/export/starexec/sandbox/solver/bin/starexec_run_complexity /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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) typed CpxTrs (5) OrderProof [LOWER BOUND(ID), 0 ms] (6) typed CpxTrs (7) RewriteLemmaProof [LOWER BOUND(ID), 487 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 94 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 71 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 116 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 61 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 98 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 48 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 228 ms] (26) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0, IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0) -> ok(0) proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok ---------------------------------------- (5) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: active, cons, s, length, take, isNatList, isNat, and, isNatIList, U11, U21, U31, proper, top They will be analysed ascendingly in the following order: cons < active s < active length < active take < active isNatList < active isNat < active and < active isNatIList < active U11 < active U21 < active U31 < active active < top cons < proper s < proper length < proper take < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper U21 < proper U31 < proper proper < top ---------------------------------------- (6) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: cons, active, s, length, take, isNatList, isNat, and, isNatIList, U11, U21, U31, proper, top They will be analysed ascendingly in the following order: cons < active s < active length < active take < active isNatList < active isNat < active and < active isNatIList < active U11 < active U21 < active U31 < active active < top cons < proper s < proper length < proper take < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper U21 < proper U31 < proper proper < top ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) Induction Base: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n5_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: cons, active, s, length, take, isNatList, isNat, and, isNatIList, U11, U21, U31, proper, top They will be analysed ascendingly in the following order: cons < active s < active length < active take < active isNatList < active isNat < active and < active isNatIList < active U11 < active U21 < active U31 < active active < top cons < proper s < proper length < proper take < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper U21 < proper U31 < proper proper < top ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: s, active, length, take, isNatList, isNat, and, isNatIList, U11, U21, U31, proper, top They will be analysed ascendingly in the following order: s < active length < active take < active isNatList < active isNat < active and < active isNatIList < active U11 < active U21 < active U31 < active active < top s < proper length < proper take < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper U21 < proper U31 < proper proper < top ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1336_0))) -> *4_0, rt in Omega(n1336_0) Induction Base: s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n1336_0, 1)))) ->_R^Omega(1) mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1336_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1336_0))) -> *4_0, rt in Omega(n1336_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: length, active, take, isNatList, isNat, and, isNatIList, U11, U21, U31, proper, top They will be analysed ascendingly in the following order: length < active take < active isNatList < active isNat < active and < active isNatIList < active U11 < active U21 < active U31 < active active < top length < proper take < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper U21 < proper U31 < proper proper < top ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1945_0))) -> *4_0, rt in Omega(n1945_0) Induction Base: length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n1945_0, 1)))) ->_R^Omega(1) mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1945_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1336_0))) -> *4_0, rt in Omega(n1336_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1945_0))) -> *4_0, rt in Omega(n1945_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: take, active, isNatList, isNat, and, isNatIList, U11, U21, U31, proper, top They will be analysed ascendingly in the following order: take < active isNatList < active isNat < active and < active isNatIList < active U11 < active U21 < active U31 < active active < top take < proper isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper U21 < proper U31 < proper proper < top ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2655_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n2655_0) Induction Base: take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2655_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2655_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1336_0))) -> *4_0, rt in Omega(n1336_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1945_0))) -> *4_0, rt in Omega(n1945_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2655_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n2655_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: isNatList, active, isNat, and, isNatIList, U11, U21, U31, proper, top They will be analysed ascendingly in the following order: isNatList < active isNat < active and < active isNatIList < active U11 < active U21 < active U31 < active active < top isNatList < proper isNat < proper and < proper isNatIList < proper U11 < proper U21 < proper U31 < proper proper < top ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4947_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4947_0) Induction Base: and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n4947_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4947_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1336_0))) -> *4_0, rt in Omega(n1336_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1945_0))) -> *4_0, rt in Omega(n1945_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2655_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n2655_0) and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4947_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4947_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: isNatIList, active, U11, U21, U31, proper, top They will be analysed ascendingly in the following order: isNatIList < active U11 < active U21 < active U31 < active active < top isNatIList < proper U11 < proper U21 < proper U31 < proper proper < top ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7331_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n7331_0) Induction Base: U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) Induction Step: U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n7331_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1) mark(U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7331_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1336_0))) -> *4_0, rt in Omega(n1336_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1945_0))) -> *4_0, rt in Omega(n1945_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2655_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n2655_0) and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4947_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4947_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7331_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n7331_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U21, active, U31, proper, top They will be analysed ascendingly in the following order: U21 < active U31 < active active < top U21 < proper U31 < proper proper < top ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9996_0))) -> *4_0, rt in Omega(n9996_0) Induction Base: U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0))) Induction Step: U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n9996_0, 1)))) ->_R^Omega(1) mark(U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9996_0)))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (24) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1336_0))) -> *4_0, rt in Omega(n1336_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1945_0))) -> *4_0, rt in Omega(n1945_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2655_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n2655_0) and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4947_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4947_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7331_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n7331_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9996_0))) -> *4_0, rt in Omega(n9996_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: U31, active, proper, top They will be analysed ascendingly in the following order: U31 < active active < top U31 < proper proper < top ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n11257_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n11257_0) Induction Base: U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) Induction Step: U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n11257_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) ->_R^Omega(1) mark(U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n11257_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))) ->_IH mark(*4_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (26) Obligation: TRS: Rules: active(zeros) -> mark(cons(0', zeros)) active(U11(tt, L)) -> mark(s(length(L))) active(U21(tt)) -> mark(nil) active(U31(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(and(tt, X)) -> mark(X) active(isNat(0')) -> mark(tt) active(isNat(length(V1))) -> mark(isNatList(V1)) active(isNat(s(V1))) -> mark(isNat(V1)) active(isNatIList(V)) -> mark(isNatList(V)) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2))) active(isNatList(take(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2))) active(length(nil)) -> mark(0') active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L)) active(take(0', IL)) -> mark(U21(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U31(and(isNatIList(IL), and(isNat(M), isNat(N))), IL, M, N)) active(cons(X1, X2)) -> cons(active(X1), X2) active(U11(X1, X2)) -> U11(active(X1), X2) active(s(X)) -> s(active(X)) active(length(X)) -> length(active(X)) active(U21(X)) -> U21(active(X)) active(U31(X1, X2, X3, X4)) -> U31(active(X1), X2, X3, X4) active(take(X1, X2)) -> take(active(X1), X2) active(take(X1, X2)) -> take(X1, active(X2)) active(and(X1, X2)) -> and(active(X1), X2) cons(mark(X1), X2) -> mark(cons(X1, X2)) U11(mark(X1), X2) -> mark(U11(X1, X2)) s(mark(X)) -> mark(s(X)) length(mark(X)) -> mark(length(X)) U21(mark(X)) -> mark(U21(X)) U31(mark(X1), X2, X3, X4) -> mark(U31(X1, X2, X3, X4)) take(mark(X1), X2) -> mark(take(X1, X2)) take(X1, mark(X2)) -> mark(take(X1, X2)) and(mark(X1), X2) -> mark(and(X1, X2)) proper(zeros) -> ok(zeros) proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) proper(0') -> ok(0') proper(U11(X1, X2)) -> U11(proper(X1), proper(X2)) proper(tt) -> ok(tt) proper(s(X)) -> s(proper(X)) proper(length(X)) -> length(proper(X)) proper(U21(X)) -> U21(proper(X)) proper(nil) -> ok(nil) proper(U31(X1, X2, X3, X4)) -> U31(proper(X1), proper(X2), proper(X3), proper(X4)) proper(take(X1, X2)) -> take(proper(X1), proper(X2)) proper(and(X1, X2)) -> and(proper(X1), proper(X2)) proper(isNat(X)) -> isNat(proper(X)) proper(isNatList(X)) -> isNatList(proper(X)) proper(isNatIList(X)) -> isNatIList(proper(X)) cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) U11(ok(X1), ok(X2)) -> ok(U11(X1, X2)) s(ok(X)) -> ok(s(X)) length(ok(X)) -> ok(length(X)) U21(ok(X)) -> ok(U21(X)) U31(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U31(X1, X2, X3, X4)) take(ok(X1), ok(X2)) -> ok(take(X1, X2)) and(ok(X1), ok(X2)) -> ok(and(X1, X2)) isNat(ok(X)) -> ok(isNat(X)) isNatList(ok(X)) -> ok(isNatList(X)) isNatIList(ok(X)) -> ok(isNatIList(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) Types: active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok zeros :: zeros:0':mark:tt:nil:ok mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok 0' :: zeros:0':mark:tt:nil:ok U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok tt :: zeros:0':mark:tt:nil:ok s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok nil :: zeros:0':mark:tt:nil:ok U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok top :: zeros:0':mark:tt:nil:ok -> top hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok hole_top2_0 :: top gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok Lemmas: cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0) s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1336_0))) -> *4_0, rt in Omega(n1336_0) length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1945_0))) -> *4_0, rt in Omega(n1945_0) take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2655_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n2655_0) and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4947_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4947_0) U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7331_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n7331_0) U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9996_0))) -> *4_0, rt in Omega(n9996_0) U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n11257_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n11257_0) Generator Equations: gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x)) The following defined symbols remain to be analysed: active, proper, top They will be analysed ascendingly in the following order: active < top proper < top