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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 14 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 619 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), 331 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 188 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 209 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 188 ms]
        (20) typed CpxTrs
        (21) RewriteLemmaProof [LOWER BOUND(ID), 130 ms]
        (22) typed CpxTrs
        (23) RewriteLemmaProof [LOWER BOUND(ID), 167 ms]
        (24) typed CpxTrs
        (25) RewriteLemmaProof [LOWER BOUND(ID), 209 ms]
        (26) typed CpxTrs
        (27) RewriteLemmaProof [LOWER BOUND(ID), 238 ms]
        (28) typed CpxTrs
        (29) RewriteLemmaProof [LOWER BOUND(ID), 169 ms]
        (30) typed CpxTrs
        (31) RewriteLemmaProof [LOWER BOUND(ID), 126 ms]
        (32) typed CpxTrs
        (33) RewriteLemmaProof [LOWER BOUND(ID), 149 ms]
        (34) typed CpxTrs
        (35) RewriteLemmaProof [LOWER BOUND(ID), 135 ms]
        (36) typed CpxTrs
        (37) RewriteLemmaProof [LOWER BOUND(ID), 132 ms]
        (38) typed CpxTrs
        (39) RewriteLemmaProof [LOWER BOUND(ID), 353 ms]
        (40) typed CpxTrs
        (41) RewriteLemmaProof [LOWER BOUND(ID), 259 ms]
        (42) typed CpxTrs
        (43) RewriteLemmaProof [LOWER BOUND(ID), 270 ms]
        (44) typed CpxTrs
        (45) RewriteLemmaProof [LOWER BOUND(ID), 230 ms]
        (46) typed CpxTrs
        (47) RewriteLemmaProof [LOWER BOUND(ID), 314 ms]
        (48) typed CpxTrs
        (49) RewriteLemmaProof [LOWER BOUND(ID), 191 ms]
        (50) typed CpxTrs
        (51) RewriteLemmaProof [LOWER BOUND(ID), 169 ms]
        (52) typed CpxTrs
        (53) RewriteLemmaProof [LOWER BOUND(ID), 119 ms]
        (54) typed CpxTrs
        (55) RewriteLemmaProof [LOWER BOUND(ID), 137 ms]
        (56) typed CpxTrs
        (57) RewriteLemmaProof [LOWER BOUND(ID), 109 ms]
        (58) typed CpxTrs
        (59) RewriteLemmaProof [LOWER BOUND(ID), 233 ms]
        (60) typed CpxTrs
        (61) RewriteLemmaProof [LOWER BOUND(ID), 265 ms]
        (62) typed CpxTrs
        (63) RewriteLemmaProof [LOWER BOUND(ID), 264 ms]
        (64) typed CpxTrs
        (65) RewriteLemmaProof [LOWER BOUND(ID), 213 ms]
        (66) typed CpxTrs
        (67) RewriteLemmaProof [LOWER BOUND(ID), 69 ms]
        (68) typed CpxTrs
        (69) RewriteLemmaProof [LOWER BOUND(ID), 116 ms]
        (70) typed CpxTrs
        (71) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
        (72) typed CpxTrs
        (73) RewriteLemmaProof [LOWER BOUND(ID), 317 ms]
        (74) typed CpxTrs
        (75) RewriteLemmaProof [LOWER BOUND(ID), 354 ms]
        (76) typed CpxTrs
        (77) RewriteLemmaProof [LOWER BOUND(ID), 323 ms]
        (78) typed CpxTrs
        (79) RewriteLemmaProof [LOWER BOUND(ID), 269 ms]
        (80) typed CpxTrs
        (81) RewriteLemmaProof [LOWER BOUND(ID), 143 ms]
        (82) typed CpxTrs
        (83) RewriteLemmaProof [LOWER BOUND(ID), 223 ms]
        (84) typed CpxTrs
        (85) RewriteLemmaProof [LOWER BOUND(ID), 266 ms]
        (86) typed CpxTrs
        (87) RewriteLemmaProof [LOWER BOUND(ID), 255 ms]
        (88) typed CpxTrs
        (89) RewriteLemmaProof [LOWER BOUND(ID), 452 ms]
        (90) typed CpxTrs
        (91) RewriteLemmaProof [LOWER BOUND(ID), 217 ms]
        (92) typed CpxTrs
        (93) RewriteLemmaProof [LOWER BOUND(ID), 235 ms]
        (94) typed CpxTrs
        (95) RewriteLemmaProof [LOWER BOUND(ID), 154 ms]
        (96) typed CpxTrs
        (97) RewriteLemmaProof [LOWER BOUND(ID), 142 ms]
        (98) typed CpxTrs
        (99) RewriteLemmaProof [LOWER BOUND(ID), 321 ms]
        (100) typed CpxTrs
        (101) RewriteLemmaProof [LOWER BOUND(ID), 328 ms]
        (102) typed CpxTrs
        (103) RewriteLemmaProof [LOWER BOUND(ID), 424 ms]
        (104) typed CpxTrs
        (105) RewriteLemmaProof [LOWER BOUND(ID), 269 ms]
        (106) typed CpxTrs
        (107) RewriteLemmaProof [LOWER BOUND(ID), 545 ms]
        (108) 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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   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(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0)) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0, IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
   active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
   active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
   active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
   active(U105(X1, X2)) -> U105(active(X1), X2)
   active(U106(X)) -> U106(active(X))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
   active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
   active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
   active(U114(X1, X2)) -> U114(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(U13(X)) -> U13(active(X))
   active(U121(X1, X2)) -> U121(active(X1), X2)
   active(U122(X)) -> U122(active(X))
   active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
   active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
   active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
   active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
   active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
   active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X1, X2)) -> U22(active(X1), X2)
   active(U23(X)) -> U23(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U33(X)) -> U33(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
   active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
   active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
   active(U45(X1, X2)) -> U45(active(X1), X2)
   active(U46(X)) -> U46(active(X))
   active(U51(X1, X2)) -> U51(active(X1), X2)
   active(U52(X)) -> U52(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X)) -> U71(active(X))
   active(U81(X)) -> U81(active(X))
   active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
   active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
   active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
   active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
   active(U95(X1, X2)) -> U95(active(X1), X2)
   active(U96(X)) -> U96(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
   U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
   U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
   U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
   U105(mark(X1), X2) -> mark(U105(X1, X2))
   U106(mark(X)) -> mark(U106(X))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
   U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
   U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
   U114(mark(X1), X2) -> mark(U114(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   U13(mark(X)) -> mark(U13(X))
   U121(mark(X1), X2) -> mark(U121(X1, X2))
   U122(mark(X)) -> mark(U122(X))
   U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
   U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
   U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
   U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
   U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
   U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U23(mark(X)) -> mark(U23(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
   U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
   U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
   U45(mark(X1), X2) -> mark(U45(X1, X2))
   U46(mark(X)) -> mark(U46(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X)) -> mark(U71(X))
   U81(mark(X)) -> mark(U81(X))
   U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
   U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
   U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
   U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
   U95(mark(X1), X2) -> mark(U95(X1, X2))
   U96(mark(X)) -> mark(U96(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
   proper(isNatKind(X)) -> isNatKind(proper(X))
   proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
   proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
   proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
   proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U106(X)) -> U106(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
   proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
   proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
   proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
   proper(U122(X)) -> U122(proper(X))
   proper(nil) -> ok(nil)
   proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
   proper(U23(X)) -> U23(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U33(X)) -> U33(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
   proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
   proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
   proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
   proper(U46(X)) -> U46(proper(X))
   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
   proper(U52(X)) -> U52(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(U71(X)) -> U71(proper(X))
   proper(U81(X)) -> U81(proper(X))
   proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
   proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
   proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
   proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
   proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
   proper(U96(X)) -> U96(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
   U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
   isNatKind(ok(X)) -> ok(isNatKind(X))
   U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
   isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
   U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
   U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U106(ok(X)) -> ok(U106(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
   U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
   U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
   U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   U13(ok(X)) -> ok(U13(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
   U122(ok(X)) -> ok(U122(X))
   U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
   U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
   U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
   U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
   U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
   U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   U23(ok(X)) -> ok(U23(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
   U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
   U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
   U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
   U46(ok(X)) -> ok(U46(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   U71(ok(X)) -> ok(U71(X))
   U81(ok(X)) -> ok(U81(X))
   U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
   U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
   U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
   U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
   U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
   U96(ok(X)) -> ok(U96(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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
   active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
   active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
   active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
   active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
   active(U106(tt)) -> mark(tt)
   active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
   active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
   active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
   active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
   active(U114(tt, L)) -> mark(s(length(L)))
   active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
   active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
   active(U122(tt)) -> mark(nil)
   active(U13(tt)) -> mark(tt)
   active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
   active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
   active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
   active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
   active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
   active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
   active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
   active(U22(tt, V1)) -> mark(U23(isNat(V1)))
   active(U23(tt)) -> mark(tt)
   active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
   active(U32(tt, V)) -> mark(U33(isNatList(V)))
   active(U33(tt)) -> mark(tt)
   active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
   active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
   active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
   active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
   active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
   active(U46(tt)) -> mark(tt)
   active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
   active(U52(tt)) -> mark(tt)
   active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
   active(U62(tt)) -> mark(tt)
   active(U71(tt)) -> mark(tt)
   active(U81(tt)) -> mark(tt)
   active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
   active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
   active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
   active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
   active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
   active(U96(tt)) -> mark(tt)
   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(isNatKind(V1), V1, V2))
   active(isNatIListKind(nil)) -> mark(tt)
   active(isNatIListKind(zeros)) -> mark(tt)
   active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
   active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
   active(isNatKind(0')) -> mark(tt)
   active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
   active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
   active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
   active(length(nil)) -> mark(0')
   active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
   active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
   active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
   active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
   active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
   active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
   active(U105(X1, X2)) -> U105(active(X1), X2)
   active(U106(X)) -> U106(active(X))
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(U12(X1, X2)) -> U12(active(X1), X2)
   active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
   active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
   active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
   active(U114(X1, X2)) -> U114(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(U13(X)) -> U13(active(X))
   active(U121(X1, X2)) -> U121(active(X1), X2)
   active(U122(X)) -> U122(active(X))
   active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
   active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
   active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
   active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
   active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
   active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
   active(take(X1, X2)) -> take(active(X1), X2)
   active(take(X1, X2)) -> take(X1, active(X2))
   active(U21(X1, X2)) -> U21(active(X1), X2)
   active(U22(X1, X2)) -> U22(active(X1), X2)
   active(U23(X)) -> U23(active(X))
   active(U31(X1, X2)) -> U31(active(X1), X2)
   active(U32(X1, X2)) -> U32(active(X1), X2)
   active(U33(X)) -> U33(active(X))
   active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
   active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
   active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
   active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
   active(U45(X1, X2)) -> U45(active(X1), X2)
   active(U46(X)) -> U46(active(X))
   active(U51(X1, X2)) -> U51(active(X1), X2)
   active(U52(X)) -> U52(active(X))
   active(U61(X1, X2)) -> U61(active(X1), X2)
   active(U62(X)) -> U62(active(X))
   active(U71(X)) -> U71(active(X))
   active(U81(X)) -> U81(active(X))
   active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
   active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
   active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
   active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
   active(U95(X1, X2)) -> U95(active(X1), X2)
   active(U96(X)) -> U96(active(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
   U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
   U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
   U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
   U105(mark(X1), X2) -> mark(U105(X1, X2))
   U106(mark(X)) -> mark(U106(X))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   U12(mark(X1), X2) -> mark(U12(X1, X2))
   U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
   U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
   U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
   U114(mark(X1), X2) -> mark(U114(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   U13(mark(X)) -> mark(U13(X))
   U121(mark(X1), X2) -> mark(U121(X1, X2))
   U122(mark(X)) -> mark(U122(X))
   U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
   U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
   U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
   U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
   U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
   U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
   take(mark(X1), X2) -> mark(take(X1, X2))
   take(X1, mark(X2)) -> mark(take(X1, X2))
   U21(mark(X1), X2) -> mark(U21(X1, X2))
   U22(mark(X1), X2) -> mark(U22(X1, X2))
   U23(mark(X)) -> mark(U23(X))
   U31(mark(X1), X2) -> mark(U31(X1, X2))
   U32(mark(X1), X2) -> mark(U32(X1, X2))
   U33(mark(X)) -> mark(U33(X))
   U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
   U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
   U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
   U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
   U45(mark(X1), X2) -> mark(U45(X1, X2))
   U46(mark(X)) -> mark(U46(X))
   U51(mark(X1), X2) -> mark(U51(X1, X2))
   U52(mark(X)) -> mark(U52(X))
   U61(mark(X1), X2) -> mark(U61(X1, X2))
   U62(mark(X)) -> mark(U62(X))
   U71(mark(X)) -> mark(U71(X))
   U81(mark(X)) -> mark(U81(X))
   U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
   U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
   U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
   U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
   U95(mark(X1), X2) -> mark(U95(X1, X2))
   U96(mark(X)) -> mark(U96(X))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
   proper(tt) -> ok(tt)
   proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
   proper(isNatKind(X)) -> isNatKind(proper(X))
   proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
   proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
   proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
   proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(U106(X)) -> U106(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
   proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
   proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
   proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
   proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(U13(X)) -> U13(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
   proper(U122(X)) -> U122(proper(X))
   proper(nil) -> ok(nil)
   proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
   proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
   proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
   proper(U23(X)) -> U23(proper(X))
   proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
   proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
   proper(U33(X)) -> U33(proper(X))
   proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
   proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
   proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
   proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
   proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
   proper(U46(X)) -> U46(proper(X))
   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
   proper(U52(X)) -> U52(proper(X))
   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
   proper(U62(X)) -> U62(proper(X))
   proper(U71(X)) -> U71(proper(X))
   proper(U81(X)) -> U81(proper(X))
   proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
   proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
   proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
   proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
   proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
   proper(U96(X)) -> U96(proper(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
   U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
   isNatKind(ok(X)) -> ok(isNatKind(X))
   U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
   isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
   U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
   U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   U106(ok(X)) -> ok(U106(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
   U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
   U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
   U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
   U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   U13(ok(X)) -> ok(U13(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
   U122(ok(X)) -> ok(U122(X))
   U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
   U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
   U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
   U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
   U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
   U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
   U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
   U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
   U23(ok(X)) -> ok(U23(X))
   U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
   U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
   U33(ok(X)) -> ok(U33(X))
   U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
   U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
   U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
   U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
   U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
   U46(ok(X)) -> ok(U46(X))
   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
   U52(ok(X)) -> ok(U52(X))
   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
   U62(ok(X)) -> ok(U62(X))
   U71(ok(X)) -> ok(U71(X))
   U81(ok(X)) -> ok(U81(X))
   U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
   U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
   U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
   U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
   U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
   U96(ok(X)) -> ok(U96(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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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, U102, isNatKind, U103, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
cons < active
U102 < active
isNatKind < active
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
cons < proper
U102 < proper
isNatKind < proper
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(6)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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, U102, isNatKind, U103, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
cons < active
U102 < active
isNatKind < active
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
cons < proper
U102 < proper
isNatKind < proper
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < 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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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, U102, isNatKind, U103, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
cons < active
U102 < active
isNatKind < active
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
cons < proper
U102 < proper
isNatKind < proper
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < 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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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:
U102, active, isNatKind, U103, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U102 < active
isNatKind < active
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U102 < proper
isNatKind < proper
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)

Induction Base:
U102(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:
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n4360_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_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).
----------------------------------------

(14)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_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, U103, isNatIListKind, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNatKind < active
U103 < active
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNatKind < proper
U103 < proper
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)

Induction Base:
U103(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:
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n12441_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_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).
----------------------------------------

(16)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_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, U104, U105, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNatIListKind < active
U104 < active
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNatIListKind < proper
U104 < proper
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)

Induction Base:
U104(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:
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n21138_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_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).
----------------------------------------

(18)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_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:
U105, active, isNat, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U105 < active
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U105 < proper
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)

Induction Base:
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n30421_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_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, U106, isNatIList, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNat < active
U106 < active
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNat < proper
U106 < proper
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)

Induction Base:
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n36329_0, 1)))) ->_R^Omega(1)
mark(U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_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, U12, U112, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNatIList < active
U12 < active
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNatIList < proper
U12 < proper
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)

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

Induction Step:
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n38371_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_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:
U112, active, U113, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U112 < active
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U112 < proper
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)

Induction Base:
U112(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:
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n44758_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_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).
----------------------------------------

(26)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_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:
U113, active, U114, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U113 < active
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U113 < proper
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)

Induction Base:
U113(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:
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n55889_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_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).
----------------------------------------

(28)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_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:
U114, active, s, length, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U114 < active
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U114 < proper
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)

Induction Base:
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n67629_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
mark(*4_0)

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

(30)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_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, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
s < active
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
s < proper
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_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, +(n75130_0, 1)))) ->_R^Omega(1)
mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_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, U13, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
length < active
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
length < proper
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_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, +(n77933_0, 1)))) ->_R^Omega(1)
mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_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:
U13, active, isNatList, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U13 < active
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U13 < proper
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)

Induction Base:
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n80837_0, 1)))) ->_R^Omega(1)
mark(U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_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).
----------------------------------------

(36)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_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, U122, U132, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
isNatList < active
U122 < active
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
isNatList < proper
U122 < proper
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)

Induction Base:
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n83913_0, 1)))) ->_R^Omega(1)
mark(U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_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).
----------------------------------------

(38)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_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:
U132, active, U133, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U132 < active
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U132 < proper
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)

Induction Base:
U132(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:
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n87019_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(U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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).
----------------------------------------

(40)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_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:
U133, active, U134, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U133 < active
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U133 < proper
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)

Induction Base:
U133(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:
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n106522_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(U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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).
----------------------------------------

(42)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_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:
U134, active, U135, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U134 < active
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U134 < proper
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)

Induction Base:
U134(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:
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n127035_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(U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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).
----------------------------------------

(44)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_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:
U135, active, U136, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U135 < active
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U135 < proper
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)

Induction Base:
U135(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:
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n148558_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(U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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).
----------------------------------------

(46)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_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:
U136, active, take, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U136 < active
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U136 < proper
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)

Induction Base:
U136(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:
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n171091_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(U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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).
----------------------------------------

(48)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_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, U22, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
take < active
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
take < proper
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(49) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_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, +(n194634_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, n194634_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).
----------------------------------------

(50)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)


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


The following defined symbols remain to be analysed:
U22, active, U23, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U22 < active
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U22 < proper
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)

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

Induction Step:
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n205982_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)


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


The following defined symbols remain to be analysed:
U23, active, U32, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U23 < active
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U23 < proper
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)

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

Induction Step:
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n217437_0, 1)))) ->_R^Omega(1)
mark(U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_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(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_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:
U32, active, U33, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U32 < active
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U32 < proper
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)

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

Induction Step:
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n222184_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_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).
----------------------------------------

(56)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_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:
U33, active, U42, U43, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U33 < active
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U33 < proper
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)

Induction Base:
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n234153_0, 1)))) ->_R^Omega(1)
mark(U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_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).
----------------------------------------

(58)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_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, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U42 < active
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U42 < proper
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)

Induction Base:
U42(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:
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n239151_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_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).
----------------------------------------

(60)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_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, U44, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U43 < active
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U43 < proper
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)

Induction Base:
U43(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:
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n259018_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_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).
----------------------------------------

(62)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_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:
U44, active, U45, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U44 < active
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U44 < proper
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)

Induction Base:
U44(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:
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n279494_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_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).
----------------------------------------

(64)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_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:
U45, active, U46, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U45 < active
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U45 < proper
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)

Induction Base:
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n300579_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_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).
----------------------------------------

(66)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_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:
U46, active, U52, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U46 < active
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U46 < proper
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)

Induction Base:
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n314274_0, 1)))) ->_R^Omega(1)
mark(U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_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).
----------------------------------------

(68)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_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:
U52, active, U62, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U52 < active
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U52 < proper
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)

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

Induction Step:
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n320120_0, 1)))) ->_R^Omega(1)
mark(U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_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).
----------------------------------------

(70)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_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, U92, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U62 < active
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U62 < proper
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)

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

Induction Step:
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n326067_0, 1)))) ->_R^Omega(1)
mark(U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_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).
----------------------------------------

(72)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_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:
U92, active, U93, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U92 < active
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U92 < proper
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(73) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)

Induction Base:
U92(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:
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n332115_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_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).
----------------------------------------

(74)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_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:
U93, active, U94, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U93 < active
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U93 < proper
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(75) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)

Induction Base:
U93(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:
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n355216_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_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).
----------------------------------------

(76)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_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:
U94, active, U95, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U94 < active
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U94 < proper
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(77) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)

Induction Base:
U94(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:
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n378926_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_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).
----------------------------------------

(78)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_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:
U95, active, U96, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U95 < active
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U95 < proper
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(79) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)

Induction Base:
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n403245_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_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).
----------------------------------------

(80)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_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:
U96, active, U11, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U96 < active
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U96 < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(81) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)

Induction Base:
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))

Induction Step:
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n419082_0, 1)))) ->_R^Omega(1)
mark(U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_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).
----------------------------------------

(82)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)


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


The following defined symbols remain to be analysed:
U11, active, U21, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U11 < active
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U11 < proper
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(83) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_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, +(n425978_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, n425978_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).
----------------------------------------

(84)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)


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


The following defined symbols remain to be analysed:
U21, active, U31, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U21 < active
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U21 < proper
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(85) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_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, +(n442329_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, n442329_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).
----------------------------------------

(86)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)


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


The following defined symbols remain to be analysed:
U31, active, U41, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U31 < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U31 < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(87) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_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, +(n458986_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, n458986_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).
----------------------------------------

(88)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)


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


The following defined symbols remain to be analysed:
U41, active, U51, U61, U71, U81, U91, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(89) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_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, +(n475949_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, n475949_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).
----------------------------------------

(90)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_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, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U51 < active
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U51 < proper
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(91) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_0)

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

Induction Step:
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n503040_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_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).
----------------------------------------

(92)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_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, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U61 < active
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U61 < proper
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(93) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n520713_0)

Induction Base:
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n520713_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_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).
----------------------------------------

(94)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n520713_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, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U71 < active
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U71 < proper
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(95) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n538692_0))) -> *4_0, rt in Omega(n538692_0)

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

Induction Step:
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n538692_0, 1)))) ->_R^Omega(1)
mark(U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n538692_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).
----------------------------------------

(96)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n520713_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n538692_0))) -> *4_0, rt in Omega(n538692_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, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U81 < active
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U81 < proper
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(97) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n546638_0))) -> *4_0, rt in Omega(n546638_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, +(n546638_0, 1)))) ->_R^Omega(1)
mark(U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n546638_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).
----------------------------------------

(98)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n520713_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n538692_0))) -> *4_0, rt in Omega(n538692_0)
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n546638_0))) -> *4_0, rt in Omega(n546638_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, U101, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U91 < active
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U91 < proper
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(99) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n554685_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n554685_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))

Induction Step:
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n554685_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n554685_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).
----------------------------------------

(100)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n520713_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n538692_0))) -> *4_0, rt in Omega(n538692_0)
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n546638_0))) -> *4_0, rt in Omega(n546638_0)
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n554685_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n554685_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:
U101, active, U111, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U101 < active
U111 < active
U121 < active
U131 < active
active < top
U101 < proper
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(101) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U101(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n583939_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n583939_0)

Induction Base:
U101(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:
U101(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n583939_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U101(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n583939_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).
----------------------------------------

(102)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n520713_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n538692_0))) -> *4_0, rt in Omega(n538692_0)
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n546638_0))) -> *4_0, rt in Omega(n546638_0)
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n554685_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n554685_0)
U101(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n583939_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n583939_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:
U111, active, U121, U131, proper, top

They will be analysed ascendingly in the following order:
U111 < active
U121 < active
U131 < active
active < top
U111 < proper
U121 < proper
U131 < proper
proper < top

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

(103) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U111(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n613802_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n613802_0)

Induction Base:
U111(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:
U111(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n613802_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
mark(U111(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n613802_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).
----------------------------------------

(104)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n520713_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n538692_0))) -> *4_0, rt in Omega(n538692_0)
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n546638_0))) -> *4_0, rt in Omega(n546638_0)
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n554685_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n554685_0)
U101(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n583939_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n583939_0)
U111(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n613802_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n613802_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:
U121, active, U131, proper, top

They will be analysed ascendingly in the following order:
U121 < active
U131 < active
active < top
U121 < proper
U131 < proper
proper < top

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

(105) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U121(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n644274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n644274_0)

Induction Base:
U121(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))

Induction Step:
U121(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n644274_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
mark(U121(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n644274_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).
----------------------------------------

(106)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n520713_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n538692_0))) -> *4_0, rt in Omega(n538692_0)
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n546638_0))) -> *4_0, rt in Omega(n546638_0)
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n554685_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n554685_0)
U101(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n583939_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n583939_0)
U111(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n613802_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n613802_0)
U121(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n644274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n644274_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:
U131, active, proper, top

They will be analysed ascendingly in the following order:
U131 < active
active < top
U131 < proper
proper < top

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

(107) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U131(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n664187_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(n664187_0)

Induction Base:
U131(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:
U131(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n664187_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(U131(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n664187_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).
----------------------------------------

(108)
Obligation:
TRS:
Rules:
active(zeros) -> mark(cons(0', zeros))
active(U101(tt, V1, V2)) -> mark(U102(isNatKind(V1), V1, V2))
active(U102(tt, V1, V2)) -> mark(U103(isNatIListKind(V2), V1, V2))
active(U103(tt, V1, V2)) -> mark(U104(isNatIListKind(V2), V1, V2))
active(U104(tt, V1, V2)) -> mark(U105(isNat(V1), V2))
active(U105(tt, V2)) -> mark(U106(isNatIList(V2)))
active(U106(tt)) -> mark(tt)
active(U11(tt, V1)) -> mark(U12(isNatIListKind(V1), V1))
active(U111(tt, L, N)) -> mark(U112(isNatIListKind(L), L, N))
active(U112(tt, L, N)) -> mark(U113(isNat(N), L, N))
active(U113(tt, L, N)) -> mark(U114(isNatKind(N), L))
active(U114(tt, L)) -> mark(s(length(L)))
active(U12(tt, V1)) -> mark(U13(isNatList(V1)))
active(U121(tt, IL)) -> mark(U122(isNatIListKind(IL)))
active(U122(tt)) -> mark(nil)
active(U13(tt)) -> mark(tt)
active(U131(tt, IL, M, N)) -> mark(U132(isNatIListKind(IL), IL, M, N))
active(U132(tt, IL, M, N)) -> mark(U133(isNat(M), IL, M, N))
active(U133(tt, IL, M, N)) -> mark(U134(isNatKind(M), IL, M, N))
active(U134(tt, IL, M, N)) -> mark(U135(isNat(N), IL, M, N))
active(U135(tt, IL, M, N)) -> mark(U136(isNatKind(N), IL, M, N))
active(U136(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
active(U21(tt, V1)) -> mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) -> mark(U23(isNat(V1)))
active(U23(tt)) -> mark(tt)
active(U31(tt, V)) -> mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) -> mark(U33(isNatList(V)))
active(U33(tt)) -> mark(tt)
active(U41(tt, V1, V2)) -> mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) -> mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) -> mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) -> mark(U45(isNat(V1), V2))
active(U45(tt, V2)) -> mark(U46(isNatIList(V2)))
active(U46(tt)) -> mark(tt)
active(U51(tt, V2)) -> mark(U52(isNatIListKind(V2)))
active(U52(tt)) -> mark(tt)
active(U61(tt, V2)) -> mark(U62(isNatIListKind(V2)))
active(U62(tt)) -> mark(tt)
active(U71(tt)) -> mark(tt)
active(U81(tt)) -> mark(tt)
active(U91(tt, V1, V2)) -> mark(U92(isNatKind(V1), V1, V2))
active(U92(tt, V1, V2)) -> mark(U93(isNatIListKind(V2), V1, V2))
active(U93(tt, V1, V2)) -> mark(U94(isNatIListKind(V2), V1, V2))
active(U94(tt, V1, V2)) -> mark(U95(isNat(V1), V2))
active(U95(tt, V2)) -> mark(U96(isNatList(V2)))
active(U96(tt)) -> mark(tt)
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(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) -> mark(tt)
active(isNatIListKind(zeros)) -> mark(tt)
active(isNatIListKind(cons(V1, V2))) -> mark(U51(isNatKind(V1), V2))
active(isNatIListKind(take(V1, V2))) -> mark(U61(isNatKind(V1), V2))
active(isNatKind(0')) -> mark(tt)
active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
active(isNatList(nil)) -> mark(tt)
active(isNatList(cons(V1, V2))) -> mark(U91(isNatKind(V1), V1, V2))
active(isNatList(take(V1, V2))) -> mark(U101(isNatKind(V1), V1, V2))
active(length(nil)) -> mark(0')
active(length(cons(N, L))) -> mark(U111(isNatList(L), L, N))
active(take(0', IL)) -> mark(U121(isNatIList(IL), IL))
active(take(s(M), cons(N, IL))) -> mark(U131(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(U101(X1, X2, X3)) -> U101(active(X1), X2, X3)
active(U102(X1, X2, X3)) -> U102(active(X1), X2, X3)
active(U103(X1, X2, X3)) -> U103(active(X1), X2, X3)
active(U104(X1, X2, X3)) -> U104(active(X1), X2, X3)
active(U105(X1, X2)) -> U105(active(X1), X2)
active(U106(X)) -> U106(active(X))
active(U11(X1, X2)) -> U11(active(X1), X2)
active(U12(X1, X2)) -> U12(active(X1), X2)
active(U111(X1, X2, X3)) -> U111(active(X1), X2, X3)
active(U112(X1, X2, X3)) -> U112(active(X1), X2, X3)
active(U113(X1, X2, X3)) -> U113(active(X1), X2, X3)
active(U114(X1, X2)) -> U114(active(X1), X2)
active(s(X)) -> s(active(X))
active(length(X)) -> length(active(X))
active(U13(X)) -> U13(active(X))
active(U121(X1, X2)) -> U121(active(X1), X2)
active(U122(X)) -> U122(active(X))
active(U131(X1, X2, X3, X4)) -> U131(active(X1), X2, X3, X4)
active(U132(X1, X2, X3, X4)) -> U132(active(X1), X2, X3, X4)
active(U133(X1, X2, X3, X4)) -> U133(active(X1), X2, X3, X4)
active(U134(X1, X2, X3, X4)) -> U134(active(X1), X2, X3, X4)
active(U135(X1, X2, X3, X4)) -> U135(active(X1), X2, X3, X4)
active(U136(X1, X2, X3, X4)) -> U136(active(X1), X2, X3, X4)
active(take(X1, X2)) -> take(active(X1), X2)
active(take(X1, X2)) -> take(X1, active(X2))
active(U21(X1, X2)) -> U21(active(X1), X2)
active(U22(X1, X2)) -> U22(active(X1), X2)
active(U23(X)) -> U23(active(X))
active(U31(X1, X2)) -> U31(active(X1), X2)
active(U32(X1, X2)) -> U32(active(X1), X2)
active(U33(X)) -> U33(active(X))
active(U41(X1, X2, X3)) -> U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) -> U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) -> U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) -> U44(active(X1), X2, X3)
active(U45(X1, X2)) -> U45(active(X1), X2)
active(U46(X)) -> U46(active(X))
active(U51(X1, X2)) -> U51(active(X1), X2)
active(U52(X)) -> U52(active(X))
active(U61(X1, X2)) -> U61(active(X1), X2)
active(U62(X)) -> U62(active(X))
active(U71(X)) -> U71(active(X))
active(U81(X)) -> U81(active(X))
active(U91(X1, X2, X3)) -> U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) -> U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) -> U93(active(X1), X2, X3)
active(U94(X1, X2, X3)) -> U94(active(X1), X2, X3)
active(U95(X1, X2)) -> U95(active(X1), X2)
active(U96(X)) -> U96(active(X))
cons(mark(X1), X2) -> mark(cons(X1, X2))
U101(mark(X1), X2, X3) -> mark(U101(X1, X2, X3))
U102(mark(X1), X2, X3) -> mark(U102(X1, X2, X3))
U103(mark(X1), X2, X3) -> mark(U103(X1, X2, X3))
U104(mark(X1), X2, X3) -> mark(U104(X1, X2, X3))
U105(mark(X1), X2) -> mark(U105(X1, X2))
U106(mark(X)) -> mark(U106(X))
U11(mark(X1), X2) -> mark(U11(X1, X2))
U12(mark(X1), X2) -> mark(U12(X1, X2))
U111(mark(X1), X2, X3) -> mark(U111(X1, X2, X3))
U112(mark(X1), X2, X3) -> mark(U112(X1, X2, X3))
U113(mark(X1), X2, X3) -> mark(U113(X1, X2, X3))
U114(mark(X1), X2) -> mark(U114(X1, X2))
s(mark(X)) -> mark(s(X))
length(mark(X)) -> mark(length(X))
U13(mark(X)) -> mark(U13(X))
U121(mark(X1), X2) -> mark(U121(X1, X2))
U122(mark(X)) -> mark(U122(X))
U131(mark(X1), X2, X3, X4) -> mark(U131(X1, X2, X3, X4))
U132(mark(X1), X2, X3, X4) -> mark(U132(X1, X2, X3, X4))
U133(mark(X1), X2, X3, X4) -> mark(U133(X1, X2, X3, X4))
U134(mark(X1), X2, X3, X4) -> mark(U134(X1, X2, X3, X4))
U135(mark(X1), X2, X3, X4) -> mark(U135(X1, X2, X3, X4))
U136(mark(X1), X2, X3, X4) -> mark(U136(X1, X2, X3, X4))
take(mark(X1), X2) -> mark(take(X1, X2))
take(X1, mark(X2)) -> mark(take(X1, X2))
U21(mark(X1), X2) -> mark(U21(X1, X2))
U22(mark(X1), X2) -> mark(U22(X1, X2))
U23(mark(X)) -> mark(U23(X))
U31(mark(X1), X2) -> mark(U31(X1, X2))
U32(mark(X1), X2) -> mark(U32(X1, X2))
U33(mark(X)) -> mark(U33(X))
U41(mark(X1), X2, X3) -> mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) -> mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) -> mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) -> mark(U44(X1, X2, X3))
U45(mark(X1), X2) -> mark(U45(X1, X2))
U46(mark(X)) -> mark(U46(X))
U51(mark(X1), X2) -> mark(U51(X1, X2))
U52(mark(X)) -> mark(U52(X))
U61(mark(X1), X2) -> mark(U61(X1, X2))
U62(mark(X)) -> mark(U62(X))
U71(mark(X)) -> mark(U71(X))
U81(mark(X)) -> mark(U81(X))
U91(mark(X1), X2, X3) -> mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) -> mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) -> mark(U93(X1, X2, X3))
U94(mark(X1), X2, X3) -> mark(U94(X1, X2, X3))
U95(mark(X1), X2) -> mark(U95(X1, X2))
U96(mark(X)) -> mark(U96(X))
proper(zeros) -> ok(zeros)
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(U101(X1, X2, X3)) -> U101(proper(X1), proper(X2), proper(X3))
proper(tt) -> ok(tt)
proper(U102(X1, X2, X3)) -> U102(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) -> isNatKind(proper(X))
proper(U103(X1, X2, X3)) -> U103(proper(X1), proper(X2), proper(X3))
proper(isNatIListKind(X)) -> isNatIListKind(proper(X))
proper(U104(X1, X2, X3)) -> U104(proper(X1), proper(X2), proper(X3))
proper(U105(X1, X2)) -> U105(proper(X1), proper(X2))
proper(isNat(X)) -> isNat(proper(X))
proper(U106(X)) -> U106(proper(X))
proper(isNatIList(X)) -> isNatIList(proper(X))
proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
proper(U111(X1, X2, X3)) -> U111(proper(X1), proper(X2), proper(X3))
proper(U112(X1, X2, X3)) -> U112(proper(X1), proper(X2), proper(X3))
proper(U113(X1, X2, X3)) -> U113(proper(X1), proper(X2), proper(X3))
proper(U114(X1, X2)) -> U114(proper(X1), proper(X2))
proper(s(X)) -> s(proper(X))
proper(length(X)) -> length(proper(X))
proper(U13(X)) -> U13(proper(X))
proper(isNatList(X)) -> isNatList(proper(X))
proper(U121(X1, X2)) -> U121(proper(X1), proper(X2))
proper(U122(X)) -> U122(proper(X))
proper(nil) -> ok(nil)
proper(U131(X1, X2, X3, X4)) -> U131(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U132(X1, X2, X3, X4)) -> U132(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U133(X1, X2, X3, X4)) -> U133(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U134(X1, X2, X3, X4)) -> U134(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U135(X1, X2, X3, X4)) -> U135(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U136(X1, X2, X3, X4)) -> U136(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))
proper(U21(X1, X2)) -> U21(proper(X1), proper(X2))
proper(U22(X1, X2)) -> U22(proper(X1), proper(X2))
proper(U23(X)) -> U23(proper(X))
proper(U31(X1, X2)) -> U31(proper(X1), proper(X2))
proper(U32(X1, X2)) -> U32(proper(X1), proper(X2))
proper(U33(X)) -> U33(proper(X))
proper(U41(X1, X2, X3)) -> U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) -> U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) -> U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) -> U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) -> U45(proper(X1), proper(X2))
proper(U46(X)) -> U46(proper(X))
proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
proper(U52(X)) -> U52(proper(X))
proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
proper(U62(X)) -> U62(proper(X))
proper(U71(X)) -> U71(proper(X))
proper(U81(X)) -> U81(proper(X))
proper(U91(X1, X2, X3)) -> U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) -> U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) -> U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2, X3)) -> U94(proper(X1), proper(X2), proper(X3))
proper(U95(X1, X2)) -> U95(proper(X1), proper(X2))
proper(U96(X)) -> U96(proper(X))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
U101(ok(X1), ok(X2), ok(X3)) -> ok(U101(X1, X2, X3))
U102(ok(X1), ok(X2), ok(X3)) -> ok(U102(X1, X2, X3))
isNatKind(ok(X)) -> ok(isNatKind(X))
U103(ok(X1), ok(X2), ok(X3)) -> ok(U103(X1, X2, X3))
isNatIListKind(ok(X)) -> ok(isNatIListKind(X))
U104(ok(X1), ok(X2), ok(X3)) -> ok(U104(X1, X2, X3))
U105(ok(X1), ok(X2)) -> ok(U105(X1, X2))
isNat(ok(X)) -> ok(isNat(X))
U106(ok(X)) -> ok(U106(X))
isNatIList(ok(X)) -> ok(isNatIList(X))
U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
U111(ok(X1), ok(X2), ok(X3)) -> ok(U111(X1, X2, X3))
U112(ok(X1), ok(X2), ok(X3)) -> ok(U112(X1, X2, X3))
U113(ok(X1), ok(X2), ok(X3)) -> ok(U113(X1, X2, X3))
U114(ok(X1), ok(X2)) -> ok(U114(X1, X2))
s(ok(X)) -> ok(s(X))
length(ok(X)) -> ok(length(X))
U13(ok(X)) -> ok(U13(X))
isNatList(ok(X)) -> ok(isNatList(X))
U121(ok(X1), ok(X2)) -> ok(U121(X1, X2))
U122(ok(X)) -> ok(U122(X))
U131(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U131(X1, X2, X3, X4))
U132(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U132(X1, X2, X3, X4))
U133(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U133(X1, X2, X3, X4))
U134(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U134(X1, X2, X3, X4))
U135(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U135(X1, X2, X3, X4))
U136(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U136(X1, X2, X3, X4))
take(ok(X1), ok(X2)) -> ok(take(X1, X2))
U21(ok(X1), ok(X2)) -> ok(U21(X1, X2))
U22(ok(X1), ok(X2)) -> ok(U22(X1, X2))
U23(ok(X)) -> ok(U23(X))
U31(ok(X1), ok(X2)) -> ok(U31(X1, X2))
U32(ok(X1), ok(X2)) -> ok(U32(X1, X2))
U33(ok(X)) -> ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) -> ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) -> ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) -> ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) -> ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) -> ok(U45(X1, X2))
U46(ok(X)) -> ok(U46(X))
U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
U52(ok(X)) -> ok(U52(X))
U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
U62(ok(X)) -> ok(U62(X))
U71(ok(X)) -> ok(U71(X))
U81(ok(X)) -> ok(U81(X))
U91(ok(X1), ok(X2), ok(X3)) -> ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) -> ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) -> ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2), ok(X3)) -> ok(U94(X1, X2, X3))
U95(ok(X1), ok(X2)) -> ok(U95(X1, X2))
U96(ok(X)) -> ok(U96(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
U101 :: zeros:0':mark:tt:nil:ok -> 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
U102 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> 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
U103 :: zeros:0':mark:tt:nil:ok -> 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
U104 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U105 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U106 :: 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
U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U111 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U112 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U113 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U114 :: 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
U13 :: 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
U121 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U122 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
nil :: zeros:0':mark:tt:nil:ok
U131 :: 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
U132 :: 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
U133 :: 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
U134 :: 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
U135 :: 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
U136 :: 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
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 -> zeros:0':mark:tt:nil:ok
U23 :: 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 -> zeros:0':mark:tt:nil:ok
U33 :: 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 -> zeros:0':mark:tt:nil:ok
U43 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U44 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U45 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U46 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U61 :: 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
U71 :: 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
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
U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U94 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U95 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
U96 :: 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)
U102(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4360_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n4360_0)
U103(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n12441_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n12441_0)
U104(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21138_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n21138_0)
U105(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n30421_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n30421_0)
U106(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n36329_0))) -> *4_0, rt in Omega(n36329_0)
U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n38371_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n38371_0)
U112(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n44758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n44758_0)
U113(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n55889_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n55889_0)
U114(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n67629_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n67629_0)
s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n75130_0))) -> *4_0, rt in Omega(n75130_0)
length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n77933_0))) -> *4_0, rt in Omega(n77933_0)
U13(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n80837_0))) -> *4_0, rt in Omega(n80837_0)
U122(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n83913_0))) -> *4_0, rt in Omega(n83913_0)
U132(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n87019_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(n87019_0)
U133(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n106522_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(n106522_0)
U134(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n127035_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(n127035_0)
U135(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n148558_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(n148558_0)
U136(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n171091_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(n171091_0)
take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n194634_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n194634_0)
U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n205982_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n205982_0)
U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n217437_0))) -> *4_0, rt in Omega(n217437_0)
U32(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n222184_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n222184_0)
U33(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n234153_0))) -> *4_0, rt in Omega(n234153_0)
U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n239151_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n239151_0)
U43(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n259018_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n259018_0)
U44(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n279494_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n279494_0)
U45(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n300579_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n300579_0)
U46(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n314274_0))) -> *4_0, rt in Omega(n314274_0)
U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n320120_0))) -> *4_0, rt in Omega(n320120_0)
U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n326067_0))) -> *4_0, rt in Omega(n326067_0)
U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n332115_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n332115_0)
U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n355216_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n355216_0)
U94(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n378926_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n378926_0)
U95(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n403245_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n403245_0)
U96(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n419082_0))) -> *4_0, rt in Omega(n419082_0)
U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n425978_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n425978_0)
U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n442329_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n442329_0)
U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n458986_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n458986_0)
U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n475949_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n475949_0)
U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n503040_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n503040_0)
U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n520713_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n520713_0)
U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n538692_0))) -> *4_0, rt in Omega(n538692_0)
U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n546638_0))) -> *4_0, rt in Omega(n546638_0)
U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n554685_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n554685_0)
U101(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n583939_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n583939_0)
U111(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n613802_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n613802_0)
U121(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n644274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n644274_0)
U131(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n664187_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(n664187_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