <=> s(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(x)) The following defined symbols remain to be analysed: cond1, diff, cond2, encArg They will be analysed ascendingly in the following order: diff = cond1 diff = cond2 diff < encArg cond1 = cond2 cond1 < encArg cond2 < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(n8410_4)) -> gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(n8410_4), rt in Omega(0) Induction Base: encArg(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(0)) ->_R^Omega(0) true Induction Step: encArg(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(+(n8410_4, 1))) ->_R^Omega(0) s(encArg(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(n8410_4))) ->_IH s(gen_true:0':false:s:cons_diff:cons_cond1:cons_cond2:cons_gt:cons_equal2_4(c8411_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)