Runtime Complexity TRS:
The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt)) → mark(tt)
active(U41(tt, V2)) → mark(U42(isNatIList(V2)))
active(U42(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatList(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt, V2)) → mark(U62(isNatIList(V2)))
active(U62(tt)) → mark(tt)
active(U71(tt, L, N)) → mark(U72(isNat(N), L))
active(U72(tt, L)) → mark(s(length(L)))
active(U81(tt)) → mark(nil)
active(U91(tt, IL, M, N)) → mark(U92(isNat(M), IL, M, N))
active(U92(tt, IL, M, N)) → mark(U93(isNat(N), IL, M, N))
active(U93(tt, IL, M, N)) → mark(cons(N, take(M, IL)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatList(V1)))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNatIList(V)) → mark(U31(isNatList(V)))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNat(V1), V2))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(isNat(V1), V2))
active(isNatList(take(V1, V2))) → mark(U61(isNat(V1), V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U71(isNatList(L), L, N))
active(take(0, IL)) → mark(U81(isNatIList(IL)))
active(take(s(M), cons(N, IL))) → mark(U91(isNatIList(IL), IL, M, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X)) → U11(active(X))
active(U21(X)) → U21(active(X))
active(U31(X)) → U31(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U42(X)) → U42(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(U62(X)) → U62(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2)) → U72(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(U81(X)) → U81(active(X))
active(U91(X1, X2, X3, X4)) → U91(active(X1), X2, X3, X4)
active(U92(X1, X2, X3, X4)) → U92(active(X1), X2, X3, X4)
active(U93(X1, X2, X3, X4)) → U93(active(X1), X2, X3, X4)
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X)) → mark(U11(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U42(mark(X)) → mark(U42(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
U62(mark(X)) → mark(U62(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2) → mark(U72(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
U81(mark(X)) → mark(U81(X))
U91(mark(X1), X2, X3, X4) → mark(U91(X1, X2, X3, X4))
U92(mark(X1), X2, X3, X4) → mark(U92(X1, X2, X3, X4))
U93(mark(X1), X2, X3, X4) → mark(U93(X1, X2, X3, X4))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X)) → U11(proper(X))
proper(tt) → ok(tt)
proper(U21(X)) → U21(proper(X))
proper(U31(X)) → U31(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U42(X)) → U42(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(U62(X)) → U62(proper(X))
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(U81(X)) → U81(proper(X))
proper(nil) → ok(nil)
proper(U91(X1, X2, X3, X4)) → U91(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U92(X1, X2, X3, X4)) → U92(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U93(X1, X2, X3, X4)) → U93(proper(X1), proper(X2), proper(X3), proper(X4))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X)) → ok(U11(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U42(ok(X)) → ok(U42(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
isNatList(ok(X)) → ok(isNatList(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
U62(ok(X)) → ok(U62(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
isNat(ok(X)) → ok(isNat(X))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
U81(ok(X)) → ok(U81(X))
U91(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U91(X1, X2, X3, X4))
U92(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U92(X1, X2, X3, X4))
U93(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U93(X1, X2, X3, X4))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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'(U11'(tt')) → mark'(tt')
active'(U21'(tt')) → mark'(tt')
active'(U31'(tt')) → mark'(tt')
active'(U41'(tt', V2)) → mark'(U42'(isNatIList'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isNatList'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt', V2)) → mark'(U62'(isNatIList'(V2)))
active'(U62'(tt')) → mark'(tt')
active'(U71'(tt', L, N)) → mark'(U72'(isNat'(N), L))
active'(U72'(tt', L)) → mark'(s'(length'(L)))
active'(U81'(tt')) → mark'(nil')
active'(U91'(tt', IL, M, N)) → mark'(U92'(isNat'(M), IL, M, N))
active'(U92'(tt', IL, M, N)) → mark'(U93'(isNat'(N), IL, M, N))
active'(U93'(tt', IL, M, N)) → mark'(cons'(N, take'(M, IL)))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(U11'(isNatList'(V1)))
active'(isNat'(s'(V1))) → mark'(U21'(isNat'(V1)))
active'(isNatIList'(V)) → mark'(U31'(isNatList'(V)))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(U41'(isNat'(V1), V2))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(U51'(isNat'(V1), V2))
active'(isNatList'(take'(V1, V2))) → mark'(U61'(isNat'(V1), V2))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U71'(isNatList'(L), L, N))
active'(take'(0', IL)) → mark'(U81'(isNatIList'(IL)))
active'(take'(s'(M), cons'(N, IL))) → mark'(U91'(isNatIList'(IL), IL, M, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X)) → U11'(active'(X))
active'(U21'(X)) → U21'(active'(X))
active'(U31'(X)) → U31'(active'(X))
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X1, X2)) → U61'(active'(X1), X2)
active'(U62'(X)) → U62'(active'(X))
active'(U71'(X1, X2, X3)) → U71'(active'(X1), X2, X3)
active'(U72'(X1, X2)) → U72'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X1, X2, X3, X4)) → U91'(active'(X1), X2, X3, X4)
active'(U92'(X1, X2, X3, X4)) → U92'(active'(X1), X2, X3, X4)
active'(U93'(X1, X2, X3, X4)) → U93'(active'(X1), X2, X3, X4)
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X)) → mark'(U11'(X))
U21'(mark'(X)) → mark'(U21'(X))
U31'(mark'(X)) → mark'(U31'(X))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X1), X2) → mark'(U61'(X1, X2))
U62'(mark'(X)) → mark'(U62'(X))
U71'(mark'(X1), X2, X3) → mark'(U71'(X1, X2, X3))
U72'(mark'(X1), X2) → mark'(U72'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X1), X2, X3, X4) → mark'(U91'(X1, X2, X3, X4))
U92'(mark'(X1), X2, X3, X4) → mark'(U92'(X1, X2, X3, X4))
U93'(mark'(X1), X2, X3, X4) → mark'(U93'(X1, X2, X3, X4))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X)) → U11'(proper'(X))
proper'(tt') → ok'(tt')
proper'(U21'(X)) → U21'(proper'(X))
proper'(U31'(X)) → U31'(proper'(X))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(U61'(X1, X2)) → U61'(proper'(X1), proper'(X2))
proper'(U62'(X)) → U62'(proper'(X))
proper'(U71'(X1, X2, X3)) → U71'(proper'(X1), proper'(X2), proper'(X3))
proper'(U72'(X1, X2)) → U72'(proper'(X1), proper'(X2))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(nil') → ok'(nil')
proper'(U91'(X1, X2, X3, X4)) → U91'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U92'(X1, X2, X3, X4)) → U92'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U93'(X1, X2, X3, X4)) → U93'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X)) → ok'(U11'(X))
U21'(ok'(X)) → ok'(U21'(X))
U31'(ok'(X)) → ok'(U31'(X))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
U61'(ok'(X1), ok'(X2)) → ok'(U61'(X1, X2))
U62'(ok'(X)) → ok'(U62'(X))
U71'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U71'(X1, X2, X3))
U72'(ok'(X1), ok'(X2)) → ok'(U72'(X1, X2))
isNat'(ok'(X)) → ok'(isNat'(X))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U91'(X1, X2, X3, X4))
U92'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U92'(X1, X2, X3, X4))
U93'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U93'(X1, X2, X3, X4))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(U11'(tt')) → mark'(tt')
active'(U21'(tt')) → mark'(tt')
active'(U31'(tt')) → mark'(tt')
active'(U41'(tt', V2)) → mark'(U42'(isNatIList'(V2)))
active'(U42'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isNatList'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt', V2)) → mark'(U62'(isNatIList'(V2)))
active'(U62'(tt')) → mark'(tt')
active'(U71'(tt', L, N)) → mark'(U72'(isNat'(N), L))
active'(U72'(tt', L)) → mark'(s'(length'(L)))
active'(U81'(tt')) → mark'(nil')
active'(U91'(tt', IL, M, N)) → mark'(U92'(isNat'(M), IL, M, N))
active'(U92'(tt', IL, M, N)) → mark'(U93'(isNat'(N), IL, M, N))
active'(U93'(tt', IL, M, N)) → mark'(cons'(N, take'(M, IL)))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(U11'(isNatList'(V1)))
active'(isNat'(s'(V1))) → mark'(U21'(isNat'(V1)))
active'(isNatIList'(V)) → mark'(U31'(isNatList'(V)))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(U41'(isNat'(V1), V2))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(U51'(isNat'(V1), V2))
active'(isNatList'(take'(V1, V2))) → mark'(U61'(isNat'(V1), V2))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U71'(isNatList'(L), L, N))
active'(take'(0', IL)) → mark'(U81'(isNatIList'(IL)))
active'(take'(s'(M), cons'(N, IL))) → mark'(U91'(isNatIList'(IL), IL, M, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X)) → U11'(active'(X))
active'(U21'(X)) → U21'(active'(X))
active'(U31'(X)) → U31'(active'(X))
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U42'(X)) → U42'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X1, X2)) → U61'(active'(X1), X2)
active'(U62'(X)) → U62'(active'(X))
active'(U71'(X1, X2, X3)) → U71'(active'(X1), X2, X3)
active'(U72'(X1, X2)) → U72'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(U81'(X)) → U81'(active'(X))
active'(U91'(X1, X2, X3, X4)) → U91'(active'(X1), X2, X3, X4)
active'(U92'(X1, X2, X3, X4)) → U92'(active'(X1), X2, X3, X4)
active'(U93'(X1, X2, X3, X4)) → U93'(active'(X1), X2, X3, X4)
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X)) → mark'(U11'(X))
U21'(mark'(X)) → mark'(U21'(X))
U31'(mark'(X)) → mark'(U31'(X))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U42'(mark'(X)) → mark'(U42'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X1), X2) → mark'(U61'(X1, X2))
U62'(mark'(X)) → mark'(U62'(X))
U71'(mark'(X1), X2, X3) → mark'(U71'(X1, X2, X3))
U72'(mark'(X1), X2) → mark'(U72'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
U81'(mark'(X)) → mark'(U81'(X))
U91'(mark'(X1), X2, X3, X4) → mark'(U91'(X1, X2, X3, X4))
U92'(mark'(X1), X2, X3, X4) → mark'(U92'(X1, X2, X3, X4))
U93'(mark'(X1), X2, X3, X4) → mark'(U93'(X1, X2, X3, X4))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X)) → U11'(proper'(X))
proper'(tt') → ok'(tt')
proper'(U21'(X)) → U21'(proper'(X))
proper'(U31'(X)) → U31'(proper'(X))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U42'(X)) → U42'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(U61'(X1, X2)) → U61'(proper'(X1), proper'(X2))
proper'(U62'(X)) → U62'(proper'(X))
proper'(U71'(X1, X2, X3)) → U71'(proper'(X1), proper'(X2), proper'(X3))
proper'(U72'(X1, X2)) → U72'(proper'(X1), proper'(X2))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(U81'(X)) → U81'(proper'(X))
proper'(nil') → ok'(nil')
proper'(U91'(X1, X2, X3, X4)) → U91'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U92'(X1, X2, X3, X4)) → U92'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(U93'(X1, X2, X3, X4)) → U93'(proper'(X1), proper'(X2), proper'(X3), proper'(X4))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X)) → ok'(U11'(X))
U21'(ok'(X)) → ok'(U21'(X))
U31'(ok'(X)) → ok'(U31'(X))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U42'(ok'(X)) → ok'(U42'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
U61'(ok'(X1), ok'(X2)) → ok'(U61'(X1, X2))
U62'(ok'(X)) → ok'(U62'(X))
U71'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U71'(X1, X2, X3))
U72'(ok'(X1), ok'(X2)) → ok'(U72'(X1, X2))
isNat'(ok'(X)) → ok'(isNat'(X))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
U81'(ok'(X)) → ok'(U81'(X))
U91'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U91'(X1, X2, X3, X4))
U92'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U92'(X1, X2, X3, X4))
U93'(ok'(X1), ok'(X2), ok'(X3), ok'(X4)) → ok'(U93'(X1, X2, X3, X4))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
U11' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
U21' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U31' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U41' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U42' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNatIList' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U51' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U52' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNatList' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U61' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
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' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U72' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNat' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U81' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
U91' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
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' → 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' → 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'
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', U42', isNatIList', U52', isNatList', U62', U72', isNat', s', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
U42' < active'
isNatIList' < active'
U52' < active'
isNatList' < active'
U62' < active'
U72' < active'
isNat' < active'
s' < active'
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
cons' < proper'
U42' < proper'
isNatIList' < proper'
U52' < proper'
isNatList' < proper'
U62' < proper'
U72' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < 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', U42', isNatIList', U52', isNatList', U62', U72', isNat', s', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', 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:
U42', active', isNatIList', U52', isNatList', U62', U72', isNat', s', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U42' < active'
isNatIList' < active'
U52' < active'
isNatList' < active'
U62' < active'
U72' < active'
isNat' < active'
s' < active'
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U42' < proper'
isNatIList' < proper'
U52' < proper'
isNatList' < proper'
U62' < proper'
U72' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5951))) → _*4, rt ∈ Ω(_n5951)

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

Induction Step:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n5952, 1)))) →_R^Ω(1)
mark'(U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n5952)))) →_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', U52', isNatList', U62', U72', isNat', s', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
isNatIList' < active'
U52' < active'
isNatList' < active'
U62' < active'
U72' < active'
isNat' < active'
s' < active'
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
isNatIList' < proper'
U52' < proper'
isNatList' < proper'
U62' < proper'
U72' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < 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:
U52', active', isNatList', U62', U72', isNat', s', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U52' < active'
isNatList' < active'
U62' < active'
U72' < active'
isNat' < active'
s' < active'
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U52' < proper'
isNatList' < proper'
U62' < proper'
U72' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n9976))) → _*4, rt ∈ Ω(_n9976)

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, +(_$n9977, 1)))) →_R^Ω(1)
mark'(U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n9977)))) →_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', U62', U72', isNat', s', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
isNatList' < active'
U62' < active'
U72' < active'
isNat' < active'
s' < active'
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
isNatList' < proper'
U62' < proper'
U72' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < 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:
U62', active', U72', isNat', s', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U62' < active'
U72' < active'
isNat' < active'
s' < active'
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U62' < proper'
U72' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n14131))) → _*4, rt ∈ Ω(_n14131)

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, +(_$n14132, 1)))) →_R^Ω(1)
mark'(U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n14132)))) →_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:
U72', active', isNat', s', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U72' < active'
isNat' < active'
s' < active'
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U72' < proper'
isNat' < proper'
s' < proper'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
U72'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n18382)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n18382)

Induction Base:
U72'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U72'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n18383, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b19959)) →_R^Ω(1)
mark'(U72'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n18383)), _gen_zeros':0':mark':tt':nil':ok'3(_b19959))) →_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', s', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
isNat' < active'
s' < active'
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
isNat' < proper'
s' < proper'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < 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:
s', active', length', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
s' < proper'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n25463))) → _*4, rt ∈ Ω(_n25463)

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, +(_$n25464, 1)))) →_R^Ω(1)
mark'(s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n25464)))) →_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', U92', U93', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
length' < active'
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
length' < proper'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n30029))) → _*4, rt ∈ Ω(_n30029)

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, +(_$n30030, 1)))) →_R^Ω(1)
mark'(length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n30030)))) →_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', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U92' < active'
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U92' < proper'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n34719)), _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(_n34719)

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), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n34720, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b39812), _gen_zeros':0':mark':tt':nil':ok'3(_c39813), _gen_zeros':0':mark':tt':nil':ok'3(_d39814)) →_R^Ω(1)
mark'(U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n34720)), _gen_zeros':0':mark':tt':nil':ok'3(_b39812), _gen_zeros':0':mark':tt':nil':ok'3(_c39813), _gen_zeros':0':mark':tt':nil':ok'3(_d39814))) →_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', take', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U93' < active'
take' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U93' < proper'
take' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n49208)), _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(_n49208)

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), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n49209, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b55381), _gen_zeros':0':mark':tt':nil':ok'3(_c55382), _gen_zeros':0':mark':tt':nil':ok'3(_d55383)) →_R^Ω(1)
mark'(U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n49209)), _gen_zeros':0':mark':tt':nil':ok'3(_b55381), _gen_zeros':0':mark':tt':nil':ok'3(_c55382), _gen_zeros':0':mark':tt':nil':ok'3(_d55383))) →_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', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

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

Proved the following rewrite lemma:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n64913)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n64913)

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, +(_$n64914, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b68542)) →_R^Ω(1)
mark'(take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n64914)), _gen_zeros':0':mark':tt':nil':ok'3(_b68542))) →_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', 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'
active' < top'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n74263))) → _*4, rt ∈ Ω(_n74263)

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

Induction Step:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n74264, 1)))) →_R^Ω(1)
mark'(U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n74264)))) →_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', 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'
active' < top'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n79918))) → _*4, rt ∈ Ω(_n79918)

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

Induction Step:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n79919, 1)))) →_R^Ω(1)
mark'(U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n79919)))) →_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', 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'
active' < top'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
U81' < proper'
U91' < proper'
proper' < top'

Proved the following rewrite lemma:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n85697))) → _*4, rt ∈ Ω(_n85697)

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

Induction Step:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n85698, 1)))) →_R^Ω(1)
mark'(U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n85698)))) →_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', proper', top'

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

Proved the following rewrite lemma:
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n91600)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n91600)

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

Induction Step:
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n91601, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b95985)) →_R^Ω(1)
mark'(U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n91601)), _gen_zeros':0':mark':tt':nil':ok'3(_b95985))) →_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', proper', top'

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

Proved the following rewrite lemma:
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n101822)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n101822)

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, +(_$n101823, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b106531)) →_R^Ω(1)
mark'(U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n101823)), _gen_zeros':0':mark':tt':nil':ok'3(_b106531))) →_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)
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5951))) → _*4, rt ∈ Ω(_n5951)
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n9976))) → _*4, rt ∈ Ω(_n9976)
U62'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n14131))) → _*4, rt ∈ Ω(_n14131)
U72'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n18382)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n18382)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n25463))) → _*4, rt ∈ Ω(_n25463)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n30029))) → _*4, rt ∈ Ω(_n30029)
U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n34719)), _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(_n34719)
U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n49208)), _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(_n49208)
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n64913)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n64913)
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n74263))) → _*4, rt ∈ Ω(_n74263)
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n79918))) → _*4, rt ∈ Ω(_n79918)
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n85697))) → _*4, rt ∈ Ω(_n85697)
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n91600)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n91600)
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n101822)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n101822)

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', proper', top'

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

Proved the following rewrite lemma:
U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n112412)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n112412)

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, +(_$n112413, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b117445)) →_R^Ω(1)
mark'(U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n112413)), _gen_zeros':0':mark':tt':nil':ok'3(_b117445))) →_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', proper', top'

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

Proved the following rewrite lemma:
U71'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n123370)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n123370)

Induction Base:
U71'(_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:
U71'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n123371, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b131511), _gen_zeros':0':mark':tt':nil':ok'3(_c131512)) →_R^Ω(1)
mark'(U71'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n123371)), _gen_zeros':0':mark':tt':nil':ok'3(_b131511), _gen_zeros':0':mark':tt':nil':ok'3(_c131512))) →_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', proper', top'

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

Proved the following rewrite lemma:
U81'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n139536))) → _*4, rt ∈ Ω(_n139536)

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, +(_$n139537, 1)))) →_R^Ω(1)
mark'(U81'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n139537)))) →_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', proper', top'

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

Proved the following rewrite lemma:
U91'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n146394)), _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(_n146394)

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), _gen_zeros':0':mark':tt':nil':ok'3(d))

Induction Step:
U91'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n146395, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b158831), _gen_zeros':0':mark':tt':nil':ok'3(_c158832), _gen_zeros':0':mark':tt':nil':ok'3(_d158833)) →_R^Ω(1)
mark'(U91'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n146395)), _gen_zeros':0':mark':tt':nil':ok'3(_b158831), _gen_zeros':0':mark':tt':nil':ok'3(_c158832), _gen_zeros':0':mark':tt':nil':ok'3(_d158833))) →_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'