Runtime Complexity TRS:
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))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

Runtime Complexity TRS:
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))

Rewrite Strategy: INNERMOST

Infered types.

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':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'

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'

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)

Induction Base:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n6, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b610)) →_R^Ω(1)
mark'(cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n6)), _gen_zeros':0':mark':tt':nil':ok'3(_b610))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U102'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n15443)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n15443)

Induction Base:
U102'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U102'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n15444, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b16942), _gen_zeros':0':mark':tt':nil':ok'3(_c16943)) →_R^Ω(1)
mark'(U102'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n15444)), _gen_zeros':0':mark':tt':nil':ok'3(_b16942), _gen_zeros':0':mark':tt':nil':ok'3(_c16943))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Could not prove a rewrite lemma for the defined symbol isNatKind'.

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U103', active', 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:
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'
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'

Proved the following rewrite lemma:
U103'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n36830)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n36830)

Induction Base:
U103'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U103'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n36831, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b38977), _gen_zeros':0':mark':tt':nil':ok'3(_c38978)) →_R^Ω(1)
mark'(U103'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n36831)), _gen_zeros':0':mark':tt':nil':ok'3(_b38977), _gen_zeros':0':mark':tt':nil':ok'3(_c38978))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Could not prove a rewrite lemma for the defined symbol isNatIListKind'.

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U104', active', 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:
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'
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'

Proved the following rewrite lemma:
U104'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n58963)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n58963)

Induction Base:
U104'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U104'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n58964, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b61758), _gen_zeros':0':mark':tt':nil':ok'3(_c61759)) →_R^Ω(1)
mark'(U104'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n58964)), _gen_zeros':0':mark':tt':nil':ok'3(_b61758), _gen_zeros':0':mark':tt':nil':ok'3(_c61759))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U105'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n81784)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n81784)

Induction Base:
U105'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U105'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n81785, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b84009)) →_R^Ω(1)
mark'(U105'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n81785)), _gen_zeros':0':mark':tt':nil':ok'3(_b84009))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Could not prove a rewrite lemma for the defined symbol isNat'.

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U106', active', 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:
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'
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'

Proved the following rewrite lemma:
U106'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n99143))) → _*4, rt ∈ Ω(_n99143)

Induction Base:
U106'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U106'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n99144, 1)))) →_R^Ω(1)
mark'(U106'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n99144)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Could not prove a rewrite lemma for the defined symbol isNatIList'.

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U12', active', 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:
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'
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'

Proved the following rewrite lemma:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n110551)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n110551)

Induction Base:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n110552, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b113316)) →_R^Ω(1)
mark'(U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n110552)), _gen_zeros':0':mark':tt':nil':ok'3(_b113316))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U112'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n128451)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n128451)

Induction Base:
U112'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U112'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n128452, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b133190), _gen_zeros':0':mark':tt':nil':ok'3(_c133191)) →_R^Ω(1)
mark'(U112'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n128452)), _gen_zeros':0':mark':tt':nil':ok'3(_b133190), _gen_zeros':0':mark':tt':nil':ok'3(_c133191))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U113'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n153452)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n153452)

Induction Base:
U113'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U113'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n153453, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b158839), _gen_zeros':0':mark':tt':nil':ok'3(_c158840)) →_R^Ω(1)
mark'(U113'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n153453)), _gen_zeros':0':mark':tt':nil':ok'3(_b158839), _gen_zeros':0':mark':tt':nil':ok'3(_c158840))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U114'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n179185)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n179185)

Induction Base:
U114'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U114'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n179186, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b183138)) →_R^Ω(1)
mark'(U114'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n179186)), _gen_zeros':0':mark':tt':nil':ok'3(_b183138))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n198443))) → _*4, rt ∈ Ω(_n198443)

Induction Base:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n198444, 1)))) →_R^Ω(1)
mark'(s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n198444)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n210799))) → _*4, rt ∈ Ω(_n210799)

Induction Base:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n210800, 1)))) →_R^Ω(1)
mark'(length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n210800)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U13'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n223279))) → _*4, rt ∈ Ω(_n223279)

Induction Base:
U13'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U13'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n223280, 1)))) →_R^Ω(1)
mark'(U13'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n223280)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Could not prove a rewrite lemma for the defined symbol isNatList'.

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))

The following defined symbols remain to be analysed:
U122', active', 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:
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'
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'

Proved the following rewrite lemma:
U122'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n236023))) → _*4, rt ∈ Ω(_n236023)

Induction Base:
U122'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U122'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n236024, 1)))) →_R^Ω(1)
mark'(U122'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n236024)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U132'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n248751)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n248751)

Induction Base:
U132'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U132'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n248752, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b259460), _gen_zeros':0':mark':tt':nil':ok'3(_c259461), _gen_zeros':0':mark':tt':nil':ok'3(_d259462)) →_R^Ω(1)
mark'(U132'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n248752)), _gen_zeros':0':mark':tt':nil':ok'3(_b259460), _gen_zeros':0':mark':tt':nil':ok'3(_c259461), _gen_zeros':0':mark':tt':nil':ok'3(_d259462))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U133'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n285273)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n285273)

Induction Base:
U133'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U133'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n285274, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b297062), _gen_zeros':0':mark':tt':nil':ok'3(_c297063), _gen_zeros':0':mark':tt':nil':ok'3(_d297064)) →_R^Ω(1)
mark'(U133'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n285274)), _gen_zeros':0':mark':tt':nil':ok'3(_b297062), _gen_zeros':0':mark':tt':nil':ok'3(_c297063), _gen_zeros':0':mark':tt':nil':ok'3(_d297064))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U134'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n323011)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n323011)

Induction Base:
U134'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U134'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n323012, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b335880), _gen_zeros':0':mark':tt':nil':ok'3(_c335881), _gen_zeros':0':mark':tt':nil':ok'3(_d335882)) →_R^Ω(1)
mark'(U134'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n323012)), _gen_zeros':0':mark':tt':nil':ok'3(_b335880), _gen_zeros':0':mark':tt':nil':ok'3(_c335881), _gen_zeros':0':mark':tt':nil':ok'3(_d335882))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U135'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n361965)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n361965)

Induction Base:
U135'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U135'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n361966, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b375914), _gen_zeros':0':mark':tt':nil':ok'3(_c375915), _gen_zeros':0':mark':tt':nil':ok'3(_d375916)) →_R^Ω(1)
mark'(U135'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n361966)), _gen_zeros':0':mark':tt':nil':ok'3(_b375914), _gen_zeros':0':mark':tt':nil':ok'3(_c375915), _gen_zeros':0':mark':tt':nil':ok'3(_d375916))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U136'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n402135)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n402135)

Induction Base:
U136'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U136'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n402136, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b417164), _gen_zeros':0':mark':tt':nil':ok'3(_c417165), _gen_zeros':0':mark':tt':nil':ok'3(_d417166)) →_R^Ω(1)
mark'(U136'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n402136)), _gen_zeros':0':mark':tt':nil':ok'3(_b417164), _gen_zeros':0':mark':tt':nil':ok'3(_c417165), _gen_zeros':0':mark':tt':nil':ok'3(_d417166))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n443521)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n443521)

Induction Base:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n443522, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b451578)) →_R^Ω(1)
mark'(take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n443522)), _gen_zeros':0':mark':tt':nil':ok'3(_b451578))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n467440)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n467440)

Induction Base:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n467441, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b475605)) →_R^Ω(1)
mark'(U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n467441)), _gen_zeros':0':mark':tt':nil':ok'3(_b475605))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n491508))) → _*4, rt ∈ Ω(_n491508)

Induction Base:
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n491509, 1)))) →_R^Ω(1)
mark'(U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n491509)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n506367)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n506367)

Induction Base:
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n506368, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b515072)) →_R^Ω(1)
mark'(U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n506368)), _gen_zeros':0':mark':tt':nil':ok'3(_b515072))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U33'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n531044))) → _*4, rt ∈ Ω(_n531044)

Induction Base:
U33'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U33'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n531045, 1)))) →_R^Ω(1)
mark'(U33'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n531045)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n546218)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n546218)

Induction Base:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n546219, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b560191), _gen_zeros':0':mark':tt':nil':ok'3(_c560192)) →_R^Ω(1)
mark'(U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n546219)), _gen_zeros':0':mark':tt':nil':ok'3(_b560191), _gen_zeros':0':mark':tt':nil':ok'3(_c560192))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n581606)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n581606)

Induction Base:
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n581607, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b596227), _gen_zeros':0':mark':tt':nil':ok'3(_c596228)) →_R^Ω(1)
mark'(U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n581607)), _gen_zeros':0':mark':tt':nil':ok'3(_b596227), _gen_zeros':0':mark':tt':nil':ok'3(_c596228))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U44'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n617726)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n617726)

Induction Base:
U44'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U44'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n617727, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b632995), _gen_zeros':0':mark':tt':nil':ok'3(_c632996)) →_R^Ω(1)
mark'(U44'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n617727)), _gen_zeros':0':mark':tt':nil':ok'3(_b632995), _gen_zeros':0':mark':tt':nil':ok'3(_c632996))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U45'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n654578)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n654578)

Induction Base:
U45'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U45'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n654579, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b665119)) →_R^Ω(1)
mark'(U45'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n654579)), _gen_zeros':0':mark':tt':nil':ok'3(_b665119))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U46'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n681349))) → _*4, rt ∈ Ω(_n681349)

Induction Base:
U46'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U46'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n681350, 1)))) →_R^Ω(1)
mark'(U46'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n681350)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n697612))) → _*4, rt ∈ Ω(_n697612)

Induction Base:
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n697613, 1)))) →_R^Ω(1)
mark'(U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n697613)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n713999))) → _*4, rt ∈ Ω(_n713999)

Induction Base:
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n714000, 1)))) →_R^Ω(1)
mark'(U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n714000)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n730510)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n730510)

Induction Base:
U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n730511, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b747885), _gen_zeros':0':mark':tt':nil':ok'3(_c747886)) →_R^Ω(1)
mark'(U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n730511)), _gen_zeros':0':mark':tt':nil':ok'3(_b747885), _gen_zeros':0':mark':tt':nil':ok'3(_c747886))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n769713)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n769713)

Induction Base:
U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n769714, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b787736), _gen_zeros':0':mark':tt':nil':ok'3(_c787737)) →_R^Ω(1)
mark'(U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n769714)), _gen_zeros':0':mark':tt':nil':ok'3(_b787736), _gen_zeros':0':mark':tt':nil':ok'3(_c787737))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U94'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n809648)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n809648)

Induction Base:
U94'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U94'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n809649, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b828319), _gen_zeros':0':mark':tt':nil':ok'3(_c828320)) →_R^Ω(1)
mark'(U94'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n809649)), _gen_zeros':0':mark':tt':nil':ok'3(_b828319), _gen_zeros':0':mark':tt':nil':ok'3(_c828320))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U95'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n850315)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n850315)

Induction Base:
U95'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U95'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n850316, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b863124)) →_R^Ω(1)
mark'(U95'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n850316)), _gen_zeros':0':mark':tt':nil':ok'3(_b863124))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n5)
U102'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n15443)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n15443)
U103'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n36830)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n36830)
U104'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n58963)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n58963)
U105'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n81784)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n81784)
U106'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n99143))) → _*4, rt ∈ Ω(_n99143)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n110551)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n110551)
U112'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n128451)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n128451)
U113'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n153452)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n153452)
U114'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n179185)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n179185)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n198443))) → _*4, rt ∈ Ω(_n198443)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n210799))) → _*4, rt ∈ Ω(_n210799)
U13'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n223279))) → _*4, rt ∈ Ω(_n223279)
U122'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n236023))) → _*4, rt ∈ Ω(_n236023)
U132'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n248751)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n248751)
U133'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n285273)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n285273)
U134'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n323011)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n323011)
U135'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n361965)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n361965)
U136'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n402135)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n402135)
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n443521)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n443521)
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n467440)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n467440)
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n491508))) → _*4, rt ∈ Ω(_n491508)
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n506367)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n506367)
U33'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n531044))) → _*4, rt ∈ Ω(_n531044)
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n546218)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n546218)
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n581606)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n581606)
U44'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n617726)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n617726)
U45'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n654578)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n654578)
U46'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n681349))) → _*4, rt ∈ Ω(_n681349)
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n697612))) → _*4, rt ∈ Ω(_n697612)
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n713999))) → _*4, rt ∈ Ω(_n713999)
U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n730510)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n730510)
U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n769713)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n769713)
U94'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n809648)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n809648)
U95'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n850315)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n850315)

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U96'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n879662))) → _*4, rt ∈ Ω(_n879662)

Induction Base:
U96'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U96'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n879663, 1)))) →_R^Ω(1)
mark'(U96'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n879663)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n897262)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n897262)

Induction Base:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n897263, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b910611)) →_R^Ω(1)
mark'(U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n897263)), _gen_zeros':0':mark':tt':nil':ok'3(_b910611))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n927218)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n927218)

Induction Base:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n927219, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b940891)) →_R^Ω(1)
mark'(U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n927219)), _gen_zeros':0':mark':tt':nil':ok'3(_b940891))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n957542)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n957542)

Induction Base:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n957543, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b971539)) →_R^Ω(1)
mark'(U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n957543)), _gen_zeros':0':mark':tt':nil':ok'3(_b971539))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n988234)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n988234)

Induction Base:
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n988235, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b1009821), _gen_zeros':0':mark':tt':nil':ok'3(_c1009822)) →_R^Ω(1)
mark'(U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n988235)), _gen_zeros':0':mark':tt':nil':ok'3(_b1009821), _gen_zeros':0':mark':tt':nil':ok'3(_c1009822))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1032171)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n1032171)

Induction Base:
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1032172, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b1046924)) →_R^Ω(1)
mark'(U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1032172)), _gen_zeros':0':mark':tt':nil':ok'3(_b1046924))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1063726)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n1063726)

Induction Base:
U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1063727, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b1078803)) →_R^Ω(1)
mark'(U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1063727)), _gen_zeros':0':mark':tt':nil':ok'3(_b1078803))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U71'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1095649))) → _*4, rt ∈ Ω(_n1095649)

Induction Base:
U71'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U71'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1095650, 1)))) →_R^Ω(1)
mark'(U71'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1095650)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U81'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1114586))) → _*4, rt ∈ Ω(_n1114586)

Induction Base:
U81'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U81'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1114587, 1)))) →_R^Ω(1)
mark'(U81'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1114587)))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U91'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1133647)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n1133647)

Induction Base:
U91'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U91'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1133648, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b1157502), _gen_zeros':0':mark':tt':nil':ok'3(_c1157503)) →_R^Ω(1)
mark'(U91'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1133648)), _gen_zeros':0':mark':tt':nil':ok'3(_b1157502), _gen_zeros':0':mark':tt':nil':ok'3(_c1157503))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U101'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1180122)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n1180122)

Induction Base:
U101'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U101'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1180123, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b1204625), _gen_zeros':0':mark':tt':nil':ok'3(_c1204626)) →_R^Ω(1)
mark'(U101'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1180123)), _gen_zeros':0':mark':tt':nil':ok'3(_b1204625), _gen_zeros':0':mark':tt':nil':ok'3(_c1204626))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U111'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1227329)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n1227329)

Induction Base:
U111'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c))

Induction Step:
U111'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1227330, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b1252480), _gen_zeros':0':mark':tt':nil':ok'3(_c1252481)) →_R^Ω(1)
mark'(U111'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1227330)), _gen_zeros':0':mark':tt':nil':ok'3(_b1252480), _gen_zeros':0':mark':tt':nil':ok'3(_c1252481))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U121'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1275268)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n1275268)

Induction Base:
U121'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U121'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1275269, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b1292397)) →_R^Ω(1)
mark'(U121'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1275269)), _gen_zeros':0':mark':tt':nil':ok'3(_b1292397))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'

Proved the following rewrite lemma:
U131'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1309526)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d)) → _*4, rt ∈ Ω(_n1309526)

Induction Base:
U131'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U131'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1309527, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b1344859), _gen_zeros':0':mark':tt':nil':ok'3(_c1344860), _gen_zeros':0':mark':tt':nil':ok'3(_d1344861)) →_R^Ω(1)
mark'(U131'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1309527)), _gen_zeros':0':mark':tt':nil':ok'3(_b1344859), _gen_zeros':0':mark':tt':nil':ok'3(_c1344860), _gen_zeros':0':mark':tt':nil':ok'3(_d1344861))) →_IH
mark'(_*4)

We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(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'