Termination w.r.t. Q of the given QTRS could not be shown:

(0) Obligation:

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(U11(tt, V1)) → MARK(U12(isNatIListKind(V1), V1))
ACTIVE(U11(tt, V1)) → U12¹(isNatIListKind(V1), V1)
ACTIVE(U11(tt, V1)) → ISNATILISTKIND(V1)
ACTIVE(U12(tt, V1)) → MARK(U13(isNatList(V1)))
ACTIVE(U12(tt, V1)) → U13¹(isNatList(V1))
ACTIVE(U12(tt, V1)) → ISNATLIST(V1)
ACTIVE(U13(tt)) → MARK(tt)
ACTIVE(U21(tt, V1)) → MARK(U22(isNatKind(V1), V1))
ACTIVE(U21(tt, V1)) → U22¹(isNatKind(V1), V1)
ACTIVE(U21(tt, V1)) → ISNATKIND(V1)
ACTIVE(U22(tt, V1)) → MARK(U23(isNat(V1)))
ACTIVE(U22(tt, V1)) → U23¹(isNat(V1))
ACTIVE(U22(tt, V1)) → ISNAT(V1)
ACTIVE(U23(tt)) → MARK(tt)
ACTIVE(U31(tt, V)) → MARK(U32(isNatIListKind(V), V))
ACTIVE(U31(tt, V)) → U32¹(isNatIListKind(V), V)
ACTIVE(U31(tt, V)) → ISNATILISTKIND(V)
ACTIVE(U32(tt, V)) → MARK(U33(isNatList(V)))
ACTIVE(U32(tt, V)) → U33¹(isNatList(V))
ACTIVE(U32(tt, V)) → ISNATLIST(V)
ACTIVE(U33(tt)) → MARK(tt)
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNatKind(V1), V1, V2))
ACTIVE(U41(tt, V1, V2)) → U42¹(isNatKind(V1), V1, V2)
ACTIVE(U41(tt, V1, V2)) → ISNATKIND(V1)
ACTIVE(U42(tt, V1, V2)) → MARK(U43(isNatIListKind(V2), V1, V2))
ACTIVE(U42(tt, V1, V2)) → U43¹(isNatIListKind(V2), V1, V2)
ACTIVE(U42(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U43(tt, V1, V2)) → MARK(U44(isNatIListKind(V2), V1, V2))
ACTIVE(U43(tt, V1, V2)) → U44¹(isNatIListKind(V2), V1, V2)
ACTIVE(U43(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U44(tt, V1, V2)) → MARK(U45(isNat(V1), V2))
ACTIVE(U44(tt, V1, V2)) → U45¹(isNat(V1), V2)
ACTIVE(U44(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U45(tt, V2)) → MARK(U46(isNatIList(V2)))
ACTIVE(U45(tt, V2)) → U46¹(isNatIList(V2))
ACTIVE(U45(tt, V2)) → ISNATILIST(V2)
ACTIVE(U46(tt)) → MARK(tt)
ACTIVE(U51(tt, V2)) → MARK(U52(isNatIListKind(V2)))
ACTIVE(U51(tt, V2)) → U52¹(isNatIListKind(V2))
ACTIVE(U51(tt, V2)) → 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(U81(tt, V1, V2)) → U82¹(isNatKind(V1), V1, V2)
ACTIVE(U81(tt, V1, V2)) → ISNATKIND(V1)
ACTIVE(U82(tt, V1, V2)) → MARK(U83(isNatIListKind(V2), V1, V2))
ACTIVE(U82(tt, V1, V2)) → U83¹(isNatIListKind(V2), V1, V2)
ACTIVE(U82(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U83(tt, V1, V2)) → MARK(U84(isNatIListKind(V2), V1, V2))
ACTIVE(U83(tt, V1, V2)) → U84¹(isNatIListKind(V2), V1, V2)
ACTIVE(U83(tt, V1, V2)) → ISNATILISTKIND(V2)
ACTIVE(U84(tt, V1, V2)) → MARK(U85(isNat(V1), V2))
ACTIVE(U84(tt, V1, V2)) → U85¹(isNat(V1), V2)
ACTIVE(U84(tt, V1, V2)) → ISNAT(V1)
ACTIVE(U85(tt, V2)) → MARK(U86(isNatList(V2)))
ACTIVE(U85(tt, V2)) → U86¹(isNatList(V2))
ACTIVE(U85(tt, V2)) → ISNATLIST(V2)
ACTIVE(U86(tt)) → MARK(tt)
ACTIVE(U91(tt, L, N)) → MARK(U92(isNatIListKind(L), L, N))
ACTIVE(U91(tt, L, N)) → U92¹(isNatIListKind(L), L, N)
ACTIVE(U91(tt, L, N)) → ISNATILISTKIND(L)
ACTIVE(U92(tt, L, N)) → MARK(U93(isNat(N), L, N))
ACTIVE(U92(tt, L, N)) → U93¹(isNat(N), L, N)
ACTIVE(U92(tt, L, N)) → ISNAT(N)
ACTIVE(U93(tt, L, N)) → MARK(U94(isNatKind(N), L))
ACTIVE(U93(tt, L, N)) → U94¹(isNatKind(N), L)
ACTIVE(U93(tt, L, N)) → ISNATKIND(N)
ACTIVE(U94(tt, L)) → MARK(s(length(L)))
ACTIVE(U94(tt, L)) → S(length(L))
ACTIVE(U94(tt, L)) → LENGTH(L)
ACTIVE(isNat(0)) → MARK(tt)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
ACTIVE(isNat(length(V1))) → U11¹(isNatIListKind(V1), V1)
ACTIVE(isNat(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(isNat(s(V1))) → U21¹(isNatKind(V1), V1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
ACTIVE(isNatIList(V)) → MARK(U31(isNatIListKind(V), V))
ACTIVE(isNatIList(V)) → U31¹(isNatIListKind(V), V)
ACTIVE(isNatIList(V)) → ISNATILISTKIND(V)
ACTIVE(isNatIList(zeros)) → MARK(tt)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(isNatKind(V1), V1, V2))
ACTIVE(isNatIList(cons(V1, V2))) → U41¹(isNatKind(V1), V1, V2)
ACTIVE(isNatIList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatIListKind(nil)) → MARK(tt)
ACTIVE(isNatIListKind(zeros)) → MARK(tt)
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(U51(isNatKind(V1), V2))
ACTIVE(isNatIListKind(cons(V1, V2))) → U51¹(isNatKind(V1), V2)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(isNatKind(0)) → MARK(tt)
ACTIVE(isNatKind(length(V1))) → MARK(U61(isNatIListKind(V1)))
ACTIVE(isNatKind(length(V1))) → U61¹(isNatIListKind(V1))
ACTIVE(isNatKind(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(isNatKind(s(V1))) → MARK(U71(isNatKind(V1)))
ACTIVE(isNatKind(s(V1))) → U71¹(isNatKind(V1))
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
ACTIVE(isNatList(nil)) → MARK(tt)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U81(isNatKind(V1), V1, V2))
ACTIVE(isNatList(cons(V1, V2))) → U81¹(isNatKind(V1), V1, V2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(length(nil)) → MARK(0)
ACTIVE(length(cons(N, L))) → MARK(U91(isNatList(L), L, N))
ACTIVE(length(cons(N, L))) → U91¹(isNatList(L), L, N)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
MARK(zeros) → ACTIVE(zeros)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(0) → ACTIVE(0)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U11(X1, X2)) → U11¹(mark(X1), X2)
MARK(U11(X1, X2)) → MARK(X1)
MARK(tt) → ACTIVE(tt)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(U12(X1, X2)) → U12¹(mark(X1), X2)
MARK(U12(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
MARK(U13(X)) → ACTIVE(U13(mark(X)))
MARK(U13(X)) → U13¹(mark(X))
MARK(U13(X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
MARK(U21(X1, X2)) → U21¹(mark(X1), X2)
MARK(U21(X1, X2)) → MARK(X1)
MARK(U22(X1, X2)) → ACTIVE(U22(mark(X1), X2))
MARK(U22(X1, X2)) → U22¹(mark(X1), X2)
MARK(U22(X1, X2)) → MARK(X1)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U23(X)) → ACTIVE(U23(mark(X)))
MARK(U23(X)) → U23¹(mark(X))
MARK(U23(X)) → MARK(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(U31(X1, X2)) → U31¹(mark(X1), X2)
MARK(U31(X1, X2)) → MARK(X1)
MARK(U32(X1, X2)) → ACTIVE(U32(mark(X1), X2))
MARK(U32(X1, X2)) → U32¹(mark(X1), X2)
MARK(U32(X1, X2)) → MARK(X1)
MARK(U33(X)) → ACTIVE(U33(mark(X)))
MARK(U33(X)) → U33¹(mark(X))
MARK(U33(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U41(X1, X2, X3)) → U41¹(mark(X1), X2, X3)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U42(X1, X2, X3)) → ACTIVE(U42(mark(X1), X2, X3))
MARK(U42(X1, X2, X3)) → U42¹(mark(X1), X2, X3)
MARK(U42(X1, X2, X3)) → MARK(X1)
MARK(U43(X1, X2, X3)) → ACTIVE(U43(mark(X1), X2, X3))
MARK(U43(X1, X2, X3)) → U43¹(mark(X1), X2, X3)
MARK(U43(X1, X2, X3)) → MARK(X1)
MARK(U44(X1, X2, X3)) → ACTIVE(U44(mark(X1), X2, X3))
MARK(U44(X1, X2, X3)) → U44¹(mark(X1), X2, X3)
MARK(U44(X1, X2, X3)) → MARK(X1)
MARK(U45(X1, X2)) → ACTIVE(U45(mark(X1), X2))
MARK(U45(X1, X2)) → U45¹(mark(X1), X2)
MARK(U45(X1, X2)) → MARK(X1)
MARK(U46(X)) → ACTIVE(U46(mark(X)))
MARK(U46(X)) → U46¹(mark(X))
MARK(U46(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U51(X1, X2)) → ACTIVE(U51(mark(X1), X2))
MARK(U51(X1, X2)) → U51¹(mark(X1), X2)
MARK(U51(X1, X2)) → MARK(X1)
MARK(U52(X)) → ACTIVE(U52(mark(X)))
MARK(U52(X)) → U52¹(mark(X))
MARK(U52(X)) → MARK(X)
MARK(U61(X)) → ACTIVE(U61(mark(X)))
MARK(U61(X)) → U61¹(mark(X))
MARK(U61(X)) → MARK(X)
MARK(U71(X)) → ACTIVE(U71(mark(X)))
MARK(U71(X)) → U71¹(mark(X))
MARK(U71(X)) → MARK(X)
MARK(U81(X1, X2, X3)) → ACTIVE(U81(mark(X1), X2, X3))
MARK(U81(X1, X2, X3)) → U81¹(mark(X1), X2, X3)
MARK(U81(X1, X2, X3)) → MARK(X1)
MARK(U82(X1, X2, X3)) → ACTIVE(U82(mark(X1), X2, X3))
MARK(U82(X1, X2, X3)) → U82¹(mark(X1), X2, X3)
MARK(U82(X1, X2, X3)) → MARK(X1)
MARK(U83(X1, X2, X3)) → ACTIVE(U83(mark(X1), X2, X3))
MARK(U83(X1, X2, X3)) → U83¹(mark(X1), X2, X3)
MARK(U83(X1, X2, X3)) → MARK(X1)
MARK(U84(X1, X2, X3)) → ACTIVE(U84(mark(X1), X2, X3))
MARK(U84(X1, X2, X3)) → U84¹(mark(X1), X2, X3)
MARK(U84(X1, X2, X3)) → MARK(X1)
MARK(U85(X1, X2)) → ACTIVE(U85(mark(X1), X2))
MARK(U85(X1, X2)) → U85¹(mark(X1), X2)
MARK(U85(X1, X2)) → MARK(X1)
MARK(U86(X)) → ACTIVE(U86(mark(X)))
MARK(U86(X)) → U86¹(mark(X))
MARK(U86(X)) → MARK(X)
MARK(U91(X1, X2, X3)) → ACTIVE(U91(mark(X1), X2, X3))
MARK(U91(X1, X2, X3)) → U91¹(mark(X1), X2, X3)
MARK(U91(X1, X2, X3)) → MARK(X1)
MARK(U92(X1, X2, X3)) → ACTIVE(U92(mark(X1), X2, X3))
MARK(U92(X1, X2, X3)) → U92¹(mark(X1), X2, X3)
MARK(U92(X1, X2, X3)) → MARK(X1)
MARK(U93(X1, X2, X3)) → ACTIVE(U93(mark(X1), X2, X3))
MARK(U93(X1, X2, X3)) → U93¹(mark(X1), X2, X3)
MARK(U93(X1, X2, X3)) → MARK(X1)
MARK(U94(X1, X2)) → ACTIVE(U94(mark(X1), X2))
MARK(U94(X1, X2)) → U94¹(mark(X1), X2)
MARK(U94(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → LENGTH(mark(X))
MARK(length(X)) → MARK(X)
MARK(nil) → ACTIVE(nil)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)
ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(active(X)) → ISNATILISTKIND(X)
U13¹(mark(X)) → U13¹(X)
U13¹(active(X)) → U13¹(X)
ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)
ISNATKIND(mark(X)) → ISNATKIND(X)
ISNATKIND(active(X)) → ISNATKIND(X)
U23¹(mark(X)) → U23¹(X)
U23¹(active(X)) → U23¹(X)
ISNAT(mark(X)) → ISNAT(X)
ISNAT(active(X)) → ISNAT(X)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)
U33¹(mark(X)) → U33¹(X)
U33¹(active(X)) → U33¹(X)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)
U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)
U43¹(mark(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, mark(X3)) → U43¹(X1, X2, X3)
U43¹(active(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, active(X3)) → U43¹(X1, X2, X3)
U44¹(mark(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, mark(X3)) → U44¹(X1, X2, X3)
U44¹(active(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, active(X3)) → U44¹(X1, X2, X3)
U45¹(mark(X1), X2) → U45¹(X1, X2)
U45¹(X1, mark(X2)) → U45¹(X1, X2)
U45¹(active(X1), X2) → U45¹(X1, X2)
U45¹(X1, active(X2)) → U45¹(X1, X2)
U46¹(mark(X)) → U46¹(X)
U46¹(active(X)) → U46¹(X)
ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)
U52¹(mark(X)) → U52¹(X)
U52¹(active(X)) → U52¹(X)
U61¹(mark(X)) → U61¹(X)
U61¹(active(X)) → U61¹(X)
U71¹(mark(X)) → U71¹(X)
U71¹(active(X)) → U71¹(X)
U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)
U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, mark(X3)) → U83¹(X1, X2, X3)
U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)
U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, mark(X3)) → U84¹(X1, X2, X3)
U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)
U85¹(mark(X1), X2) → U85¹(X1, X2)
U85¹(X1, mark(X2)) → U85¹(X1, X2)
U85¹(active(X1), X2) → U85¹(X1, X2)
U85¹(X1, active(X2)) → U85¹(X1, X2)
U86¹(mark(X)) → U86¹(X)
U86¹(active(X)) → U86¹(X)
U91¹(mark(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, mark(X3)) → U91¹(X1, X2, X3)
U91¹(active(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, active(X3)) → U91¹(X1, X2, X3)
U92¹(mark(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, mark(X3)) → U92¹(X1, X2, X3)
U92¹(active(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, active(X3)) → U92¹(X1, X2, X3)
U93¹(mark(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, mark(X2), X3) → U93¹(X1, X2, X3)
U93¹(X1, X2, mark(X3)) → U93¹(X1, X2, X3)
U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, active(X2), X3) → U93¹(X1, X2, X3)
U93¹(X1, X2, active(X3)) → U93¹(X1, X2, X3)
U94¹(mark(X1), X2) → U94¹(X1, X2)
U94¹(X1, mark(X2)) → U94¹(X1, X2)
U94¹(active(X1), X2) → U94¹(X1, X2)
U94¹(X1, active(X2)) → U94¹(X1, X2)
S(mark(X)) → S(X)
S(active(X)) → S(X)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 38 SCCs with 111 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LENGTH(active(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(active(X)) → LENGTH(X)
LENGTH(mark(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1)  =  LENGTH(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

LENGTH₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(7) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(8) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(9) TRUE

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

S(active(X)) → S(X)
S(mark(X)) → S(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)
S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  S(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

S₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(12) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(14) TRUE

(15) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U94¹(X1, mark(X2)) → U94¹(X1, X2)
U94¹(mark(X1), X2) → U94¹(X1, X2)
U94¹(active(X1), X2) → U94¹(X1, X2)
U94¹(X1, active(X2)) → U94¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U94¹(mark(X1), X2) → U94¹(X1, X2)
U94¹(active(X1), X2) → U94¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U94¹(x1, x2)  =  U94¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U94^1₁ > active₁
zeros > cons > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > tt > s > active₁
zeros > cons > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s > active₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U51 > isNatIListKind₁ > mark₁ > tt > s > active₁
zeros > cons > U51 > isNatIListKind₁ > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U51 > U52 > mark₁ > tt > s > active₁
zeros > cons > U51 > U52 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U91 > isNatIListKind₁ > mark₁ > tt > s > active₁
zeros > cons > U91 > isNatIListKind₁ > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > tt > s > active₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > tt > length₁ > 0 > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > mark₁ > tt > s > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > mark₁ > tt > length₁ > 0 > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > s > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > length₁ > 0 > active₁
isNatKind > isNatIListKind₁ > mark₁ > tt > s > active₁
isNatKind > isNatIListKind₁ > mark₁ > tt > length₁ > 0 > active₁
isNatKind > U61 > mark₁ > tt > s > active₁
isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
isNatKind > U71 > mark₁ > tt > s > active₁
isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
isNat > U11 > U12 > U13 > mark₁ > tt > s > active₁
isNat > U11 > U12 > U13 > mark₁ > tt > length₁ > 0 > active₁
isNat > U11 > isNatIListKind₁ > mark₁ > tt > s > active₁
isNat > U11 > isNatIListKind₁ > mark₁ > tt > length₁ > 0 > active₁
isNat > U21 > U22 > U23 > mark₁ > tt > s > active₁
isNat > U21 > U22 > U23 > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > mark₁ > tt > s > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U31₁ > U32 > U33 > mark₁ > tt > s > active₁
isNatIList₁ > U31₁ > U32 > U33 > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > length₁ > 0 > active₁
nil > mark₁ > tt > s > active₁
nil > mark₁ > tt > length₁ > 0 > active₁

The following usable rules [FROCOS05] were oriented:

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

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U94¹(X1, mark(X2)) → U94¹(X1, X2)
U94¹(X1, active(X2)) → U94¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U94¹(X1, mark(X2)) → U94¹(X1, X2)
U94¹(X1, active(X2)) → U94¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U94¹(x1, x2)  =  U94¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U51 > U52 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
zeros > 0 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNatList > U81 > isNatKind₁ > U61 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNatList > U81 > isNatKind₁ > U71 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNat > U11 > U12 > U13 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNat > U21 > U22 > U23 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNat > U21 > isNatKind₁ > U61 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNat > U21 > isNatKind₁ > U71 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNatIList₁ > U31 > U32 > U33 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
length > 0 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
length > U11 > U12 > U13 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
length > U61 > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
length > U91 > isNatIListKind₁ > mark₁ > active₁ > U94^1₂
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > U94^1₂
nil > tt > isNatIListKind₁ > mark₁ > active₁ > U94^1₂

The following usable rules [FROCOS05] were oriented:

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

(19) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(20) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(21) TRUE

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U93¹(X1, mark(X2), X3) → U93¹(X1, X2, X3)
U93¹(mark(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, X2, mark(X3)) → U93¹(X1, X2, X3)
U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, active(X2), X3) → U93¹(X1, X2, X3)
U93¹(X1, X2, active(X3)) → U93¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(X1, X2, mark(X3)) → U93¹(X1, X2, X3)
U93¹(X1, X2, active(X3)) → U93¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  x1
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > tt > mark₁ > active₁ > cons₁
isNatIList > U31 > isNatIListKind > U51 > U52 > tt > mark₁ > active₁ > cons₁
isNatIList > U31 > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList > U41 > isNatKind > isNatIListKind > U51 > U52 > tt > mark₁ > active₁ > cons₁
isNatIList > U41 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList > U41 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > mark₁ > active₁ > cons₁
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > tt > mark₁ > active₁ > cons₁
length > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > U11 > isNatIListKind > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > isNatKind > isNatIListKind > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > isNatIListKind > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > isNatIListKind > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
nil > 0 > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(24) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U93¹(X1, mark(X2), X3) → U93¹(X1, X2, X3)
U93¹(mark(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(X1, active(X2), X3) → U93¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(X1, mark(X2), X3) → U93¹(X1, X2, X3)
U93¹(X1, active(X2), X3) → U93¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3)  =  U93¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s > active₁
zeros > 0 > tt > U93 > U94 > mark₁ > s > active₁
isNat > U11 > U12 > U13 > tt > U93 > U94 > mark₁ > s > active₁
isNat > U11 > isNatIListKind > isNatKind > U61 > tt > U93 > U94 > mark₁ > s > active₁
isNat > U11 > isNatIListKind > isNatKind > U71 > tt > U93 > U94 > mark₁ > s > active₁
isNat > U11 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s > active₁
isNat > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > s > active₁
isNat > U21 > isNatKind > U61 > tt > U93 > U94 > mark₁ > s > active₁
isNat > U21 > isNatKind > U71 > tt > U93 > U94 > mark₁ > s > active₁
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > isNatIListKind > isNatKind > U61 > tt > U93 > U94 > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > isNatIListKind > isNatKind > U71 > tt > U93 > U94 > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > s > active₁
length > 0 > tt > U93 > U94 > mark₁ > s > active₁
length > U11 > U12 > U13 > tt > U93 > U94 > mark₁ > s > active₁
length > U11 > isNatIListKind > isNatKind > U61 > tt > U93 > U94 > mark₁ > s > active₁
length > U11 > isNatIListKind > isNatKind > U71 > tt > U93 > U94 > mark₁ > s > active₁
length > U11 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s > active₁
length > isNatList > U81 > isNatKind > U61 > tt > U93 > U94 > mark₁ > s > active₁
length > isNatList > U81 > isNatKind > U71 > tt > U93 > U94 > mark₁ > s > active₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > s > active₁
length > U91 > U92 > U93 > U94 > mark₁ > s > active₁
nil > 0 > tt > U93 > U94 > mark₁ > s > active₁

The following usable rules [FROCOS05] were oriented:

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

(26) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U93¹(mark(X1), X2, X3) → U93¹(X1, X2, X3)
U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(mark(X1), X2, X3) → U93¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3)  =  U93¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  x1
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  x1
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > U92 > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U92 > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > isNat > U11 > U12 > U13 > tt > U92 > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > isNat > U21 > U22 > U23 > tt > U92 > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > U92 > U93 > U94 > mark₁ > s
zeros > 0 > tt > U92 > U93 > U94 > mark₁ > s
isNatKind > isNatIListKind > U51 > U52 > tt > U92 > U93 > U94 > mark₁ > s
isNatKind > U61 > tt > U92 > U93 > U94 > mark₁ > s
isNatKind > U71 > tt > U92 > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U92 > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U92 > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U92 > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > isNat > U11 > U12 > U13 > tt > U92 > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > isNat > U21 > U22 > U23 > tt > U92 > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U92 > U93 > U94 > mark₁ > s
nil > 0 > tt > U92 > U93 > U94 > mark₁ > s

The following usable rules [FROCOS05] were oriented:

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

(28) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(active(X1), X2, X3) → U93¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U93¹(x1, x2, x3)  =  x1
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > tt > mark₁ > cons₁ > active₁
isNatList > U81 > U82 > U83 > isNatIListKind > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatList > U81 > U82 > U83 > isNatIListKind > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatList > U81 > U82 > U83 > isNatIListKind > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > cons₁ > active₁
length > 0 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
nil > 0 > tt > mark₁ > cons₁ > active₁

The following usable rules [FROCOS05] were oriented:

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

(30) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(31) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(32) TRUE

(33) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)
U92¹(mark(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, mark(X3)) → U92¹(X1, X2, X3)
U92¹(active(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, active(X3)) → U92¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(mark(X1), X2, X3) → U92¹(X1, X2, X3)
U92¹(active(X1), X2, X3) → U92¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3)  =  U92¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U92^1₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > active₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > tt > cons₁
length > 0 > mark₁ > isNat₁ > active₁ > cons₁
length > 0 > mark₁ > isNat₁ > tt > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > tt > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > tt > cons₁
nil > mark₁ > isNat₁ > active₁ > cons₁
nil > mark₁ > isNat₁ > tt > cons₁

The following usable rules [FROCOS05] were oriented:

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

(35) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, mark(X3)) → U92¹(X1, X2, X3)
U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, active(X3)) → U92¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(X1, X2, mark(X3)) → U92¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11(x2)
tt  =  tt
U12(x1, x2)  =  U12(x2)
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  x1
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U51 > U52 > tt > U93 > U94 > mark₁ > s
zeros > cons > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > length₁
zeros > 0 > tt > U93 > U94 > mark₁ > s
zeros > 0 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > length₁
nil > 0 > tt > U93 > U94 > mark₁ > s
nil > 0 > tt > U93 > U94 > mark₁ > length₁

The following usable rules [FROCOS05] were oriented:

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

(37) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, active(X3)) → U92¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(X1, mark(X2), X3) → U92¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3)  =  U92¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  x1
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  x1
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > isNatKind₁ > tt
zeros > cons > U51 > U52 > mark₁ > tt
zeros > 0 > mark₁ > tt
isNatList > U81 > isNatKind₁ > tt
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt
isNatIList > U31 > U32 > U33 > mark₁ > tt
isNatIList > U41 > isNatKind₁ > tt
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt
length > 0 > mark₁ > tt
length > U11 > U12 > U13 > mark₁ > tt
length > U61 > mark₁ > tt
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > tt
length > U91 > U92 > U93 > U94 > s > U21 > isNatKind₁ > tt
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt
nil > 0 > mark₁ > tt

The following usable rules [FROCOS05] were oriented:

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

(39) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)
U92¹(X1, X2, active(X3)) → U92¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(X1, X2, active(X3)) → U92¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3)  =  U92¹(x3)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U81 > isNatKind > isNatIListKind > U51 > U52 > tt > mark₁ > active₁
zeros > cons > U81 > isNatKind > U61 > tt > mark₁ > active₁
zeros > cons > U81 > isNatKind > U71 > tt > mark₁ > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > tt > mark₁ > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > isNatIListKind > U51 > U52 > tt > mark₁ > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > tt > mark₁ > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > isNatList₁ > tt > mark₁ > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁
zeros > cons > U91 > U92 > isNat > U11 > U12 > U13 > tt > mark₁ > active₁
zeros > cons > U91 > U92 > isNat > isNatIListKind > U51 > U52 > tt > mark₁ > active₁
zeros > cons > U91 > U92 > isNat > U21 > U22 > U23 > tt > mark₁ > active₁
zeros > cons > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > active₁
zeros > 0 > tt > mark₁ > active₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > mark₁ > active₁
isNatIList₁ > U31 > U32 > isNatList₁ > tt > mark₁ > active₁
isNatIList₁ > U31 > U32 > U33 > tt > mark₁ > active₁
isNatIList₁ > U41 > isNatKind > isNatIListKind > U51 > U52 > tt > mark₁ > active₁
isNatIList₁ > U41 > isNatKind > U61 > tt > mark₁ > active₁
isNatIList₁ > U41 > isNatKind > U71 > tt > mark₁ > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > mark₁ > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > mark₁ > active₁
length > 0 > tt > mark₁ > active₁
length > isNatList₁ > tt > mark₁ > active₁
length > U61 > tt > mark₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > tt > mark₁ > active₁
length > U91 > U92 > isNat > isNatIListKind > U51 > U52 > tt > mark₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > tt > mark₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > active₁
nil > mark₁ > active₁

The following usable rules [FROCOS05] were oriented:

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

(41) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(X1, active(X2), X3) → U92¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U92¹(x1, x2, x3)  =  U92¹(x1, x2)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > s > mark₁ > cons > active₁ > U92^1₂
zeros > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U11 > U12 > U13 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U11 > U12 > U13 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U21 > U22 > U23 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U21 > U22 > U23 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U21 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U21 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNat > U21 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNat > U21 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNatIList > U31 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNatIList > U31 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNatIList > U31 > U32 > U33 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNatIList > U31 > U32 > U33 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U92^1₂
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > mark₁ > cons > active₁ > U92^1₂
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U11 > U12 > U13 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U11 > U12 > U13 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U91 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U92^1₂
length > U91 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U92^1₂
length > U91 > U92 > U93 > U94 > s > mark₁ > cons > active₁ > U92^1₂
length > U91 > U92 > U93 > U94 > s > mark₁ > 0 > active₁ > U92^1₂
nil > mark₁ > cons > active₁ > U92^1₂
nil > mark₁ > 0 > active₁ > U92^1₂

The following usable rules [FROCOS05] were oriented:

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

(43) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(44) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(45) TRUE

(46) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(mark(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, mark(X3)) → U91¹(X1, X2, X3)
U91¹(active(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, active(X3)) → U91¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(mark(X1), X2, X3) → U91¹(X1, X2, X3)
U91¹(active(X1), X2, X3) → U91¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2, x3)  =  U91¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U91^1₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > active₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > tt > cons₁
length > 0 > mark₁ > isNat₁ > active₁ > cons₁
length > 0 > mark₁ > isNat₁ > tt > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > tt > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > tt > cons₁
nil > mark₁ > isNat₁ > active₁ > cons₁
nil > mark₁ > isNat₁ > tt > cons₁

The following usable rules [FROCOS05] were oriented:

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

(48) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, mark(X3)) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, active(X3)) → U91¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(X1, X2, mark(X3)) → U91¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11(x2)
tt  =  tt
U12(x1, x2)  =  U12(x2)
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  x1
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > length₁
zeros > 0 > tt > U93 > U94 > mark₁ > s
zeros > 0 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > length₁
nil > 0 > tt > U93 > U94 > mark₁ > s
nil > 0 > tt > U93 > U94 > mark₁ > length₁

The following usable rules [FROCOS05] were oriented:

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

(50) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, X2, active(X3)) → U91¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(X1, X2, active(X3)) → U91¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2, x3)  =  U91¹(x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > tt > mark₁ > cons₁ > active₁
zeros > 0 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > isNat₁ > active₁
length > 0 > tt > mark₁ > cons₁ > active₁
length > 0 > tt > mark₁ > isNat₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > cons₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > isNat₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U61 > tt > mark₁ > cons₁ > active₁
length > U61 > tt > mark₁ > isNat₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > isNat₁ > active₁
nil > 0 > tt > mark₁ > cons₁ > active₁
nil > 0 > tt > mark₁ > isNat₁ > active₁

The following usable rules [FROCOS05] were oriented:

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

(52) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(X1, mark(X2), X3) → U91¹(X1, X2, X3)
U91¹(X1, active(X2), X3) → U91¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U91¹(x1, x2, x3)  =  U91¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U91^1₃ > active₁
zeros > cons > isNatList₁ > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > isNatList₁ > isNatKind > isNatIListKind₁ > tt > s > active₁
zeros > cons > isNatList₁ > isNatKind > U61 > mark₁ > s > active₁
zeros > cons > isNatList₁ > isNatKind > U61 > tt > s > active₁
zeros > cons > isNatList₁ > isNatKind > U71 > mark₁ > s > active₁
zeros > cons > isNatList₁ > isNatKind > U71 > tt > s > active₁
zeros > cons > U41 > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U41 > isNatKind > isNatIListKind₁ > tt > s > active₁
zeros > cons > U41 > isNatKind > U61 > mark₁ > s > active₁
zeros > cons > U41 > isNatKind > U61 > tt > s > active₁
zeros > cons > U41 > isNatKind > U71 > mark₁ > s > active₁
zeros > cons > U41 > isNatKind > U71 > tt > s > active₁
zeros > cons > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U41 > U42 > U43 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > s > active₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > active₁
zeros > cons > U51 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U51 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U51 > U52 > mark₁ > s > active₁
zeros > cons > U51 > U52 > tt > s > active₁
zeros > cons > U81 > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U81 > isNatKind > isNatIListKind₁ > tt > s > active₁
zeros > cons > U81 > isNatKind > U61 > mark₁ > s > active₁
zeros > cons > U81 > isNatKind > U61 > tt > s > active₁
zeros > cons > U81 > isNatKind > U71 > mark₁ > s > active₁
zeros > cons > U81 > isNatKind > U71 > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > active₁
zeros > cons > U91 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U91 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s > active₁
zeros > 0 > mark₁ > s > active₁
zeros > 0 > tt > s > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > mark₁ > s > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > tt > s > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > mark₁ > s > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > s > active₁
isNatIList₁ > U41 > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
isNatIList₁ > U41 > isNatKind > isNatIListKind₁ > tt > s > active₁
isNatIList₁ > U41 > isNatKind > U61 > mark₁ > s > active₁
isNatIList₁ > U41 > isNatKind > U61 > tt > s > active₁
isNatIList₁ > U41 > isNatKind > U71 > mark₁ > s > active₁
isNatIList₁ > U41 > isNatKind > U71 > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > active₁
length > 0 > mark₁ > s > active₁
length > 0 > tt > s > active₁
length > U11 > U12 > U13 > mark₁ > s > active₁
length > U11 > U12 > U13 > tt > s > active₁
length > U11 > isNatIListKind₁ > mark₁ > s > active₁
length > U11 > isNatIListKind₁ > tt > s > active₁
length > U61 > mark₁ > s > active₁
length > U61 > tt > s > active₁
length > U91 > isNatIListKind₁ > mark₁ > s > active₁
length > U91 > isNatIListKind₁ > tt > s > active₁
length > U91 > U92 > U93 > U94 > mark₁ > s > active₁
nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(54) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(55) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(56) TRUE

(57) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U86¹(active(X)) → U86¹(X)
U86¹(mark(X)) → U86¹(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U86¹(active(X)) → U86¹(X)
U86¹(mark(X)) → U86¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U86¹(x1)  =  U86¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U86^1₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(59) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(60) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(61) TRUE

(62) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U85¹(X1, mark(X2)) → U85¹(X1, X2)
U85¹(mark(X1), X2) → U85¹(X1, X2)
U85¹(active(X1), X2) → U85¹(X1, X2)
U85¹(X1, active(X2)) → U85¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U85¹(mark(X1), X2) → U85¹(X1, X2)
U85¹(active(X1), X2) → U85¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U85¹(x1, x2)  =  U85¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82(x2)
U83(x1, x2, x3)  =  U83(x2)
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons₁ > active₁
zeros > mark₁ > 0 > active₁
zeros > mark₁ > tt > active₁
zeros > mark₁ > tt > s
zeros > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > cons₁ > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > 0 > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > s
isNatIListKind₁ > U51 > U52 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > nil > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > 0 > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > s
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > 0 > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > nil > active₁
length > U61 > mark₁ > cons₁ > active₁
length > U61 > mark₁ > 0 > active₁
length > U61 > mark₁ > tt > active₁
length > U61 > mark₁ > tt > s
length > U61 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > s
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > s
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
length > U91 > U92 > U93 > U94 > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > mark₁ > 0 > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > s
length > U91 > U92 > U93 > U94 > mark₁ > nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(64) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U85¹(X1, mark(X2)) → U85¹(X1, X2)
U85¹(X1, active(X2)) → U85¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U85¹(X1, mark(X2)) → U85¹(X1, X2)
U85¹(X1, active(X2)) → U85¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U85¹(x1, x2)  =  U85¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > U85^1₂ > cons₁
zeros > 0 > mark₁ > active₁ > cons₁
zeros > tt > mark₁ > U85^1₂ > cons₁
zeros > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > U85^1₂ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > U85^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > U85^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > U85^1₂ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > mark₁ > U85^1₂ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > U85^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > U85^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > U85^1₂ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > U85^1₂ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > U85^1₂ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > U85^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > U85^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > U85^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > U85^1₂ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > cons₁
nil > 0 > mark₁ > U85^1₂ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > tt > mark₁ > U85^1₂ > cons₁
nil > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(66) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(67) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(68) TRUE

(69) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)
U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, mark(X3)) → U84¹(X1, X2, X3)
U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(mark(X1), X2, X3) → U84¹(X1, X2, X3)
U84¹(active(X1), X2, X3) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  U84¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U84^1₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > active₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > tt > cons₁
length > 0 > mark₁ > isNat₁ > active₁ > cons₁
length > 0 > mark₁ > isNat₁ > tt > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > tt > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > tt > cons₁
nil > mark₁ > isNat₁ > active₁ > cons₁
nil > mark₁ > isNat₁ > tt > cons₁

The following usable rules [FROCOS05] were oriented:

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

(71) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, mark(X3)) → U84¹(X1, X2, X3)
U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(X1, X2, mark(X3)) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11(x2)
tt  =  tt
U12(x1, x2)  =  U12(x2)
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  x1
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U51 > U52 > tt > U93 > U94 > mark₁ > s
zeros > cons > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > length₁
zeros > 0 > tt > U93 > U94 > mark₁ > s
zeros > 0 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > length₁
nil > 0 > tt > U93 > U94 > mark₁ > s
nil > 0 > tt > U93 > U94 > mark₁ > length₁

The following usable rules [FROCOS05] were oriented:

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

(73) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(X1, X2, active(X3)) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  U84¹(x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > tt > mark₁ > cons₁ > active₁
zeros > 0 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > isNat₁ > active₁
length > 0 > tt > mark₁ > cons₁ > active₁
length > 0 > tt > mark₁ > isNat₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > cons₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > isNat₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U61 > tt > mark₁ > cons₁ > active₁
length > U61 > tt > mark₁ > isNat₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > isNat₁ > active₁
nil > 0 > tt > mark₁ > cons₁ > active₁
nil > 0 > tt > mark₁ > isNat₁ > active₁

The following usable rules [FROCOS05] were oriented:

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

(75) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)
U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(X1, mark(X2), X3) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  U84¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x1
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > isNatKind₁
zeros > 0 > tt > isNatKind₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind > U51 > U52 > tt > isNatKind₁
isNatList₁ > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > mark₁ > isNatKind₁
isNatList₁ > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > tt > isNatKind₁
isNatList₁ > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > mark₁ > isNatKind₁
isNatList₁ > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > tt > isNatKind₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNatKind₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > isNatKind₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > isNatKind₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNatKind₁
isNatIList₁ > U31 > U32 > U33 > tt > isNatKind₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind > U51 > U52 > tt > isNatKind₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > isNat > U11 > U12 > U13 > mark₁ > isNatKind₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > isNat > U11 > U12 > U13 > tt > isNatKind₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > isNat > U21 > U22 > U23 > mark₁ > isNatKind₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > isNat > U21 > U22 > U23 > tt > isNatKind₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNatKind₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > tt > isNatKind₁
length > 0 > mark₁ > isNatKind₁
length > 0 > tt > isNatKind₁
length > U61 > mark₁ > isNatKind₁
length > U61 > tt > isNatKind₁
length > U91 > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁
length > U91 > isNatIListKind > U51 > U52 > tt > isNatKind₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > isNatKind₁
length > U91 > U92 > isNat > U11 > U12 > U13 > tt > isNatKind₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > isNatKind₁
length > U91 > U92 > isNat > U21 > U22 > U23 > tt > isNatKind₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > isNatKind₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > isNatKind₁
nil > mark₁ > isNatKind₁
nil > tt > isNatKind₁

The following usable rules [FROCOS05] were oriented:

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

(77) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U84¹(X1, active(X2), X3) → U84¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U84¹(x1, x2, x3)  =  U84¹(x1, x2)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > s > mark₁ > cons > active₁ > U84^1₂
zeros > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U11 > U12 > U13 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U11 > U12 > U13 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U21 > U22 > U23 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U21 > U22 > U23 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U21 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U21 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNat > U21 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNat > U21 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U31 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U31 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U31 > U32 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U31 > U32 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U31 > U32 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U31 > U32 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U31 > U32 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U31 > U32 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U31 > U32 > U33 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U31 > U32 > U33 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U84^1₂
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > mark₁ > cons > active₁ > U84^1₂
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U11 > U12 > U13 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U11 > U12 > U13 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U91 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U84^1₂
length > U91 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U84^1₂
length > U91 > U92 > U93 > U94 > s > mark₁ > cons > active₁ > U84^1₂
length > U91 > U92 > U93 > U94 > s > mark₁ > 0 > active₁ > U84^1₂
nil > mark₁ > cons > active₁ > U84^1₂
nil > mark₁ > 0 > active₁ > U84^1₂

The following usable rules [FROCOS05] were oriented:

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

(79) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(80) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(81) TRUE

(82) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)
U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, mark(X3)) → U83¹(X1, X2, X3)
U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(83) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(mark(X1), X2, X3) → U83¹(X1, X2, X3)
U83¹(active(X1), X2, X3) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  U83¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U83^1₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > active₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > tt > cons₁
length > 0 > mark₁ > isNat₁ > active₁ > cons₁
length > 0 > mark₁ > isNat₁ > tt > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > tt > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > tt > cons₁
nil > mark₁ > isNat₁ > active₁ > cons₁
nil > mark₁ > isNat₁ > tt > cons₁

The following usable rules [FROCOS05] were oriented:

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

(84) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, mark(X3)) → U83¹(X1, X2, X3)
U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(85) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(X1, X2, mark(X3)) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11(x2)
tt  =  tt
U12(x1, x2)  =  U12(x2)
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  x1
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U51 > U52 > tt > U93 > U94 > mark₁ > s
zeros > cons > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > length₁
zeros > 0 > tt > U93 > U94 > mark₁ > s
zeros > 0 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > length₁
nil > 0 > tt > U93 > U94 > mark₁ > s
nil > 0 > tt > U93 > U94 > mark₁ > length₁

The following usable rules [FROCOS05] were oriented:

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

(86) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(87) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(X1, X2, active(X3)) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  U83¹(x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > tt > mark₁ > cons₁ > active₁
zeros > 0 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > isNat₁ > active₁
length > 0 > tt > mark₁ > cons₁ > active₁
length > 0 > tt > mark₁ > isNat₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > cons₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > isNat₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U61 > tt > mark₁ > cons₁ > active₁
length > U61 > tt > mark₁ > isNat₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > isNat₁ > active₁
nil > 0 > tt > mark₁ > cons₁ > active₁
nil > 0 > tt > mark₁ > isNat₁ > active₁

The following usable rules [FROCOS05] were oriented:

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

(88) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)
U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(89) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(X1, mark(X2), X3) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  U83¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons₁
zeros > mark₁ > 0
zeros > mark₁ > tt
zeros > mark₁ > s
isNat > U11 > U12 > U13 > mark₁ > cons₁
isNat > U11 > U12 > U13 > mark₁ > 0
isNat > U11 > U12 > U13 > mark₁ > tt
isNat > U11 > U12 > U13 > mark₁ > s
isNat > U21 > U22 > U23 > mark₁ > cons₁
isNat > U21 > U22 > U23 > mark₁ > 0
isNat > U21 > U22 > U23 > mark₁ > tt
isNat > U21 > U22 > U23 > mark₁ > s
isNat > isNatKind > isNatIListKind > U51 > U52 > mark₁ > cons₁
isNat > isNatKind > isNatIListKind > U51 > U52 > mark₁ > 0
isNat > isNatKind > isNatIListKind > U51 > U52 > mark₁ > tt
isNat > isNatKind > isNatIListKind > U51 > U52 > mark₁ > s
isNat > isNatKind > U61 > mark₁ > cons₁
isNat > isNatKind > U61 > mark₁ > 0
isNat > isNatKind > U61 > mark₁ > tt
isNat > isNatKind > U61 > mark₁ > s
isNat > isNatKind > U71 > mark₁ > cons₁
isNat > isNatKind > U71 > mark₁ > 0
isNat > isNatKind > U71 > mark₁ > tt
isNat > isNatKind > U71 > mark₁ > s
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > cons₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > 0
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > tt
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > s
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > 0
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt
isNatIList₁ > U31 > U32 > U33 > mark₁ > s
isNatIList₁ > U41₁ > isNatKind > isNatIListKind > U51 > U52 > mark₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > isNatIListKind > U51 > U52 > mark₁ > 0
isNatIList₁ > U41₁ > isNatKind > isNatIListKind > U51 > U52 > mark₁ > tt
isNatIList₁ > U41₁ > isNatKind > isNatIListKind > U51 > U52 > mark₁ > s
isNatIList₁ > U41₁ > isNatKind > U61 > mark₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > mark₁ > 0
isNatIList₁ > U41₁ > isNatKind > U61 > mark₁ > tt
isNatIList₁ > U41₁ > isNatKind > U61 > mark₁ > s
isNatIList₁ > U41₁ > isNatKind > U71 > mark₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > mark₁ > 0
isNatIList₁ > U41₁ > isNatKind > U71 > mark₁ > tt
isNatIList₁ > U41₁ > isNatKind > U71 > mark₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind > U51 > U52 > mark₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind > U51 > U52 > mark₁ > 0
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind > U51 > U52 > mark₁ > tt
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind > U51 > U52 > mark₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > mark₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > mark₁ > 0
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > mark₁ > tt
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > mark₁ > s
length > U11 > U12 > U13 > mark₁ > cons₁
length > U11 > U12 > U13 > mark₁ > 0
length > U11 > U12 > U13 > mark₁ > tt
length > U11 > U12 > U13 > mark₁ > s
length > isNatList > U81 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > cons₁
length > isNatList > U81 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > 0
length > isNatList > U81 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > tt
length > isNatList > U81 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > s
length > isNatList > U81 > isNatKind > U61 > mark₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > mark₁ > 0
length > isNatList > U81 > isNatKind > U61 > mark₁ > tt
length > isNatList > U81 > isNatKind > U61 > mark₁ > s
length > isNatList > U81 > isNatKind > U71 > mark₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > mark₁ > 0
length > isNatList > U81 > isNatKind > U71 > mark₁ > tt
length > isNatList > U81 > isNatKind > U71 > mark₁ > s
length > isNatList > U81 > U82 > U83 > isNatIListKind > U51 > U52 > mark₁ > cons₁
length > isNatList > U81 > U82 > U83 > isNatIListKind > U51 > U52 > mark₁ > 0
length > isNatList > U81 > U82 > U83 > isNatIListKind > U51 > U52 > mark₁ > tt
length > isNatList > U81 > U82 > U83 > isNatIListKind > U51 > U52 > mark₁ > s
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > 0
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > s
length > U91 > isNatIListKind > U51 > U52 > mark₁ > cons₁
length > U91 > isNatIListKind > U51 > U52 > mark₁ > 0
length > U91 > isNatIListKind > U51 > U52 > mark₁ > tt
length > U91 > isNatIListKind > U51 > U52 > mark₁ > s
length > U91 > U92 > U93 > U94 > mark₁ > cons₁
length > U91 > U92 > U93 > U94 > mark₁ > 0
length > U91 > U92 > U93 > U94 > mark₁ > tt
length > U91 > U92 > U93 > U94 > mark₁ > s
nil > mark₁ > cons₁
nil > mark₁ > 0
nil > mark₁ > tt
nil > mark₁ > s

The following usable rules [FROCOS05] were oriented:

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

(90) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(91) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(X1, active(X2), X3) → U83¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U83¹(x1, x2, x3)  =  U83¹(x1, x2)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > s > mark₁ > cons > active₁ > U83^1₂
zeros > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U11 > U12 > U13 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U11 > U12 > U13 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U21 > U22 > U23 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U21 > U22 > U23 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U21 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U21 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNat > U21 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNat > U21 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNatIList > U31 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNatIList > U31 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNatIList > U31 > U32 > U33 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNatIList > U31 > U32 > U33 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U83^1₂
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > mark₁ > cons > active₁ > U83^1₂
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U11 > U12 > U13 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U11 > U12 > U13 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U91 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁ > U83^1₂
length > U91 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁ > U83^1₂
length > U91 > U92 > U93 > U94 > s > mark₁ > cons > active₁ > U83^1₂
length > U91 > U92 > U93 > U94 > s > mark₁ > 0 > active₁ > U83^1₂
nil > mark₁ > cons > active₁ > U83^1₂
nil > mark₁ > 0 > active₁ > U83^1₂

The following usable rules [FROCOS05] were oriented:

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

(92) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(93) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(94) TRUE

(95) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(96) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, X2, mark(X3)) → U82¹(X1, X2, X3)
U82¹(X1, X2, active(X3)) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U82^1₁ > cons₁
zeros > 0 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
zeros > 0 > mark₁ > nil > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > nil > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil > cons₁
isNatIList₁ > U41₁ > U42₁ > U43 > U44 > U45 > U46 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43 > U44 > U45 > U46 > mark₁ > nil > cons₁
length > 0 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > 0 > mark₁ > nil > cons₁
length > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > nil > cons₁
length > U61 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U61 > mark₁ > nil > cons₁
length > U91 > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > isNatIListKind > U51 > U52 > mark₁ > nil > cons₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > nil > cons₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > nil > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > nil > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > nil > cons₁

The following usable rules [FROCOS05] were oriented:

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

(97) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(mark(X1), X2, X3) → U82¹(X1, X2, X3)
U82¹(active(X1), X2, X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > isNatKind₁ > isNatIListKind₁ > active₁ > tt
zeros > cons > isNatKind₁ > U71 > mark₁ > active₁ > tt
zeros > cons > U51 > isNatIListKind₁ > active₁ > tt
zeros > cons > U51 > U52 > mark₁ > active₁ > tt
zeros > cons > U91 > isNatIListKind₁ > active₁ > tt
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > active₁ > tt
zeros > cons > U91 > U92 > U93 > U94 > s > active₁ > tt
zeros > 0 > mark₁ > active₁ > tt
isNatList₁ > U81 > isNatKind₁ > isNatIListKind₁ > active₁ > tt
isNatList₁ > U81 > isNatKind₁ > U71 > mark₁ > active₁ > tt
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > active₁ > tt
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > tt
isNat > U11 > U12 > U13 > mark₁ > active₁ > tt
isNat > U11 > isNatIListKind₁ > active₁ > tt
isNat > U21 > U22 > U23 > mark₁ > active₁ > tt
isNat > isNatKind₁ > isNatIListKind₁ > active₁ > tt
isNat > isNatKind₁ > U71 > mark₁ > active₁ > tt
isNatIList₁ > U31₁ > isNatIListKind₁ > active₁ > tt
isNatIList₁ > U31₁ > U32 > U33 > mark₁ > active₁ > tt
isNatIList₁ > U41 > isNatKind₁ > isNatIListKind₁ > active₁ > tt
isNatIList₁ > U41 > isNatKind₁ > U71 > mark₁ > active₁ > tt
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > active₁ > tt
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > active₁ > tt
length > 0 > mark₁ > active₁ > tt
length > U11 > U12 > U13 > mark₁ > active₁ > tt
length > U11 > isNatIListKind₁ > active₁ > tt
length > U61 > mark₁ > active₁ > tt
length > U91 > isNatIListKind₁ > active₁ > tt
length > U91 > U92 > U93 > U94 > mark₁ > active₁ > tt
length > U91 > U92 > U93 > U94 > s > active₁ > tt
nil > 0 > mark₁ > active₁ > tt

The following usable rules [FROCOS05] were oriented:

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

(99) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(X1, mark(X2), X3) → U82¹(X1, X2, X3)
U82¹(X1, active(X2), X3) → U82¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U82¹(x1, x2, x3)  =  U82¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  x1
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > tt > isNatList₁ > active₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > isNatList₁ > active₁
zeros > cons > U51 > isNatIListKind₁ > mark₁ > tt > isNatList₁ > active₁
zeros > cons > U51 > U52 > mark₁ > tt > isNatList₁ > active₁
zeros > cons > U81 > U82 > U83 > isNatIListKind₁ > mark₁ > tt > isNatList₁ > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > isNatList₁ > active₁
zeros > cons > U91 > isNatIListKind₁ > mark₁ > tt > isNatList₁ > active₁
zeros > cons > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > tt > isNatList₁ > active₁
zeros > 0 > mark₁ > tt > isNatList₁ > active₁
isNatKind > isNatIListKind₁ > mark₁ > tt > isNatList₁ > active₁
isNatKind > U61 > mark₁ > tt > isNatList₁ > active₁
isNatKind > U71 > mark₁ > tt > isNatList₁ > active₁
isNatIList₁ > U31 > isNatIListKind₁ > mark₁ > tt > isNatList₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > isNatList₁ > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > tt > isNatList₁ > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > isNatList₁ > active₁
length > 0 > mark₁ > tt > isNatList₁ > active₁
length > U11 > U12 > U13 > mark₁ > tt > isNatList₁ > active₁
length > U11 > isNatIListKind₁ > mark₁ > tt > isNatList₁ > active₁
length > U61 > mark₁ > tt > isNatList₁ > active₁
length > U91 > isNatIListKind₁ > mark₁ > tt > isNatList₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > tt > isNatList₁ > active₁
nil > 0 > mark₁ > tt > isNatList₁ > active₁

The following usable rules [FROCOS05] were oriented:

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

(101) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(102) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(103) TRUE

(104) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(105) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(X1, mark(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, mark(X3)) → U81¹(X1, X2, X3)
U81¹(X1, active(X2), X3) → U81¹(X1, X2, X3)
U81¹(X1, X2, active(X3)) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U81^1₂ > cons₁
zeros > 0 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
zeros > 0 > mark₁ > nil > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > nil > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil > cons₁
isNatIList₁ > U41₁ > U42₁ > U43 > U44 > U45 > U46 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43 > U44 > U45 > U46 > mark₁ > nil > cons₁
length > 0 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > 0 > mark₁ > nil > cons₁
length > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > nil > cons₁
length > U61 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U61 > mark₁ > nil > cons₁
length > U91 > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > isNatIListKind > U51 > U52 > mark₁ > nil > cons₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > nil > cons₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > nil > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > nil > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > active₁ > tt > isNatKind₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > nil > cons₁

The following usable rules [FROCOS05] were oriented:

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

(106) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)
U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(mark(X1), X2, X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  x1
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82(x2)
U83(x1, x2, x3)  =  U83(x2)
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93(x1)
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > U94 > mark₁ > 0
zeros > tt > U94 > mark₁ > U93₁
zeros > tt > U94 > mark₁ > s
isNatIListKind > isNatKind₁ > U61 > tt > U94 > mark₁ > 0
isNatIListKind > isNatKind₁ > U61 > tt > U94 > mark₁ > U93₁
isNatIListKind > isNatKind₁ > U61 > tt > U94 > mark₁ > s
isNatIListKind > isNatKind₁ > U71 > tt > U94 > mark₁ > 0
isNatIListKind > isNatKind₁ > U71 > tt > U94 > mark₁ > U93₁
isNatIListKind > isNatKind₁ > U71 > tt > U94 > mark₁ > s
isNatIListKind > U51 > U52 > tt > U94 > mark₁ > 0
isNatIListKind > U51 > U52 > tt > U94 > mark₁ > U93₁
isNatIListKind > U51 > U52 > tt > U94 > mark₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U94 > mark₁ > 0
isNatIList₁ > U31 > U32 > U33 > tt > U94 > mark₁ > U93₁
isNatIList₁ > U31 > U32 > U33 > tt > U94 > mark₁ > s
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > U94 > mark₁ > 0
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > U94 > mark₁ > U93₁
isNatIList₁ > U41 > isNatKind₁ > U61 > tt > U94 > mark₁ > s
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > U94 > mark₁ > 0
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > U94 > mark₁ > U93₁
isNatIList₁ > U41 > isNatKind₁ > U71 > tt > U94 > mark₁ > s
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U94 > mark₁ > 0
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U94 > mark₁ > U93₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > U94 > mark₁ > s
length > isNatList₁ > U81₁ > isNatKind₁ > U61 > tt > U94 > mark₁ > 0
length > isNatList₁ > U81₁ > isNatKind₁ > U61 > tt > U94 > mark₁ > U93₁
length > isNatList₁ > U81₁ > isNatKind₁ > U61 > tt > U94 > mark₁ > s
length > isNatList₁ > U81₁ > isNatKind₁ > U71 > tt > U94 > mark₁ > 0
length > isNatList₁ > U81₁ > isNatKind₁ > U71 > tt > U94 > mark₁ > U93₁
length > isNatList₁ > U81₁ > isNatKind₁ > U71 > tt > U94 > mark₁ > s
length > isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > tt > U94 > mark₁ > 0
length > isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > tt > U94 > mark₁ > U93₁
length > isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > tt > U94 > mark₁ > s
length > U91 > U92 > isNat > U11 > U12 > U13 > tt > U94 > mark₁ > 0
length > U91 > U92 > isNat > U11 > U12 > U13 > tt > U94 > mark₁ > U93₁
length > U91 > U92 > isNat > U11 > U12 > U13 > tt > U94 > mark₁ > s
length > U91 > U92 > isNat > U21 > U22 > U23 > tt > U94 > mark₁ > 0
length > U91 > U92 > isNat > U21 > U22 > U23 > tt > U94 > mark₁ > U93₁
length > U91 > U92 > isNat > U21 > U22 > U23 > tt > U94 > mark₁ > s
nil > tt > U94 > mark₁ > 0
nil > tt > U94 > mark₁ > U93₁
nil > tt > U94 > mark₁ > s

The following usable rules [FROCOS05] were oriented:

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

(108) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(109) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(active(X1), X2, X3) → U81¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U81¹(x1, x2, x3)  =  U81¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > tt > s > mark₁ > cons > active₁
zeros > tt > s > mark₁ > 0 > active₁
isNat > U11 > U12 > U13 > tt > s > mark₁ > cons > active₁
isNat > U11 > U12 > U13 > tt > s > mark₁ > 0 > active₁
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > cons > active₁
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > cons > active₁
isNat > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁
isNat > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > cons > active₁
isNat > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > 0 > active₁
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁
isNat > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁
isNat > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁
isNat > U21 > U22 > U23 > tt > s > mark₁ > cons > active₁
isNat > U21 > U22 > U23 > tt > s > mark₁ > 0 > active₁
isNat > U21 > isNatKind > U61 > tt > s > mark₁ > cons > active₁
isNat > U21 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁
isNat > U21 > isNatKind > U71 > tt > s > mark₁ > cons > active₁
isNat > U21 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁
isNatIList > U31 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁
isNatIList > U31 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁
isNatIList > U31 > U32 > U33 > tt > s > mark₁ > cons > active₁
isNatIList > U31 > U32 > U33 > tt > s > mark₁ > 0 > active₁
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁
isNatIList > U41 > U42 > U43 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > mark₁ > cons > active₁
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > mark₁ > 0 > active₁
length > U11 > U12 > U13 > tt > s > mark₁ > cons > active₁
length > U11 > U12 > U13 > tt > s > mark₁ > 0 > active₁
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > cons > active₁
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U61 > tt > s > mark₁ > 0 > active₁
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > cons > active₁
length > U11 > U12 > isNatList₁ > U81 > isNatKind > U71 > tt > s > mark₁ > 0 > active₁
length > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > cons > active₁
length > U11 > U12 > isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > mark₁ > 0 > active₁
length > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁
length > U11 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁
length > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁
length > U11 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > cons > active₁
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > s > mark₁ > 0 > active₁
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > cons > active₁
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > s > mark₁ > 0 > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > cons > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > s > mark₁ > 0 > active₁
length > U91 > U92 > U93 > U94 > s > mark₁ > cons > active₁
length > U91 > U92 > U93 > U94 > s > mark₁ > 0 > active₁
nil > mark₁ > cons > active₁
nil > mark₁ > 0 > active₁

The following usable rules [FROCOS05] were oriented:

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

(110) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(111) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(112) TRUE

(113) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U71¹(active(X)) → U71¹(X)
U71¹(mark(X)) → U71¹(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(114) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(active(X)) → U71¹(X)
U71¹(mark(X)) → U71¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1)  =  U71¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U71^1₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(115) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(116) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(117) TRUE

(118) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U61¹(active(X)) → U61¹(X)
U61¹(mark(X)) → U61¹(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(119) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(active(X)) → U61¹(X)
U61¹(mark(X)) → U61¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1)  =  U61¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U61^1₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(120) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(121) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(122) TRUE

(123) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U52¹(active(X)) → U52¹(X)
U52¹(mark(X)) → U52¹(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(124) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U52¹(active(X)) → U52¹(X)
U52¹(mark(X)) → U52¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1)  =  U52¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U52^1₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(125) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(126) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(127) TRUE

(128) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(129) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(mark(X1), X2) → U51¹(X1, X2)
U51¹(active(X1), X2) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > cons₁ > active₁
zeros > 0 > mark₁ > isNatKind₁ > tt > active₁
zeros > 0 > mark₁ > nil
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > cons₁ > active₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNatKind₁ > tt > active₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > nil
isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
isNat > U11 > U12 > U13 > mark₁ > isNatKind₁ > tt > active₁
isNat > U11 > U12 > U13 > mark₁ > nil
isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
isNat > U21 > U22 > U23 > mark₁ > isNatKind₁ > tt > active₁
isNat > U21 > U22 > U23 > mark₁ > nil
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁ > tt > active₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > nil
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNatKind₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil
isNatIList₁ > U41₁ > U42 > isNatIListKind > U51 > U52 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > isNatIListKind > U51 > U52 > mark₁ > nil
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > isNatKind₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > nil
length > 0 > mark₁ > cons₁ > active₁
length > 0 > mark₁ > isNatKind₁ > tt > active₁
length > 0 > mark₁ > nil
length > U11 > U12 > U13 > mark₁ > cons₁ > active₁
length > U11 > U12 > U13 > mark₁ > isNatKind₁ > tt > active₁
length > U11 > U12 > U13 > mark₁ > nil
length > isNatIListKind > U51 > U52 > mark₁ > cons₁ > active₁
length > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁ > tt > active₁
length > isNatIListKind > U51 > U52 > mark₁ > nil
length > U61 > mark₁ > cons₁ > active₁
length > U61 > mark₁ > isNatKind₁ > tt > active₁
length > U61 > mark₁ > nil
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > isNatKind₁ > tt > active₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

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

(130) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(131) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U51¹(X1, mark(X2)) → U51¹(X1, X2)
U51¹(X1, active(X2)) → U51¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U51¹(x1, x2)  =  U51¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > U51^1₂ > cons₁
zeros > 0 > mark₁ > active₁ > cons₁
zeros > tt > mark₁ > U51^1₂ > cons₁
zeros > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > U51^1₂ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > U51^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > U51^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > U51^1₂ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > mark₁ > U51^1₂ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > U51^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > U51^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > U51^1₂ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > U51^1₂ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > U51^1₂ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > U51^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > U51^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > U51^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > U51^1₂ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > cons₁
nil > 0 > mark₁ > U51^1₂ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > tt > mark₁ > U51^1₂ > cons₁
nil > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(132) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(133) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(134) TRUE

(135) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATILIST(active(X)) → ISNATILIST(X)
ISNATILIST(mark(X)) → ISNATILIST(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(136) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILIST(active(X)) → ISNATILIST(X)
ISNATILIST(mark(X)) → ISNATILIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILIST(x1)  =  ISNATILIST(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

ISNATILIST₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(137) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(138) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(139) TRUE

(140) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U46¹(active(X)) → U46¹(X)
U46¹(mark(X)) → U46¹(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(141) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U46¹(active(X)) → U46¹(X)
U46¹(mark(X)) → U46¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U46¹(x1)  =  U46¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U46^1₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(142) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(143) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(144) TRUE

(145) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U45¹(X1, mark(X2)) → U45¹(X1, X2)
U45¹(mark(X1), X2) → U45¹(X1, X2)
U45¹(active(X1), X2) → U45¹(X1, X2)
U45¹(X1, active(X2)) → U45¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(146) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U45¹(mark(X1), X2) → U45¹(X1, X2)
U45¹(active(X1), X2) → U45¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U45¹(x1, x2)  =  U45¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82(x2)
U83(x1, x2, x3)  =  U83(x2)
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons₁ > active₁
zeros > mark₁ > 0 > active₁
zeros > mark₁ > tt > active₁
zeros > mark₁ > tt > s
zeros > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > cons₁ > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > 0 > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > s
isNatIListKind₁ > U51 > U52 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > nil > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > 0 > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > s
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > 0 > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > nil > active₁
length > U61 > mark₁ > cons₁ > active₁
length > U61 > mark₁ > 0 > active₁
length > U61 > mark₁ > tt > active₁
length > U61 > mark₁ > tt > s
length > U61 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > s
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > s
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
length > U91 > U92 > U93 > U94 > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > mark₁ > 0 > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > s
length > U91 > U92 > U93 > U94 > mark₁ > nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(147) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U45¹(X1, mark(X2)) → U45¹(X1, X2)
U45¹(X1, active(X2)) → U45¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(148) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U45¹(X1, mark(X2)) → U45¹(X1, X2)
U45¹(X1, active(X2)) → U45¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U45¹(x1, x2)  =  U45¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > U45^1₂ > cons₁
zeros > 0 > mark₁ > active₁ > cons₁
zeros > tt > mark₁ > U45^1₂ > cons₁
zeros > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > U45^1₂ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > U45^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > U45^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > U45^1₂ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > mark₁ > U45^1₂ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > U45^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > U45^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > U45^1₂ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > U45^1₂ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > U45^1₂ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > U45^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > U45^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > U45^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > U45^1₂ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > cons₁
nil > 0 > mark₁ > U45^1₂ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > tt > mark₁ > U45^1₂ > cons₁
nil > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(149) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(150) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(151) TRUE

(152) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)
U44¹(mark(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, mark(X3)) → U44¹(X1, X2, X3)
U44¹(active(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, active(X3)) → U44¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(153) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U44¹(mark(X1), X2, X3) → U44¹(X1, X2, X3)
U44¹(active(X1), X2, X3) → U44¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U44¹(x1, x2, x3)  =  U44¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U44^1₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > active₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > tt > cons₁
length > 0 > mark₁ > isNat₁ > active₁ > cons₁
length > 0 > mark₁ > isNat₁ > tt > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > tt > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > tt > cons₁
nil > mark₁ > isNat₁ > active₁ > cons₁
nil > mark₁ > isNat₁ > tt > cons₁

The following usable rules [FROCOS05] were oriented:

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

(154) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, mark(X3)) → U44¹(X1, X2, X3)
U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, active(X3)) → U44¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(155) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U44¹(X1, X2, mark(X3)) → U44¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U44¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  x1
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U51 > U52 > tt > mark₁ > s
zeros > cons > U51 > U52 > tt > mark₁ > nil > 0
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > s
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > nil > 0
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > s
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > nil > 0
zeros > cons > U91 > U92 > U93 > isNatKind > U61 > tt > mark₁ > s
zeros > cons > U91 > U92 > U93 > isNatKind > U61 > tt > mark₁ > nil > 0
zeros > cons > U91 > U92 > U93 > isNatKind > U71 > tt > mark₁ > s
zeros > cons > U91 > U92 > U93 > isNatKind > U71 > tt > mark₁ > nil > 0
isNatList > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > s
isNatList > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > nil > 0
isNatList > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > s
isNatList > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > nil > 0
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > s
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > nil > 0
isNat > U11 > U12 > U13 > tt > mark₁ > s
isNat > U11 > U12 > U13 > tt > mark₁ > nil > 0
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > s
isNat > U11 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > nil > 0
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > s
isNat > U11 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > nil > 0
isNat > U21 > U22 > U23 > tt > mark₁ > s
isNat > U21 > U22 > U23 > tt > mark₁ > nil > 0
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > s
isNatIList > U31 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > nil > 0
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > s
isNatIList > U31 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > nil > 0
isNatIList > U31 > U32 > U33 > tt > mark₁ > s
isNatIList > U31 > U32 > U33 > tt > mark₁ > nil > 0
isNatIList > U41 > isNatKind > U61 > tt > mark₁ > s
isNatIList > U41 > isNatKind > U61 > tt > mark₁ > nil > 0
isNatIList > U41 > isNatKind > U71 > tt > mark₁ > s
isNatIList > U41 > isNatKind > U71 > tt > mark₁ > nil > 0
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > mark₁ > s
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > tt > mark₁ > nil > 0
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > s
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > nil > 0
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > s
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > nil > 0
length > U91 > U92 > U93 > isNatKind > U61 > tt > mark₁ > s
length > U91 > U92 > U93 > isNatKind > U61 > tt > mark₁ > nil > 0
length > U91 > U92 > U93 > isNatKind > U71 > tt > mark₁ > s
length > U91 > U92 > U93 > isNatKind > U71 > tt > mark₁ > nil > 0

The following usable rules [FROCOS05] were oriented:

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

(156) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, active(X3)) → U44¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(157) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U44¹(X1, mark(X2), X3) → U44¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U44¹(x1, x2, x3)  =  U44¹(x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  x1
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  x1
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > isNatKind₁ > tt
zeros > cons > U51 > U52 > mark₁ > tt
zeros > 0 > mark₁ > tt
isNatList > U81 > isNatKind₁ > tt
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt
isNatIList > U31 > U32 > U33 > mark₁ > tt
isNatIList > U41 > isNatKind₁ > tt
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt
length > 0 > mark₁ > tt
length > U11 > U12 > U13 > mark₁ > tt
length > U61 > mark₁ > tt
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > tt
length > U91 > U92 > U93 > U94 > s > U21 > isNatKind₁ > tt
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt
nil > 0 > mark₁ > tt

The following usable rules [FROCOS05] were oriented:

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

(158) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)
U44¹(X1, X2, active(X3)) → U44¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(159) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U44¹(X1, X2, active(X3)) → U44¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U44¹(x1, x2, x3)  =  x3
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > 0
zeros > cons > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > tt > length₁
zeros > cons > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > s
zeros > cons > U51 > U52 > mark₁ > active₁ > 0
zeros > cons > U51 > U52 > mark₁ > active₁ > tt > length₁
zeros > cons > U51 > U52 > mark₁ > s
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > active₁ > 0
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > active₁ > tt > length₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s
isNat > U11 > U12 > U13 > mark₁ > active₁ > 0
isNat > U11 > U12 > U13 > mark₁ > active₁ > tt > length₁
isNat > U11 > U12 > U13 > mark₁ > s
isNat > U11 > isNatIListKind > U51 > U52 > mark₁ > active₁ > 0
isNat > U11 > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > length₁
isNat > U11 > isNatIListKind > U51 > U52 > mark₁ > s
isNat > U21 > U22 > U23 > mark₁ > active₁ > 0
isNat > U21 > U22 > U23 > mark₁ > active₁ > tt > length₁
isNat > U21 > U22 > U23 > mark₁ > s
isNat > U21 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > active₁ > 0
isNat > U21 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > length₁
isNat > U21 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > s
isNat > U21 > isNatKind > U61 > mark₁ > active₁ > 0
isNat > U21 > isNatKind > U61 > mark₁ > active₁ > tt > length₁
isNat > U21 > isNatKind > U61 > mark₁ > s
isNat > U21 > isNatKind > U71 > mark₁ > active₁ > 0
isNat > U21 > isNatKind > U71 > mark₁ > active₁ > tt > length₁
isNat > U21 > isNatKind > U71 > mark₁ > s
isNatIList > U31 > isNatIListKind > U51 > U52 > mark₁ > active₁ > 0
isNatIList > U31 > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > length₁
isNatIList > U31 > isNatIListKind > U51 > U52 > mark₁ > s
isNatIList > U31 > U32 > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > 0
isNatIList > U31 > U32 > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > tt > length₁
isNatIList > U31 > U32 > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > s
isNatIList > U31 > U32 > U33 > mark₁ > active₁ > 0
isNatIList > U31 > U32 > U33 > mark₁ > active₁ > tt > length₁
isNatIList > U31 > U32 > U33 > mark₁ > s
isNatIList > U41 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > active₁ > 0
isNatIList > U41 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > length₁
isNatIList > U41 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > s
isNatIList > U41 > isNatKind > U61 > mark₁ > active₁ > 0
isNatIList > U41 > isNatKind > U61 > mark₁ > active₁ > tt > length₁
isNatIList > U41 > isNatKind > U61 > mark₁ > s
isNatIList > U41 > isNatKind > U71 > mark₁ > active₁ > 0
isNatIList > U41 > isNatKind > U71 > mark₁ > active₁ > tt > length₁
isNatIList > U41 > isNatKind > U71 > mark₁ > s
isNatIList > U41 > U42 > U43 > isNatIListKind > U51 > U52 > mark₁ > active₁ > 0
isNatIList > U41 > U42 > U43 > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > length₁
isNatIList > U41 > U42 > U43 > isNatIListKind > U51 > U52 > mark₁ > s
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > active₁ > 0
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > active₁ > tt > length₁
isNatIList > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > s
nil > mark₁ > active₁ > 0
nil > mark₁ > active₁ > tt > length₁
nil > mark₁ > s

The following usable rules [FROCOS05] were oriented:

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

(160) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(161) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U44¹(X1, active(X2), X3) → U44¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U44¹(x1, x2, x3)  =  U44¹(x2)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U44^1₁ > active₁
zeros > 0 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > cons₁ > active₁
length > 0 > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
nil > 0 > mark₁ > cons₁ > active₁
nil > tt > mark₁ > cons₁ > active₁

The following usable rules [FROCOS05] were oriented:

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

(162) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(163) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(164) TRUE

(165) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(mark(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, mark(X3)) → U43¹(X1, X2, X3)
U43¹(active(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, active(X3)) → U43¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(166) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(mark(X1), X2, X3) → U43¹(X1, X2, X3)
U43¹(active(X1), X2, X3) → U43¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U43¹(x1, x2, x3)  =  U43¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U43^1₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > active₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > tt > cons₁
length > 0 > mark₁ > isNat₁ > active₁ > cons₁
length > 0 > mark₁ > isNat₁ > tt > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > tt > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > tt > cons₁
nil > mark₁ > isNat₁ > active₁ > cons₁
nil > mark₁ > isNat₁ > tt > cons₁

The following usable rules [FROCOS05] were oriented:

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

(167) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, mark(X3)) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, active(X3)) → U43¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(168) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(X1, X2, mark(X3)) → U43¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U43¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11(x2)
tt  =  tt
U12(x1, x2)  =  U12(x2)
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  x1
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > length₁
zeros > 0 > tt > U93 > U94 > mark₁ > s
zeros > 0 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > length₁
nil > 0 > tt > U93 > U94 > mark₁ > s
nil > 0 > tt > U93 > U94 > mark₁ > length₁

The following usable rules [FROCOS05] were oriented:

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

(169) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, X2, active(X3)) → U43¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(170) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(X1, X2, active(X3)) → U43¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U43¹(x1, x2, x3)  =  U43¹(x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > tt > mark₁ > cons₁ > active₁
zeros > 0 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > isNat₁ > active₁
length > 0 > tt > mark₁ > cons₁ > active₁
length > 0 > tt > mark₁ > isNat₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > cons₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > isNat₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U61 > tt > mark₁ > cons₁ > active₁
length > U61 > tt > mark₁ > isNat₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > isNat₁ > active₁
nil > 0 > tt > mark₁ > cons₁ > active₁
nil > 0 > tt > mark₁ > isNat₁ > active₁

The following usable rules [FROCOS05] were oriented:

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

(171) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(172) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(X1, mark(X2), X3) → U43¹(X1, X2, X3)
U43¹(X1, active(X2), X3) → U43¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U43¹(x1, x2, x3)  =  U43¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U43^1₃ > active₁
zeros > cons > isNatList₁ > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > isNatList₁ > isNatKind > isNatIListKind₁ > tt > s > active₁
zeros > cons > isNatList₁ > isNatKind > U61 > mark₁ > s > active₁
zeros > cons > isNatList₁ > isNatKind > U61 > tt > s > active₁
zeros > cons > isNatList₁ > isNatKind > U71 > mark₁ > s > active₁
zeros > cons > isNatList₁ > isNatKind > U71 > tt > s > active₁
zeros > cons > U41 > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U41 > isNatKind > isNatIListKind₁ > tt > s > active₁
zeros > cons > U41 > isNatKind > U61 > mark₁ > s > active₁
zeros > cons > U41 > isNatKind > U61 > tt > s > active₁
zeros > cons > U41 > isNatKind > U71 > mark₁ > s > active₁
zeros > cons > U41 > isNatKind > U71 > tt > s > active₁
zeros > cons > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U41 > U42 > U43 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > s > active₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > active₁
zeros > cons > U51 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U51 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U51 > U52 > mark₁ > s > active₁
zeros > cons > U51 > U52 > tt > s > active₁
zeros > cons > U81 > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U81 > isNatKind > isNatIListKind₁ > tt > s > active₁
zeros > cons > U81 > isNatKind > U61 > mark₁ > s > active₁
zeros > cons > U81 > isNatKind > U61 > tt > s > active₁
zeros > cons > U81 > isNatKind > U71 > mark₁ > s > active₁
zeros > cons > U81 > isNatKind > U71 > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > active₁
zeros > cons > U91 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U91 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s > active₁
zeros > 0 > mark₁ > s > active₁
zeros > 0 > tt > s > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > mark₁ > s > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > tt > s > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > mark₁ > s > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > s > active₁
isNatIList₁ > U41 > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
isNatIList₁ > U41 > isNatKind > isNatIListKind₁ > tt > s > active₁
isNatIList₁ > U41 > isNatKind > U61 > mark₁ > s > active₁
isNatIList₁ > U41 > isNatKind > U61 > tt > s > active₁
isNatIList₁ > U41 > isNatKind > U71 > mark₁ > s > active₁
isNatIList₁ > U41 > isNatKind > U71 > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > active₁
length > 0 > mark₁ > s > active₁
length > 0 > tt > s > active₁
length > U11 > U12 > U13 > mark₁ > s > active₁
length > U11 > U12 > U13 > tt > s > active₁
length > U11 > isNatIListKind₁ > mark₁ > s > active₁
length > U11 > isNatIListKind₁ > tt > s > active₁
length > U61 > mark₁ > s > active₁
length > U61 > tt > s > active₁
length > U91 > isNatIListKind₁ > mark₁ > s > active₁
length > U91 > isNatIListKind₁ > tt > s > active₁
length > U91 > U92 > U93 > U94 > mark₁ > s > active₁
nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(173) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(174) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(175) TRUE

(176) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(177) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(mark(X1), X2, X3) → U42¹(X1, X2, X3)
U42¹(active(X1), X2, X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U42^1₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > active₁ > cons₁
zeros > 0 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNat₁ > tt > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > isNat₁ > tt > cons₁
length > 0 > mark₁ > isNat₁ > active₁ > cons₁
length > 0 > mark₁ > isNat₁ > tt > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > isNat₁ > tt > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U61 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > isNatKind₁ > U71 > mark₁ > isNat₁ > tt > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > isNat₁ > tt > cons₁
nil > mark₁ > isNat₁ > active₁ > cons₁
nil > mark₁ > isNat₁ > tt > cons₁

The following usable rules [FROCOS05] were oriented:

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

(178) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(179) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, X2, mark(X3)) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11(x2)
tt  =  tt
U12(x1, x2)  =  U12(x2)
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  x1
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U11₁ > U12₁ > U13 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > isNat₁ > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > s
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > s
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > length₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > length₁
zeros > 0 > tt > U93 > U94 > mark₁ > s
zeros > 0 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > isNatKind₁ > U61 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > isNatKind₁ > U71 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > length₁
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > length₁
nil > 0 > tt > U93 > U94 > mark₁ > s
nil > 0 > tt > U93 > U94 > mark₁ > length₁

The following usable rules [FROCOS05] were oriented:

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

(180) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(181) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, X2, active(X3)) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > tt > mark₁ > cons₁ > active₁
zeros > 0 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > cons₁ > active₁
isNatList₁ > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U61 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > isNatKind > U71 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > isNat₁ > active₁
length > 0 > tt > mark₁ > cons₁ > active₁
length > 0 > tt > mark₁ > isNat₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > cons₁ > active₁
length > U11 > U12 > U13 > tt > mark₁ > isNat₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U11 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U61 > tt > mark₁ > cons₁ > active₁
length > U61 > tt > mark₁ > isNat₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > cons₁ > active₁
length > U91 > isNatIListKind₁ > U51 > U52 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > mark₁ > isNat₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > mark₁ > isNat₁ > active₁
nil > 0 > tt > mark₁ > cons₁ > active₁
nil > 0 > tt > mark₁ > isNat₁ > active₁

The following usable rules [FROCOS05] were oriented:

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

(182) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(183) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(X1, mark(X2), X3) → U42¹(X1, X2, X3)
U42¹(X1, active(X2), X3) → U42¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U42¹(x1, x2, x3)  =  U42¹(x1, x2, x3)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U42^1₃ > active₁
zeros > cons > isNatList₁ > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > isNatList₁ > isNatKind > isNatIListKind₁ > tt > s > active₁
zeros > cons > isNatList₁ > isNatKind > U61 > mark₁ > s > active₁
zeros > cons > isNatList₁ > isNatKind > U61 > tt > s > active₁
zeros > cons > isNatList₁ > isNatKind > U71 > mark₁ > s > active₁
zeros > cons > isNatList₁ > isNatKind > U71 > tt > s > active₁
zeros > cons > U41 > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U41 > isNatKind > isNatIListKind₁ > tt > s > active₁
zeros > cons > U41 > isNatKind > U61 > mark₁ > s > active₁
zeros > cons > U41 > isNatKind > U61 > tt > s > active₁
zeros > cons > U41 > isNatKind > U71 > mark₁ > s > active₁
zeros > cons > U41 > isNatKind > U71 > tt > s > active₁
zeros > cons > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U41 > U42 > U43 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > s > active₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > active₁
zeros > cons > U51 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U51 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U51 > U52 > mark₁ > s > active₁
zeros > cons > U51 > U52 > tt > s > active₁
zeros > cons > U81 > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U81 > isNatKind > isNatIListKind₁ > tt > s > active₁
zeros > cons > U81 > isNatKind > U61 > mark₁ > s > active₁
zeros > cons > U81 > isNatKind > U61 > tt > s > active₁
zeros > cons > U81 > isNatKind > U71 > mark₁ > s > active₁
zeros > cons > U81 > isNatKind > U71 > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U11 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > tt > s > active₁
zeros > cons > U91 > isNatIListKind₁ > mark₁ > s > active₁
zeros > cons > U91 > isNatIListKind₁ > tt > s > active₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > s > active₁
zeros > 0 > mark₁ > s > active₁
zeros > 0 > tt > s > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > mark₁ > s > active₁
isNatIList₁ > U31₁ > isNatIListKind₁ > tt > s > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > mark₁ > s > active₁
isNatIList₁ > U31₁ > U32₁ > U33 > tt > s > active₁
isNatIList₁ > U41 > isNatKind > isNatIListKind₁ > mark₁ > s > active₁
isNatIList₁ > U41 > isNatKind > isNatIListKind₁ > tt > s > active₁
isNatIList₁ > U41 > isNatKind > U61 > mark₁ > s > active₁
isNatIList₁ > U41 > isNatKind > U61 > tt > s > active₁
isNatIList₁ > U41 > isNatKind > U71 > mark₁ > s > active₁
isNatIList₁ > U41 > isNatKind > U71 > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > s > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > tt > s > active₁
length > 0 > mark₁ > s > active₁
length > 0 > tt > s > active₁
length > U11 > U12 > U13 > mark₁ > s > active₁
length > U11 > U12 > U13 > tt > s > active₁
length > U11 > isNatIListKind₁ > mark₁ > s > active₁
length > U11 > isNatIListKind₁ > tt > s > active₁
length > U61 > mark₁ > s > active₁
length > U61 > tt > s > active₁
length > U91 > isNatIListKind₁ > mark₁ > s > active₁
length > U91 > isNatIListKind₁ > tt > s > active₁
length > U91 > U92 > U93 > U94 > mark₁ > s > active₁
nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(184) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(185) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(186) TRUE

(187) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(188) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, X2, mark(X3)) → U41¹(X1, X2, X3)
U41¹(X1, X2, active(X3)) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  x3
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > active₁ > cons₁
zeros > 0 > tt > active₁ > cons₁
isNatList > U81 > isNatKind > isNatIListKind > U51 > U52 > mark₁ > active₁ > cons₁
isNatList > U81 > isNatKind > isNatIListKind > U51 > U52 > tt > active₁ > cons₁
isNatList > U81 > isNatKind > U61 > mark₁ > active₁ > cons₁
isNatList > U81 > isNatKind > U61 > tt > active₁ > cons₁
isNatList > U81 > isNatKind > U71 > mark₁ > active₁ > cons₁
isNatList > U81 > isNatKind > U71 > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > isNatIListKind > U51 > U52 > mark₁ > active₁ > cons₁
isNatList > U81 > U82 > U83 > isNatIListKind > U51 > U52 > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > mark₁ > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > isNat > U11 > U12 > U13 > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > mark₁ > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > isNat > U21 > U22 > U23 > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > active₁ > cons₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > active₁ > cons₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > tt > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > isNatIListKind > U51 > U52 > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > isNatIListKind > U51 > U52 > tt > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > isNatIListKind > U51 > U52 > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > isNatIListKind > U51 > U52 > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > U11 > U12 > U13 > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > U11 > U12 > U13 > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > U21 > U22 > U23 > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > U21 > U22 > U23 > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > active₁ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > 0 > tt > active₁ > cons₁
length > U61 > mark₁ > active₁ > cons₁
length > U61 > tt > active₁ > cons₁
length > U91 > isNatIListKind > U51 > U52 > mark₁ > active₁ > cons₁
length > U91 > isNatIListKind > U51 > U52 > tt > active₁ > cons₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > active₁ > cons₁
length > U91 > U92 > isNat > U11 > U12 > U13 > tt > active₁ > cons₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > active₁ > cons₁
length > U91 > U92 > isNat > U21 > U22 > U23 > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > tt > active₁ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > 0 > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(189) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(190) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(mark(X1), X2, X3) → U41¹(X1, X2, X3)
U41¹(active(X1), X2, X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  x1
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > tt > length₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > active₁ > s > length₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > active₁ > tt > length₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > active₁ > s > length₁
isNat > U11 > U12 > U13 > mark₁ > active₁ > tt > length₁
isNat > U11 > U12 > U13 > mark₁ > active₁ > s > length₁
isNat > U21 > U22 > U23 > mark₁ > active₁ > tt > length₁
isNat > U21 > U22 > U23 > mark₁ > active₁ > s > length₁
isNatIList₁ > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > length₁
isNatIList₁ > isNatKind₁ > isNatIListKind > U51 > U52 > mark₁ > active₁ > s > length₁
isNatIList₁ > isNatKind₁ > U61 > mark₁ > active₁ > tt > length₁
isNatIList₁ > isNatKind₁ > U61 > mark₁ > active₁ > s > length₁
isNatIList₁ > isNatKind₁ > U71 > mark₁ > active₁ > tt > length₁
isNatIList₁ > isNatKind₁ > U71 > mark₁ > active₁ > s > length₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > active₁ > tt > length₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > active₁ > s > length₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > mark₁ > active₁ > tt > length₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind > U51 > U52 > mark₁ > active₁ > s > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > active₁ > tt > length₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > active₁ > s > length₁
nil > 0 > mark₁ > active₁ > tt > length₁
nil > 0 > mark₁ > active₁ > s > length₁

The following usable rules [FROCOS05] were oriented:

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

(191) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(192) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(X1, mark(X2), X3) → U41¹(X1, X2, X3)
U41¹(X1, active(X2), X3) → U41¹(X1, X2, X3)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  x2
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > cons > active₁
zeros > 0 > mark₁ > tt > active₁
isNatIListKind > isNatKind > U61 > mark₁ > cons > active₁
isNatIListKind > isNatKind > U61 > mark₁ > tt > active₁
isNatIListKind > isNatKind > U71 > mark₁ > cons > active₁
isNatIListKind > isNatKind > U71 > mark₁ > tt > active₁
isNatIListKind > U51 > U52 > mark₁ > cons > active₁
isNatIListKind > U51 > U52 > mark₁ > tt > active₁
isNatList > U81 > isNatKind > U61 > mark₁ > cons > active₁
isNatList > U81 > isNatKind > U61 > mark₁ > tt > active₁
isNatList > U81 > isNatKind > U71 > mark₁ > cons > active₁
isNatList > U81 > isNatKind > U71 > mark₁ > tt > active₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > cons > active₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁
isNatIList₁ > U41 > isNatKind > U61 > mark₁ > cons > active₁
isNatIList₁ > U41 > isNatKind > U61 > mark₁ > tt > active₁
isNatIList₁ > U41 > isNatKind > U71 > mark₁ > cons > active₁
isNatIList₁ > U41 > isNatKind > U71 > mark₁ > tt > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > cons > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > active₁
length > 0 > mark₁ > cons > active₁
length > 0 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > cons > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > cons > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > isNatKind > U61 > mark₁ > cons > active₁
length > U91 > U92 > isNat > U21 > isNatKind > U61 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > isNatKind > U71 > mark₁ > cons > active₁
length > U91 > U92 > isNat > U21 > isNatKind > U71 > mark₁ > tt > active₁
length > U91 > U92 > U93 > isNatKind > U61 > mark₁ > cons > active₁
length > U91 > U92 > U93 > isNatKind > U61 > mark₁ > tt > active₁
length > U91 > U92 > U93 > isNatKind > U71 > mark₁ > cons > active₁
length > U91 > U92 > U93 > isNatKind > U71 > mark₁ > tt > active₁
length > U91 > U92 > U93 > U94 > s > mark₁ > cons > active₁
length > U91 > U92 > U93 > U94 > s > mark₁ > tt > active₁
nil > 0 > mark₁ > cons > active₁
nil > 0 > mark₁ > tt > active₁

The following usable rules [FROCOS05] were oriented:

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

(193) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(194) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(195) TRUE

(196) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U33¹(active(X)) → U33¹(X)
U33¹(mark(X)) → U33¹(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(197) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(active(X)) → U33¹(X)
U33¹(mark(X)) → U33¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1)  =  U33¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U33^1₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(198) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(199) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(200) TRUE

(201) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(202) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(X1, mark(X2)) → U32¹(X1, X2)
U32¹(X1, active(X2)) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  U32¹(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82(x2)
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > U32^1₁ > s
zeros > 0 > mark₁ > cons₁ > active₁ > s
isNatIListKind > isNatKind > U61 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatIListKind > isNatKind > U61 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatIListKind > isNatKind > U71 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatIListKind > isNatKind > U71 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatIListKind > U51 > U52 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatList₁ > U81₁ > isNatKind > U61 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatList₁ > U81₁ > isNatKind > U61 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatList₁ > U81₁ > isNatKind > U71 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatList₁ > U81₁ > isNatKind > U71 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatList₁ > U81₁ > U82₁ > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatList₁ > U81₁ > U82₁ > U83 > U84 > U85 > U86 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatIList₁ > U31 > U32 > U33 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > U11 > U12 > U13 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > U11 > U12 > U13 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > U21 > U22 > U23 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > isNatKind > U61 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > isNatKind > U61 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > isNatKind > U71 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > isNat > isNatKind > U71 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > U93 > U94 > mark₁ > U32^1₁ > s
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
length > 0 > mark₁ > U32^1₁ > s
length > 0 > mark₁ > cons₁ > active₁ > s
length > U11 > U12 > U13 > tt > U93 > U94 > mark₁ > U32^1₁ > s
length > U11 > U12 > U13 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
length > U61 > tt > U93 > U94 > mark₁ > U32^1₁ > s
length > U61 > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s
length > U91 > U92 > U93 > U94 > mark₁ > U32^1₁ > s
length > U91 > U92 > U93 > U94 > mark₁ > cons₁ > active₁ > s
nil > 0 > mark₁ > U32^1₁ > s
nil > 0 > mark₁ > cons₁ > active₁ > s
nil > tt > U93 > U94 > mark₁ > U32^1₁ > s
nil > tt > U93 > U94 > mark₁ > cons₁ > active₁ > s

The following usable rules [FROCOS05] were oriented:

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

(203) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U32¹(mark(X1), X2) → U32¹(X1, X2)
U32¹(active(X1), X2) → U32¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(204) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(mark(X1), X2) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  x1
zeros  =  zeros
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32(x2)
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > cons > U41 > isNatKind > U61 > mark₁ > tt > isNat₁
zeros > cons > U41 > isNatKind > U61 > mark₁ > nil
zeros > cons > U41 > isNatKind > U71 > mark₁ > tt > isNat₁
zeros > cons > U41 > isNatKind > U71 > mark₁ > nil
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > isNat₁
zeros > cons > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > nil
zeros > cons > U51 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > isNat₁
zeros > cons > U51 > isNatIListKind₁ > isNatKind > U61 > mark₁ > nil
zeros > cons > U51 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > isNat₁
zeros > cons > U51 > isNatIListKind₁ > isNatKind > U71 > mark₁ > nil
zeros > cons > U51 > U52 > mark₁ > tt > isNat₁
zeros > cons > U51 > U52 > mark₁ > nil
zeros > cons > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > isNat₁
zeros > cons > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U61 > mark₁ > nil
zeros > cons > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > isNat₁
zeros > cons > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U71 > mark₁ > nil
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > isNat₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > nil
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > isNat₁
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U61 > mark₁ > nil
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > isNat₁
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U71 > mark₁ > nil
zeros > cons > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > tt > isNat₁
zeros > cons > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > nil
zeros > cons > U91 > U92 > U93 > U94 > s > U21 > isNatKind > U61 > mark₁ > tt > isNat₁
zeros > cons > U91 > U92 > U93 > U94 > s > U21 > isNatKind > U61 > mark₁ > nil
zeros > cons > U91 > U92 > U93 > U94 > s > U21 > isNatKind > U71 > mark₁ > tt > isNat₁
zeros > cons > U91 > U92 > U93 > U94 > s > U21 > isNatKind > U71 > mark₁ > nil
zeros > 0 > mark₁ > tt > isNat₁
zeros > 0 > mark₁ > nil
isNatList > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > isNat₁
isNatList > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U61 > mark₁ > nil
isNatList > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > isNat₁
isNatList > U81 > U82 > U83 > isNatIListKind₁ > isNatKind > U71 > mark₁ > nil
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > isNat₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > nil
isNatIList₁ > U31₁ > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > isNat₁
isNatIList₁ > U31₁ > isNatIListKind₁ > isNatKind > U61 > mark₁ > nil
isNatIList₁ > U31₁ > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > isNat₁
isNatIList₁ > U31₁ > isNatIListKind₁ > isNatKind > U71 > mark₁ > nil
isNatIList₁ > U31₁ > U32₁ > U33 > mark₁ > tt > isNat₁
isNatIList₁ > U31₁ > U32₁ > U33 > mark₁ > nil
isNatIList₁ > U41 > isNatKind > U61 > mark₁ > tt > isNat₁
isNatIList₁ > U41 > isNatKind > U61 > mark₁ > nil
isNatIList₁ > U41 > isNatKind > U71 > mark₁ > tt > isNat₁
isNatIList₁ > U41 > isNatKind > U71 > mark₁ > nil
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > isNat₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > nil
length > 0 > mark₁ > tt > isNat₁
length > 0 > mark₁ > nil
length > U11 > U12 > U13 > mark₁ > tt > isNat₁
length > U11 > U12 > U13 > mark₁ > nil
length > U91 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > isNat₁
length > U91 > isNatIListKind₁ > isNatKind > U61 > mark₁ > nil
length > U91 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > isNat₁
length > U91 > isNatIListKind₁ > isNatKind > U71 > mark₁ > nil
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > tt > isNat₁
length > U91 > U92 > U93 > U94 > s > U21 > U22 > U23 > mark₁ > nil
length > U91 > U92 > U93 > U94 > s > U21 > isNatKind > U61 > mark₁ > tt > isNat₁
length > U91 > U92 > U93 > U94 > s > U21 > isNatKind > U61 > mark₁ > nil
length > U91 > U92 > U93 > U94 > s > U21 > isNatKind > U71 > mark₁ > tt > isNat₁
length > U91 > U92 > U93 > U94 > s > U21 > isNatKind > U71 > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

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

(205) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U32¹(active(X1), X2) → U32¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(206) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U32¹(active(X1), X2) → U32¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U32¹(x1, x2)  =  U32¹(x1)
active(x1)  =  active(x1)
zeros  =  zeros
mark(x1)  =  mark(x1)
cons(x1, x2)  =  cons
0  =  0
U11(x1, x2)  =  U11(x2)
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length(x1)
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U32^1₁ > active₁
zeros > cons > U51 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s > active₁
zeros > cons > U51 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U51 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s > active₁
zeros > cons > U51 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U51 > U52 > mark₁ > tt > s > active₁
zeros > cons > U51 > U52 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U81 > U82 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s > active₁
zeros > cons > U81 > U82 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U81 > U82 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s > active₁
zeros > cons > U81 > U82 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > s > active₁
zeros > cons > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s > active₁
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s > active₁
zeros > cons > U91 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U91 > U92 > U93 > isNatKind > U61 > mark₁ > tt > s > active₁
zeros > cons > U91 > U92 > U93 > isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U91 > U92 > U93 > isNatKind > U71 > mark₁ > tt > s > active₁
zeros > cons > U91 > U92 > U93 > isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > tt > s > active₁
zeros > cons > U91 > U92 > U93 > U94 > mark₁ > tt > length₁ > 0 > active₁
isNatList > U81 > U82 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s > active₁
isNatList > U81 > U82 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
isNatList > U81 > U82 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s > active₁
isNatList > U81 > U82 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > s > active₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > length₁ > 0 > active₁
isNat₁ > U11₁ > U12 > U13 > mark₁ > tt > s > active₁
isNat₁ > U11₁ > U12 > U13 > mark₁ > tt > length₁ > 0 > active₁
isNat₁ > U11₁ > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s > active₁
isNat₁ > U11₁ > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
isNat₁ > U11₁ > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s > active₁
isNat₁ > U11₁ > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > s > active₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > length₁ > 0 > active₁
isNat₁ > U21 > isNatKind > U61 > mark₁ > tt > s > active₁
isNat₁ > U21 > isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
isNat₁ > U21 > isNatKind > U71 > mark₁ > tt > s > active₁
isNat₁ > U21 > isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U31 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s > active₁
isNatIList₁ > U31 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U31 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s > active₁
isNatIList₁ > U31 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > s > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > length₁ > 0 > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s > active₁
isNatIList₁ > U41 > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > length₁ > 0 > active₁
nil > mark₁ > tt > s > active₁
nil > mark₁ > tt > length₁ > 0 > active₁

The following usable rules [FROCOS05] were oriented:

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

(207) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(208) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(209) TRUE

(210) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(211) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(mark(X1), X2) → U31¹(X1, X2)
U31¹(active(X1), X2) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  U31¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82(x2)
U83(x1, x2, x3)  =  U83(x2)
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons₁ > active₁
zeros > mark₁ > 0 > active₁
zeros > mark₁ > tt > active₁
zeros > mark₁ > tt > s
zeros > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > cons₁ > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > 0 > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > s
isNatIListKind₁ > U51 > U52 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > nil > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > 0 > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > s
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > 0 > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > nil > active₁
length > U61 > mark₁ > cons₁ > active₁
length > U61 > mark₁ > 0 > active₁
length > U61 > mark₁ > tt > active₁
length > U61 > mark₁ > tt > s
length > U61 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > s
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > s
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
length > U91 > U92 > U93 > U94 > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > mark₁ > 0 > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > s
length > U91 > U92 > U93 > U94 > mark₁ > nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(212) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(213) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(X1, mark(X2)) → U31¹(X1, X2)
U31¹(X1, active(X2)) → U31¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U31¹(x1, x2)  =  U31¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > U31^1₂ > cons₁
zeros > 0 > mark₁ > active₁ > cons₁
zeros > tt > mark₁ > U31^1₂ > cons₁
zeros > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > U31^1₂ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > U31^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > U31^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > U31^1₂ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > mark₁ > U31^1₂ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > U31^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > U31^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > U31^1₂ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > U31^1₂ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > U31^1₂ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > U31^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > U31^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > U31^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > U31^1₂ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > cons₁
nil > 0 > mark₁ > U31^1₂ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > tt > mark₁ > U31^1₂ > cons₁
nil > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(214) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(215) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(216) TRUE

(217) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(218) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  ISNAT(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

ISNAT₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(219) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(220) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(221) TRUE

(222) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U23¹(active(X)) → U23¹(X)
U23¹(mark(X)) → U23¹(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(223) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U23¹(active(X)) → U23¹(X)
U23¹(mark(X)) → U23¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1)  =  U23¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U23^1₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(224) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(225) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(226) TRUE

(227) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATKIND(active(X)) → ISNATKIND(X)
ISNATKIND(mark(X)) → ISNATKIND(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(228) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATKIND(active(X)) → ISNATKIND(X)
ISNATKIND(mark(X)) → ISNATKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  ISNATKIND(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

ISNATKIND₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(229) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(230) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(231) TRUE

(232) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(233) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(mark(X1), X2) → U22¹(X1, X2)
U22¹(active(X1), X2) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82(x2)
U83(x1, x2, x3)  =  U83(x2)
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons₁ > active₁
zeros > mark₁ > 0 > active₁
zeros > mark₁ > tt > active₁
zeros > mark₁ > tt > s
zeros > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > cons₁ > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > 0 > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > s
isNatIListKind₁ > U51 > U52 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > nil > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > 0 > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > s
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > 0 > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > nil > active₁
length > U61 > mark₁ > cons₁ > active₁
length > U61 > mark₁ > 0 > active₁
length > U61 > mark₁ > tt > active₁
length > U61 > mark₁ > tt > s
length > U61 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > s
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > s
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
length > U91 > U92 > U93 > U94 > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > mark₁ > 0 > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > s
length > U91 > U92 > U93 > U94 > mark₁ > nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(234) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(235) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U22¹(X1, mark(X2)) → U22¹(X1, X2)
U22¹(X1, active(X2)) → U22¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U22¹(x1, x2)  =  U22¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > U22^1₂ > cons₁
zeros > 0 > mark₁ > active₁ > cons₁
zeros > tt > mark₁ > U22^1₂ > cons₁
zeros > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > U22^1₂ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > U22^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > U22^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > U22^1₂ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > mark₁ > U22^1₂ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > U22^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > U22^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > U22^1₂ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > U22^1₂ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > U22^1₂ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > U22^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > U22^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > U22^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > U22^1₂ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > cons₁
nil > 0 > mark₁ > U22^1₂ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > tt > mark₁ > U22^1₂ > cons₁
nil > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(236) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(237) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(238) TRUE

(239) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(240) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(mark(X1), X2) → U21¹(X1, X2)
U21¹(active(X1), X2) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82(x2)
U83(x1, x2, x3)  =  U83(x2)
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons₁ > active₁
zeros > mark₁ > 0 > active₁
zeros > mark₁ > tt > active₁
zeros > mark₁ > tt > s
zeros > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > cons₁ > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > 0 > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > s
isNatIListKind₁ > U51 > U52 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > nil > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > 0 > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > s
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > 0 > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > nil > active₁
length > U61 > mark₁ > cons₁ > active₁
length > U61 > mark₁ > 0 > active₁
length > U61 > mark₁ > tt > active₁
length > U61 > mark₁ > tt > s
length > U61 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > s
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > s
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
length > U91 > U92 > U93 > U94 > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > mark₁ > 0 > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > s
length > U91 > U92 > U93 > U94 > mark₁ > nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(241) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(242) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(X1, mark(X2)) → U21¹(X1, X2)
U21¹(X1, active(X2)) → U21¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U21¹(x1, x2)  =  U21¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > U21^1₂ > cons₁
zeros > 0 > mark₁ > active₁ > cons₁
zeros > tt > mark₁ > U21^1₂ > cons₁
zeros > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > U21^1₂ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > U21^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > U21^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > U21^1₂ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > mark₁ > U21^1₂ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > U21^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > U21^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > U21^1₂ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > U21^1₂ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > U21^1₂ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > U21^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > U21^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > U21^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > U21^1₂ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > cons₁
nil > 0 > mark₁ > U21^1₂ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > tt > mark₁ > U21^1₂ > cons₁
nil > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(243) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(244) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(245) TRUE

(246) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATLIST(active(X)) → ISNATLIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(247) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(active(X)) → ISNATLIST(X)
ISNATLIST(mark(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1)  =  ISNATLIST(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

ISNATLIST₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(248) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(249) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(250) TRUE

(251) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U13¹(active(X)) → U13¹(X)
U13¹(mark(X)) → U13¹(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(252) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(active(X)) → U13¹(X)
U13¹(mark(X)) → U13¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1)  =  U13¹(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

U13^1₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(253) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(254) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(255) TRUE

(256) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ISNATILISTKIND(active(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(257) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(active(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILISTKIND(x1)  =  ISNATILISTKIND(x1)
active(x1)  =  active(x1)
mark(x1)  =  mark(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  x1
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

ISNATILISTKIND₁ > cons₁
zeros > 0 > mark₁ > tt > active₁ > cons₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > tt > active₁ > cons₁
isNat₁ > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
isNat₁ > U21 > U22 > U23 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > isNatIListKind > U51 > U52 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44 > U45 > U46 > mark₁ > tt > active₁ > cons₁
length > 0 > mark₁ > tt > active₁ > cons₁
length > U11 > U12 > U13 > mark₁ > tt > active₁ > cons₁
length > U61 > mark₁ > tt > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > tt > active₁ > cons₁
nil > mark₁ > tt > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(258) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(259) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(260) TRUE

(261) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(262) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(mark(X1), X2) → U12¹(X1, X2)
U12¹(active(X1), X2) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2)  =  U12¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind(x1)
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > cons₁ > active₁
zeros > 0 > mark₁ > isNatKind₁ > tt > active₁
zeros > 0 > mark₁ > nil
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > cons₁ > active₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > isNatKind₁ > tt > active₁
isNatList > U81 > U82 > U83 > U84 > U85 > U86 > mark₁ > nil
isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
isNat > U11 > U12 > U13 > mark₁ > isNatKind₁ > tt > active₁
isNat > U11 > U12 > U13 > mark₁ > nil
isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
isNat > U21 > U22 > U23 > mark₁ > isNatKind₁ > tt > active₁
isNat > U21 > U22 > U23 > mark₁ > nil
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁ > tt > active₁
isNatIList₁ > U31 > isNatIListKind > U51 > U52 > mark₁ > nil
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > isNatKind₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil
isNatIList₁ > U41₁ > U42 > isNatIListKind > U51 > U52 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > isNatIListKind > U51 > U52 > mark₁ > nil
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > isNatKind₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > nil
length > 0 > mark₁ > cons₁ > active₁
length > 0 > mark₁ > isNatKind₁ > tt > active₁
length > 0 > mark₁ > nil
length > U11 > U12 > U13 > mark₁ > cons₁ > active₁
length > U11 > U12 > U13 > mark₁ > isNatKind₁ > tt > active₁
length > U11 > U12 > U13 > mark₁ > nil
length > isNatIListKind > U51 > U52 > mark₁ > cons₁ > active₁
length > isNatIListKind > U51 > U52 > mark₁ > isNatKind₁ > tt > active₁
length > isNatIListKind > U51 > U52 > mark₁ > nil
length > U61 > mark₁ > cons₁ > active₁
length > U61 > mark₁ > isNatKind₁ > tt > active₁
length > U61 > mark₁ > nil
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > isNatKind₁ > tt > active₁
length > U91 > U92 > U93 > U94 > s > U71 > mark₁ > nil

The following usable rules [FROCOS05] were oriented:

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

(263) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(264) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(X1, mark(X2)) → U12¹(X1, X2)
U12¹(X1, active(X2)) → U12¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U12¹(x1, x2)  =  U12¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > U12^1₂ > cons₁
zeros > 0 > mark₁ > active₁ > cons₁
zeros > tt > mark₁ > U12^1₂ > cons₁
zeros > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > U12^1₂ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > U12^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > U12^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > U12^1₂ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > mark₁ > U12^1₂ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > U12^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > U12^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > U12^1₂ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > U12^1₂ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > U12^1₂ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > U12^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > U12^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > U12^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > U12^1₂ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > cons₁
nil > 0 > mark₁ > U12^1₂ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > tt > mark₁ > U12^1₂ > cons₁
nil > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(265) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(266) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(267) TRUE

(268) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(269) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(mark(X1), X2) → U11¹(X1, X2)
U11¹(active(X1), X2) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2)  =  U11¹(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82(x2)
U83(x1, x2, x3)  =  U83(x2)
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons₁ > active₁
zeros > mark₁ > 0 > active₁
zeros > mark₁ > tt > active₁
zeros > mark₁ > tt > s
zeros > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > cons₁ > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > 0 > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > s
isNatIListKind₁ > U51 > U52 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > nil > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > 0 > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > s
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > 0 > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > nil > active₁
length > U61 > mark₁ > cons₁ > active₁
length > U61 > mark₁ > 0 > active₁
length > U61 > mark₁ > tt > active₁
length > U61 > mark₁ > tt > s
length > U61 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > s
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > s
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
length > U91 > U92 > U93 > U94 > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > mark₁ > 0 > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > s
length > U91 > U92 > U93 > U94 > mark₁ > nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(270) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(271) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(X1, mark(X2)) → U11¹(X1, X2)
U11¹(X1, active(X2)) → U11¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U11¹(x1, x2)  =  U11¹(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > U11^1₂ > cons₁
zeros > 0 > mark₁ > active₁ > cons₁
zeros > tt > mark₁ > U11^1₂ > cons₁
zeros > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > U11^1₂ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > U11^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > U11^1₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > U11^1₂ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > mark₁ > U11^1₂ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > U11^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > U11^1₂ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > U11^1₂ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > U11^1₂ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > U11^1₂ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > U11^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > U11^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > U11^1₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > U11^1₂ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > cons₁
nil > 0 > mark₁ > U11^1₂ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > tt > mark₁ > U11^1₂ > cons₁
nil > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(272) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(273) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(274) TRUE

(275) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(276) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList(x1)
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat
U31(x1, x2)  =  U31
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42
U43(x1, x2, x3)  =  U43
U44(x1, x2, x3)  =  U44
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81(x2)
U82(x1, x2, x3)  =  U82(x2)
U83(x1, x2, x3)  =  U83(x2)
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > mark₁ > cons₁ > active₁
zeros > mark₁ > 0 > active₁
zeros > mark₁ > tt > active₁
zeros > mark₁ > tt > s
zeros > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U61 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatIListKind₁ > isNatKind > U71 > mark₁ > tt > s
isNatIListKind₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > cons₁ > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > 0 > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > active₁
isNatIListKind₁ > U51 > U52 > mark₁ > tt > s
isNatIListKind₁ > U51 > U52 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U61 > mark₁ > nil > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > active₁
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > tt > s
isNatList₁ > U81₁ > isNatKind > U71 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > cons₁ > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > 0 > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > active₁
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > tt > s
isNatList₁ > U81₁ > U82₁ > U83₁ > U84 > U85 > U86 > mark₁ > nil > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > cons₁ > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > 0 > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > active₁
isNatIList₁ > U31 > U32 > U33 > mark₁ > tt > s
isNatIList₁ > U31 > U32 > U33 > mark₁ > nil > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > cons₁ > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > 0 > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > active₁
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > tt > s
isNatIList₁ > U41₁ > U42 > U43 > U44 > U45 > U46 > mark₁ > nil > active₁
length > U61 > mark₁ > cons₁ > active₁
length > U61 > mark₁ > 0 > active₁
length > U61 > mark₁ > tt > active₁
length > U61 > mark₁ > tt > s
length > U61 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > tt > s
length > U91 > U92 > isNat > U11 > U12 > U13 > mark₁ > nil > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > cons₁ > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > 0 > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > active₁
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > tt > s
length > U91 > U92 > isNat > U21 > U22 > U23 > mark₁ > nil > active₁
length > U91 > U92 > U93 > U94 > mark₁ > cons₁ > active₁
length > U91 > U92 > U93 > U94 > mark₁ > 0 > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > active₁
length > U91 > U92 > U93 > U94 > mark₁ > tt > s
length > U91 > U92 > U93 > U94 > mark₁ > nil > active₁

The following usable rules [FROCOS05] were oriented:

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

(277) Obligation:

Q DP problem:
The TRS P consists of the following rules:

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(278) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1, x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
zeros  =  zeros
cons(x1, x2)  =  cons(x1)
0  =  0
U11(x1, x2)  =  U11
tt  =  tt
U12(x1, x2)  =  U12
isNatIListKind(x1)  =  isNatIListKind(x1)
U13(x1)  =  U13
isNatList(x1)  =  isNatList
U21(x1, x2)  =  U21
U22(x1, x2)  =  U22
isNatKind(x1)  =  isNatKind
U23(x1)  =  U23
isNat(x1)  =  isNat(x1)
U31(x1, x2)  =  U31(x2)
U32(x1, x2)  =  U32
U33(x1)  =  U33
U41(x1, x2, x3)  =  U41(x2)
U42(x1, x2, x3)  =  U42(x2)
U43(x1, x2, x3)  =  U43(x2)
U44(x1, x2, x3)  =  U44(x2)
U45(x1, x2)  =  U45
U46(x1)  =  U46
isNatIList(x1)  =  isNatIList(x1)
U51(x1, x2)  =  U51
U52(x1)  =  U52
U61(x1)  =  U61
U71(x1)  =  U71
U81(x1, x2, x3)  =  U81
U82(x1, x2, x3)  =  U82
U83(x1, x2, x3)  =  U83
U84(x1, x2, x3)  =  U84
U85(x1, x2)  =  U85
U86(x1)  =  U86
U91(x1, x2, x3)  =  U91
U92(x1, x2, x3)  =  U92
U93(x1, x2, x3)  =  U93
U94(x1, x2)  =  U94
s(x1)  =  s
length(x1)  =  length
nil  =  nil

Lexicographic Path Order [LPO].
Precedence:

zeros > 0 > mark₁ > CONS₂ > cons₁
zeros > 0 > mark₁ > active₁ > cons₁
zeros > tt > mark₁ > CONS₂ > cons₁
zeros > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > CONS₂ > cons₁
isNatIList₁ > U31₁ > U32 > U33 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > CONS₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > CONS₂ > cons₁
isNatIList₁ > U41₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > CONS₂ > cons₁
isNatIList₁ > U41₁ > U42₁ > U43₁ > U44₁ > U45 > U46 > tt > mark₁ > active₁ > cons₁
length > 0 > mark₁ > CONS₂ > cons₁
length > 0 > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > CONS₂ > cons₁
length > isNatIListKind₁ > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > CONS₂ > cons₁
length > isNatIListKind₁ > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > CONS₂ > cons₁
length > isNatIListKind₁ > U51 > U52 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > CONS₂ > cons₁
length > isNatList > U81 > isNatKind > U61 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > CONS₂ > cons₁
length > isNatList > U81 > isNatKind > U71 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > CONS₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U11 > U12 > U13 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > CONS₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > isNat₁ > U21 > U22 > U23 > tt > mark₁ > active₁ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > CONS₂ > cons₁
length > isNatList > U81 > U82 > U83 > U84 > U85 > U86 > tt > mark₁ > active₁ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > CONS₂ > cons₁
length > U91 > U92 > U93 > U94 > s > mark₁ > active₁ > cons₁
nil > 0 > mark₁ > CONS₂ > cons₁
nil > 0 > mark₁ > active₁ > cons₁
nil > tt > mark₁ > CONS₂ > cons₁
nil > tt > mark₁ > active₁ > cons₁

The following usable rules [FROCOS05] were oriented:

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

(279) Obligation:

Q DP problem:
P is empty.
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))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(U13(X)) → active(U13(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X1, X2)) → active(U22(mark(X1), X2))
mark(isNatKind(X)) → active(isNatKind(X))
mark(U23(X)) → active(U23(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X1, X2)) → active(U32(mark(X1), X2))
mark(U33(X)) → active(U33(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2, X3)) → active(U42(mark(X1), X2, X3))
mark(U43(X1, X2, X3)) → active(U43(mark(X1), X2, X3))
mark(U44(X1, X2, X3)) → active(U44(mark(X1), X2, X3))
mark(U45(X1, X2)) → active(U45(mark(X1), X2))
mark(U46(X)) → active(U46(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2)) → active(U51(mark(X1), X2))
mark(U52(X)) → active(U52(mark(X)))
mark(U61(X)) → active(U61(mark(X)))
mark(U71(X)) → active(U71(mark(X)))
mark(U81(X1, X2, X3)) → active(U81(mark(X1), X2, X3))
mark(U82(X1, X2, X3)) → active(U82(mark(X1), X2, X3))
mark(U83(X1, X2, X3)) → active(U83(mark(X1), X2, X3))
mark(U84(X1, X2, X3)) → active(U84(mark(X1), X2, X3))
mark(U85(X1, X2)) → active(U85(mark(X1), X2))
mark(U86(X)) → active(U86(mark(X)))
mark(U91(X1, X2, X3)) → active(U91(mark(X1), X2, X3))
mark(U92(X1, X2, X3)) → active(U92(mark(X1), X2, X3))
mark(U93(X1, X2, X3)) → active(U93(mark(X1), X2, X3))
mark(U94(X1, X2)) → active(U94(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
U13(mark(X)) → U13(X)
U13(active(X)) → U13(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X1), X2) → U22(X1, X2)
U22(X1, mark(X2)) → U22(X1, X2)
U22(active(X1), X2) → U22(X1, X2)
U22(X1, active(X2)) → U22(X1, X2)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)
U23(mark(X)) → U23(X)
U23(active(X)) → U23(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X1), X2) → U32(X1, X2)
U32(X1, mark(X2)) → U32(X1, X2)
U32(active(X1), X2) → U32(X1, X2)
U32(X1, active(X2)) → U32(X1, X2)
U33(mark(X)) → U33(X)
U33(active(X)) → U33(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, mark(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, mark(X3)) → U42(X1, X2, X3)
U42(active(X1), X2, X3) → U42(X1, X2, X3)
U42(X1, active(X2), X3) → U42(X1, X2, X3)
U42(X1, X2, active(X3)) → U42(X1, X2, X3)
U43(mark(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, mark(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, mark(X3)) → U43(X1, X2, X3)
U43(active(X1), X2, X3) → U43(X1, X2, X3)
U43(X1, active(X2), X3) → U43(X1, X2, X3)
U43(X1, X2, active(X3)) → U43(X1, X2, X3)
U44(mark(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, mark(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, mark(X3)) → U44(X1, X2, X3)
U44(active(X1), X2, X3) → U44(X1, X2, X3)
U44(X1, active(X2), X3) → U44(X1, X2, X3)
U44(X1, X2, active(X3)) → U44(X1, X2, X3)
U45(mark(X1), X2) → U45(X1, X2)
U45(X1, mark(X2)) → U45(X1, X2)
U45(active(X1), X2) → U45(X1, X2)
U45(X1, active(X2)) → U45(X1, X2)
U46(mark(X)) → U46(X)
U46(active(X)) → U46(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2) → U51(X1, X2)
U51(X1, mark(X2)) → U51(X1, X2)
U51(active(X1), X2) → U51(X1, X2)
U51(X1, active(X2)) → U51(X1, X2)
U52(mark(X)) → U52(X)
U52(active(X)) → U52(X)
U61(mark(X)) → U61(X)
U61(active(X)) → U61(X)
U71(mark(X)) → U71(X)
U71(active(X)) → U71(X)
U81(mark(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, mark(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, mark(X3)) → U81(X1, X2, X3)
U81(active(X1), X2, X3) → U81(X1, X2, X3)
U81(X1, active(X2), X3) → U81(X1, X2, X3)
U81(X1, X2, active(X3)) → U81(X1, X2, X3)
U82(mark(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, mark(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, mark(X3)) → U82(X1, X2, X3)
U82(active(X1), X2, X3) → U82(X1, X2, X3)
U82(X1, active(X2), X3) → U82(X1, X2, X3)
U82(X1, X2, active(X3)) → U82(X1, X2, X3)
U83(mark(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, mark(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, mark(X3)) → U83(X1, X2, X3)
U83(active(X1), X2, X3) → U83(X1, X2, X3)
U83(X1, active(X2), X3) → U83(X1, X2, X3)
U83(X1, X2, active(X3)) → U83(X1, X2, X3)
U84(mark(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, mark(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, mark(X3)) → U84(X1, X2, X3)
U84(active(X1), X2, X3) → U84(X1, X2, X3)
U84(X1, active(X2), X3) → U84(X1, X2, X3)
U84(X1, X2, active(X3)) → U84(X1, X2, X3)
U85(mark(X1), X2) → U85(X1, X2)
U85(X1, mark(X2)) → U85(X1, X2)
U85(active(X1), X2) → U85(X1, X2)
U85(X1, active(X2)) → U85(X1, X2)
U86(mark(X)) → U86(X)
U86(active(X)) → U86(X)
U91(mark(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, mark(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, mark(X3)) → U91(X1, X2, X3)
U91(active(X1), X2, X3) → U91(X1, X2, X3)
U91(X1, active(X2), X3) → U91(X1, X2, X3)
U91(X1, X2, active(X3)) → U91(X1, X2, X3)
U92(mark(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, mark(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, mark(X3)) → U92(X1, X2, X3)
U92(active(X1), X2, X3) → U92(X1, X2, X3)
U92(X1, active(X2), X3) → U92(X1, X2, X3)
U92(X1, X2, active(X3)) → U92(X1, X2, X3)
U93(mark(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, mark(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, mark(X3)) → U93(X1, X2, X3)
U93(active(X1), X2, X3) → U93(X1, X2, X3)
U93(X1, active(X2), X3) → U93(X1, X2, X3)
U93(X1, X2, active(X3)) → U93(X1, X2, X3)
U94(mark(X1), X2) → U94(X1, X2)
U94(X1, mark(X2)) → U94(X1, X2)
U94(active(X1), X2) → U94(X1, X2)
U94(X1, active(X2)) → U94(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(280) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(281) TRUE

(282) Obligation:

Q DP problem:
The TRS P consists of the following rules:

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.