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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 326 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), 416 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), 121 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 484 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: active(f(X)) -> mark(if(X, c, f(true))) active(if(true, X, Y)) -> mark(X) active(if(false, X, Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1, X2, X3)) -> if(active(X1), X2, X3) active(if(X1, X2, X3)) -> if(X1, active(X2), X3) f(mark(X)) -> mark(f(X)) if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) if(X1, mark(X2), X3) -> mark(if(X1, X2, X3)) proper(f(X)) -> f(proper(X)) proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) proper(c) -> ok(c) proper(true) -> ok(true) proper(false) -> ok(false) f(ok(X)) -> ok(f(X)) if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) 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(mark(x_1)) -> mark(encArg(x_1)) encArg(c) -> c encArg(true) -> true encArg(false) -> false encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(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) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_c -> c encode_true -> true encode_false -> false 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(X)) -> mark(if(X, c, f(true))) active(if(true, X, Y)) -> mark(X) active(if(false, X, Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1, X2, X3)) -> if(active(X1), X2, X3) active(if(X1, X2, X3)) -> if(X1, active(X2), X3) f(mark(X)) -> mark(f(X)) if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) if(X1, mark(X2), X3) -> mark(if(X1, X2, X3)) proper(f(X)) -> f(proper(X)) proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) proper(c) -> ok(c) proper(true) -> ok(true) proper(false) -> ok(false) f(ok(X)) -> ok(f(X)) if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(c) -> c encArg(true) -> true encArg(false) -> false encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(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) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_c -> c encode_true -> true encode_false -> false 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(X)) -> mark(if(X, c, f(true))) active(if(true, X, Y)) -> mark(X) active(if(false, X, Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1, X2, X3)) -> if(active(X1), X2, X3) active(if(X1, X2, X3)) -> if(X1, active(X2), X3) f(mark(X)) -> mark(f(X)) if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) if(X1, mark(X2), X3) -> mark(if(X1, X2, X3)) proper(f(X)) -> f(proper(X)) proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) proper(c) -> ok(c) proper(true) -> ok(true) proper(false) -> ok(false) f(ok(X)) -> ok(f(X)) if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(c) -> c encArg(true) -> true encArg(false) -> false encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(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) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_c -> c encode_true -> true encode_false -> false 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(X)) -> mark(if(X, c, f(true))) active(if(true, X, Y)) -> mark(X) active(if(false, X, Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1, X2, X3)) -> if(active(X1), X2, X3) active(if(X1, X2, X3)) -> if(X1, active(X2), X3) f(mark(X)) -> mark(f(X)) if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) if(X1, mark(X2), X3) -> mark(if(X1, X2, X3)) proper(f(X)) -> f(proper(X)) proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) proper(c) -> ok(c) proper(true) -> ok(true) proper(false) -> ok(false) f(ok(X)) -> ok(f(X)) if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) The (relative) TRS S consists of the following rules: encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(c) -> c encArg(true) -> true encArg(false) -> false encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(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) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_c -> c encode_true -> true encode_false -> false 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(X)) -> mark(if(X, c, f(true))) active(if(true, X, Y)) -> mark(X) active(if(false, X, Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1, X2, X3)) -> if(active(X1), X2, X3) active(if(X1, X2, X3)) -> if(X1, active(X2), X3) f(mark(X)) -> mark(f(X)) if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) if(X1, mark(X2), X3) -> mark(if(X1, X2, X3)) proper(f(X)) -> f(proper(X)) proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) proper(c) -> ok(c) proper(true) -> ok(true) proper(false) -> ok(false) f(ok(X)) -> ok(f(X)) if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(c) -> c encArg(true) -> true encArg(false) -> false encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(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) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_c -> c encode_true -> true encode_false -> false 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 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encArg :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top hole_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top1_0 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0 :: Nat -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, if, f, proper, top, encArg They will be analysed ascendingly in the following order: if < active f < active active < top active < encArg if < proper if < encArg f < proper f < encArg proper < top proper < encArg top < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: active(f(X)) -> mark(if(X, c, f(true))) active(if(true, X, Y)) -> mark(X) active(if(false, X, Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1, X2, X3)) -> if(active(X1), X2, X3) active(if(X1, X2, X3)) -> if(X1, active(X2), X3) f(mark(X)) -> mark(f(X)) if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) if(X1, mark(X2), X3) -> mark(if(X1, X2, X3)) proper(f(X)) -> f(proper(X)) proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) proper(c) -> ok(c) proper(true) -> ok(true) proper(false) -> ok(false) f(ok(X)) -> ok(f(X)) if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(c) -> c encArg(true) -> true encArg(false) -> false encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(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) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_c -> c encode_true -> true encode_false -> false 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 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encArg :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top hole_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top1_0 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0 :: Nat -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top Generator Equations: gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(0) <=> c gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(x, 1)) <=> mark(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(x)) The following defined symbols remain to be analysed: if, active, f, proper, top, encArg They will be analysed ascendingly in the following order: if < active f < active active < top active < encArg if < proper if < encArg f < proper f < encArg proper < top proper < encArg top < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: if(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, n4_0)), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(b), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(c)) -> *3_0, rt in Omega(n4_0) Induction Base: if(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, 0)), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(b), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(c)) Induction Step: if(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, +(n4_0, 1))), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(b), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(c)) ->_R^Omega(1) mark(if(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, n4_0)), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(b), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(c))) ->_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(X)) -> mark(if(X, c, f(true))) active(if(true, X, Y)) -> mark(X) active(if(false, X, Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1, X2, X3)) -> if(active(X1), X2, X3) active(if(X1, X2, X3)) -> if(X1, active(X2), X3) f(mark(X)) -> mark(f(X)) if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) if(X1, mark(X2), X3) -> mark(if(X1, X2, X3)) proper(f(X)) -> f(proper(X)) proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) proper(c) -> ok(c) proper(true) -> ok(true) proper(false) -> ok(false) f(ok(X)) -> ok(f(X)) if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(c) -> c encArg(true) -> true encArg(false) -> false encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(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) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_c -> c encode_true -> true encode_false -> false 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 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encArg :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top hole_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top1_0 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0 :: Nat -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top Generator Equations: gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(0) <=> c gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(x, 1)) <=> mark(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(x)) The following defined symbols remain to be analysed: if, active, f, proper, top, encArg They will be analysed ascendingly in the following order: if < active f < active active < top active < encArg if < proper if < 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(X)) -> mark(if(X, c, f(true))) active(if(true, X, Y)) -> mark(X) active(if(false, X, Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1, X2, X3)) -> if(active(X1), X2, X3) active(if(X1, X2, X3)) -> if(X1, active(X2), X3) f(mark(X)) -> mark(f(X)) if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) if(X1, mark(X2), X3) -> mark(if(X1, X2, X3)) proper(f(X)) -> f(proper(X)) proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) proper(c) -> ok(c) proper(true) -> ok(true) proper(false) -> ok(false) f(ok(X)) -> ok(f(X)) if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(c) -> c encArg(true) -> true encArg(false) -> false encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(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) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_c -> c encode_true -> true encode_false -> false 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 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encArg :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top hole_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top1_0 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0 :: Nat -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top Lemmas: if(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, n4_0)), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(b), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(c)) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(0) <=> c gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(x, 1)) <=> mark(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if: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 ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: f(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, n2042_0))) -> *3_0, rt in Omega(n2042_0) Induction Base: f(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, 0))) Induction Step: f(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, +(n2042_0, 1)))) ->_R^Omega(1) mark(f(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, n2042_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). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: active(f(X)) -> mark(if(X, c, f(true))) active(if(true, X, Y)) -> mark(X) active(if(false, X, Y)) -> mark(Y) active(f(X)) -> f(active(X)) active(if(X1, X2, X3)) -> if(active(X1), X2, X3) active(if(X1, X2, X3)) -> if(X1, active(X2), X3) f(mark(X)) -> mark(f(X)) if(mark(X1), X2, X3) -> mark(if(X1, X2, X3)) if(X1, mark(X2), X3) -> mark(if(X1, X2, X3)) proper(f(X)) -> f(proper(X)) proper(if(X1, X2, X3)) -> if(proper(X1), proper(X2), proper(X3)) proper(c) -> ok(c) proper(true) -> ok(true) proper(false) -> ok(false) f(ok(X)) -> ok(f(X)) if(ok(X1), ok(X2), ok(X3)) -> ok(if(X1, X2, X3)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) encArg(mark(x_1)) -> mark(encArg(x_1)) encArg(c) -> c encArg(true) -> true encArg(false) -> false encArg(ok(x_1)) -> ok(encArg(x_1)) encArg(cons_active(x_1)) -> active(encArg(x_1)) encArg(cons_f(x_1)) -> f(encArg(x_1)) encArg(cons_if(x_1, x_2, x_3)) -> if(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) -> f(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_if(x_1, x_2, x_3) -> if(encArg(x_1), encArg(x_2), encArg(x_3)) encode_c -> c encode_true -> true encode_false -> false 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 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encArg :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top cons_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_active :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_f :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_mark :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_if :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_c :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_true :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_false :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_proper :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_ok :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top encode_top :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top hole_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top1_0 :: c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0 :: Nat -> c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top Lemmas: if(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, n4_0)), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(b), gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(c)) -> *3_0, rt in Omega(n4_0) f(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(1, n2042_0))) -> *3_0, rt in Omega(n2042_0) Generator Equations: gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(0) <=> c gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(x, 1)) <=> mark(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if: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 ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(n3115_0)) -> gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(n3115_0), rt in Omega(0) Induction Base: encArg(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(0)) ->_R^Omega(0) c Induction Step: encArg(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(+(n3115_0, 1))) ->_R^Omega(0) mark(encArg(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(n3115_0))) ->_IH mark(gen_c:true:mark:false:ok:cons_active:cons_f:cons_if:cons_proper:cons_top2_0(c3116_0)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (20) BOUNDS(1, INF)