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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2)) → U42(active(X1), X2)
active(U43(X)) → U43(active(X))
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U53(X)) → U53(active(X))
active(U61(X1, X2)) → U61(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
active(and(X1, X2)) → and(active(X1), X2)
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2) → mark(U42(X1, X2))
U43(mark(X)) → mark(U43(X))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2) → mark(U52(X1, X2))
U53(mark(X)) → mark(U53(X))
U61(mark(X1), X2) → mark(U61(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
and(mark(X1), X2) → mark(and(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2)) → U42(proper(X1), proper(X2))
proper(U43(X)) → U43(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U53(X)) → U53(proper(X))
proper(U61(X1, X2)) → U61(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2)) → ok(U42(X1, X2))
U43(ok(X)) → ok(U43(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U53(ok(X)) → ok(U53(X))
U61(ok(X1), ok(X2)) → ok(U61(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
isNatKind(ok(X)) → ok(isNatKind(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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', V1)) → mark'(U12'(isNatList'(V1)))
active'(U12'(tt')) → mark'(tt')
active'(U21'(tt', V1)) → mark'(U22'(isNat'(V1)))
active'(U22'(tt')) → mark'(tt')
active'(U31'(tt', V)) → mark'(U32'(isNatList'(V)))
active'(U32'(tt')) → mark'(tt')
active'(U41'(tt', V1, V2)) → mark'(U42'(isNat'(V1), V2))
active'(U42'(tt', V2)) → mark'(U43'(isNatIList'(V2)))
active'(U43'(tt')) → mark'(tt')
active'(U51'(tt', V1, V2)) → mark'(U52'(isNat'(V1), V2))
active'(U52'(tt', V2)) → mark'(U53'(isNatList'(V2)))
active'(U53'(tt')) → mark'(tt')
active'(U61'(tt', L)) → mark'(s'(length'(L)))
active'(and'(tt', X)) → mark'(X)
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(U11'(isNatIListKind'(V1), V1))
active'(isNat'(s'(V1))) → mark'(U21'(isNatKind'(V1), V1))
active'(isNatIList'(V)) → mark'(U31'(isNatIListKind'(V), V))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(U41'(and'(isNatKind'(V1), isNatIListKind'(V2)), V1, V2))
active'(isNatIListKind'(nil')) → mark'(tt')
active'(isNatIListKind'(zeros')) → mark'(tt')
active'(isNatIListKind'(cons'(V1, V2))) → mark'(and'(isNatKind'(V1), isNatIListKind'(V2)))
active'(isNatKind'(0')) → mark'(tt')
active'(isNatKind'(length'(V1))) → mark'(isNatIListKind'(V1))
active'(isNatKind'(s'(V1))) → mark'(isNatKind'(V1))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(U51'(and'(isNatKind'(V1), isNatIListKind'(V2)), V1, V2))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U61'(and'(and'(isNatList'(L), isNatIListKind'(L)), and'(isNat'(N), isNatKind'(N))), L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X)) → U12'(active'(X))
active'(U21'(X1, X2)) → U21'(active'(X1), X2)
active'(U22'(X)) → U22'(active'(X))
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U32'(X)) → U32'(active'(X))
active'(U41'(X1, X2, X3)) → U41'(active'(X1), X2, X3)
active'(U42'(X1, X2)) → U42'(active'(X1), X2)
active'(U43'(X)) → U43'(active'(X))
active'(U51'(X1, X2, X3)) → U51'(active'(X1), X2, X3)
active'(U52'(X1, X2)) → U52'(active'(X1), X2)
active'(U53'(X)) → U53'(active'(X))
active'(U61'(X1, X2)) → U61'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(and'(X1, X2)) → and'(active'(X1), X2)
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X)) → mark'(U12'(X))
U21'(mark'(X1), X2) → mark'(U21'(X1, X2))
U22'(mark'(X)) → mark'(U22'(X))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U32'(mark'(X)) → mark'(U32'(X))
U41'(mark'(X1), X2, X3) → mark'(U41'(X1, X2, X3))
U42'(mark'(X1), X2) → mark'(U42'(X1, X2))
U43'(mark'(X)) → mark'(U43'(X))
U51'(mark'(X1), X2, X3) → mark'(U51'(X1, X2, X3))
U52'(mark'(X1), X2) → mark'(U52'(X1, X2))
U53'(mark'(X)) → mark'(U53'(X))
U61'(mark'(X1), X2) → mark'(U61'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X)) → U12'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(U21'(X1, X2)) → U21'(proper'(X1), proper'(X2))
proper'(U22'(X)) → U22'(proper'(X))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X)) → U32'(proper'(X))
proper'(U41'(X1, X2, X3)) → U41'(proper'(X1), proper'(X2), proper'(X3))
proper'(U42'(X1, X2)) → U42'(proper'(X1), proper'(X2))
proper'(U43'(X)) → U43'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(U51'(X1, X2, X3)) → U51'(proper'(X1), proper'(X2), proper'(X3))
proper'(U52'(X1, X2)) → U52'(proper'(X1), proper'(X2))
proper'(U53'(X)) → U53'(proper'(X))
proper'(U61'(X1, X2)) → U61'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(isNatIListKind'(X)) → isNatIListKind'(proper'(X))
proper'(isNatKind'(X)) → isNatKind'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X)) → ok'(U12'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
U21'(ok'(X1), ok'(X2)) → ok'(U21'(X1, X2))
U22'(ok'(X)) → ok'(U22'(X))
isNat'(ok'(X)) → ok'(isNat'(X))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U32'(ok'(X)) → ok'(U32'(X))
U41'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U41'(X1, X2, X3))
U42'(ok'(X1), ok'(X2)) → ok'(U42'(X1, X2))
U43'(ok'(X)) → ok'(U43'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
U51'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U51'(X1, X2, X3))
U52'(ok'(X1), ok'(X2)) → ok'(U52'(X1, X2))
U53'(ok'(X)) → ok'(U53'(X))
U61'(ok'(X1), ok'(X2)) → ok'(U61'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isNatIListKind'(ok'(X)) → ok'(isNatIListKind'(X))
isNatKind'(ok'(X)) → ok'(isNatKind'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(U11'(tt', V1)) → mark'(U12'(isNatList'(V1)))
active'(U12'(tt')) → mark'(tt')
active'(U21'(tt', V1)) → mark'(U22'(isNat'(V1)))
active'(U22'(tt')) → mark'(tt')
active'(U31'(tt', V)) → mark'(U32'(isNatList'(V)))
active'(U32'(tt')) → mark'(tt')
active'(U41'(tt', V1, V2)) → mark'(U42'(isNat'(V1), V2))
active'(U42'(tt', V2)) → mark'(U43'(isNatIList'(V2)))
active'(U43'(tt')) → mark'(tt')
active'(U51'(tt', V1, V2)) → mark'(U52'(isNat'(V1), V2))
active'(U52'(tt', V2)) → mark'(U53'(isNatList'(V2)))
active'(U53'(tt')) → mark'(tt')
active'(U61'(tt', L)) → mark'(s'(length'(L)))
active'(and'(tt', X)) → mark'(X)
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(U11'(isNatIListKind'(V1), V1))
active'(isNat'(s'(V1))) → mark'(U21'(isNatKind'(V1), V1))
active'(isNatIList'(V)) → mark'(U31'(isNatIListKind'(V), V))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(U41'(and'(isNatKind'(V1), isNatIListKind'(V2)), V1, V2))
active'(isNatIListKind'(nil')) → mark'(tt')
active'(isNatIListKind'(zeros')) → mark'(tt')
active'(isNatIListKind'(cons'(V1, V2))) → mark'(and'(isNatKind'(V1), isNatIListKind'(V2)))
active'(isNatKind'(0')) → mark'(tt')
active'(isNatKind'(length'(V1))) → mark'(isNatIListKind'(V1))
active'(isNatKind'(s'(V1))) → mark'(isNatKind'(V1))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(U51'(and'(isNatKind'(V1), isNatIListKind'(V2)), V1, V2))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U61'(and'(and'(isNatList'(L), isNatIListKind'(L)), and'(isNat'(N), isNatKind'(N))), L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X)) → U12'(active'(X))
active'(U21'(X1, X2)) → U21'(active'(X1), X2)
active'(U22'(X)) → U22'(active'(X))
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U32'(X)) → U32'(active'(X))
active'(U41'(X1, X2, X3)) → U41'(active'(X1), X2, X3)
active'(U42'(X1, X2)) → U42'(active'(X1), X2)
active'(U43'(X)) → U43'(active'(X))
active'(U51'(X1, X2, X3)) → U51'(active'(X1), X2, X3)
active'(U52'(X1, X2)) → U52'(active'(X1), X2)
active'(U53'(X)) → U53'(active'(X))
active'(U61'(X1, X2)) → U61'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
active'(and'(X1, X2)) → and'(active'(X1), X2)
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X)) → mark'(U12'(X))
U21'(mark'(X1), X2) → mark'(U21'(X1, X2))
U22'(mark'(X)) → mark'(U22'(X))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U32'(mark'(X)) → mark'(U32'(X))
U41'(mark'(X1), X2, X3) → mark'(U41'(X1, X2, X3))
U42'(mark'(X1), X2) → mark'(U42'(X1, X2))
U43'(mark'(X)) → mark'(U43'(X))
U51'(mark'(X1), X2, X3) → mark'(U51'(X1, X2, X3))
U52'(mark'(X1), X2) → mark'(U52'(X1, X2))
U53'(mark'(X)) → mark'(U53'(X))
U61'(mark'(X1), X2) → mark'(U61'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X)) → U12'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(U21'(X1, X2)) → U21'(proper'(X1), proper'(X2))
proper'(U22'(X)) → U22'(proper'(X))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X)) → U32'(proper'(X))
proper'(U41'(X1, X2, X3)) → U41'(proper'(X1), proper'(X2), proper'(X3))
proper'(U42'(X1, X2)) → U42'(proper'(X1), proper'(X2))
proper'(U43'(X)) → U43'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(U51'(X1, X2, X3)) → U51'(proper'(X1), proper'(X2), proper'(X3))
proper'(U52'(X1, X2)) → U52'(proper'(X1), proper'(X2))
proper'(U53'(X)) → U53'(proper'(X))
proper'(U61'(X1, X2)) → U61'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(isNatIListKind'(X)) → isNatIListKind'(proper'(X))
proper'(isNatKind'(X)) → isNatKind'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X)) → ok'(U12'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
U21'(ok'(X1), ok'(X2)) → ok'(U21'(X1, X2))
U22'(ok'(X)) → ok'(U22'(X))
isNat'(ok'(X)) → ok'(isNat'(X))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U32'(ok'(X)) → ok'(U32'(X))
U41'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U41'(X1, X2, X3))
U42'(ok'(X1), ok'(X2)) → ok'(U42'(X1, X2))
U43'(ok'(X)) → ok'(U43'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
U51'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U51'(X1, X2, X3))
U52'(ok'(X1), ok'(X2)) → ok'(U52'(X1, X2))
U53'(ok'(X)) → ok'(U53'(X))
U61'(ok'(X1), ok'(X2)) → ok'(U61'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isNatIListKind'(ok'(X)) → ok'(isNatIListKind'(X))
isNatKind'(ok'(X)) → ok'(isNatKind'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
U11' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNatList' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U21' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U22' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNat' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U31' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U32' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U41' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U42' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U43' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNatIList' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U51' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U52' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U53' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U61' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
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'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNatIListKind' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNatKind' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':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', U12', isNatList', U22', isNat', U32', U42', U43', isNatIList', U52', U53', s', length', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

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

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

Proved the following rewrite lemma:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5189))) → _*4, rt ∈ Ω(_n5189)

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

Induction Step:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n5190, 1)))) →_R^Ω(1)
mark'(U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n5190)))) →_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', U22', isNat', U32', U42', U43', isNatIList', U52', U53', s', length', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

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

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

Proved the following rewrite lemma:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n8680))) → _*4, rt ∈ Ω(_n8680)

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

Induction Step:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n8681, 1)))) →_R^Ω(1)
mark'(U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n8681)))) →_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', U32', U42', U43', isNatIList', U52', U53', s', length', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

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

They will be analysed ascendingly in the following order:
U32' < active'
U42' < active'
U43' < active'
isNatIList' < active'
U52' < active'
U53' < active'
s' < active'
length' < active'
U11' < active'
isNatIListKind' < active'
U21' < active'
isNatKind' < active'
U31' < active'
U41' < active'
and' < active'
U51' < active'
U61' < active'
active' < top'
U32' < proper'
U42' < proper'
U43' < proper'
isNatIList' < proper'
U52' < proper'
U53' < proper'
s' < proper'
length' < proper'
U11' < proper'
isNatIListKind' < proper'
U21' < proper'
isNatKind' < proper'
U31' < proper'
U41' < proper'
and' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n12301))) → _*4, rt ∈ Ω(_n12301)

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

Induction Step:
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n12302, 1)))) →_R^Ω(1)
mark'(U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n12302)))) →_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', isNatIList', U52', U53', s', length', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

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

Proved the following rewrite lemma:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n16018)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n16018)

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

Induction Step:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n16019, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b17595)) →_R^Ω(1)
mark'(U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n16019)), _gen_zeros':0':mark':tt':nil':ok'3(_b17595))) →_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', isNatIList', U52', U53', s', length', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

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

Proved the following rewrite lemma:
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n22293))) → _*4, rt ∈ Ω(_n22293)

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

Induction Step:
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n22294, 1)))) →_R^Ω(1)
mark'(U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n22294)))) →_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', U53', s', length', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
isNatIList' < active'
U52' < active'
U53' < active'
s' < active'
length' < active'
U11' < active'
isNatIListKind' < active'
U21' < active'
isNatKind' < active'
U31' < active'
U41' < active'
and' < active'
U51' < active'
U61' < active'
active' < top'
isNatIList' < proper'
U52' < proper'
U53' < proper'
s' < proper'
length' < proper'
U11' < proper'
isNatIListKind' < proper'
U21' < proper'
isNatKind' < proper'
U31' < proper'
U41' < proper'
and' < proper'
U51' < proper'
U61' < 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', U53', s', length', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U52' < active'
U53' < active'
s' < active'
length' < active'
U11' < active'
isNatIListKind' < active'
U21' < active'
isNatKind' < active'
U31' < active'
U41' < active'
and' < active'
U51' < active'
U61' < active'
active' < top'
U52' < proper'
U53' < proper'
s' < proper'
length' < proper'
U11' < proper'
isNatIListKind' < proper'
U21' < proper'
isNatKind' < proper'
U31' < proper'
U41' < proper'
and' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n26375)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n26375)

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

Induction Step:
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n26376, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b28492)) →_R^Ω(1)
mark'(U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n26376)), _gen_zeros':0':mark':tt':nil':ok'3(_b28492))) →_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:
U53', active', s', length', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

They will be analysed ascendingly in the following order:
U53' < active'
s' < active'
length' < active'
U11' < active'
isNatIListKind' < active'
U21' < active'
isNatKind' < active'
U31' < active'
U41' < active'
and' < active'
U51' < active'
U61' < active'
active' < top'
U53' < proper'
s' < proper'
length' < proper'
U11' < proper'
isNatIListKind' < proper'
U21' < proper'
isNatKind' < proper'
U31' < proper'
U41' < proper'
and' < proper'
U51' < proper'
U61' < proper'
proper' < top'

Proved the following rewrite lemma:
U53'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n33259))) → _*4, rt ∈ Ω(_n33259)

Induction Base:
U53'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U53'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n33260, 1)))) →_R^Ω(1)
mark'(U53'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n33260)))) →_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', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

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

Proved the following rewrite lemma:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n37606))) → _*4, rt ∈ Ω(_n37606)

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, +(_$n37607, 1)))) →_R^Ω(1)
mark'(s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n37607)))) →_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', U11', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

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

Proved the following rewrite lemma:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n42077))) → _*4, rt ∈ Ω(_n42077)

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, +(_$n42078, 1)))) →_R^Ω(1)
mark'(length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n42078)))) →_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', isNatIListKind', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

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

Proved the following rewrite lemma:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n46672)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n46672)

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, +(_$n46673, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b49761)) →_R^Ω(1)
mark'(U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n46673)), _gen_zeros':0':mark':tt':nil':ok'3(_b49761))) →_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', U21', isNatKind', U31', U41', and', U51', U61', proper', top'

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

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

Proved the following rewrite lemma:
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n54735)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n54735)

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, +(_$n54736, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b58148)) →_R^Ω(1)
mark'(U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n54736)), _gen_zeros':0':mark':tt':nil':ok'3(_b58148))) →_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', U31', U41', and', U51', U61', proper', top'

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

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

Proved the following rewrite lemma:
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n63176)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n63176)

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, +(_$n63177, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b66913)) →_R^Ω(1)
mark'(U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n63177)), _gen_zeros':0':mark':tt':nil':ok'3(_b66913))) →_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', and', U51', U61', proper', top'

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

Proved the following rewrite lemma:
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n71887)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n71887)

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, +(_$n71888, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b78084), _gen_zeros':0':mark':tt':nil':ok'3(_c78085)) →_R^Ω(1)
mark'(U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n71888)), _gen_zeros':0':mark':tt':nil':ok'3(_b78084), _gen_zeros':0':mark':tt':nil':ok'3(_c78085))) →_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)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5189))) → _*4, rt ∈ Ω(_n5189)
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n8680))) → _*4, rt ∈ Ω(_n8680)
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n12301))) → _*4, rt ∈ Ω(_n12301)
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n16018)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n16018)
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n22293))) → _*4, rt ∈ Ω(_n22293)
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n26375)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n26375)
U53'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n33259))) → _*4, rt ∈ Ω(_n33259)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n37606))) → _*4, rt ∈ Ω(_n37606)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n42077))) → _*4, rt ∈ Ω(_n42077)
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n46672)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n46672)
U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n54735)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n54735)
U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n63176)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n63176)
U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n71887)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n71887)

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:
and', active', U51', U61', proper', top'

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

Proved the following rewrite lemma:
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n84867)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n84867)

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

Induction Step:
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n84868, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b89360)) →_R^Ω(1)
mark'(and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n84868)), _gen_zeros':0':mark':tt':nil':ok'3(_b89360))) →_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', proper', top'

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

Proved the following rewrite lemma:
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n94441)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n94441)

Induction Base:
U51'(_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:
U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n94442, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b101772), _gen_zeros':0':mark':tt':nil':ok'3(_c101773)) →_R^Ω(1)
mark'(U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n94442)), _gen_zeros':0':mark':tt':nil':ok'3(_b101772), _gen_zeros':0':mark':tt':nil':ok'3(_c101773))) →_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', proper', top'

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

Proved the following rewrite lemma:
U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n108698)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n108698)

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, +(_$n108699, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b113947)) →_R^Ω(1)
mark'(U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n108699)), _gen_zeros':0':mark':tt':nil':ok'3(_b113947))) →_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'