/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 258 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), 451 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), 408 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: active(f(b, c, x)) -> mark(f(x, x, x)) active(f(x, y, z)) -> f(x, y, active(z)) active(d) -> m(b) f(x, y, mark(z)) -> mark(f(x, y, z)) active(d) -> mark(c) proper(b) -> ok(b) proper(c) -> ok(c) proper(d) -> ok(d) proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z)) f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z)) top(mark(x)) -> top(proper(x)) top(ok(x)) -> top(active(x)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(b) -> b encArg(c) -> c encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(d) -> d encArg(m(x_1)) -> m(encArg(x_1)) encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_m(x_1) -> m(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(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: active(f(b, c, x)) -> mark(f(x, x, x)) active(f(x, y, z)) -> f(x, y, active(z)) active(d) -> m(b) f(x, y, mark(z)) -> mark(f(x, y, z)) active(d) -> mark(c) proper(b) -> ok(b) proper(c) -> ok(c) proper(d) -> ok(d) proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z)) f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z)) top(mark(x)) -> top(proper(x)) top(ok(x)) -> top(active(x)) The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(c) -> c encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(d) -> d encArg(m(x_1)) -> m(encArg(x_1)) encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_m(x_1) -> m(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(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: active(f(b, c, x)) -> mark(f(x, x, x)) active(f(x, y, z)) -> f(x, y, active(z)) active(d) -> m(b) f(x, y, mark(z)) -> mark(f(x, y, z)) active(d) -> mark(c) proper(b) -> ok(b) proper(c) -> ok(c) proper(d) -> ok(d) proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z)) f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z)) top(mark(x)) -> top(proper(x)) top(ok(x)) -> top(active(x)) The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(c) -> c encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(d) -> d encArg(m(x_1)) -> m(encArg(x_1)) encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_m(x_1) -> m(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(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: active(f(b, c, x)) -> mark(f(x, x, x)) active(f(x, y, z)) -> f(x, y, active(z)) active(d) -> m(b) f(x, y, mark(z)) -> mark(f(x, y, z)) active(d) -> mark(c) proper(b) -> ok(b) proper(c) -> ok(c) proper(d) -> ok(d) proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z)) f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z)) top(mark(x)) -> top(proper(x)) top(ok(x)) -> top(active(x)) The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(c) -> c encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(d) -> d encArg(m(x_1)) -> m(encArg(x_1)) encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_m(x_1) -> m(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: active(f(b, c, x)) -> mark(f(x, x, x)) active(f(x, y, z)) -> f(x, y, active(z)) active(d) -> m(b) f(x, y, mark(z)) -> mark(f(x, y, z)) active(d) -> mark(c) proper(b) -> ok(b) proper(c) -> ok(c) proper(d) -> ok(d) proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z)) f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z)) top(mark(x)) -> top(proper(x)) top(ok(x)) -> top(active(x)) encArg(b) -> b encArg(c) -> c encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(d) -> d encArg(m(x_1)) -> m(encArg(x_1)) encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_m(x_1) -> m(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Types: active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top b :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top c :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top mark :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top d :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top m :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top ok :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encArg :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_b :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_c :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_mark :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_d :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_m :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_ok :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top hole_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top1_0 :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0 :: Nat -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, f, proper, top, encArg They will be analysed ascendingly in the following order: f < active active < top active < encArg f < proper f < encArg proper < top proper < encArg top < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: active(f(b, c, x)) -> mark(f(x, x, x)) active(f(x, y, z)) -> f(x, y, active(z)) active(d) -> m(b) f(x, y, mark(z)) -> mark(f(x, y, z)) active(d) -> mark(c) proper(b) -> ok(b) proper(c) -> ok(c) proper(d) -> ok(d) proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z)) f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z)) top(mark(x)) -> top(proper(x)) top(ok(x)) -> top(active(x)) encArg(b) -> b encArg(c) -> c encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(d) -> d encArg(m(x_1)) -> m(encArg(x_1)) encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_m(x_1) -> m(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Types: active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top b :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top c :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top mark :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top d :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top m :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top ok :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encArg :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_b :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_c :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_mark :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_d :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_m :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_ok :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top hole_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top1_0 :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0 :: Nat -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top Generator Equations: gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(0) <=> b gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(+(x, 1)) <=> mark(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(x)) The following defined symbols remain to be analysed: f, active, proper, top, encArg They will be analysed ascendingly in the following order: f < active active < top active < encArg f < proper f < encArg proper < top proper < encArg top < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: f(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(a), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(b), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Induction Base: f(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(a), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(b), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(+(1, 0))) Induction Step: f(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(a), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(b), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) mark(f(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(a), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(b), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(+(1, n4_0)))) ->_IH mark(*3_0) 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: active(f(b, c, x)) -> mark(f(x, x, x)) active(f(x, y, z)) -> f(x, y, active(z)) active(d) -> m(b) f(x, y, mark(z)) -> mark(f(x, y, z)) active(d) -> mark(c) proper(b) -> ok(b) proper(c) -> ok(c) proper(d) -> ok(d) proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z)) f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z)) top(mark(x)) -> top(proper(x)) top(ok(x)) -> top(active(x)) encArg(b) -> b encArg(c) -> c encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(d) -> d encArg(m(x_1)) -> m(encArg(x_1)) encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_m(x_1) -> m(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Types: active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top b :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top c :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top mark :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top d :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top m :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top ok :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encArg :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_b :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_c :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_mark :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_d :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_m :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_ok :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top hole_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top1_0 :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0 :: Nat -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top Generator Equations: gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(0) <=> b gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(+(x, 1)) <=> mark(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(x)) The following defined symbols remain to be analysed: f, active, proper, top, encArg They will be analysed ascendingly in the following order: f < active active < top active < encArg f < proper f < encArg proper < top proper < encArg top < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: active(f(b, c, x)) -> mark(f(x, x, x)) active(f(x, y, z)) -> f(x, y, active(z)) active(d) -> m(b) f(x, y, mark(z)) -> mark(f(x, y, z)) active(d) -> mark(c) proper(b) -> ok(b) proper(c) -> ok(c) proper(d) -> ok(d) proper(f(x, y, z)) -> f(proper(x), proper(y), proper(z)) f(ok(x), ok(y), ok(z)) -> ok(f(x, y, z)) top(mark(x)) -> top(proper(x)) top(ok(x)) -> top(active(x)) encArg(b) -> b encArg(c) -> c encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(d) -> d encArg(m(x_1)) -> m(encArg(x_1)) encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_proper(x_1)) -> proper(encArg(x_1)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encode_active(x_1) -> active(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_mark(x_1) -> mark(encArg(x_1)) encode_d -> d encode_m(x_1) -> m(encArg(x_1)) encode_proper(x_1) -> proper(encArg(x_1)) encode_ok(x_1) -> ok(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) Types: active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top b :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top c :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top mark :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top d :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top m :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top ok :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encArg :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top cons_top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_active :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_f :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_b :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_c :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_mark :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_d :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_m :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_proper :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_ok :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top encode_top :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top hole_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top1_0 :: b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0 :: Nat -> b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top Lemmas: f(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(a), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(b), gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(0) <=> b gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(+(x, 1)) <=> mark(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(x)) The following defined symbols remain to be analysed: active, proper, top, encArg They will be analysed ascendingly in the following order: active < top active < encArg proper < top proper < encArg top < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(n1874_0)) -> gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(n1874_0), rt in Omega(0) Induction Base: encArg(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(0)) ->_R^Omega(0) b Induction Step: encArg(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(+(n1874_0, 1))) ->_R^Omega(0) mark(encArg(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(n1874_0))) ->_IH mark(gen_b:c:mark:d:m:ok:cons_active:cons_f:cons_proper:cons_top2_0(c1875_0)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)