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

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

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 40 SCCs with 132 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

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.

ISNATILIST(ok(X)) → ISNATILIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILIST(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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:

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

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.

ISNAT(ok(X)) → ISNAT(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNAT(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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:

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

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.

ISNATKIND(ok(X)) → ISNATKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATKIND(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(17) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(18) PisEmptyProof (EQUIVALENT transformation)

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

(19) TRUE

(20) Obligation:

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

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

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATLIST(ok(X)) → ISNATLIST(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATLIST(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(22) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(23) PisEmptyProof (EQUIVALENT transformation)

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

(24) TRUE

(25) Obligation:

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

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

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ISNATILISTKIND(ok(X)) → ISNATILISTKIND(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ISNATILISTKIND(x1)  =  x1
ok(x1)  =  ok(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(27) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

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

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

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

LENGTH(ok(X)) → LENGTH(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LENGTH(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

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

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(34) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) Obligation:

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

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

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.

S(ok(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

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

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.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(41) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(42) PisEmptyProof (EQUIVALENT transformation)

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

(43) TRUE

(44) Obligation:

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

U94¹(ok(X1), ok(X2)) → U94¹(X1, X2)
U94¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U94¹(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U94^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

U94¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U94¹(mark(X1), X2) → U94¹(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U94^1₂, mark₁]

Status:

U94^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(48) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(49) PisEmptyProof (EQUIVALENT transformation)

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

(50) TRUE

(51) Obligation:

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

U93¹(ok(X1), ok(X2), ok(X3)) → U93¹(X1, X2, X3)
U93¹(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U93¹(ok(X1), ok(X2), ok(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, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U93^1₃, ok₁]

Status:

U93^1₃: [3,2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(53) Obligation:

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

U93¹(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(54) 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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U93^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(55) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(56) PisEmptyProof (EQUIVALENT transformation)

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

(57) TRUE

(58) Obligation:

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

U92¹(ok(X1), ok(X2), ok(X3)) → U92¹(X1, X2, X3)
U92¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U92¹(ok(X1), ok(X2), ok(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, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U92^1₃, ok₁]

Status:

ok₁: [1]
U92^1₃: [3,2,1]

The following usable rules [FROCOS05] were oriented: none

(60) Obligation:

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

U92¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(61) 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)

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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]
U92^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(62) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(63) PisEmptyProof (EQUIVALENT transformation)

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

(64) TRUE

(65) Obligation:

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

U91¹(ok(X1), ok(X2), ok(X3)) → U91¹(X1, X2, X3)
U91¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(66) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U91¹(ok(X1), ok(X2), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U91^1₃, ok₁]

Status:

U91^1₃: [3,2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(67) Obligation:

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

U91¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(68) 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)

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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U91^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(69) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(70) PisEmptyProof (EQUIVALENT transformation)

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

(71) TRUE

(72) Obligation:

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

U86¹(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(73) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U86¹(ok(X)) → U86¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U86¹(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(74) Obligation:

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

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

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

(75) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(76) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(77) PisEmptyProof (EQUIVALENT transformation)

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

(78) TRUE

(79) Obligation:

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

U85¹(ok(X1), ok(X2)) → U85¹(X1, X2)
U85¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(80) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U85¹(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U85^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(81) Obligation:

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

U85¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(82) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U85^1₂, mark₁]

Status:

U85^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(83) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(84) PisEmptyProof (EQUIVALENT transformation)

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

(85) TRUE

(86) Obligation:

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

U84¹(ok(X1), ok(X2), ok(X3)) → U84¹(X1, X2, X3)
U84¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U84¹(ok(X1), ok(X2), ok(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, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U84^1₃, ok₁]

Status:

ok₁: [1]
U84^1₃: [3,2,1]

The following usable rules [FROCOS05] were oriented: none

(88) Obligation:

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

U84¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U84¹(mark(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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U84^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(90) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(91) PisEmptyProof (EQUIVALENT transformation)

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

(92) TRUE

(93) Obligation:

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

U83¹(ok(X1), ok(X2), ok(X3)) → U83¹(X1, X2, X3)
U83¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U83¹(ok(X1), ok(X2), ok(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, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U83^1₃, ok₁]

Status:

U83^1₃: [3,2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(95) Obligation:

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

U83¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U83¹(mark(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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U83^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(97) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(98) PisEmptyProof (EQUIVALENT transformation)

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

(99) TRUE

(100) Obligation:

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

U82¹(ok(X1), ok(X2), ok(X3)) → U82¹(X1, X2, X3)
U82¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(101) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U82¹(ok(X1), ok(X2), ok(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, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U82^1₃, ok₁]

Status:

ok₁: [1]
U82^1₃: [3,2,1]

The following usable rules [FROCOS05] were oriented: none

(102) Obligation:

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

U82¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(103) 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)

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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]
U82^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(104) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(105) PisEmptyProof (EQUIVALENT transformation)

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

(106) TRUE

(107) Obligation:

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

U81¹(ok(X1), ok(X2), ok(X3)) → U81¹(X1, X2, X3)
U81¹(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U81¹(ok(X1), ok(X2), ok(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, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U81^1₃, ok₁]

Status:

U81^1₃: [3,2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(109) Obligation:

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

U81¹(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(110) 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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U81^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(111) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(112) PisEmptyProof (EQUIVALENT transformation)

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

(113) TRUE

(114) Obligation:

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

U71¹(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(115) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U71¹(ok(X)) → U71¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U71¹(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(116) Obligation:

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

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

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

(117) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(118) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(119) PisEmptyProof (EQUIVALENT transformation)

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

(120) TRUE

(121) Obligation:

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

U61¹(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(122) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U61¹(ok(X)) → U61¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U61¹(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(123) Obligation:

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

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

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.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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:

U52¹(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U52¹(ok(X)) → U52¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U52¹(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(130) Obligation:

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

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

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.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

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

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:

U51¹(ok(X1), ok(X2)) → U51¹(X1, X2)
U51¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U51¹(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U51^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(137) Obligation:

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

U51¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(138) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U51^1₂, mark₁]

Status:

U51^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(139) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(140) PisEmptyProof (EQUIVALENT transformation)

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

(141) TRUE

(142) Obligation:

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

U46¹(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(143) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U46¹(ok(X)) → U46¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U46¹(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(144) Obligation:

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

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

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

(145) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(146) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(147) PisEmptyProof (EQUIVALENT transformation)

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

(148) TRUE

(149) Obligation:

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

U45¹(ok(X1), ok(X2)) → U45¹(X1, X2)
U45¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(150) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U45¹(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U45^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(151) Obligation:

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

U45¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(152) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U45^1₂, mark₁]

Status:

U45^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(153) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(154) PisEmptyProof (EQUIVALENT transformation)

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

(155) TRUE

(156) Obligation:

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

U44¹(ok(X1), ok(X2), ok(X3)) → U44¹(X1, X2, X3)
U44¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(ok(X1), ok(X2), ok(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, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U44^1₃, ok₁]

Status:

ok₁: [1]
U44^1₃: [3,2,1]

The following usable rules [FROCOS05] were oriented: none

(158) Obligation:

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

U44¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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(mark(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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]
U44^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(160) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(161) PisEmptyProof (EQUIVALENT transformation)

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

(162) TRUE

(163) Obligation:

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

U43¹(ok(X1), ok(X2), ok(X3)) → U43¹(X1, X2, X3)
U43¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(164) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U43¹(ok(X1), ok(X2), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U43^1₃, ok₁]

Status:

U43^1₃: [3,2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(165) Obligation:

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

U43¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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)

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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]
U43^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(167) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(168) PisEmptyProof (EQUIVALENT transformation)

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

(169) TRUE

(170) Obligation:

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

U42¹(ok(X1), ok(X2), ok(X3)) → U42¹(X1, X2, X3)
U42¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(171) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U42¹(ok(X1), ok(X2), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U42^1₃, ok₁]

Status:

U42^1₃: [3,2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(172) Obligation:

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

U42¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(173) 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)

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)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]
U42^1₁: [1]

The following usable rules [FROCOS05] were oriented: none

(174) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(175) PisEmptyProof (EQUIVALENT transformation)

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

(176) TRUE

(177) Obligation:

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

U41¹(ok(X1), ok(X2), ok(X3)) → U41¹(X1, X2, X3)
U41¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(178) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U41¹(ok(X1), ok(X2), ok(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)  =  U41¹(x1, x2, x3)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[U41^1₃, ok₁]

Status:

ok₁: [1]
U41^1₃: [3,2,1]

The following usable rules [FROCOS05] were oriented: none

(179) Obligation:

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

U41¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(180) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U41¹(x1, x2, x3)  =  U41¹(x1)
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U41^1₁: [1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(181) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(182) PisEmptyProof (EQUIVALENT transformation)

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

(183) TRUE

(184) Obligation:

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

U33¹(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(185) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U33¹(ok(X)) → U33¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U33¹(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(186) Obligation:

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

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

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

(187) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(188) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(189) PisEmptyProof (EQUIVALENT transformation)

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

(190) TRUE

(191) Obligation:

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

U32¹(ok(X1), ok(X2)) → U32¹(X1, X2)
U32¹(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U32¹(ok(X1), ok(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, x2)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U32^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(193) Obligation:

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

U32¹(mark(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(194) 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: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U32^1₂, mark₁]

Status:

U32^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(195) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(196) PisEmptyProof (EQUIVALENT transformation)

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

(197) TRUE

(198) Obligation:

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

U31¹(ok(X1), ok(X2)) → U31¹(X1, X2)
U31¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(199) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U31¹(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U31^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(200) Obligation:

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

U31¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(201) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U31^1₂, mark₁]

Status:

U31^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(202) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(203) PisEmptyProof (EQUIVALENT transformation)

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

(204) TRUE

(205) Obligation:

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

U23¹(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U23¹(ok(X)) → U23¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U23¹(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(207) Obligation:

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

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

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

(208) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(209) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(210) PisEmptyProof (EQUIVALENT transformation)

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

(211) TRUE

(212) Obligation:

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

U22¹(ok(X1), ok(X2)) → U22¹(X1, X2)
U22¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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.

U22¹(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U22^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(214) Obligation:

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

U22¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(215) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U22^1₂, mark₁]

Status:

U22^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(216) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(217) PisEmptyProof (EQUIVALENT transformation)

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

(218) TRUE

(219) Obligation:

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

U21¹(ok(X1), ok(X2)) → U21¹(X1, X2)
U21¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(220) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U21¹(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U21^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(221) Obligation:

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

U21¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(222) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U21^1₂, mark₁]

Status:

U21^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(223) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(224) PisEmptyProof (EQUIVALENT transformation)

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

(225) TRUE

(226) Obligation:

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

U13¹(ok(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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(227) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U13¹(ok(X)) → U13¹(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U13¹(x1)  =  x1
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(228) Obligation:

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

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

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

(229) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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)  =  x1
mark(x1)  =  mark(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(230) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(231) PisEmptyProof (EQUIVALENT transformation)

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

(232) TRUE

(233) Obligation:

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

U12¹(ok(X1), ok(X2)) → U12¹(X1, X2)
U12¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(234) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U12¹(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U12^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(235) Obligation:

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

U12¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(236) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U12^1₂, mark₁]

Status:

U12^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(237) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(238) PisEmptyProof (EQUIVALENT transformation)

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

(239) TRUE

(240) Obligation:

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

U11¹(ok(X1), ok(X2)) → U11¹(X1, X2)
U11¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(241) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U11¹(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U11^1₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(242) Obligation:

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

U11¹(mark(X1), 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(243) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[U11^1₂, mark₁]

Status:

U11^1₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(244) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(245) PisEmptyProof (EQUIVALENT transformation)

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

(246) TRUE

(247) Obligation:

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

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

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

(248) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(ok(X1), ok(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)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

CONS₂: [2,1]
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(249) Obligation:

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

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

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

(250) 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)

The remaining pairs can at least be oriented weakly.
Used ordering: Lexicographic path order with status [LPO].
Quasi-Precedence:

[CONS₂, mark₁]

Status:

CONS₂: [2,1]
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(251) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(252) PisEmptyProof (EQUIVALENT transformation)

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

(253) TRUE

(254) Obligation:

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

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

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

(255) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X1)
PROPER(U11(X1, X2)) → PROPER(X2)
PROPER(U12(X1, X2)) → PROPER(X1)
PROPER(U12(X1, X2)) → PROPER(X2)
PROPER(isNatIListKind(X)) → PROPER(X)
PROPER(U21(X1, X2)) → PROPER(X1)
PROPER(U21(X1, X2)) → PROPER(X2)
PROPER(U22(X1, X2)) → PROPER(X1)
PROPER(U22(X1, X2)) → PROPER(X2)
PROPER(U31(X1, X2)) → PROPER(X1)
PROPER(U31(X1, X2)) → PROPER(X2)
PROPER(U32(X1, X2)) → PROPER(X1)
PROPER(U32(X1, X2)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X1)
PROPER(U41(X1, X2, X3)) → PROPER(X2)
PROPER(U41(X1, X2, X3)) → PROPER(X3)
PROPER(U42(X1, X2, X3)) → PROPER(X1)
PROPER(U42(X1, X2, X3)) → PROPER(X2)
PROPER(U42(X1, X2, X3)) → PROPER(X3)
PROPER(U43(X1, X2, X3)) → PROPER(X1)
PROPER(U43(X1, X2, X3)) → PROPER(X2)
PROPER(U43(X1, X2, X3)) → PROPER(X3)
PROPER(U44(X1, X2, X3)) → PROPER(X1)
PROPER(U44(X1, X2, X3)) → PROPER(X2)
PROPER(U44(X1, X2, X3)) → PROPER(X3)
PROPER(U45(X1, X2)) → PROPER(X1)
PROPER(U45(X1, X2)) → PROPER(X2)
PROPER(U51(X1, X2)) → PROPER(X1)
PROPER(U51(X1, X2)) → PROPER(X2)
PROPER(U81(X1, X2, X3)) → PROPER(X1)
PROPER(U81(X1, X2, X3)) → PROPER(X2)
PROPER(U81(X1, X2, X3)) → PROPER(X3)
PROPER(U82(X1, X2, X3)) → PROPER(X1)
PROPER(U82(X1, X2, X3)) → PROPER(X2)
PROPER(U82(X1, X2, X3)) → PROPER(X3)
PROPER(U83(X1, X2, X3)) → PROPER(X1)
PROPER(U83(X1, X2, X3)) → PROPER(X2)
PROPER(U83(X1, X2, X3)) → PROPER(X3)
PROPER(U84(X1, X2, X3)) → PROPER(X1)
PROPER(U84(X1, X2, X3)) → PROPER(X2)
PROPER(U84(X1, X2, X3)) → PROPER(X3)
PROPER(U85(X1, X2)) → PROPER(X1)
PROPER(U85(X1, X2)) → PROPER(X2)
PROPER(U91(X1, X2, X3)) → PROPER(X1)
PROPER(U91(X1, X2, X3)) → PROPER(X2)
PROPER(U91(X1, X2, X3)) → PROPER(X3)
PROPER(U92(X1, X2, X3)) → PROPER(X1)
PROPER(U92(X1, X2, X3)) → PROPER(X2)
PROPER(U92(X1, X2, X3)) → PROPER(X3)
PROPER(U93(X1, X2, X3)) → PROPER(X1)
PROPER(U93(X1, X2, X3)) → PROPER(X2)
PROPER(U93(X1, X2, X3)) → PROPER(X3)
PROPER(U94(X1, X2)) → PROPER(X1)
PROPER(U94(X1, X2)) → PROPER(X2)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

cons₂ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U11₂ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U31₂ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U32₂ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U41₃ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U42₃ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U44₃ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U45₂ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U51₂ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U81₃ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U82₃ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U84₃ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U92₃ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U93₃ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]
U94₂ > [PROPER₁, U12₂, isNatIListKind₁, U21₂, U22₂, U43₃, U83₃, U85₂, U91₃]

Status:

U82₃: [2,3,1]
U41₃: [2,3,1]
U32₂: [2,1]
U12₂: [2,1]
U43₃: [3,2,1]
isNatIListKind₁: [1]
U21₂: [2,1]
U51₂: [2,1]
U42₃: [2,1,3]
U93₃: [3,2,1]
U85₂: [1,2]
U45₂: [2,1]
PROPER₁: [1]
U31₂: [1,2]
U83₃: [2,3,1]
U22₂: [2,1]
U81₃: [2,1,3]
U11₂: [2,1]
U94₂: [2,1]
cons₂: [2,1]
U91₃: [2,3,1]
U44₃: [3,2,1]
U92₃: [3,2,1]
U84₃: [2,3,1]

The following usable rules [FROCOS05] were oriented: none

(256) Obligation:

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

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

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.

PROPER(U13(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
U13(x1)  =  U13(x1)
isNatList(x1)  =  x1
isNatKind(x1)  =  x1
U23(x1)  =  x1
isNat(x1)  =  x1
U33(x1)  =  x1
U46(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  x1
U61(x1)  =  x1
U71(x1)  =  x1
U86(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

PROPER₁: [1]
U13₁: [1]

The following usable rules [FROCOS05] were oriented: none

(258) Obligation:

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

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

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

(259) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(s(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNatKind(x1)  =  x1
U23(x1)  =  x1
isNat(x1)  =  x1
U33(x1)  =  x1
U46(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  x1
U61(x1)  =  x1
U71(x1)  =  x1
U86(x1)  =  x1
s(x1)  =  s(x1)
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

PROPER₁: [1]
s₁: [1]

The following usable rules [FROCOS05] were oriented: none

(260) Obligation:

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

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

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

(261) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U46(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
isNatList(x1)  =  x1
isNatKind(x1)  =  x1
U23(x1)  =  x1
isNat(x1)  =  x1
U33(x1)  =  x1
U46(x1)  =  U46(x1)
isNatIList(x1)  =  x1
U52(x1)  =  x1
U61(x1)  =  x1
U71(x1)  =  x1
U86(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U46₁: [1]

The following usable rules [FROCOS05] were oriented: none

(262) Obligation:

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

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

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

(263) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U71(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNatKind(x1)  =  x1
U23(x1)  =  x1
isNat(x1)  =  x1
U33(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  x1
U61(x1)  =  x1
U71(x1)  =  U71(x1)
U86(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

PROPER₁: [1]
U71₁: [1]

The following usable rules [FROCOS05] were oriented: none

(264) Obligation:

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

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

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

(265) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U23(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
isNatList(x1)  =  x1
isNatKind(x1)  =  x1
U23(x1)  =  U23(x1)
isNat(x1)  =  x1
U33(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  x1
U61(x1)  =  x1
U86(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U23₁: [1]

The following usable rules [FROCOS05] were oriented: none

(266) Obligation:

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

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

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

(267) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U33(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNatKind(x1)  =  x1
isNat(x1)  =  x1
U33(x1)  =  U33(x1)
isNatIList(x1)  =  x1
U52(x1)  =  x1
U61(x1)  =  x1
U86(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

PROPER₁: [1]
U33₁: [1]

The following usable rules [FROCOS05] were oriented: none

(268) Obligation:

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

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

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.

PROPER(U52(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNatKind(x1)  =  x1
isNat(x1)  =  x1
isNatIList(x1)  =  x1
U52(x1)  =  U52(x1)
U61(x1)  =  x1
U86(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

PROPER₁: [1]
U52₁: [1]

The following usable rules [FROCOS05] were oriented: none

(270) Obligation:

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

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

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.

PROPER(U61(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNatKind(x1)  =  x1
isNat(x1)  =  x1
isNatIList(x1)  =  x1
U61(x1)  =  U61(x1)
U86(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

PROPER₁: [1]
U61₁: [1]

The following usable rules [FROCOS05] were oriented: none

(272) Obligation:

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

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

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

(273) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(U86(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNatKind(x1)  =  x1
isNat(x1)  =  x1
isNatIList(x1)  =  x1
U86(x1)  =  U86(x1)
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

PROPER₁: [1]
U86₁: [1]

The following usable rules [FROCOS05] were oriented: none

(274) Obligation:

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

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

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

(275) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isNatKind(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  x1
isNatKind(x1)  =  isNatKind(x1)
isNat(x1)  =  x1
isNatIList(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

PROPER₁: [1]
isNatKind₁: [1]

The following usable rules [FROCOS05] were oriented: none

(276) Obligation:

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

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

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

(277) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isNat(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
isNatList(x1)  =  x1
isNat(x1)  =  isNat(x1)
isNatIList(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

isNat₁: [1]

The following usable rules [FROCOS05] were oriented: none

(278) Obligation:

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

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

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

(279) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isNatList(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
isNatList(x1)  =  isNatList(x1)
isNatIList(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

PROPER₁: [1]
isNatList₁: [1]

The following usable rules [FROCOS05] were oriented: none

(280) Obligation:

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

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

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

(281) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(isNatIList(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
isNatIList(x1)  =  isNatIList(x1)
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

isNatIList₁: [1]

The following usable rules [FROCOS05] were oriented: none

(282) Obligation:

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

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

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

(283) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(length(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  x1
length(x1)  =  length(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

length₁: [1]

The following usable rules [FROCOS05] were oriented: none

(284) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(285) PisEmptyProof (EQUIVALENT transformation)

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

(286) TRUE

(287) Obligation:

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

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

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

(288) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U42(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U43(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U44(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U46(X)) → ACTIVE(X)
ACTIVE(U85(X1, X2)) → ACTIVE(X1)

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

Lexicographic path order with status [LPO].
Quasi-Precedence:

[ACTIVE₁, U42₃, U43₃, U44₃, U46₁, U85₂]

Status:

U42₃: [2,1,3]
U44₃: [3,2,1]
U46₁: [1]
U85₂: [2,1]
U43₃: [3,2,1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(289) Obligation:

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

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

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

(290) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U11(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(U12(X1, X2)) → ACTIVE(X1)
ACTIVE(U21(X1, X2)) → ACTIVE(X1)
ACTIVE(U22(X1, X2)) → ACTIVE(X1)
ACTIVE(U31(X1, X2)) → ACTIVE(X1)
ACTIVE(U32(X1, X2)) → ACTIVE(X1)
ACTIVE(U41(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U45(X1, X2)) → ACTIVE(X1)
ACTIVE(U51(X1, X2)) → ACTIVE(X1)
ACTIVE(U81(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U82(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U83(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U84(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U91(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U92(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U93(X1, X2, X3)) → ACTIVE(X1)
ACTIVE(U94(X1, X2)) → ACTIVE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U11(x1, x2)  =  U11(x1, x2)
cons(x1, x2)  =  cons(x1, x2)
U12(x1, x2)  =  U12(x1, x2)
U13(x1)  =  x1
U21(x1, x2)  =  U21(x1, x2)
U22(x1, x2)  =  U22(x1, x2)
U23(x1)  =  x1
U31(x1, x2)  =  U31(x1, x2)
U32(x1, x2)  =  U32(x1, x2)
U33(x1)  =  x1
U41(x1, x2, x3)  =  U41(x1, x2, x3)
U45(x1, x2)  =  U45(x1, x2)
U51(x1, x2)  =  U51(x1, x2)
U52(x1)  =  x1
U61(x1)  =  x1
U71(x1)  =  x1
U81(x1, x2, x3)  =  U81(x1, x2, x3)
U82(x1, x2, x3)  =  U82(x1, x2, x3)
U83(x1, x2, x3)  =  U83(x1, x2, x3)
U84(x1, x2, x3)  =  U84(x1, x2, x3)
U86(x1)  =  x1
U91(x1, x2, x3)  =  U91(x1, x2, x3)
U92(x1, x2, x3)  =  U92(x1, x2, x3)
U93(x1, x2, x3)  =  U93(x1, x2, x3)
U94(x1, x2)  =  U94(x1, x2)
s(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

[ACTIVE₁, U11₂, cons₂, U12₂, U21₂, U31₂, U32₂, U41₃, U81₃, U93₃]

Status:

U82₃: [3,2,1]
U31₂: [2,1]
U83₃: [3,2,1]
U22₂: [2,1]
U41₃: [3,2,1]
U32₂: [2,1]
U81₃: [3,2,1]
U12₂: [2,1]
U11₂: [2,1]
ACTIVE₁: [1]
U21₂: [2,1]
U51₂: [2,1]
U94₂: [2,1]
cons₂: [2,1]
U91₃: [3,2,1]
U93₃: [3,2,1]
U92₃: [3,2,1]
U45₂: [2,1]
U84₃: [3,2,1]

The following usable rules [FROCOS05] were oriented: none

(291) Obligation:

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

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

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

(292) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U52(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U13(x1)  =  x1
U23(x1)  =  x1
U33(x1)  =  x1
U52(x1)  =  U52(x1)
U61(x1)  =  x1
U71(x1)  =  x1
U86(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U52₁: [1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(293) Obligation:

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

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

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

(294) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U71(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U13(x1)  =  x1
U23(x1)  =  x1
U33(x1)  =  x1
U61(x1)  =  x1
U71(x1)  =  U71(x1)
U86(x1)  =  x1
s(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U71₁: [1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(295) Obligation:

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

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

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

(296) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U86(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U13(x1)  =  x1
U23(x1)  =  x1
U33(x1)  =  x1
U61(x1)  =  x1
U86(x1)  =  U86(x1)
s(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U86₁: [1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(297) Obligation:

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

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

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

(298) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(s(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U13(x1)  =  x1
U23(x1)  =  x1
U33(x1)  =  x1
U61(x1)  =  x1
s(x1)  =  s(x1)
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

s₁: [1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(299) Obligation:

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

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

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

(300) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U23(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U13(x1)  =  x1
U23(x1)  =  U23(x1)
U33(x1)  =  x1
U61(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

ACTIVE₁: [1]
U23₁: [1]

The following usable rules [FROCOS05] were oriented: none

(301) Obligation:

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

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

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

(302) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U33(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U13(x1)  =  x1
U33(x1)  =  U33(x1)
U61(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U33₁: [1]

The following usable rules [FROCOS05] were oriented: none

(303) Obligation:

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

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

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

(304) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U13(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
U13(x1)  =  U13(x1)
U61(x1)  =  x1
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U13₁: [1]
ACTIVE₁: [1]

The following usable rules [FROCOS05] were oriented: none

(305) Obligation:

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

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

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

(306) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(U61(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
U61(x1)  =  U61(x1)
length(x1)  =  x1

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

U61₁: [1]

The following usable rules [FROCOS05] were oriented: none

(307) Obligation:

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

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

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

(308) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(length(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  x1
length(x1)  =  length(x1)

Lexicographic path order with status [LPO].
Quasi-Precedence:

trivial

Status:

length₁: [1]

The following usable rules [FROCOS05] were oriented: none

(309) 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))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U33(X)) → U33(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(active(X1), X2, X3)
active(U43(X1, X2, X3)) → U43(active(X1), X2, X3)
active(U44(X1, X2, X3)) → U44(active(X1), X2, X3)
active(U45(X1, X2)) → U45(active(X1), X2)
active(U46(X)) → U46(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X)) → U52(active(X))
active(U61(X)) → U61(active(X))
active(U71(X)) → U71(active(X))
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(U83(X1, X2, X3)) → U83(active(X1), X2, X3)
active(U84(X1, X2, X3)) → U84(active(X1), X2, X3)
active(U85(X1, X2)) → U85(active(X1), X2)
active(U86(X)) → U86(active(X))
active(U91(X1, X2, X3)) → U91(active(X1), X2, X3)
active(U92(X1, X2, X3)) → U92(active(X1), X2, X3)
active(U93(X1, X2, X3)) → U93(active(X1), X2, X3)
active(U94(X1, X2)) → U94(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U33(mark(X)) → mark(U33(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(X1, X2, X3))
U43(mark(X1), X2, X3) → mark(U43(X1, X2, X3))
U44(mark(X1), X2, X3) → mark(U44(X1, X2, X3))
U45(mark(X1), X2) → mark(U45(X1, X2))
U46(mark(X)) → mark(U46(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X)) → mark(U52(X))
U61(mark(X)) → mark(U61(X))
U71(mark(X)) → mark(U71(X))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
U83(mark(X1), X2, X3) → mark(U83(X1, X2, X3))
U84(mark(X1), X2, X3) → mark(U84(X1, X2, X3))
U85(mark(X1), X2) → mark(U85(X1, X2))
U86(mark(X)) → mark(U86(X))
U91(mark(X1), X2, X3) → mark(U91(X1, X2, X3))
U92(mark(X1), X2, X3) → mark(U92(X1, X2, X3))
U93(mark(X1), X2, X3) → mark(U93(X1, X2, X3))
U94(mark(X1), X2) → mark(U94(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNatIListKind(X)) → isNatIListKind(proper(X))
proper(U13(X)) → U13(proper(X))
proper(isNatList(X)) → isNatList(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U23(X)) → U23(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U33(X)) → U33(proper(X))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(U43(X1, X2, X3)) → U43(proper(X1), proper(X2), proper(X3))
proper(U44(X1, X2, X3)) → U44(proper(X1), proper(X2), proper(X3))
proper(U45(X1, X2)) → U45(proper(X1), proper(X2))
proper(U46(X)) → U46(proper(X))
proper(isNatIList(X)) → isNatIList(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X)) → U52(proper(X))
proper(U61(X)) → U61(proper(X))
proper(U71(X)) → U71(proper(X))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(U83(X1, X2, X3)) → U83(proper(X1), proper(X2), proper(X3))
proper(U84(X1, X2, X3)) → U84(proper(X1), proper(X2), proper(X3))
proper(U85(X1, X2)) → U85(proper(X1), proper(X2))
proper(U86(X)) → U86(proper(X))
proper(U91(X1, X2, X3)) → U91(proper(X1), proper(X2), proper(X3))
proper(U92(X1, X2, X3)) → U92(proper(X1), proper(X2), proper(X3))
proper(U93(X1, X2, X3)) → U93(proper(X1), proper(X2), proper(X3))
proper(U94(X1, X2)) → U94(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNatIListKind(ok(X)) → ok(isNatIListKind(X))
U13(ok(X)) → ok(U13(X))
isNatList(ok(X)) → ok(isNatList(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
U23(ok(X)) → ok(U23(X))
isNat(ok(X)) → ok(isNat(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U33(ok(X)) → ok(U33(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
U43(ok(X1), ok(X2), ok(X3)) → ok(U43(X1, X2, X3))
U44(ok(X1), ok(X2), ok(X3)) → ok(U44(X1, X2, X3))
U45(ok(X1), ok(X2)) → ok(U45(X1, X2))
U46(ok(X)) → ok(U46(X))
isNatIList(ok(X)) → ok(isNatIList(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X)) → ok(U52(X))
U61(ok(X)) → ok(U61(X))
U71(ok(X)) → ok(U71(X))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
U83(ok(X1), ok(X2), ok(X3)) → ok(U83(X1, X2, X3))
U84(ok(X1), ok(X2), ok(X3)) → ok(U84(X1, X2, X3))
U85(ok(X1), ok(X2)) → ok(U85(X1, X2))
U86(ok(X)) → ok(U86(X))
U91(ok(X1), ok(X2), ok(X3)) → ok(U91(X1, X2, X3))
U92(ok(X1), ok(X2), ok(X3)) → ok(U92(X1, X2, X3))
U93(ok(X1), ok(X2), ok(X3)) → ok(U93(X1, X2, X3))
U94(ok(X1), ok(X2)) → ok(U94(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(310) PisEmptyProof (EQUIVALENT transformation)

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

(311) TRUE

(312) Obligation:

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

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

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