/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The 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), 459 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), 107 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 134 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 103 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 134 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 91 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 131 ms]
        (24) typed CpxTrs
        (25) RewriteLemmaProof [LOWER BOUND(ID), 51 ms]
        (26) typed CpxTrs
        (27) RewriteLemmaProof [LOWER BOUND(ID), 98 ms]
        (28) typed CpxTrs
        (29) RewriteLemmaProof [LOWER BOUND(ID), 108 ms]
        (30) typed CpxTrs
        (31) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
        (32) typed CpxTrs
        (33) RewriteLemmaProof [LOWER BOUND(ID), 124 ms]
        (34) typed CpxTrs
        (35) RewriteLemmaProof [LOWER BOUND(ID), 177 ms]
        (36) typed CpxTrs
        (37) RewriteLemmaProof [LOWER BOUND(ID), 155 ms]
        (38) typed CpxTrs
        (39) RewriteLemmaProof [LOWER BOUND(ID), 153 ms]
        (40) typed CpxTrs
        (41) RewriteLemmaProof [LOWER BOUND(ID), 176 ms]
        (42) typed CpxTrs
        (43) RewriteLemmaProof [LOWER BOUND(ID), 161 ms]
        (44) typed CpxTrs
        (45) RewriteLemmaProof [LOWER BOUND(ID), 121 ms]
        (46) typed CpxTrs
        (47) RewriteLemmaProof [LOWER BOUND(ID), 231 ms]
        (48) typed CpxTrs
        (49) RewriteLemmaProof [LOWER BOUND(ID), 196 ms]
        (50) typed CpxTrs
        (51) RewriteLemmaProof [LOWER BOUND(ID), 164 ms]
        (52) typed CpxTrs
        (53) RewriteLemmaProof [LOWER BOUND(ID), 61 ms]
        (54) typed CpxTrs
        (55) RewriteLemmaProof [LOWER BOUND(ID), 250 ms]
        (56) 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, V1)) -> mark(U12(isNatList(V1)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatList(V)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
   active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
   active(U63(tt)) -> mark(tt)
   active(U71(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
   active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
   active(take(0, IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
   active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X)) -> U12(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X)) -> U32(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
   active(U62(X1, X2)) -> U62(active(X1), X2)
   active(U63(X)) -> U63(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(U81(X)) -> U81(active(X))
   active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X)) -> mark(U12(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X)) -> mark(U32(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
   U62(mark(X1), X2) -> mark(U62(X1, X2))
   U63(mark(X)) -> mark(U63(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   U81(mark(X)) -> mark(U81(X))
   U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X)) -> U12(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X)) -> U32(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
   proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
   proper(U63(X)) -> U63(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(U81(X)) -> U81(proper(X))
   proper(nil) -> ok(nil)
   proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
   proper(isNatKind(X)) -> isNatKind(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X)) -> ok(U12(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   isNat(ok(X)) -> ok(isNat(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X)) -> ok(U32(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
   U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
   U63(ok(X)) -> ok(U63(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   U81(ok(X)) -> ok(U81(X))
   U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
   isNatKind(ok(X)) -> ok(isNatKind(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, V1)) -> mark(U12(isNatList(V1)))
   active(U12(tt)) -> mark(tt)
   active(U21(tt, V1)) -> mark(U22(isNat(V1)))
   active(U22(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatList(V)))
   active(U32(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
   active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
   active(U43(tt)) -> mark(tt)
   active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
   active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
   active(U53(tt)) -> mark(tt)
   active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
   active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
   active(U63(tt)) -> mark(tt)
   active(U71(tt, L)) -> mark(s(length(L)))
   active(U81(tt)) -> mark(nil)
   active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0')) -> mark(tt)
   active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
   active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
   active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
   active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
   active(isNatKind(0')) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
   active(isNatKind(s(V1))) -> mark(isNatKind(V1))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
   active(length(nil)) -> mark(0')
   active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
   active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
   active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X)) -> U12(active(X))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X)) -> U22(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X)) -> U32(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2)) -> U42(active(X1), X2)
   active(U43(X)) -> U43(active(X))
   active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
   active(U52(X1, X2)) -> U52(active(X1), X2)
   active(U53(X)) -> U53(active(X))
   active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
   active(U62(X1, X2)) -> U62(active(X1), X2)
   active(U63(X)) -> U63(active(X))
   active(U71(X1, X2)) -> U71(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(U81(X)) -> U81(active(X))
   active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(and(X1, X2)) -> and(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X)) -> mark(U12(X))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X)) -> mark(U22(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X)) -> mark(U32(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2) -> mark(U42(X1, X2))
   U43(mark(X)) -> mark(U43(X))
   U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
   U52(mark(X1), X2) -> mark(U52(X1, X2))
   U53(mark(X)) -> mark(U53(X))
   U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
   U62(mark(X1), X2) -> mark(U62(X1, X2))
   U63(mark(X)) -> mark(U63(X))
   U71(mark(X1), X2) -> mark(U71(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   U81(mark(X)) -> mark(U81(X))
   U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(U12(X)) -> U12(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X)) -> U22(proper(X))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X)) -> U32(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
   proper(U43(X)) -> U43(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
   proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
   proper(U53(X)) -> U53(proper(X))
   proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
   proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
   proper(U63(X)) -> U63(proper(X))
   proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(U81(X)) -> U81(proper(X))
   proper(nil) -> ok(nil)
   proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
   proper(isNatKind(X)) -> isNatKind(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X)) -> ok(U12(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X)) -> ok(U22(X))
   isNat(ok(X)) -> ok(isNat(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X)) -> ok(U32(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
   U43(ok(X)) -> ok(U43(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
   U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
   U53(ok(X)) -> ok(U53(X))
   U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
   U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
   U63(ok(X)) -> ok(U63(X))
   U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   U81(ok(X)) -> ok(U81(X))
   U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
   isNatKind(ok(X)) -> ok(isNatKind(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, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, cons, U12, isNatList, U22, isNat, U32, U42, U43, isNatIList, U52, U53, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
cons < active
U12 < active
isNatList < active
U22 < active
isNat < active
U32 < active
U42 < active
U43 < active
isNatIList < active
U52 < active
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
cons < proper
U12 < proper
isNatList < proper
U22 < proper
isNat < proper
U32 < proper
U42 < proper
U43 < proper
isNatIList < proper
U52 < proper
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(6)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, U12, isNatList, U22, isNat, U32, U42, U43, isNatIList, U52, U53, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
cons < active
U12 < active
isNatList < active
U22 < active
isNat < active
U32 < active
U42 < active
U43 < active
isNatIList < active
U52 < active
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
cons < proper
U12 < proper
isNatList < proper
U22 < proper
isNat < proper
U32 < proper
U42 < proper
U43 < proper
isNatIList < proper
U52 < proper
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < 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, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, U12, isNatList, U22, isNat, U32, U42, U43, isNatIList, U52, U53, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
cons < active
U12 < active
isNatList < active
U22 < active
isNat < active
U32 < active
U42 < active
U43 < active
isNatIList < active
U52 < active
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
cons < proper
U12 < proper
isNatList < proper
U22 < proper
isNat < proper
U32 < proper
U42 < proper
U43 < proper
isNatIList < proper
U52 < proper
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < 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, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
U12, active, isNatList, U22, isNat, U32, U42, U43, isNatIList, U52, U53, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U12 < active
isNatList < active
U22 < active
isNat < active
U32 < active
U42 < active
U43 < active
isNatIList < active
U52 < active
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U12 < proper
isNatList < proper
U22 < proper
isNat < proper
U32 < proper
U42 < proper
U43 < proper
isNatIList < proper
U52 < proper
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)

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

Induction Step:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2540_0, 1)))) ->_R^Omega(1)
mark(U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_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, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_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, U22, isNat, U32, U42, U43, isNatIList, U52, U53, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
isNatList < active
U22 < active
isNat < active
U32 < active
U42 < active
U43 < active
isNatIList < active
U52 < active
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
isNatList < proper
U22 < proper
isNat < proper
U32 < proper
U42 < proper
U43 < proper
isNatIList < proper
U52 < proper
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)

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

Induction Step:
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n3419_0, 1)))) ->_R^Omega(1)
mark(U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_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, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_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, U32, U42, U43, isNatIList, U52, U53, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
isNat < active
U32 < active
U42 < active
U43 < active
isNatIList < active
U52 < active
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
isNat < proper
U32 < proper
U42 < proper
U43 < proper
isNatIList < proper
U52 < proper
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)

Induction Base:
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n4402_0, 1)))) ->_R^Omega(1)
mark(U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0)))) ->_IH
mark(*4_0)

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

(18)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_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, U43, isNatIList, U52, U53, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U42 < active
U43 < active
isNatIList < active
U52 < active
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U42 < proper
U43 < proper
isNatIList < proper
U52 < proper
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)

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

Induction Step:
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n5471_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
mark(*4_0)

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

(20)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_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:
U43, active, isNatIList, U52, U53, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U43 < active
isNatIList < active
U52 < active
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U43 < proper
isNatIList < proper
U52 < proper
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)

Induction Base:
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n8936_0, 1)))) ->_R^Omega(1)
mark(U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_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, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_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, U53, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
isNatIList < active
U52 < active
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
isNatIList < proper
U52 < proper
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)

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

Induction Step:
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n10282_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
mark(*4_0)

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

(24)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_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:
U53, active, U62, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U53 < active
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U53 < proper
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)

Induction Base:
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n14261_0, 1)))) ->_R^Omega(1)
mark(U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_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, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_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:
U62, active, U63, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U62 < active
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U62 < proper
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_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, +(n15832_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, n15832_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).
----------------------------------------

(28)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_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:
U63, active, s, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U63 < active
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U63 < proper
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)

Induction Base:
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n20325_0, 1)))) ->_R^Omega(1)
mark(U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_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).
----------------------------------------

(30)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
s, active, length, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
s < active
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
s < proper
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_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, +(n22147_0, 1)))) ->_R^Omega(1)
mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_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).
----------------------------------------

(32)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
length, active, take, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
length < active
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
length < proper
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_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, +(n24070_0, 1)))) ->_R^Omega(1)
mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_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).
----------------------------------------

(34)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)


Generator Equations:
gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))


The following defined symbols remain to be analysed:
take, active, U11, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
take < active
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
take < proper
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)

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

Induction Step:
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n26094_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_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).
----------------------------------------

(36)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_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, isNatIListKind, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U11 < active
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U11 < proper
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)

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

Induction Step:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n31716_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_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).
----------------------------------------

(38)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_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:
isNatIListKind, active, U21, isNatKind, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
isNatIListKind < active
U21 < active
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
isNatIListKind < proper
U21 < proper
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_0)

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

Induction Step:
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n37503_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_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).
----------------------------------------

(40)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_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:
isNatKind, active, U31, U41, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
isNatKind < active
U31 < active
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
isNatKind < proper
U31 < proper
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n43601_0)

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

Induction Step:
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n43601_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_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).
----------------------------------------

(42)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n43601_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, and, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U41 < active
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U41 < proper
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n49942_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n49942_0)

Induction Base:
U41(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:
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n49942_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n49942_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).
----------------------------------------

(44)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n43601_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n49942_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n49942_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:
and, active, U51, U61, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
and < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
and < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n60511_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n60511_0)

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

Induction Step:
and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n60511_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n60511_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).
----------------------------------------

(46)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n43601_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n49942_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n49942_0)
and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n60511_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n60511_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, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67562_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n67562_0)

Induction Base:
U51(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:
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n67562_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67562_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).
----------------------------------------

(48)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n43601_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n49942_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n49942_0)
and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n60511_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n60511_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67562_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n67562_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, U71, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U61 < active
U71 < active
U81 < active
U91 < active
active < top
U61 < proper
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(49) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n79202_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n79202_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, +(n79202_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, n79202_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).
----------------------------------------

(50)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n43601_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n49942_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n49942_0)
and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n60511_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n60511_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67562_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n67562_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n79202_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n79202_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:
U71, active, U81, U91, proper, top

They will be analysed ascendingly in the following order:
U71 < active
U81 < active
U91 < active
active < top
U71 < proper
U81 < proper
U91 < proper
proper < top

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n91451_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n91451_0)

Induction Base:
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n91451_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n91451_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).
----------------------------------------

(52)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n43601_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n49942_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n49942_0)
and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n60511_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n60511_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67562_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n67562_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n79202_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n79202_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n91451_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n91451_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:
U81, active, U91, proper, top

They will be analysed ascendingly in the following order:
U81 < active
U91 < active
active < top
U81 < proper
U91 < proper
proper < top

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n99616_0))) -> *4_0, rt in Omega(n99616_0)

Induction Base:
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n99616_0, 1)))) ->_R^Omega(1)
mark(U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n99616_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).
----------------------------------------

(54)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n43601_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n49942_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n49942_0)
and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n60511_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n60511_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67562_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n67562_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n79202_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n79202_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n91451_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n91451_0)
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n99616_0))) -> *4_0, rt in Omega(n99616_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:
U91, active, proper, top

They will be analysed ascendingly in the following order:
U91 < active
active < top
U91 < proper
proper < top

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

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n103238_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n103238_0)

Induction Base:
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))

Induction Step:
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n103238_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) ->_R^Omega(1)
mark(U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n103238_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))) ->_IH
mark(*4_0)

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

(56)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U11(tt, V1)) -> mark(U12(isNatList(V1)))
active(U12(tt)) -> mark(tt)
active(U21(tt, V1)) -> mark(U22(isNat(V1)))
active(U22(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatList(V)))
active(U32(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNat(V1), V2))
active(U42(tt, V2)) -> mark(U43(isNatIList(V2)))
active(U43(tt)) -> mark(tt)
active(U51(tt, V1, V2)) -> mark(U52(isNat(V1), V2))
active(U52(tt, V2)) -> mark(U53(isNatList(V2)))
active(U53(tt)) -> mark(tt)
active(U61(tt, V1, V2)) -> mark(U62(isNat(V1), V2))
active(U62(tt, V2)) -> mark(U63(isNatIList(V2)))
active(U63(tt)) -> mark(tt)
active(U71(tt, L)) -> mark(s(length(L)))
active(U81(tt)) -> mark(nil)
active(U91(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(and(tt, X)) -> mark(X)
active(isNat(0')) -> mark(tt)
active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) -> mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) -> mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) -> mark(tt)
active(isNatIList(cons(V1, V2))) -> mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatIListKind(take(V1, V2))) -> mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(isNatIListKind(V1))
active(isNatKind(s(V1))) -> mark(isNatKind(V1))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(take(0', IL)) -> mark(U81(and(isNatIList(IL), isNatIListKind(IL))))
active(take(s(M), cons(N, IL))) -> mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X)) -> U12(active(X))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X)) -> U22(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X)) -> U32(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2)) -> U42(active(X1), X2)
active(U43(X)) -> U43(active(X))
active(U51(X1, X2, X3)) -> U51(active(X1), X2, X3)
active(U52(X1, X2)) -> U52(active(X1), X2)
active(U53(X)) -> U53(active(X))
active(U61(X1, X2, X3)) -> U61(active(X1), X2, X3)
active(U62(X1, X2)) -> U62(active(X1), X2)
active(U63(X)) -> U63(active(X))
active(U71(X1, X2)) -> U71(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(and(X1, X2)) -> and(active(X1), X2)
cons(mark(X1), X2) -> mark(cons(X1, X2))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X)) -> mark(U12(X))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X)) -> mark(U22(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X)) -> mark(U32(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2) -> mark(U42(X1, X2))
U43(mark(X)) -> mark(U43(X))
U51(mark(X1), X2, X3) -> mark(U51(X1, X2, X3))
U52(mark(X1), X2) -> mark(U52(X1, X2))
U53(mark(X)) -> mark(U53(X))
U61(mark(X1), X2, X3) -> mark(U61(X1, X2, X3))
U62(mark(X1), X2) -> mark(U62(X1, X2))
U63(mark(X)) -> mark(U63(X))
U71(mark(X1), X2) -> mark(U71(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
and(mark(X1), X2) -> mark(and(X1, X2))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(tt) -> ok(tt)
proper(U12(X)) -> U12(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X)) -> U22(proper(X))
proper(isNat(X)) -> isNat(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X)) -> U32(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) -> U42(proper(X1), proper(X2))
proper(U43(X)) -> U43(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U51(X1, X2, X3)) -> U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) -> U52(proper(X1), proper(X2))
proper(U53(X)) -> U53(proper(X))
proper(U61(X1, X2, X3)) -> U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2)) -> U62(proper(X1), proper(X2))
proper(U63(X)) -> U63(proper(X))
proper(U71(X1, X2)) -> U71(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(nil) -> ok(nil)
proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(and(X1, X2)) -> and(proper(X1), proper(X2))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(isNatKind(X)) -> isNatKind(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X)) -> ok(U12(X))
isNatList(ok(X)) -> ok(isNatList(X))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X)) -> ok(U22(X))
isNat(ok(X)) -> ok(isNat(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X)) -> ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) -> ok(U42(X1, X2))
U43(ok(X)) -> ok(U43(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) -> ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) -> ok(U52(X1, X2))
U53(ok(X)) -> ok(U53(X))
U61(ok(X1), ok(X2), ok(X3)) -> ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2)) -> ok(U62(X1, X2))
U63(ok(X)) -> ok(U63(X))
U71(ok(X1), ok(X2)) -> ok(U71(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
and(ok(X1), ok(X2)) -> ok(and(X1, X2))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
isNatKind(ok(X)) -> ok(isNatKind(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
zeros :: zeros:0':mark:tt:nil:ok
mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
0' :: zeros:0':mark:tt:nil:ok
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
tt :: zeros:0':mark:tt:nil:ok
U12 :: 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
U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U22 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: 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 -> zeros:0':mark:tt:nil:ok
U32 :: 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 -> zeros:0':mark:tt:nil:ok
U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U43 :: 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 -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U53 :: 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
U63 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
and :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatIListKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNatKind :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
top :: zeros:0':mark:tt:nil:ok -> top
hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
hole_top2_0 :: top
gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok


Lemmas:
cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2540_0))) -> *4_0, rt in Omega(n2540_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3419_0))) -> *4_0, rt in Omega(n3419_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4402_0))) -> *4_0, rt in Omega(n4402_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5471_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5471_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n8936_0))) -> *4_0, rt in Omega(n8936_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10282_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n10282_0)
U53(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14261_0))) -> *4_0, rt in Omega(n14261_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n15832_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n15832_0)
U63(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n20325_0))) -> *4_0, rt in Omega(n20325_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n22147_0))) -> *4_0, rt in Omega(n22147_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n24070_0))) -> *4_0, rt in Omega(n24070_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n26094_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n26094_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n31716_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n31716_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37503_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n37503_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n43601_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n43601_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n49942_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n49942_0)
and(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n60511_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n60511_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67562_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n67562_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n79202_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n79202_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n91451_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n91451_0)
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n99616_0))) -> *4_0, rt in Omega(n99616_0)
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n103238_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n103238_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