/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 438 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), 102 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 105 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 129 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 72 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 123 ms]
        (24) typed CpxTrs
        (25) RewriteLemmaProof [LOWER BOUND(ID), 112 ms]
        (26) typed CpxTrs
        (27) RewriteLemmaProof [LOWER BOUND(ID), 118 ms]
        (28) typed CpxTrs
        (29) RewriteLemmaProof [LOWER BOUND(ID), 137 ms]
        (30) typed CpxTrs
        (31) RewriteLemmaProof [LOWER BOUND(ID), 111 ms]
        (32) typed CpxTrs
        (33) RewriteLemmaProof [LOWER BOUND(ID), 154 ms]
        (34) 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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
   active(U62(tt, L)) -> mark(s(length(L)))
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X)) -> U11(active(X))
   active(U21(X)) -> U21(active(X))
   active(U31(X)) -> U31(active(X))
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U42(X)) -> U42(active(X))
   active(U51(X1, X2)) -> U51(active(X1), X2)
   active(U52(X)) -> U52(active(X))
   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
   active(U62(X1, X2)) -> U62(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X)) -> mark(U11(X))
   U21(mark(X)) -> mark(U21(X))
   U31(mark(X)) -> mark(U31(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U42(mark(X)) -> mark(U42(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
   U62(mark(X1), X2) -> mark(U62(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X)) -> U11(proper(X))
   proper(tt) -> ok(tt)
   proper(U21(X)) -> U21(proper(X))
   proper(U31(X)) -> U31(proper(X))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U42(X)) -> U42(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
   proper(U52(X)) -> U52(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
   proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X)) -> ok(U11(X))
   U21(ok(X)) -> ok(U21(X))
   U31(ok(X)) -> ok(U31(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U42(ok(X)) -> ok(U42(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
   U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(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)) -> mark(tt)
   active(U21(tt)) -> mark(tt)
   active(U31(tt)) -> mark(tt)
   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
   active(U42(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
   active(U62(tt, L)) -> mark(s(length(L)))
   active(isNat(0')) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
   active(isNatIList(V)) -> mark(U31(isNatList(V)))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
   active(length(nil)) -> mark(0')
   active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X)) -> U11(active(X))
   active(U21(X)) -> U21(active(X))
   active(U31(X)) -> U31(active(X))
   active(U41(X1, X2)) -> U41(active(X1), X2)
   active(U42(X)) -> U42(active(X))
   active(U51(X1, X2)) -> U51(active(X1), X2)
   active(U52(X)) -> U52(active(X))
   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
   active(U62(X1, X2)) -> U62(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X)) -> mark(U11(X))
   U21(mark(X)) -> mark(U21(X))
   U31(mark(X)) -> mark(U31(X))
   U41(mark(X1), X2) -> mark(U41(X1, X2))
   U42(mark(X)) -> mark(U42(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
   U62(mark(X1), X2) -> mark(U62(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(U11(X)) -> U11(proper(X))
   proper(tt) -> ok(tt)
   proper(U21(X)) -> U21(proper(X))
   proper(U31(X)) -> U31(proper(X))
   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
   proper(U42(X)) -> U42(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
   proper(U52(X)) -> U52(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
   proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X)) -> ok(U11(X))
   U21(ok(X)) -> ok(U21(X))
   U31(ok(X)) -> ok(U31(X))
   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
   U42(ok(X)) -> ok(U42(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
   U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(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)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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, U42, isNatIList, U52, isNatList, U62, isNat, s, length, U11, U21, U31, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
cons < active
U42 < active
isNatIList < active
U52 < active
isNatList < active
U62 < active
isNat < active
s < active
length < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
cons < proper
U42 < proper
isNatIList < proper
U52 < proper
isNatList < proper
U62 < proper
isNat < proper
s < proper
length < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(6)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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, U42, isNatIList, U52, isNatList, U62, isNat, s, length, U11, U21, U31, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
cons < active
U42 < active
isNatIList < active
U52 < active
isNatList < active
U62 < active
isNat < active
s < active
length < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
cons < proper
U42 < proper
isNatIList < proper
U52 < proper
isNatList < proper
U62 < proper
isNat < proper
s < proper
length < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < 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)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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, U42, isNatIList, U52, isNatList, U62, isNat, s, length, U11, U21, U31, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
cons < active
U42 < active
isNatIList < active
U52 < active
isNatList < active
U62 < active
isNat < active
s < active
length < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
cons < proper
U42 < proper
isNatIList < proper
U52 < proper
isNatList < proper
U62 < proper
isNat < proper
s < proper
length < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < 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)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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:
U42, active, isNatIList, U52, isNatList, U62, isNat, s, length, U11, U21, U31, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
U42 < active
isNatIList < active
U52 < active
isNatList < active
U62 < active
isNat < active
s < active
length < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
U42 < proper
isNatIList < proper
U52 < proper
isNatList < proper
U62 < proper
isNat < proper
s < proper
length < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)

Induction Base:
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n1560_0, 1)))) ->_R^Omega(1)
mark(U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_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)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_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, U52, isNatList, U62, isNat, s, length, U11, U21, U31, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
isNatIList < active
U52 < active
isNatList < active
U62 < active
isNat < active
s < active
length < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
isNatIList < proper
U52 < proper
isNatList < proper
U62 < proper
isNat < proper
s < proper
length < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)

Induction Base:
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2229_0, 1)))) ->_R^Omega(1)
mark(U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_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)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_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, U62, isNat, s, length, U11, U21, U31, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
isNatList < active
U62 < active
isNat < active
s < active
length < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
isNatList < proper
U62 < proper
isNat < proper
s < proper
length < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_0)

Induction Base:
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n3002_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_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)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_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:
isNat, active, s, length, U11, U21, U31, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
isNat < active
s < active
length < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
isNat < proper
s < proper
length < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_0))) -> *4_0, rt in Omega(n5299_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, +(n5299_0, 1)))) ->_R^Omega(1)
mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_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).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_0))) -> *4_0, rt in Omega(n5299_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, U11, U21, U31, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
length < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
length < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n6308_0))) -> *4_0, rt in Omega(n6308_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, +(n6308_0, 1)))) ->_R^Omega(1)
mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n6308_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).
----------------------------------------

(22)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_0))) -> *4_0, rt in Omega(n5299_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n6308_0))) -> *4_0, rt in Omega(n6308_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, U31, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7418_0))) -> *4_0, rt in Omega(n7418_0)

Induction Base:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n7418_0, 1)))) ->_R^Omega(1)
mark(U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7418_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)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_0))) -> *4_0, rt in Omega(n5299_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n6308_0))) -> *4_0, rt in Omega(n6308_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7418_0))) -> *4_0, rt in Omega(n7418_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, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
active < top
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8629_0))) -> *4_0, rt in Omega(n8629_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, +(n8629_0, 1)))) ->_R^Omega(1)
mark(U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8629_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).
----------------------------------------

(26)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_0))) -> *4_0, rt in Omega(n5299_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n6308_0))) -> *4_0, rt in Omega(n6308_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7418_0))) -> *4_0, rt in Omega(n7418_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8629_0))) -> *4_0, rt in Omega(n8629_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, U41, U51, U61, proper, top

They will be analysed ascendingly in the following order:
U31 < active
U41 < active
U51 < active
U61 < active
active < top
U31 < proper
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9941_0))) -> *4_0, rt in Omega(n9941_0)

Induction Base:
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n9941_0, 1)))) ->_R^Omega(1)
mark(U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9941_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).
----------------------------------------

(28)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_0))) -> *4_0, rt in Omega(n5299_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n6308_0))) -> *4_0, rt in Omega(n6308_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7418_0))) -> *4_0, rt in Omega(n7418_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8629_0))) -> *4_0, rt in Omega(n8629_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9941_0))) -> *4_0, rt in Omega(n9941_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:
U41, active, U51, U61, proper, top

They will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
active < top
U41 < proper
U51 < proper
U61 < proper
proper < top

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n11354_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n11354_0)

Induction Base:
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n11354_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n11354_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).
----------------------------------------

(30)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_0))) -> *4_0, rt in Omega(n5299_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n6308_0))) -> *4_0, rt in Omega(n6308_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7418_0))) -> *4_0, rt in Omega(n7418_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8629_0))) -> *4_0, rt in Omega(n8629_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9941_0))) -> *4_0, rt in Omega(n9941_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n11354_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n11354_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:
U51, active, U61, proper, top

They will be analysed ascendingly in the following order:
U51 < active
U61 < active
active < top
U51 < proper
U61 < proper
proper < top

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14977_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14977_0)

Induction Base:
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n14977_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14977_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).
----------------------------------------

(32)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_0))) -> *4_0, rt in Omega(n5299_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n6308_0))) -> *4_0, rt in Omega(n6308_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7418_0))) -> *4_0, rt in Omega(n7418_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8629_0))) -> *4_0, rt in Omega(n8629_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9941_0))) -> *4_0, rt in Omega(n9941_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n11354_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n11354_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14977_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14977_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:
U61, active, proper, top

They will be analysed ascendingly in the following order:
U61 < active
active < top
U61 < proper
proper < top

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18906_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n18906_0)

Induction Base:
U61(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))

Induction Step:
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n18906_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18906_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(34)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt)) -> mark(tt)
active(U21(tt)) -> mark(tt)
active(U31(tt)) -> mark(tt)
active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
active(U42(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, L, N)) -> mark(U62(isNat(N), L))
active(U62(tt, L)) -> mark(s(length(L)))
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
active(isNat(s(V1))) -> mark(U21(isNat(V1)))
active(isNatIList(V)) -> mark(U31(isNatList(V)))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U61(isNatList(L), L, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X)) -> U11(active(X))
active(U21(X)) -> U21(active(X))
active(U31(X)) -> U31(active(X))
active(U41(X1, X2)) -> U41(active(X1), X2)
active(U42(X)) -> U42(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X)) -> mark(U11(X))
U21(mark(X)) -> mark(U21(X))
U31(mark(X)) -> mark(U31(X))
U41(mark(X1), X2) -> mark(U41(X1, X2))
U42(mark(X)) -> mark(U42(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X)) -> U11(proper(X))
proper(tt) -> ok(tt)
proper(U21(X)) -> U21(proper(X))
proper(U31(X)) -> U31(proper(X))
proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
proper(U42(X)) -> U42(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(nil) -> ok(nil)
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X)) -> ok(U11(X))
U21(ok(X)) -> ok(U21(X))
U31(ok(X)) -> ok(U31(X))
U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
U42(ok(X)) -> ok(U42(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
isNatList(ok(X)) -> ok(isNatList(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(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
tt :: zeros:0':mark:tt:nil:ok
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U42 :: 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
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: 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
U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U62 :: 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
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
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)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1560_0))) -> *4_0, rt in Omega(n1560_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2229_0))) -> *4_0, rt in Omega(n2229_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3002_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n3002_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5299_0))) -> *4_0, rt in Omega(n5299_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n6308_0))) -> *4_0, rt in Omega(n6308_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7418_0))) -> *4_0, rt in Omega(n7418_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8629_0))) -> *4_0, rt in Omega(n8629_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9941_0))) -> *4_0, rt in Omega(n9941_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n11354_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n11354_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14977_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14977_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18906_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n18906_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