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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, V1)) → mark(U12(isNatIListKind(V1), V1))
active(U12(tt, V1)) → mark(U13(isNatList(V1)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatIListKind(V), V))
active(U32(tt, V)) → mark(U33(isNatList(V)))
active(U33(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNatKind(V1), V1, V2))
active(U42(tt, V1, V2)) → mark(U43(isNatIListKind(V2), V1, V2))
active(U43(tt, V1, V2)) → mark(U44(isNatIListKind(V2), V1, V2))
active(U44(tt, V1, V2)) → mark(U45(isNat(V1), V2))
active(U45(tt, V2)) → mark(U46(isNatIList(V2)))
active(U46(tt)) → mark(tt)
active(U51(tt, V2)) → mark(U52(isNatIListKind(V2)))
active(U52(tt)) → mark(tt)
active(U61(tt)) → mark(tt)
active(U71(tt)) → mark(tt)
active(U81(tt, V1, V2)) → mark(U82(isNatKind(V1), V1, V2))
active(U82(tt, V1, V2)) → mark(U83(isNatIListKind(V2), V1, V2))
active(U83(tt, V1, V2)) → mark(U84(isNatIListKind(V2), V1, V2))
active(U84(tt, V1, V2)) → mark(U85(isNat(V1), V2))
active(U85(tt, V2)) → mark(U86(isNatList(V2)))
active(U86(tt)) → mark(tt)
active(U91(tt, L, N)) → mark(U92(isNatIListKind(L), L, N))
active(U92(tt, L, N)) → mark(U93(isNat(N), L, N))
active(U93(tt, L, N)) → mark(U94(isNatKind(N), L))
active(U94(tt, L)) → mark(s(length(L)))
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(isNatKind(V1), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(U51(isNatKind(V1), V2))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(U61(isNatIListKind(V1)))
active(isNatKind(s(V1))) → mark(U71(isNatKind(V1)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U81(isNatKind(V1), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U91(isNatList(L), L, N))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
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(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(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(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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'(isNatIListKind'(V1), V1))
active'(U12'(tt', V1)) → mark'(U13'(isNatList'(V1)))
active'(U13'(tt')) → mark'(tt')
active'(U21'(tt', V1)) → mark'(U22'(isNatKind'(V1), V1))
active'(U22'(tt', V1)) → mark'(U23'(isNat'(V1)))
active'(U23'(tt')) → mark'(tt')
active'(U31'(tt', V)) → mark'(U32'(isNatIListKind'(V), V))
active'(U32'(tt', V)) → mark'(U33'(isNatList'(V)))
active'(U33'(tt')) → mark'(tt')
active'(U41'(tt', V1, V2)) → mark'(U42'(isNatKind'(V1), V1, V2))
active'(U42'(tt', V1, V2)) → mark'(U43'(isNatIListKind'(V2), V1, V2))
active'(U43'(tt', V1, V2)) → mark'(U44'(isNatIListKind'(V2), V1, V2))
active'(U44'(tt', V1, V2)) → mark'(U45'(isNat'(V1), V2))
active'(U45'(tt', V2)) → mark'(U46'(isNatIList'(V2)))
active'(U46'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isNatIListKind'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt', V1, V2)) → mark'(U82'(isNatKind'(V1), V1, V2))
active'(U82'(tt', V1, V2)) → mark'(U83'(isNatIListKind'(V2), V1, V2))
active'(U83'(tt', V1, V2)) → mark'(U84'(isNatIListKind'(V2), V1, V2))
active'(U84'(tt', V1, V2)) → mark'(U85'(isNat'(V1), V2))
active'(U85'(tt', V2)) → mark'(U86'(isNatList'(V2)))
active'(U86'(tt')) → mark'(tt')
active'(U91'(tt', L, N)) → mark'(U92'(isNatIListKind'(L), L, N))
active'(U92'(tt', L, N)) → mark'(U93'(isNat'(N), L, N))
active'(U93'(tt', L, N)) → mark'(U94'(isNatKind'(N), L))
active'(U94'(tt', L)) → mark'(s'(length'(L)))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(U11'(isNatIListKind'(V1), V1))
active'(isNat'(s'(V1))) → mark'(U21'(isNatKind'(V1), V1))
active'(isNatIList'(V)) → mark'(U31'(isNatIListKind'(V), V))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(U41'(isNatKind'(V1), V1, V2))
active'(isNatIListKind'(nil')) → mark'(tt')
active'(isNatIListKind'(zeros')) → mark'(tt')
active'(isNatIListKind'(cons'(V1, V2))) → mark'(U51'(isNatKind'(V1), V2))
active'(isNatKind'(0')) → mark'(tt')
active'(isNatKind'(length'(V1))) → mark'(U61'(isNatIListKind'(V1)))
active'(isNatKind'(s'(V1))) → mark'(U71'(isNatKind'(V1)))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(U81'(isNatKind'(V1), V1, V2))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U91'(isNatList'(L), L, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(U13'(X)) → U13'(active'(X))
active'(U21'(X1, X2)) → U21'(active'(X1), X2)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U23'(X)) → U23'(active'(X))
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U33'(X)) → U33'(active'(X))
active'(U41'(X1, X2, X3)) → U41'(active'(X1), X2, X3)
active'(U42'(X1, X2, X3)) → U42'(active'(X1), X2, X3)
active'(U43'(X1, X2, X3)) → U43'(active'(X1), X2, X3)
active'(U44'(X1, X2, X3)) → U44'(active'(X1), X2, X3)
active'(U45'(X1, X2)) → U45'(active'(X1), X2)
active'(U46'(X)) → U46'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X1, X2, X3)) → U81'(active'(X1), X2, X3)
active'(U82'(X1, X2, X3)) → U82'(active'(X1), X2, X3)
active'(U83'(X1, X2, X3)) → U83'(active'(X1), X2, X3)
active'(U84'(X1, X2, X3)) → U84'(active'(X1), X2, X3)
active'(U85'(X1, X2)) → U85'(active'(X1), X2)
active'(U86'(X)) → U86'(active'(X))
active'(U91'(X1, X2, X3)) → U91'(active'(X1), X2, X3)
active'(U92'(X1, X2, X3)) → U92'(active'(X1), X2, X3)
active'(U93'(X1, X2, X3)) → U93'(active'(X1), X2, X3)
active'(U94'(X1, X2)) → U94'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
U13'(mark'(X)) → mark'(U13'(X))
U21'(mark'(X1), X2) → mark'(U21'(X1, X2))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U23'(mark'(X)) → mark'(U23'(X))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U33'(mark'(X)) → mark'(U33'(X))
U41'(mark'(X1), X2, X3) → mark'(U41'(X1, X2, X3))
U42'(mark'(X1), X2, X3) → mark'(U42'(X1, X2, X3))
U43'(mark'(X1), X2, X3) → mark'(U43'(X1, X2, X3))
U44'(mark'(X1), X2, X3) → mark'(U44'(X1, X2, X3))
U45'(mark'(X1), X2) → mark'(U45'(X1, X2))
U46'(mark'(X)) → mark'(U46'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X1), X2, X3) → mark'(U81'(X1, X2, X3))
U82'(mark'(X1), X2, X3) → mark'(U82'(X1, X2, X3))
U83'(mark'(X1), X2, X3) → mark'(U83'(X1, X2, X3))
U84'(mark'(X1), X2, X3) → mark'(U84'(X1, X2, X3))
U85'(mark'(X1), X2) → mark'(U85'(X1, X2))
U86'(mark'(X)) → mark'(U86'(X))
U91'(mark'(X1), X2, X3) → mark'(U91'(X1, X2, X3))
U92'(mark'(X1), X2, X3) → mark'(U92'(X1, X2, X3))
U93'(mark'(X1), X2, X3) → mark'(U93'(X1, X2, X3))
U94'(mark'(X1), X2) → mark'(U94'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
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'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(isNatIListKind'(X)) → isNatIListKind'(proper'(X))
proper'(U13'(X)) → U13'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(U21'(X1, X2)) → U21'(proper'(X1), proper'(X2))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(isNatKind'(X)) → isNatKind'(proper'(X))
proper'(U23'(X)) → U23'(proper'(X))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U33'(X)) → U33'(proper'(X))
proper'(U41'(X1, X2, X3)) → U41'(proper'(X1), proper'(X2), proper'(X3))
proper'(U42'(X1, X2, X3)) → U42'(proper'(X1), proper'(X2), proper'(X3))
proper'(U43'(X1, X2, X3)) → U43'(proper'(X1), proper'(X2), proper'(X3))
proper'(U44'(X1, X2, X3)) → U44'(proper'(X1), proper'(X2), proper'(X3))
proper'(U45'(X1, X2)) → U45'(proper'(X1), proper'(X2))
proper'(U46'(X)) → U46'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X1, X2, X3)) → U81'(proper'(X1), proper'(X2), proper'(X3))
proper'(U82'(X1, X2, X3)) → U82'(proper'(X1), proper'(X2), proper'(X3))
proper'(U83'(X1, X2, X3)) → U83'(proper'(X1), proper'(X2), proper'(X3))
proper'(U84'(X1, X2, X3)) → U84'(proper'(X1), proper'(X2), proper'(X3))
proper'(U85'(X1, X2)) → U85'(proper'(X1), proper'(X2))
proper'(U86'(X)) → U86'(proper'(X))
proper'(U91'(X1, X2, X3)) → U91'(proper'(X1), proper'(X2), proper'(X3))
proper'(U92'(X1, X2, X3)) → U92'(proper'(X1), proper'(X2), proper'(X3))
proper'(U93'(X1, X2, X3)) → U93'(proper'(X1), proper'(X2), proper'(X3))
proper'(U94'(X1, X2)) → U94'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(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'(X1), ok'(X2)) → ok'(U12'(X1, X2))
isNatIListKind'(ok'(X)) → ok'(isNatIListKind'(X))
U13'(ok'(X)) → ok'(U13'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
U21'(ok'(X1), ok'(X2)) → ok'(U21'(X1, X2))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
isNatKind'(ok'(X)) → ok'(isNatKind'(X))
U23'(ok'(X)) → ok'(U23'(X))
isNat'(ok'(X)) → ok'(isNat'(X))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U33'(ok'(X)) → ok'(U33'(X))
U41'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U41'(X1, X2, X3))
U42'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U42'(X1, X2, X3))
U43'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U43'(X1, X2, X3))
U44'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U44'(X1, X2, X3))
U45'(ok'(X1), ok'(X2)) → ok'(U45'(X1, X2))
U46'(ok'(X)) → ok'(U46'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U81'(X1, X2, X3))
U82'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U82'(X1, X2, X3))
U83'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U83'(X1, X2, X3))
U84'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U84'(X1, X2, X3))
U85'(ok'(X1), ok'(X2)) → ok'(U85'(X1, X2))
U86'(ok'(X)) → ok'(U86'(X))
U91'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U91'(X1, X2, X3))
U92'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U92'(X1, X2, X3))
U93'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U93'(X1, X2, X3))
U94'(ok'(X1), ok'(X2)) → ok'(U94'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(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'(isNatIListKind'(V1), V1))
active'(U12'(tt', V1)) → mark'(U13'(isNatList'(V1)))
active'(U13'(tt')) → mark'(tt')
active'(U21'(tt', V1)) → mark'(U22'(isNatKind'(V1), V1))
active'(U22'(tt', V1)) → mark'(U23'(isNat'(V1)))
active'(U23'(tt')) → mark'(tt')
active'(U31'(tt', V)) → mark'(U32'(isNatIListKind'(V), V))
active'(U32'(tt', V)) → mark'(U33'(isNatList'(V)))
active'(U33'(tt')) → mark'(tt')
active'(U41'(tt', V1, V2)) → mark'(U42'(isNatKind'(V1), V1, V2))
active'(U42'(tt', V1, V2)) → mark'(U43'(isNatIListKind'(V2), V1, V2))
active'(U43'(tt', V1, V2)) → mark'(U44'(isNatIListKind'(V2), V1, V2))
active'(U44'(tt', V1, V2)) → mark'(U45'(isNat'(V1), V2))
active'(U45'(tt', V2)) → mark'(U46'(isNatIList'(V2)))
active'(U46'(tt')) → mark'(tt')
active'(U51'(tt', V2)) → mark'(U52'(isNatIListKind'(V2)))
active'(U52'(tt')) → mark'(tt')
active'(U61'(tt')) → mark'(tt')
active'(U71'(tt')) → mark'(tt')
active'(U81'(tt', V1, V2)) → mark'(U82'(isNatKind'(V1), V1, V2))
active'(U82'(tt', V1, V2)) → mark'(U83'(isNatIListKind'(V2), V1, V2))
active'(U83'(tt', V1, V2)) → mark'(U84'(isNatIListKind'(V2), V1, V2))
active'(U84'(tt', V1, V2)) → mark'(U85'(isNat'(V1), V2))
active'(U85'(tt', V2)) → mark'(U86'(isNatList'(V2)))
active'(U86'(tt')) → mark'(tt')
active'(U91'(tt', L, N)) → mark'(U92'(isNatIListKind'(L), L, N))
active'(U92'(tt', L, N)) → mark'(U93'(isNat'(N), L, N))
active'(U93'(tt', L, N)) → mark'(U94'(isNatKind'(N), L))
active'(U94'(tt', L)) → mark'(s'(length'(L)))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(length'(V1))) → mark'(U11'(isNatIListKind'(V1), V1))
active'(isNat'(s'(V1))) → mark'(U21'(isNatKind'(V1), V1))
active'(isNatIList'(V)) → mark'(U31'(isNatIListKind'(V), V))
active'(isNatIList'(zeros')) → mark'(tt')
active'(isNatIList'(cons'(V1, V2))) → mark'(U41'(isNatKind'(V1), V1, V2))
active'(isNatIListKind'(nil')) → mark'(tt')
active'(isNatIListKind'(zeros')) → mark'(tt')
active'(isNatIListKind'(cons'(V1, V2))) → mark'(U51'(isNatKind'(V1), V2))
active'(isNatKind'(0')) → mark'(tt')
active'(isNatKind'(length'(V1))) → mark'(U61'(isNatIListKind'(V1)))
active'(isNatKind'(s'(V1))) → mark'(U71'(isNatKind'(V1)))
active'(isNatList'(nil')) → mark'(tt')
active'(isNatList'(cons'(V1, V2))) → mark'(U81'(isNatKind'(V1), V1, V2))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U91'(isNatList'(L), L, N))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(U13'(X)) → U13'(active'(X))
active'(U21'(X1, X2)) → U21'(active'(X1), X2)
active'(U22'(X1, X2)) → U22'(active'(X1), X2)
active'(U23'(X)) → U23'(active'(X))
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U32'(X1, X2)) → U32'(active'(X1), X2)
active'(U33'(X)) → U33'(active'(X))
active'(U41'(X1, X2, X3)) → U41'(active'(X1), X2, X3)
active'(U42'(X1, X2, X3)) → U42'(active'(X1), X2, X3)
active'(U43'(X1, X2, X3)) → U43'(active'(X1), X2, X3)
active'(U44'(X1, X2, X3)) → U44'(active'(X1), X2, X3)
active'(U45'(X1, X2)) → U45'(active'(X1), X2)
active'(U46'(X)) → U46'(active'(X))
active'(U51'(X1, X2)) → U51'(active'(X1), X2)
active'(U52'(X)) → U52'(active'(X))
active'(U61'(X)) → U61'(active'(X))
active'(U71'(X)) → U71'(active'(X))
active'(U81'(X1, X2, X3)) → U81'(active'(X1), X2, X3)
active'(U82'(X1, X2, X3)) → U82'(active'(X1), X2, X3)
active'(U83'(X1, X2, X3)) → U83'(active'(X1), X2, X3)
active'(U84'(X1, X2, X3)) → U84'(active'(X1), X2, X3)
active'(U85'(X1, X2)) → U85'(active'(X1), X2)
active'(U86'(X)) → U86'(active'(X))
active'(U91'(X1, X2, X3)) → U91'(active'(X1), X2, X3)
active'(U92'(X1, X2, X3)) → U92'(active'(X1), X2, X3)
active'(U93'(X1, X2, X3)) → U93'(active'(X1), X2, X3)
active'(U94'(X1, X2)) → U94'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
U13'(mark'(X)) → mark'(U13'(X))
U21'(mark'(X1), X2) → mark'(U21'(X1, X2))
U22'(mark'(X1), X2) → mark'(U22'(X1, X2))
U23'(mark'(X)) → mark'(U23'(X))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U32'(mark'(X1), X2) → mark'(U32'(X1, X2))
U33'(mark'(X)) → mark'(U33'(X))
U41'(mark'(X1), X2, X3) → mark'(U41'(X1, X2, X3))
U42'(mark'(X1), X2, X3) → mark'(U42'(X1, X2, X3))
U43'(mark'(X1), X2, X3) → mark'(U43'(X1, X2, X3))
U44'(mark'(X1), X2, X3) → mark'(U44'(X1, X2, X3))
U45'(mark'(X1), X2) → mark'(U45'(X1, X2))
U46'(mark'(X)) → mark'(U46'(X))
U51'(mark'(X1), X2) → mark'(U51'(X1, X2))
U52'(mark'(X)) → mark'(U52'(X))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X)) → mark'(U71'(X))
U81'(mark'(X1), X2, X3) → mark'(U81'(X1, X2, X3))
U82'(mark'(X1), X2, X3) → mark'(U82'(X1, X2, X3))
U83'(mark'(X1), X2, X3) → mark'(U83'(X1, X2, X3))
U84'(mark'(X1), X2, X3) → mark'(U84'(X1, X2, X3))
U85'(mark'(X1), X2) → mark'(U85'(X1, X2))
U86'(mark'(X)) → mark'(U86'(X))
U91'(mark'(X1), X2, X3) → mark'(U91'(X1, X2, X3))
U92'(mark'(X1), X2, X3) → mark'(U92'(X1, X2, X3))
U93'(mark'(X1), X2, X3) → mark'(U93'(X1, X2, X3))
U94'(mark'(X1), X2) → mark'(U94'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
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'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(isNatIListKind'(X)) → isNatIListKind'(proper'(X))
proper'(U13'(X)) → U13'(proper'(X))
proper'(isNatList'(X)) → isNatList'(proper'(X))
proper'(U21'(X1, X2)) → U21'(proper'(X1), proper'(X2))
proper'(U22'(X1, X2)) → U22'(proper'(X1), proper'(X2))
proper'(isNatKind'(X)) → isNatKind'(proper'(X))
proper'(U23'(X)) → U23'(proper'(X))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X1, X2)) → U32'(proper'(X1), proper'(X2))
proper'(U33'(X)) → U33'(proper'(X))
proper'(U41'(X1, X2, X3)) → U41'(proper'(X1), proper'(X2), proper'(X3))
proper'(U42'(X1, X2, X3)) → U42'(proper'(X1), proper'(X2), proper'(X3))
proper'(U43'(X1, X2, X3)) → U43'(proper'(X1), proper'(X2), proper'(X3))
proper'(U44'(X1, X2, X3)) → U44'(proper'(X1), proper'(X2), proper'(X3))
proper'(U45'(X1, X2)) → U45'(proper'(X1), proper'(X2))
proper'(U46'(X)) → U46'(proper'(X))
proper'(isNatIList'(X)) → isNatIList'(proper'(X))
proper'(U51'(X1, X2)) → U51'(proper'(X1), proper'(X2))
proper'(U52'(X)) → U52'(proper'(X))
proper'(U61'(X)) → U61'(proper'(X))
proper'(U71'(X)) → U71'(proper'(X))
proper'(U81'(X1, X2, X3)) → U81'(proper'(X1), proper'(X2), proper'(X3))
proper'(U82'(X1, X2, X3)) → U82'(proper'(X1), proper'(X2), proper'(X3))
proper'(U83'(X1, X2, X3)) → U83'(proper'(X1), proper'(X2), proper'(X3))
proper'(U84'(X1, X2, X3)) → U84'(proper'(X1), proper'(X2), proper'(X3))
proper'(U85'(X1, X2)) → U85'(proper'(X1), proper'(X2))
proper'(U86'(X)) → U86'(proper'(X))
proper'(U91'(X1, X2, X3)) → U91'(proper'(X1), proper'(X2), proper'(X3))
proper'(U92'(X1, X2, X3)) → U92'(proper'(X1), proper'(X2), proper'(X3))
proper'(U93'(X1, X2, X3)) → U93'(proper'(X1), proper'(X2), proper'(X3))
proper'(U94'(X1, X2)) → U94'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(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'(X1), ok'(X2)) → ok'(U12'(X1, X2))
isNatIListKind'(ok'(X)) → ok'(isNatIListKind'(X))
U13'(ok'(X)) → ok'(U13'(X))
isNatList'(ok'(X)) → ok'(isNatList'(X))
U21'(ok'(X1), ok'(X2)) → ok'(U21'(X1, X2))
U22'(ok'(X1), ok'(X2)) → ok'(U22'(X1, X2))
isNatKind'(ok'(X)) → ok'(isNatKind'(X))
U23'(ok'(X)) → ok'(U23'(X))
isNat'(ok'(X)) → ok'(isNat'(X))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U32'(ok'(X1), ok'(X2)) → ok'(U32'(X1, X2))
U33'(ok'(X)) → ok'(U33'(X))
U41'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U41'(X1, X2, X3))
U42'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U42'(X1, X2, X3))
U43'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U43'(X1, X2, X3))
U44'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U44'(X1, X2, X3))
U45'(ok'(X1), ok'(X2)) → ok'(U45'(X1, X2))
U46'(ok'(X)) → ok'(U46'(X))
isNatIList'(ok'(X)) → ok'(isNatIList'(X))
U51'(ok'(X1), ok'(X2)) → ok'(U51'(X1, X2))
U52'(ok'(X)) → ok'(U52'(X))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X)) → ok'(U71'(X))
U81'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U81'(X1, X2, X3))
U82'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U82'(X1, X2, X3))
U83'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U83'(X1, X2, X3))
U84'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U84'(X1, X2, X3))
U85'(ok'(X1), ok'(X2)) → ok'(U85'(X1, X2))
U86'(ok'(X)) → ok'(U86'(X))
U91'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U91'(X1, X2, X3))
U92'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U92'(X1, X2, X3))
U93'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U93'(X1, X2, X3))
U94'(ok'(X1), ok'(X2)) → ok'(U94'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(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' → zeros':0':mark':tt':nil':ok'
isNatIListKind' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U13' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNatList' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U21' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U22' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
isNatKind' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U23' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
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' → zeros':0':mark':tt':nil':ok'
U33' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U41' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U42' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U43' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U44' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U45' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U46' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
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'
U61' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U71' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U81' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U82' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U83' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U84' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U85' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U86' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U91' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U92' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U93' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
U94' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
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'
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', isNatIListKind', U13', isNatList', U22', isNatKind', U23', isNat', U32', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
cons' < active'
U12' < active'
isNatIListKind' < active'
U13' < active'
isNatList' < active'
U22' < active'
isNatKind' < active'
U23' < active'
isNat' < active'
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
cons' < proper'
U12' < proper'
isNatIListKind' < proper'
U13' < proper'
isNatList' < proper'
U22' < proper'
isNatKind' < proper'
U23' < proper'
isNat' < proper'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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', U12', isNatIListKind', U13', isNatList', U22', isNatKind', U23', isNat', U32', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', 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:
U12', active', isNatIListKind', U13', isNatList', U22', isNatKind', U23', isNat', U32', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U12' < active'
isNatIListKind' < active'
U13' < active'
isNatList' < active'
U22' < active'
isNatKind' < active'
U23' < active'
isNat' < active'
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U12' < proper'
isNatIListKind' < proper'
U13' < proper'
isNatList' < proper'
U22' < proper'
isNatKind' < proper'
U23' < proper'
isNat' < proper'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n9554)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n9554)

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

Induction Step:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n9555, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b10483)) →_R^Ω(1)
mark'(U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n9555)), _gen_zeros':0':mark':tt':nil':ok'3(_b10483))) →_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', U13', isNatList', U22', isNatKind', U23', isNat', U32', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
isNatIListKind' < active'
U13' < active'
isNatList' < active'
U22' < active'
isNatKind' < active'
U23' < active'
isNat' < active'
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
isNatIListKind' < proper'
U13' < proper'
isNatList' < proper'
U22' < proper'
isNatKind' < proper'
U23' < proper'
isNat' < proper'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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 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:
U13', active', isNatList', U22', isNatKind', U23', isNat', U32', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U13' < active'
isNatList' < active'
U22' < active'
isNatKind' < active'
U23' < active'
isNat' < active'
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U13' < proper'
isNatList' < proper'
U22' < proper'
isNatKind' < proper'
U23' < proper'
isNat' < proper'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U13'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n19497))) → _*4, rt ∈ Ω(_n19497)

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

Induction Step:
U13'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n19498, 1)))) →_R^Ω(1)
mark'(U13'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n19498)))) →_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', isNatKind', U23', isNat', U32', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
isNatList' < active'
U22' < active'
isNatKind' < active'
U23' < active'
isNat' < active'
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
isNatList' < proper'
U22' < proper'
isNatKind' < proper'
U23' < proper'
isNat' < proper'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U22', active', isNatKind', U23', isNat', U32', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U22' < active'
isNatKind' < active'
U23' < active'
isNat' < active'
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U22' < proper'
isNatKind' < proper'
U23' < proper'
isNat' < proper'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n26114)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n26114)

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

Induction Step:
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n26115, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b27583)) →_R^Ω(1)
mark'(U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n26115)), _gen_zeros':0':mark':tt':nil':ok'3(_b27583))) →_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', U23', isNat', U32', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
isNatKind' < active'
U23' < active'
isNat' < active'
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
isNatKind' < proper'
U23' < proper'
isNat' < proper'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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 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:
U23', active', isNat', U32', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U23' < active'
isNat' < active'
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U23' < proper'
isNat' < proper'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n36682))) → _*4, rt ∈ Ω(_n36682)

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

Induction Step:
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n36683, 1)))) →_R^Ω(1)
mark'(U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n36683)))) →_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', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
isNat' < active'
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
isNat' < proper'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U32', active', U33', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U32' < active'
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U32' < proper'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n43630)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n43630)

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

Induction Step:
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n43631, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b45639)) →_R^Ω(1)
mark'(U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n43631)), _gen_zeros':0':mark':tt':nil':ok'3(_b45639))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U33', active', U42', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U33' < active'
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U33' < proper'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U33'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n54765))) → _*4, rt ∈ Ω(_n54765)

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

Induction Step:
U33'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n54766, 1)))) →_R^Ω(1)
mark'(U33'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n54766)))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U42', active', U43', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U42' < active'
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U42' < proper'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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, _n61980)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n61980)

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

Induction Step:
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n61981, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b65909), _gen_zeros':0':mark':tt':nil':ok'3(_c65910)) →_R^Ω(1)
mark'(U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n61981)), _gen_zeros':0':mark':tt':nil':ok'3(_b65909), _gen_zeros':0':mark':tt':nil':ok'3(_c65910))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U43', active', U44', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U43' < active'
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U43' < proper'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n78243)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n78243)

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

Induction Step:
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n78244, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b82820), _gen_zeros':0':mark':tt':nil':ok'3(_c82821)) →_R^Ω(1)
mark'(U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n78244)), _gen_zeros':0':mark':tt':nil':ok'3(_b82820), _gen_zeros':0':mark':tt':nil':ok'3(_c82821))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U44', active', U45', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U44' < active'
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U44' < proper'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U44'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n95238)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n95238)

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

Induction Step:
U44'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n95239, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b100463), _gen_zeros':0':mark':tt':nil':ok'3(_c100464)) →_R^Ω(1)
mark'(U44'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n95239)), _gen_zeros':0':mark':tt':nil':ok'3(_b100463), _gen_zeros':0':mark':tt':nil':ok'3(_c100464))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U45', active', U46', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U45' < active'
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U45' < proper'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U45'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n112965)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n112965)

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

Induction Step:
U45'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n112966, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b116810)) →_R^Ω(1)
mark'(U45'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n112966)), _gen_zeros':0':mark':tt':nil':ok'3(_b116810))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U46', active', isNatIList', U52', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U46' < active'
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U46' < proper'
isNatIList' < proper'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U46'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n126194))) → _*4, rt ∈ Ω(_n126194)

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

Induction Step:
U46'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n126195, 1)))) →_R^Ω(1)
mark'(U46'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n126195)))) →_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', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
isNatIList' < active'
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < 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'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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', U82', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U52' < active'
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U52' < proper'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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, _n134620))) → _*4, rt ∈ Ω(_n134620)

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, +(_$n134621, 1)))) →_R^Ω(1)
mark'(U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n134621)))) →_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:
U82', active', U83', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U82' < active'
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U82' < proper'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U82'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n143048)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n143048)

Induction Base:
U82'(_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:
U82'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n143049, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b150055), _gen_zeros':0':mark':tt':nil':ok'3(_c150056)) →_R^Ω(1)
mark'(U82'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n143049)), _gen_zeros':0':mark':tt':nil':ok'3(_b150055), _gen_zeros':0':mark':tt':nil':ok'3(_c150056))) →_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:
U83', active', U84', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U83' < active'
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U83' < proper'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U83'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n162768)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n162768)

Induction Base:
U83'(_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:
U83'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n162769, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b170423), _gen_zeros':0':mark':tt':nil':ok'3(_c170424)) →_R^Ω(1)
mark'(U83'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n162769)), _gen_zeros':0':mark':tt':nil':ok'3(_b170423), _gen_zeros':0':mark':tt':nil':ok'3(_c170424))) →_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:
U84', active', U85', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U84' < active'
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U84' < proper'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U84'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n183220)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n183220)

Induction Base:
U84'(_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:
U84'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n183221, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b191523), _gen_zeros':0':mark':tt':nil':ok'3(_c191524)) →_R^Ω(1)
mark'(U84'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n183221)), _gen_zeros':0':mark':tt':nil':ok'3(_b191523), _gen_zeros':0':mark':tt':nil':ok'3(_c191524))) →_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:
U85', active', U86', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U85' < active'
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U85' < proper'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U85'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n204404)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n204404)

Induction Base:
U85'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))

Induction Step:
U85'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n204405, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b210301)) →_R^Ω(1)
mark'(U85'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n204405)), _gen_zeros':0':mark':tt':nil':ok'3(_b210301))) →_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:
U86', active', U92', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U86' < active'
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U86' < proper'
U92' < proper'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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:
U86'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n219968))) → _*4, rt ∈ Ω(_n219968)

Induction Base:
U86'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))

Induction Step:
U86'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n219969, 1)))) →_R^Ω(1)
mark'(U86'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n219969)))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U92', active', U93', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U92' < active'
U93' < active'
U94' < active'
s' < active'
length' < 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'
U94' < proper'
s' < proper'
length' < 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, _n229485)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n229485)

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

Induction Step:
U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n229486, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b239246), _gen_zeros':0':mark':tt':nil':ok'3(_c239247)) →_R^Ω(1)
mark'(U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n229486)), _gen_zeros':0':mark':tt':nil':ok'3(_b239246), _gen_zeros':0':mark':tt':nil':ok'3(_c239247))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U93', active', U94', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U93' < active'
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U93' < proper'
U94' < proper'
s' < proper'
length' < 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, _n252304)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n252304)

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

Induction Step:
U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n252305, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b262713), _gen_zeros':0':mark':tt':nil':ok'3(_c262714)) →_R^Ω(1)
mark'(U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n252305)), _gen_zeros':0':mark':tt':nil':ok'3(_b262713), _gen_zeros':0':mark':tt':nil':ok'3(_c262714))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U94', active', s', length', U11', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
U94' < active'
s' < active'
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
U94' < proper'
s' < proper'
length' < 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:
U94'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n275855)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n275855)

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

Induction Step:
U94'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n275856, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b283156)) →_R^Ω(1)
mark'(U94'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n275856)), _gen_zeros':0':mark':tt':nil':ok'3(_b283156))) →_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', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
length' < 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'
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, _n293018))) → _*4, rt ∈ Ω(_n293018)

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, +(_$n293019, 1)))) →_R^Ω(1)
mark'(s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n293019)))) →_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', U21', U31', U41', U51', U61', U71', U81', U91', proper', top'

They will be analysed ascendingly in the following order:
length' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
U81' < active'
U91' < active'
active' < top'
length' < 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, _n303366))) → _*4, rt ∈ Ω(_n303366)

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, +(_$n303367, 1)))) →_R^Ω(1)
mark'(length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n303367)))) →_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, _n9554)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n9554)
U13'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n19497))) → _*4, rt ∈ Ω(_n19497)
U22'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n26114)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n26114)
U23'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n36682))) → _*4, rt ∈ Ω(_n36682)
U32'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n43630)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n43630)
U33'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n54765))) → _*4, rt ∈ Ω(_n54765)
U42'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n61980)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n61980)
U43'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n78243)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n78243)
U44'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n95238)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n95238)
U45'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n112965)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n112965)
U46'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n126194))) → _*4, rt ∈ Ω(_n126194)
U52'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n134620))) → _*4, rt ∈ Ω(_n134620)
U82'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n143048)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n143048)
U83'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n162768)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n162768)
U84'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n183220)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n183220)
U85'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n204404)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n204404)
U86'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n219968))) → _*4, rt ∈ Ω(_n219968)
U92'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n229485)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n229485)
U93'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n252304)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n252304)
U94'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n275855)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n275855)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n293018))) → _*4, rt ∈ Ω(_n293018)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n303366))) → _*4, rt ∈ Ω(_n303366)

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, _n313838)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n313838)

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, +(_$n313839, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b321895)) →_R^Ω(1)
mark'(U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n313839)), _gen_zeros':0':mark':tt':nil':ok'3(_b321895))) →_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, _n331851)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n331851)

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, +(_$n331852, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b340232)) →_R^Ω(1)
mark'(U21'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n331852)), _gen_zeros':0':mark':tt':nil':ok'3(_b340232))) →_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, _n350232)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n350232)

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, +(_$n350233, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b358937)) →_R^Ω(1)
mark'(U31'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n350233)), _gen_zeros':0':mark':tt':nil':ok'3(_b358937))) →_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, _n368981)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n368981)

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, +(_$n368982, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b382630), _gen_zeros':0':mark':tt':nil':ok'3(_c382631)) →_R^Ω(1)
mark'(U41'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n368982)), _gen_zeros':0':mark':tt':nil':ok'3(_b382630), _gen_zeros':0':mark':tt':nil':ok'3(_c382631))) →_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, _n396160)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(_n396160)

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, +(_$n396161, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b405621)) →_R^Ω(1)
mark'(U51'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n396161)), _gen_zeros':0':mark':tt':nil':ok'3(_b405621))) →_IH
mark'(_*4)

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

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

The following defined symbols remain to be analysed:
U61', active', U71', U81', U91', 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, _n415772))) → _*4, rt ∈ Ω(_n415772)

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

Induction Step:
U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n415773, 1)))) →_R^Ω(1)
mark'(U61'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n415773)))) →_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, _n427390))) → _*4, rt ∈ Ω(_n427390)

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

Induction Step:
U71'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n427391, 1)))) →_R^Ω(1)
mark'(U71'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n427391)))) →_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, _n439132)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n439132)

Induction Base:
U81'(_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:
U81'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n439133, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b454563), _gen_zeros':0':mark':tt':nil':ok'3(_c454564)) →_R^Ω(1)
mark'(U81'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n439133)), _gen_zeros':0':mark':tt':nil':ok'3(_b454563), _gen_zeros':0':mark':tt':nil':ok'3(_c454564))) →_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, _n468304)), _gen_zeros':0':mark':tt':nil':ok'3(b), _gen_zeros':0':mark':tt':nil':ok'3(c)) → _*4, rt ∈ Ω(_n468304)

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

Induction Step:
U91'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n468305, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b484383), _gen_zeros':0':mark':tt':nil':ok'3(_c484384)) →_R^Ω(1)
mark'(U91'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n468305)), _gen_zeros':0':mark':tt':nil':ok'3(_b484383), _gen_zeros':0':mark':tt':nil':ok'3(_c484384))) →_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'