/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 509 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), 146 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 96 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 85 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 174 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 202 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 140 ms]
        (24) typed CpxTrs
        (25) RewriteLemmaProof [LOWER BOUND(ID), 140 ms]
        (26) typed CpxTrs
        (27) RewriteLemmaProof [LOWER BOUND(ID), 211 ms]
        (28) 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(U12(tt, L))
   active(U12(tt, L)) -> mark(s(length(L)))
   active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
   active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
   active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(tt, L))
   active(take(0, IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
   active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
   active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
   U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
   U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
   U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
   U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   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(U12(tt, L))
   active(U12(tt, L)) -> mark(s(length(L)))
   active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
   active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
   active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(length(nil)) -> mark(0')
   active(length(cons(N, L))) -> mark(U11(tt, L))
   active(take(0', IL)) -> mark(nil)
   active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
   active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
   active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
   U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
   U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
   U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
   U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   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(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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, U12, s, length, U22, U23, take, U11, U21, proper, top

They will be analysed ascendingly in the following order:
cons < active
U12 < active
s < active
length < active
U22 < active
U23 < active
take < active
U11 < active
U21 < active
active < top
cons < proper
U12 < proper
s < proper
length < proper
U22 < proper
U23 < proper
take < proper
U11 < proper
U21 < proper
proper < top

----------------------------------------

(6)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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, U12, s, length, U22, U23, take, U11, U21, proper, top

They will be analysed ascendingly in the following order:
cons < active
U12 < active
s < active
length < active
U22 < active
U23 < active
take < active
U11 < active
U21 < active
active < top
cons < proper
U12 < proper
s < proper
length < proper
U22 < proper
U23 < proper
take < proper
U11 < proper
U21 < 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(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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, U12, s, length, U22, U23, take, U11, U21, proper, top

They will be analysed ascendingly in the following order:
cons < active
U12 < active
s < active
length < active
U22 < active
U23 < active
take < active
U11 < active
U21 < active
active < top
cons < proper
U12 < proper
s < proper
length < proper
U22 < proper
U23 < proper
take < proper
U11 < proper
U21 < 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(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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:
U12, active, s, length, U22, U23, take, U11, U21, proper, top

They will be analysed ascendingly in the following order:
U12 < active
s < active
length < active
U22 < active
U23 < active
take < active
U11 < active
U21 < active
active < top
U12 < proper
s < proper
length < proper
U22 < proper
U23 < proper
take < proper
U11 < proper
U21 < proper
proper < top

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)

Induction Base:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n1196_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_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).
----------------------------------------

(14)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_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, U22, U23, take, U11, U21, proper, top

They will be analysed ascendingly in the following order:
s < active
length < active
U22 < active
U23 < active
take < active
U11 < active
U21 < active
active < top
s < proper
length < proper
U22 < proper
U23 < proper
take < proper
U11 < proper
U21 < proper
proper < top

----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_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, +(n2693_0, 1)))) ->_R^Omega(1)
mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_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(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_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, U22, U23, take, U11, U21, proper, top

They will be analysed ascendingly in the following order:
length < active
U22 < active
U23 < active
take < active
U11 < active
U21 < active
active < top
length < proper
U22 < proper
U23 < proper
take < proper
U11 < proper
U21 < proper
proper < top

----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_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, +(n3422_0, 1)))) ->_R^Omega(1)
mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_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).
----------------------------------------

(18)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_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:
U22, active, U23, take, U11, U21, proper, top

They will be analysed ascendingly in the following order:
U22 < active
U23 < active
take < active
U11 < active
U21 < active
active < top
U22 < proper
U23 < proper
take < proper
U11 < proper
U21 < proper
proper < top

----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)

Induction Base:
U22(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:
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n4252_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(U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_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:
U23, active, take, U11, U21, proper, top

They will be analysed ascendingly in the following order:
U23 < active
take < active
U11 < active
U21 < active
active < top
U23 < proper
take < proper
U11 < proper
U21 < proper
proper < top

----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_0)

Induction Base:
U23(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:
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n9047_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(U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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).
----------------------------------------

(22)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_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, U11, U21, proper, top

They will be analysed ascendingly in the following order:
take < active
U11 < active
U21 < active
active < top
take < proper
U11 < proper
U21 < proper
proper < top

----------------------------------------

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14852_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14852_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, +(n14852_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, n14852_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).
----------------------------------------

(24)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14852_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14852_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:
U11, active, U21, proper, top

They will be analysed ascendingly in the following order:
U11 < active
U21 < active
active < top
U11 < proper
U21 < proper
proper < top

----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n18274_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, +(n18274_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, n18274_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).
----------------------------------------

(26)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14852_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14852_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n18274_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, proper, top

They will be analysed ascendingly in the following order:
U21 < active
active < top
U21 < proper
proper < top

----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21803_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(n21803_0)

Induction Base:
U21(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:
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n21803_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(U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21803_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).
----------------------------------------

(28)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, L)) -> mark(U12(tt, L))
active(U12(tt, L)) -> mark(s(length(L)))
active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U11(tt, L))
active(take(0', IL)) -> mark(nil)
active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(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(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
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
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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 -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: 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
U23 :: 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
nil :: 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)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14852_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14852_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n18274_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21803_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(n21803_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