Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → U611(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
ACTIVE(U22(tt)) → MARK(tt)
ACTIVE(length(cons(N, L))) → MARK(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
CONS(X1, mark(X2)) → CONS(X1, X2)
ACTIVE(isNatIListKind(nil)) → MARK(tt)
MARK(U22(X)) → U221(mark(X))
U111(X1, mark(X2)) → U111(X1, X2)
ACTIVE(U52(tt, V2)) → U531(isNatList(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(length(cons(N, L))) → ISNATILISTKIND(L)
U121(active(X)) → U121(X)
U311(mark(X1), X2) → U311(X1, X2)
U311(X1, mark(X2)) → U311(X1, X2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(U61(tt, L)) → LENGTH(L)
MARK(U32(X)) → MARK(X)
ACTIVE(length(cons(N, L))) → AND(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATILISTKIND(V2)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(U42(tt, V2)) → ISNATILIST(V2)
U511(X1, active(X2), X3) → U511(X1, X2, X3)
MARK(U11(X1, X2)) → U111(mark(X1), X2)
ACTIVE(U51(tt, V1, V2)) → U521(isNat(V1), V2)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U31(X1, X2)) → U311(mark(X1), X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)
U211(X1, mark(X2)) → U211(X1, X2)
U611(X1, active(X2)) → U611(X1, X2)
LENGTH(active(X)) → LENGTH(X)
U411(X1, active(X2), X3) → U411(X1, X2, X3)
ACTIVE(U11(tt, V1)) → U121(isNatList(V1))
S(mark(X)) → S(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNat(V1), V2))
MARK(U61(X1, X2)) → U611(mark(X1), X2)
ACTIVE(isNatList(nil)) → MARK(tt)
MARK(U43(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(U12(tt)) → MARK(tt)
U421(X1, mark(X2)) → U421(X1, X2)
U121(mark(X)) → U121(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U41(tt, V1, V2)) → ISNAT(V1)
U431(mark(X)) → U431(X)
ACTIVE(U52(tt, V2)) → MARK(U53(isNatList(V2)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
U611(X1, mark(X2)) → U611(X1, X2)
ACTIVE(zeros) → CONS(0, zeros)
MARK(U43(X)) → ACTIVE(U43(mark(X)))
ACTIVE(U11(tt, V1)) → ISNATLIST(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
U531(active(X)) → U531(X)
ACTIVE(U51(tt, V1, V2)) → ISNAT(V1)
U311(X1, active(X2)) → U311(X1, X2)
S(active(X)) → S(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
U321(active(X)) → U321(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U61(tt, L)) → MARK(s(length(L)))
AND(X1, active(X2)) → AND(X1, X2)
MARK(U12(X)) → ACTIVE(U12(mark(X)))
MARK(U51(X1, X2, X3)) → U511(mark(X1), X2, X3)
ACTIVE(isNatIList(V)) → ISNATILISTKIND(V)
MARK(U53(X)) → U531(mark(X))
ACTIVE(isNat(length(V1))) → U111(isNatIListKind(V1), V1)
AND(active(X1), X2) → AND(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNat(V1), V2))
MARK(U52(X1, X2)) → MARK(X1)
U221(mark(X)) → U221(X)
ACTIVE(U21(tt, V1)) → U221(isNat(V1))
ACTIVE(isNatIListKind(zeros)) → MARK(tt)
ACTIVE(U41(tt, V1, V2)) → U421(isNat(V1), V2)
U511(active(X1), X2, X3) → U511(X1, X2, X3)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
U521(X1, active(X2)) → U521(X1, X2)
MARK(U42(X1, X2)) → U421(mark(X1), X2)
ACTIVE(isNatIList(V)) → MARK(U31(isNatIListKind(V), V))
ACTIVE(isNatIListKind(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
U611(mark(X1), X2) → U611(X1, X2)
MARK(U12(X)) → U121(mark(X))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(and(X1, X2)) → AND(mark(X1), X2)
U511(X1, mark(X2), X3) → U511(X1, X2, X3)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
MARK(U53(X)) → ACTIVE(U53(mark(X)))
ISNAT(active(X)) → ISNAT(X)
ACTIVE(isNat(s(V1))) → U211(isNatKind(V1), V1)
U421(X1, active(X2)) → U421(X1, X2)
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(zeros) → MARK(cons(0, zeros))
U411(X1, mark(X2), X3) → U411(X1, X2, X3)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ISNATKIND(active(X)) → ISNATKIND(X)
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNatIList(V2)))
ACTIVE(U32(tt)) → MARK(tt)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNatIListKind(L))
ACTIVE(isNatIList(zeros)) → MARK(tt)
MARK(length(X)) → MARK(X)
U311(active(X1), X2) → U311(X1, X2)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
U511(X1, X2, active(X3)) → U511(X1, X2, X3)
ACTIVE(isNatList(cons(V1, V2))) → U511(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ACTIVE(isNatKind(length(V1))) → ISNATILISTKIND(V1)
U211(mark(X1), X2) → U211(X1, X2)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
U521(active(X1), X2) → U521(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(U21(tt, V1)) → ISNAT(V1)
U221(active(X)) → U221(X)
ACTIVE(U31(tt, V)) → ISNATLIST(V)
ISNATILIST(mark(X)) → ISNATILIST(X)
U321(mark(X)) → U321(X)
U421(active(X1), X2) → U421(X1, X2)
ACTIVE(and(tt, X)) → MARK(X)
U611(active(X1), X2) → U611(X1, X2)
MARK(U43(X)) → U431(mark(X))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILISTKIND(V2)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(0)) → MARK(tt)
CONS(X1, active(X2)) → CONS(X1, X2)
ACTIVE(isNat(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatKind(N))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U53(tt)) → MARK(tt)
U411(active(X1), X2, X3) → U411(X1, X2, X3)
LENGTH(mark(X)) → LENGTH(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
U111(active(X1), X2) → U111(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(isNatList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ISNAT(mark(X)) → ISNAT(X)
ACTIVE(length(nil)) → MARK(0)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U61(X1, X2)) → MARK(X1)
AND(mark(X1), X2) → AND(X1, X2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(U61(tt, L)) → S(length(L))
U531(mark(X)) → U531(X)
MARK(tt) → ACTIVE(tt)
ACTIVE(isNatIList(cons(V1, V2))) → U411(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U32(X)) → U321(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
AND(X1, mark(X2)) → AND(X1, X2)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
U421(mark(X1), X2) → U421(X1, X2)
ACTIVE(length(cons(N, L))) → ISNATKIND(N)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U53(X)) → MARK(X)
MARK(U21(X1, X2)) → U211(mark(X1), X2)
U431(active(X)) → U431(X)
ISNATILISTKIND(active(X)) → ISNATILISTKIND(X)
MARK(U41(X1, X2, X3)) → U411(mark(X1), X2, X3)
ACTIVE(U11(tt, V1)) → MARK(U12(isNatList(V1)))
ACTIVE(U42(tt, V2)) → U431(isNatIList(V2))
ACTIVE(U52(tt, V2)) → ISNATLIST(V2)
MARK(U22(X)) → MARK(X)
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(and(isNatKind(V1), isNatIListKind(V2)))
MARK(zeros) → ACTIVE(zeros)
MARK(length(X)) → LENGTH(mark(X))
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ISNATLIST(active(X)) → ISNATLIST(X)
U111(mark(X1), X2) → U111(X1, X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U22(X)) → ACTIVE(U22(mark(X)))
MARK(U12(X)) → MARK(X)
U211(X1, active(X2)) → U211(X1, X2)
ACTIVE(U31(tt, V)) → U321(isNatList(V))
MARK(U31(X1, X2)) → MARK(X1)
ISNATLIST(mark(X)) → ISNATLIST(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
U521(X1, mark(X2)) → U521(X1, X2)
U411(X1, X2, active(X3)) → U411(X1, X2, X3)
U511(X1, X2, mark(X3)) → U511(X1, X2, X3)
ACTIVE(U31(tt, V)) → MARK(U32(isNatList(V)))
ACTIVE(isNatIList(V)) → U311(isNatIListKind(V), V)
ACTIVE(U43(tt)) → MARK(tt)
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U521(mark(X1), X2) → U521(X1, X2)
MARK(s(X)) → S(mark(X))
ISNATKIND(mark(X)) → ISNATKIND(X)
ACTIVE(isNatKind(length(V1))) → MARK(isNatIListKind(V1))
ACTIVE(isNatKind(0)) → MARK(tt)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
MARK(U52(X1, X2)) → U521(mark(X1), X2)
MARK(0) → ACTIVE(0)
U411(X1, X2, mark(X3)) → U411(X1, X2, X3)
ISNATILIST(active(X)) → ISNATILIST(X)
MARK(nil) → ACTIVE(nil)

The TRS R consists of the following rules:

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

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → U611(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)
ACTIVE(U22(tt)) → MARK(tt)
ACTIVE(length(cons(N, L))) → MARK(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
CONS(X1, mark(X2)) → CONS(X1, X2)
ACTIVE(isNatIListKind(nil)) → MARK(tt)
MARK(U22(X)) → U221(mark(X))
U111(X1, mark(X2)) → U111(X1, X2)
ACTIVE(U52(tt, V2)) → U531(isNatList(V2))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(length(cons(N, L))) → ISNATILISTKIND(L)
U121(active(X)) → U121(X)
U311(mark(X1), X2) → U311(X1, X2)
U311(X1, mark(X2)) → U311(X1, X2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(U61(tt, L)) → LENGTH(L)
MARK(U32(X)) → MARK(X)
ACTIVE(length(cons(N, L))) → AND(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N)))
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATILISTKIND(V2)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(U42(tt, V2)) → ISNATILIST(V2)
U511(X1, active(X2), X3) → U511(X1, X2, X3)
MARK(U11(X1, X2)) → U111(mark(X1), X2)
ACTIVE(U51(tt, V1, V2)) → U521(isNat(V1), V2)
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(U31(X1, X2)) → U311(mark(X1), X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)
U211(X1, mark(X2)) → U211(X1, X2)
U611(X1, active(X2)) → U611(X1, X2)
LENGTH(active(X)) → LENGTH(X)
U411(X1, active(X2), X3) → U411(X1, X2, X3)
ACTIVE(U11(tt, V1)) → U121(isNatList(V1))
S(mark(X)) → S(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNat(V1), V2))
MARK(U61(X1, X2)) → U611(mark(X1), X2)
ACTIVE(isNatList(nil)) → MARK(tt)
MARK(U43(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(U12(tt)) → MARK(tt)
U421(X1, mark(X2)) → U421(X1, X2)
U121(mark(X)) → U121(X)
MARK(isNat(X)) → ACTIVE(isNat(X))
ACTIVE(U41(tt, V1, V2)) → ISNAT(V1)
U431(mark(X)) → U431(X)
ACTIVE(U52(tt, V2)) → MARK(U53(isNatList(V2)))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
U611(X1, mark(X2)) → U611(X1, X2)
ACTIVE(zeros) → CONS(0, zeros)
MARK(U43(X)) → ACTIVE(U43(mark(X)))
ACTIVE(U11(tt, V1)) → ISNATLIST(V1)
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
U531(active(X)) → U531(X)
ACTIVE(U51(tt, V1, V2)) → ISNAT(V1)
U311(X1, active(X2)) → U311(X1, X2)
S(active(X)) → S(X)
MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
U321(active(X)) → U321(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U61(tt, L)) → MARK(s(length(L)))
AND(X1, active(X2)) → AND(X1, X2)
MARK(U12(X)) → ACTIVE(U12(mark(X)))
MARK(U51(X1, X2, X3)) → U511(mark(X1), X2, X3)
ACTIVE(isNatIList(V)) → ISNATILISTKIND(V)
MARK(U53(X)) → U531(mark(X))
ACTIVE(isNat(length(V1))) → U111(isNatIListKind(V1), V1)
AND(active(X1), X2) → AND(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNat(V1), V2))
MARK(U52(X1, X2)) → MARK(X1)
U221(mark(X)) → U221(X)
ACTIVE(U21(tt, V1)) → U221(isNat(V1))
ACTIVE(isNatIListKind(zeros)) → MARK(tt)
ACTIVE(U41(tt, V1, V2)) → U421(isNat(V1), V2)
U511(active(X1), X2, X3) → U511(X1, X2, X3)
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
U521(X1, active(X2)) → U521(X1, X2)
MARK(U42(X1, X2)) → U421(mark(X1), X2)
ACTIVE(isNatIList(V)) → MARK(U31(isNatIListKind(V), V))
ACTIVE(isNatIListKind(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
U611(mark(X1), X2) → U611(X1, X2)
MARK(U12(X)) → U121(mark(X))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(and(X1, X2)) → AND(mark(X1), X2)
U511(X1, mark(X2), X3) → U511(X1, X2, X3)
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
MARK(U53(X)) → ACTIVE(U53(mark(X)))
ISNAT(active(X)) → ISNAT(X)
ACTIVE(isNat(s(V1))) → U211(isNatKind(V1), V1)
U421(X1, active(X2)) → U421(X1, X2)
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(zeros) → MARK(cons(0, zeros))
U411(X1, mark(X2), X3) → U411(X1, X2, X3)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
ISNATKIND(active(X)) → ISNATKIND(X)
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
ACTIVE(U42(tt, V2)) → MARK(U43(isNatIList(V2)))
ACTIVE(U32(tt)) → MARK(tt)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNatIListKind(L))
ACTIVE(isNatIList(zeros)) → MARK(tt)
MARK(length(X)) → MARK(X)
U311(active(X1), X2) → U311(X1, X2)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
U511(X1, X2, active(X3)) → U511(X1, X2, X3)
ACTIVE(isNatList(cons(V1, V2))) → U511(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
ACTIVE(isNatKind(length(V1))) → ISNATILISTKIND(V1)
U211(mark(X1), X2) → U211(X1, X2)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
U521(active(X1), X2) → U521(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
ACTIVE(isNatIListKind(cons(V1, V2))) → ISNATKIND(V1)
ACTIVE(U21(tt, V1)) → ISNAT(V1)
U221(active(X)) → U221(X)
ACTIVE(U31(tt, V)) → ISNATLIST(V)
ISNATILIST(mark(X)) → ISNATILIST(X)
U321(mark(X)) → U321(X)
U421(active(X1), X2) → U421(X1, X2)
ACTIVE(and(tt, X)) → MARK(X)
U611(active(X1), X2) → U611(X1, X2)
MARK(U43(X)) → U431(mark(X))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILISTKIND(V2)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(isNat(0)) → MARK(tt)
CONS(X1, active(X2)) → CONS(X1, X2)
ACTIVE(isNat(length(V1))) → ISNATILISTKIND(V1)
ACTIVE(length(cons(N, L))) → AND(isNat(N), isNatKind(N))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U53(tt)) → MARK(tt)
U411(active(X1), X2, X3) → U411(X1, X2, X3)
LENGTH(mark(X)) → LENGTH(X)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
U111(active(X1), X2) → U111(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(isNatList(cons(V1, V2))) → AND(isNatKind(V1), isNatIListKind(V2))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ISNAT(mark(X)) → ISNAT(X)
ACTIVE(length(nil)) → MARK(0)
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U61(X1, X2)) → MARK(X1)
AND(mark(X1), X2) → AND(X1, X2)
ACTIVE(isNatList(cons(V1, V2))) → ISNATILISTKIND(V2)
ACTIVE(U61(tt, L)) → S(length(L))
U531(mark(X)) → U531(X)
MARK(tt) → ACTIVE(tt)
ACTIVE(isNatIList(cons(V1, V2))) → U411(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)
MARK(U32(X)) → U321(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
AND(X1, mark(X2)) → AND(X1, X2)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
U421(mark(X1), X2) → U421(X1, X2)
ACTIVE(length(cons(N, L))) → ISNATKIND(N)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U53(X)) → MARK(X)
MARK(U21(X1, X2)) → U211(mark(X1), X2)
U431(active(X)) → U431(X)
ISNATILISTKIND(active(X)) → ISNATILISTKIND(X)
MARK(U41(X1, X2, X3)) → U411(mark(X1), X2, X3)
ACTIVE(U11(tt, V1)) → MARK(U12(isNatList(V1)))
ACTIVE(U42(tt, V2)) → U431(isNatIList(V2))
ACTIVE(U52(tt, V2)) → ISNATLIST(V2)
MARK(U22(X)) → MARK(X)
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(and(isNatKind(V1), isNatIListKind(V2)))
MARK(zeros) → ACTIVE(zeros)
MARK(length(X)) → LENGTH(mark(X))
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → ISNATKIND(V1)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ISNATLIST(active(X)) → ISNATLIST(X)
U111(mark(X1), X2) → U111(X1, X2)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U22(X)) → ACTIVE(U22(mark(X)))
MARK(U12(X)) → MARK(X)
U211(X1, active(X2)) → U211(X1, X2)
ACTIVE(U31(tt, V)) → U321(isNatList(V))
MARK(U31(X1, X2)) → MARK(X1)
ISNATLIST(mark(X)) → ISNATLIST(X)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
U521(X1, mark(X2)) → U521(X1, X2)
U411(X1, X2, active(X3)) → U411(X1, X2, X3)
U511(X1, X2, mark(X3)) → U511(X1, X2, X3)
ACTIVE(U31(tt, V)) → MARK(U32(isNatList(V)))
ACTIVE(isNatIList(V)) → U311(isNatIListKind(V), V)
ACTIVE(U43(tt)) → MARK(tt)
ACTIVE(isNatKind(s(V1))) → ISNATKIND(V1)
U521(mark(X1), X2) → U521(X1, X2)
MARK(s(X)) → S(mark(X))
ISNATKIND(mark(X)) → ISNATKIND(X)
ACTIVE(isNatKind(length(V1))) → MARK(isNatIListKind(V1))
ACTIVE(isNatKind(0)) → MARK(tt)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
MARK(U52(X1, X2)) → U521(mark(X1), X2)
MARK(0) → ACTIVE(0)
U411(X1, X2, mark(X3)) → U411(X1, X2, X3)
ISNATILIST(active(X)) → ISNATILIST(X)
MARK(nil) → ACTIVE(nil)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 23 SCCs with 76 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ISNATKIND(active(X)) → ISNATKIND(X)
ISNATKIND(mark(X)) → ISNATKIND(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 9/4 + x_1   
POL(ISNATKIND(x1)) = (1/2)x_1   
POL(mark(x1)) = 1/2 + (3/2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ISNATILISTKIND(active(X)) → ISNATILISTKIND(X)
ISNATILISTKIND(mark(X)) → ISNATILISTKIND(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 9/4 + x_1   
POL(ISNATILISTKIND(x1)) = (1/2)x_1   
POL(mark(x1)) = 1/2 + (3/2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


AND(mark(X1), X2) → AND(X1, X2)
AND(active(X1), X2) → AND(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
AND(X1, active(X2)) → AND(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 1/4 + (2)x_1   
POL(AND(x1, x2)) = (1/4)x_1 + (1/2)x_2   
POL(mark(x1)) = 13/4 + (2)x_1   
The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(LENGTH(x1)) = (1/2)x_1   
POL(active(x1)) = 1/2 + (3/2)x_1   
POL(mark(x1)) = 9/4 + x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S(mark(X)) → S(X)
S(active(X)) → S(X)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


S(mark(X)) → S(X)
S(active(X)) → S(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 9/4 + x_1   
POL(mark(x1)) = 1/2 + (3/2)x_1   
POL(S(x1)) = (1/2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U611(X1, mark(X2)) → U611(X1, X2)
U611(X1, active(X2)) → U611(X1, X2)
U611(active(X1), X2) → U611(X1, X2)
U611(mark(X1), X2) → U611(X1, X2)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U611(X1, mark(X2)) → U611(X1, X2)
U611(mark(X1), X2) → U611(X1, X2)
The remaining pairs can at least be oriented weakly.

U611(X1, active(X2)) → U611(X1, X2)
U611(active(X1), X2) → U611(X1, X2)
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = (2)x_1   
POL(U611(x1, x2)) = (3/2)x_1 + (4)x_2   
POL(mark(x1)) = 3/2 + x_1   
The value of delta used in the strict ordering is 9/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U611(active(X1), X2) → U611(X1, X2)
U611(X1, active(X2)) → U611(X1, X2)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U611(active(X1), X2) → U611(X1, X2)
U611(X1, active(X2)) → U611(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 4 + (9/4)x_1   
POL(U611(x1, x2)) = (3)x_1 + (1/2)x_2   
The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U531(mark(X)) → U531(X)
U531(active(X)) → U531(X)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U531(mark(X)) → U531(X)
U531(active(X)) → U531(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 9/4 + x_1   
POL(mark(x1)) = 1/2 + (3/2)x_1   
POL(U531(x1)) = (1/2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U521(active(X1), X2) → U521(X1, X2)
U521(mark(X1), X2) → U521(X1, X2)
U521(X1, active(X2)) → U521(X1, X2)
U521(X1, mark(X2)) → U521(X1, X2)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U521(active(X1), X2) → U521(X1, X2)
U521(mark(X1), X2) → U521(X1, X2)
U521(X1, active(X2)) → U521(X1, X2)
U521(X1, mark(X2)) → U521(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(U521(x1, x2)) = (1/4)x_1 + (1/2)x_2   
POL(active(x1)) = 1/4 + (2)x_1   
POL(mark(x1)) = 13/4 + (2)x_1   
The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U511(X1, X2, mark(X3)) → U511(X1, X2, X3)
U511(X1, mark(X2), X3) → U511(X1, X2, X3)
U511(X1, X2, active(X3)) → U511(X1, X2, X3)
U511(X1, active(X2), X3) → U511(X1, X2, X3)
U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(active(X1), X2, X3) → U511(X1, X2, X3)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U511(X1, X2, mark(X3)) → U511(X1, X2, X3)
U511(X1, mark(X2), X3) → U511(X1, X2, X3)
U511(X1, X2, active(X3)) → U511(X1, X2, X3)
U511(X1, active(X2), X3) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.

U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(active(X1), X2, X3) → U511(X1, X2, X3)
Used ordering: Polynomial interpretation [25,35]:

POL(U511(x1, x2, x3)) = (15/4)x_2 + (4)x_3   
POL(active(x1)) = 1/4 + (5/4)x_1   
POL(mark(x1)) = 1/4 + (5/4)x_1   
The value of delta used in the strict ordering is 15/16.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(active(X1), X2, X3) → U511(X1, X2, X3)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U511(mark(X1), X2, X3) → U511(X1, X2, X3)
U511(active(X1), X2, X3) → U511(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(U511(x1, x2, x3)) = (1/2)x_1   
POL(active(x1)) = 1/2 + (3/2)x_1   
POL(mark(x1)) = 9/4 + x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ISNATILIST(mark(X)) → ISNATILIST(X)
ISNATILIST(active(X)) → ISNATILIST(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 1/2 + (3/2)x_1   
POL(ISNATILIST(x1)) = (1/2)x_1   
POL(mark(x1)) = 9/4 + x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U431(active(X)) → U431(X)
U431(mark(X)) → U431(X)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U431(active(X)) → U431(X)
U431(mark(X)) → U431(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 1/2 + (3/2)x_1   
POL(mark(x1)) = 9/4 + x_1   
POL(U431(x1)) = (1/2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U421(mark(X1), X2) → U421(X1, X2)
U421(active(X1), X2) → U421(X1, X2)
U421(X1, mark(X2)) → U421(X1, X2)
U421(X1, active(X2)) → U421(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 13/4 + (2)x_1   
POL(U421(x1, x2)) = (1/2)x_1 + (1/4)x_2   
POL(mark(x1)) = 7/4 + x_1   
The value of delta used in the strict ordering is 7/16.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U411(X1, X2, active(X3)) → U411(X1, X2, X3)
U411(X1, active(X2), X3) → U411(X1, X2, X3)
U411(X1, X2, mark(X3)) → U411(X1, X2, X3)
U411(X1, mark(X2), X3) → U411(X1, X2, X3)
U411(active(X1), X2, 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(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(isNatKind(X)) → active(isNatKind(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U411(X1, X2, active(X3)) → U411(X1, X2, X3)
U411(X1, active(X2), X3) → U411(X1, X2, X3)
U411(X1, X2, mark(X3)) → U411(X1, X2, X3)
U411(X1, mark(X2), X3) → U411(X1, X2, X3)
U411(active(X1), X2, X3) → U411(X1, X2, X3)
U411(mark(X1), X2, X3) → U411(X1, X2, X3)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 1/4 + (4)x_1   
POL(mark(x1)) = 1/4 + (4)x_1   
POL(U411(x1, x2, x3)) = (5/4)x_1 + (4)x_2 + (1/4)x_3   
The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U321(active(X)) → U321(X)
U321(mark(X)) → U321(X)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U321(active(X)) → U321(X)
U321(mark(X)) → U321(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 1/2 + (3/2)x_1   
POL(U321(x1)) = (1/2)x_1   
POL(mark(x1)) = 9/4 + x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U311(active(X1), X2) → U311(X1, X2)
U311(mark(X1), X2) → U311(X1, X2)
The remaining pairs can at least be oriented weakly.

U311(X1, active(X2)) → U311(X1, X2)
U311(X1, mark(X2)) → U311(X1, X2)
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 4 + (9/4)x_1   
POL(U311(x1, x2)) = x_1   
POL(mark(x1)) = 1/4 + (2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U311(X1, active(X2)) → U311(X1, X2)
U311(X1, mark(X2)) → U311(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 9/4 + x_1   
POL(U311(x1, x2)) = (1/2)x_2   
POL(mark(x1)) = 1/2 + (3/2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 9/4 + x_1   
POL(mark(x1)) = 1/2 + (3/2)x_1   
POL(ISNAT(x1)) = (1/2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U221(active(X)) → U221(X)
U221(mark(X)) → U221(X)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U221(active(X)) → U221(X)
U221(mark(X)) → U221(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 9/4 + x_1   
POL(U221(x1)) = (1/2)x_1   
POL(mark(x1)) = 1/2 + (3/2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U211(X1, active(X2)) → U211(X1, X2)
U211(active(X1), X2) → U211(X1, X2)
U211(X1, mark(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(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(isNatKind(X)) → active(isNatKind(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U211(X1, active(X2)) → U211(X1, X2)
U211(X1, mark(X2)) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.

U211(active(X1), X2) → U211(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 4 + (4)x_1   
POL(U211(x1, x2)) = (4)x_2   
POL(mark(x1)) = 1/4 + (3/2)x_1   
The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U211(active(X1), 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(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(isNatKind(X)) → active(isNatKind(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U211(active(X1), X2) → U211(X1, X2)
U211(mark(X1), X2) → U211(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 9/4 + x_1   
POL(U211(x1, x2)) = (1/2)x_1   
POL(mark(x1)) = 1/2 + (3/2)x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ISNATLIST(mark(X)) → ISNATLIST(X)
ISNATLIST(active(X)) → ISNATLIST(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(ISNATLIST(x1)) = (1/2)x_1   
POL(active(x1)) = 1/2 + (3/2)x_1   
POL(mark(x1)) = 9/4 + x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U121(mark(X)) → U121(X)
U121(active(X)) → U121(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 1/2 + (3/2)x_1   
POL(U121(x1)) = (1/2)x_1   
POL(mark(x1)) = 9/4 + x_1   
The value of delta used in the strict ordering is 1/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

U111(X1, mark(X2)) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U111(active(X1), 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(isNatList(V1)))
active(U12(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, V)) → mark(U32(isNatList(V)))
active(U32(tt)) → mark(tt)
active(U41(tt, V1, V2)) → mark(U42(isNat(V1), V2))
active(U42(tt, V2)) → mark(U43(isNatIList(V2)))
active(U43(tt)) → mark(tt)
active(U51(tt, V1, V2)) → mark(U52(isNat(V1), V2))
active(U52(tt, V2)) → mark(U53(isNatList(V2)))
active(U53(tt)) → mark(tt)
active(U61(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(U11(isNatIListKind(V1), V1))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatIList(V)) → mark(U31(isNatIListKind(V), V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(isNatIListKind(nil)) → mark(tt)
active(isNatIListKind(zeros)) → mark(tt)
active(isNatIListKind(cons(V1, V2))) → mark(and(isNatKind(V1), isNatIListKind(V2)))
active(isNatKind(0)) → mark(tt)
active(isNatKind(length(V1))) → mark(isNatIListKind(V1))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X)) → active(U12(mark(X)))
mark(isNatList(X)) → active(isNatList(X))
mark(U21(X1, X2)) → active(U21(mark(X1), X2))
mark(U22(X)) → active(U22(mark(X)))
mark(isNat(X)) → active(isNat(X))
mark(U31(X1, X2)) → active(U31(mark(X1), X2))
mark(U32(X)) → active(U32(mark(X)))
mark(U41(X1, X2, X3)) → active(U41(mark(X1), X2, X3))
mark(U42(X1, X2)) → active(U42(mark(X1), X2))
mark(U43(X)) → active(U43(mark(X)))
mark(isNatIList(X)) → active(isNatIList(X))
mark(U51(X1, X2, X3)) → active(U51(mark(X1), X2, X3))
mark(U52(X1, X2)) → active(U52(mark(X1), X2))
mark(U53(X)) → active(U53(mark(X)))
mark(U61(X1, X2)) → active(U61(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatIListKind(X)) → active(isNatIListKind(X))
mark(isNatKind(X)) → active(isNatKind(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X)) → U12(X)
U12(active(X)) → U12(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(active(X1), X2) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U32(mark(X)) → U32(X)
U32(active(X)) → U32(X)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U42(X1, active(X2)) → U42(X1, X2)
U43(mark(X)) → U43(X)
U43(active(X)) → U43(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U53(mark(X)) → U53(X)
U53(active(X)) → U53(X)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)
and(mark(X1), X2) → and(X1, X2)
and(X1, mark(X2)) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
isNatIListKind(mark(X)) → isNatIListKind(X)
isNatIListKind(active(X)) → isNatIListKind(X)
isNatKind(mark(X)) → isNatKind(X)
isNatKind(active(X)) → isNatKind(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U111(X1, mark(X2)) → U111(X1, X2)
U111(X1, active(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 1/4 + (2)x_1   
POL(U111(x1, x2)) = (1/4)x_1 + (1/2)x_2   
POL(mark(x1)) = 13/4 + (2)x_1   
The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


CONS(X1, active(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
Used ordering: Polynomial interpretation [25,35]:

POL(active(x1)) = 3/2 + x_1   
POL(CONS(x1, x2)) = (3/2)x_1 + (4)x_2   
POL(mark(x1)) = (2)x_1   
The value of delta used in the strict ordering is 9/4.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [25,35]:

POL(CONS(x1, x2)) = (1/2)x_1 + (3)x_2   
POL(mark(x1)) = 4 + (9/4)x_1   
The value of delta used in the strict ordering is 2.
The following usable rules [17] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
ACTIVE(U42(tt, V2)) → MARK(U43(isNatIList(V2)))
ACTIVE(U52(tt, V2)) → MARK(U53(isNatList(V2)))
MARK(length(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
MARK(U43(X)) → ACTIVE(U43(mark(X)))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
MARK(U53(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U41(X1, X2, X3)) → MARK(X1)
ACTIVE(U11(tt, V1)) → MARK(U12(isNatList(V1)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U61(tt, L)) → MARK(s(length(L)))
MARK(U32(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(and(isNatKind(V1), isNatIListKind(V2)))
MARK(U12(X)) → ACTIVE(U12(mark(X)))
MARK(zeros) → ACTIVE(zeros)
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNat(V1), V2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
MARK(U22(X)) → ACTIVE(U22(mark(X)))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
ACTIVE(isNatIList(V)) → MARK(U31(isNatIListKind(V), V))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isNatList(V)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U32(X)) → ACTIVE(U32(mark(X)))
ACTIVE(isNatKind(length(V1))) → MARK(isNatIListKind(V1))
MARK(U53(X)) → ACTIVE(U53(mark(X)))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
Used ordering: Polynomial interpretation [25,35]:

POL(U52(x1, x2)) = 1   
POL(U41(x1, x2, x3)) = 1   
POL(mark(x1)) = 0   
POL(and(x1, x2)) = 1   
POL(U21(x1, x2)) = 1   
POL(isNatIListKind(x1)) = 1   
POL(active(x1)) = 3 + (1/4)x_1   
POL(tt) = 0   
POL(isNatList(x1)) = 1   
POL(zeros) = 1   
POL(isNatIList(x1)) = 1   
POL(U12(x1)) = 1   
POL(s(x1)) = 1/4   
POL(isNat(x1)) = 1   
POL(U51(x1, x2, x3)) = 1   
POL(nil) = 3/4   
POL(U31(x1, x2)) = 1   
POL(U42(x1, x2)) = 1   
POL(isNatKind(x1)) = 1   
POL(U11(x1, x2)) = 1   
POL(0) = 3/2   
POL(U43(x1)) = 1   
POL(ACTIVE(x1)) = (1/4)x_1   
POL(cons(x1, x2)) = 0   
POL(U22(x1)) = 1   
POL(MARK(x1)) = 1/4   
POL(U53(x1)) = 1   
POL(U61(x1, x2)) = 1   
POL(U32(x1)) = 1   
POL(length(x1)) = 1   
The value of delta used in the strict ordering is 3/16.
The following usable rules [17] were oriented:

U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
isNatIListKind(active(X)) → isNatIListKind(X)
isNatIListKind(mark(X)) → isNatIListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U22(X)) → ACTIVE(U22(mark(X)))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
ACTIVE(U42(tt, V2)) → MARK(U43(isNatIList(V2)))
ACTIVE(U52(tt, V2)) → MARK(U53(isNatList(V2)))
MARK(length(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
MARK(U43(X)) → ACTIVE(U43(mark(X)))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U53(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U41(X1, X2, X3)) → MARK(X1)
ACTIVE(U11(tt, V1)) → MARK(U12(isNatList(V1)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U32(X)) → MARK(X)
ACTIVE(U61(tt, L)) → MARK(s(length(L)))
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(and(isNatKind(V1), isNatIListKind(V2)))
MARK(U22(X)) → MARK(X)
MARK(U12(X)) → ACTIVE(U12(mark(X)))
MARK(zeros) → ACTIVE(zeros)
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNat(V1), V2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
ACTIVE(isNatIList(V)) → MARK(U31(isNatIListKind(V), V))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isNatList(V)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(U53(X)) → ACTIVE(U53(mark(X)))
ACTIVE(isNatKind(length(V1))) → MARK(isNatIListKind(V1))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
Used ordering: Polynomial interpretation [25,35]:

POL(U52(x1, x2)) = 1/4   
POL(U41(x1, x2, x3)) = 1/4   
POL(mark(x1)) = 0   
POL(and(x1, x2)) = 1/4   
POL(U21(x1, x2)) = 1/4   
POL(isNatIListKind(x1)) = 1/4   
POL(active(x1)) = 7/2   
POL(tt) = 0   
POL(isNatList(x1)) = 1/4   
POL(zeros) = 1/4   
POL(isNatIList(x1)) = 1/4   
POL(U12(x1)) = 1/4   
POL(s(x1)) = 0   
POL(isNat(x1)) = 1/4   
POL(U51(x1, x2, x3)) = 1/4   
POL(nil) = 15/4   
POL(U31(x1, x2)) = 1/4   
POL(U42(x1, x2)) = 1/4   
POL(isNatKind(x1)) = 1/4   
POL(U11(x1, x2)) = 1/4   
POL(0) = 4   
POL(U43(x1)) = 1/4   
POL(ACTIVE(x1)) = (2)x_1   
POL(cons(x1, x2)) = (4)x_2   
POL(U22(x1)) = 0   
POL(MARK(x1)) = 1/2   
POL(U53(x1)) = 1/4   
POL(U61(x1, x2)) = 1/4   
POL(U32(x1)) = 1/4   
POL(length(x1)) = 1/4   
The value of delta used in the strict ordering is 1/2.
The following usable rules [17] were oriented:

U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U22(mark(X)) → U22(X)
U22(active(X)) → U22(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
isNatIListKind(active(X)) → isNatIListKind(X)
isNatIListKind(mark(X)) → isNatIListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U43(X)) → ACTIVE(U43(mark(X)))
MARK(U12(X)) → ACTIVE(U12(mark(X)))
MARK(U32(X)) → ACTIVE(U32(mark(X)))
MARK(U53(X)) → ACTIVE(U53(mark(X)))
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
ACTIVE(U42(tt, V2)) → MARK(U43(isNatIList(V2)))
ACTIVE(U52(tt, V2)) → MARK(U53(isNatList(V2)))
MARK(length(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
MARK(U53(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U41(X1, X2, X3)) → MARK(X1)
ACTIVE(U11(tt, V1)) → MARK(U12(isNatList(V1)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U61(tt, L)) → MARK(s(length(L)))
MARK(U32(X)) → MARK(X)
MARK(U22(X)) → MARK(X)
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(and(isNatKind(V1), isNatIListKind(V2)))
MARK(zeros) → ACTIVE(zeros)
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNat(V1), V2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U42(X1, X2)) → MARK(X1)
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U31(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
ACTIVE(isNatIList(V)) → MARK(U31(isNatIListKind(V), V))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isNatList(V)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isNatKind(length(V1))) → MARK(isNatIListKind(V1))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
Used ordering: Polynomial interpretation [25,35]:

POL(U52(x1, x2)) = 1/2   
POL(U41(x1, x2, x3)) = 1/2   
POL(mark(x1)) = 0   
POL(and(x1, x2)) = 1/2   
POL(U21(x1, x2)) = 1/2   
POL(isNatIListKind(x1)) = 1/2   
POL(active(x1)) = 5/4   
POL(tt) = 0   
POL(isNatList(x1)) = 1/2   
POL(zeros) = 1/2   
POL(isNatIList(x1)) = 1/2   
POL(U12(x1)) = 0   
POL(s(x1)) = 4   
POL(isNat(x1)) = 1/2   
POL(U51(x1, x2, x3)) = 1/2   
POL(nil) = 7/4   
POL(U31(x1, x2)) = 1/2   
POL(U42(x1, x2)) = 1/2   
POL(isNatKind(x1)) = 1/2   
POL(U11(x1, x2)) = 1/2   
POL(0) = 7/2   
POL(U43(x1)) = 0   
POL(ACTIVE(x1)) = x_1   
POL(cons(x1, x2)) = (1/2)x_2   
POL(U22(x1)) = 15/4 + (3/2)x_1   
POL(MARK(x1)) = 1/2   
POL(U53(x1)) = 0   
POL(U61(x1, x2)) = 1/2   
POL(U32(x1)) = 0   
POL(length(x1)) = 1/2   
The value of delta used in the strict ordering is 1/2.
The following usable rules [17] were oriented:

U21(active(X1), X2) → U21(X1, X2)
U21(mark(X1), X2) → U21(X1, X2)
U21(X1, mark(X2)) → U21(X1, X2)
U21(X1, active(X2)) → U21(X1, X2)
isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
U31(active(X1), X2) → U31(X1, X2)
U31(X1, active(X2)) → U31(X1, X2)
U31(mark(X1), X2) → U31(X1, X2)
U31(X1, mark(X2)) → U31(X1, X2)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
U32(active(X)) → U32(X)
U32(mark(X)) → U32(X)
U11(mark(X1), X2) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U12(active(X)) → U12(X)
U12(mark(X)) → U12(X)
U52(mark(X1), X2) → U52(X1, X2)
U52(X1, active(X2)) → U52(X1, X2)
U52(X1, mark(X2)) → U52(X1, X2)
U52(active(X1), X2) → U52(X1, X2)
U53(active(X)) → U53(X)
U53(mark(X)) → U53(X)
U61(active(X1), X2) → U61(X1, X2)
U61(X1, mark(X2)) → U61(X1, X2)
U61(mark(X1), X2) → U61(X1, X2)
U61(X1, active(X2)) → U61(X1, X2)
U41(X1, X2, active(X3)) → U41(X1, X2, X3)
U41(X1, X2, mark(X3)) → U41(X1, X2, X3)
U41(X1, active(X2), X3) → U41(X1, X2, X3)
U41(X1, mark(X2), X3) → U41(X1, X2, X3)
U41(mark(X1), X2, X3) → U41(X1, X2, X3)
U41(active(X1), X2, X3) → U41(X1, X2, X3)
U42(X1, active(X2)) → U42(X1, X2)
U42(mark(X1), X2) → U42(X1, X2)
U42(X1, mark(X2)) → U42(X1, X2)
U42(active(X1), X2) → U42(X1, X2)
U43(active(X)) → U43(X)
U43(mark(X)) → U43(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
U51(active(X1), X2, X3) → U51(X1, X2, X3)
U51(mark(X1), X2, X3) → U51(X1, X2, X3)
U51(X1, mark(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, mark(X3)) → U51(X1, X2, X3)
U51(X1, active(X2), X3) → U51(X1, X2, X3)
U51(X1, X2, active(X3)) → U51(X1, X2, X3)
isNatIListKind(active(X)) → isNatIListKind(X)
isNatIListKind(mark(X)) → isNatIListKind(X)
and(X1, mark(X2)) → and(X1, X2)
and(X1, active(X2)) → and(X1, X2)
and(mark(X1), X2) → and(X1, X2)
and(active(X1), X2) → and(X1, X2)
length(active(X)) → length(X)
length(mark(X)) → length(X)
isNatKind(active(X)) → isNatKind(X)
isNatKind(mark(X)) → isNatKind(X)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(U41(X1, X2, X3)) → MARK(X1)
MARK(U32(X)) → MARK(X)
MARK(U42(X1, X2)) → MARK(X1)
MARK(U31(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
ACTIVE(U42(tt, V2)) → MARK(U43(isNatIList(V2)))
ACTIVE(U52(tt, V2)) → MARK(U53(isNatList(V2)))
MARK(length(X)) → MARK(X)
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U53(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
ACTIVE(U11(tt, V1)) → MARK(U12(isNatList(V1)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U61(tt, L)) → MARK(s(length(L)))
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(and(isNatKind(V1), isNatIListKind(V2)))
MARK(U22(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNat(V1), V2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
ACTIVE(isNatIList(V)) → MARK(U31(isNatIListKind(V), V))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U31(tt, V)) → MARK(U32(isNatList(V)))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isNatKind(length(V1))) → MARK(isNatIListKind(V1))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNat(V1), V2))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
Used ordering: Polynomial interpretation [25,35]:

POL(U52(x1, x2)) = x_1   
POL(U41(x1, x2, x3)) = 1/4 + (2)x_1   
POL(mark(x1)) = x_1   
POL(and(x1, x2)) = (4)x_1 + (4)x_2   
POL(U21(x1, x2)) = (4)x_1   
POL(isNatIListKind(x1)) = 0   
POL(active(x1)) = x_1   
POL(tt) = 0   
POL(isNatList(x1)) = 0   
POL(zeros) = 0   
POL(isNatIList(x1)) = 1/4   
POL(U12(x1)) = (2)x_1   
POL(s(x1)) = (4)x_1   
POL(isNat(x1)) = 0   
POL(U51(x1, x2, x3)) = (4)x_1   
POL(nil) = 0   
POL(U31(x1, x2)) = 1/4 + x_1   
POL(U42(x1, x2)) = 1/4 + (4)x_1   
POL(isNatKind(x1)) = 0   
POL(U11(x1, x2)) = (4)x_1   
POL(0) = 0   
POL(U43(x1)) = x_1   
POL(ACTIVE(x1)) = (1/4)x_1   
POL(cons(x1, x2)) = (4)x_1 + (4)x_2   
POL(U22(x1)) = x_1   
POL(MARK(x1)) = (1/4)x_1   
POL(U53(x1)) = x_1   
POL(U61(x1, x2)) = x_1 + (4)x_2   
POL(U32(x1)) = 1/4 + (4)x_1   
POL(length(x1)) = x_1   
The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


ACTIVE(isNatIList(V)) → MARK(U31(isNatIListKind(V), V))
The remaining pairs can at least be oriented weakly.

MARK(U52(X1, X2)) → MARK(X1)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U12(X)) → MARK(X)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
ACTIVE(U42(tt, V2)) → MARK(U43(isNatIList(V2)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isNatList(V2)))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
MARK(length(X)) → MARK(X)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
ACTIVE(U31(tt, V)) → MARK(U32(isNatList(V)))
MARK(and(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U53(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isNatKind(length(V1))) → MARK(isNatIListKind(V1))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ACTIVE(U11(tt, V1)) → MARK(U12(isNatList(V1)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U61(tt, L)) → MARK(s(length(L)))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNat(V1), V2))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
MARK(U61(X1, X2)) → MARK(X1)
MARK(U43(X)) → MARK(X)
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(and(isNatKind(V1), isNatIListKind(V2)))
MARK(U22(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U21(X1, X2)) → MARK(X1)
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNat(V1), V2))
ACTIVE(and(tt, X)) → MARK(X)
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
Used ordering: Polynomial interpretation [25,35]:

POL(U52(x1, x2)) = (2)x_1   
POL(U41(x1, x2, x3)) = 1/2 + (1/4)x_2 + (1/2)x_3   
POL(mark(x1)) = x_1   
POL(and(x1, x2)) = (4)x_1 + x_2   
POL(U21(x1, x2)) = (4)x_1   
POL(isNatIListKind(x1)) = 0   
POL(active(x1)) = x_1   
POL(tt) = 0   
POL(isNatList(x1)) = 0   
POL(zeros) = 0   
POL(isNatIList(x1)) = 1/2 + (1/4)x_1   
POL(U12(x1)) = x_1   
POL(s(x1)) = (2)x_1   
POL(isNat(x1)) = 0   
POL(U51(x1, x2, x3)) = (4)x_1   
POL(nil) = 1/4   
POL(U31(x1, x2)) = (1/4)x_2   
POL(U42(x1, x2)) = 1/2 + (1/2)x_2   
POL(U11(x1, x2)) = x_1   
POL(isNatKind(x1)) = 0   
POL(0) = 0   
POL(U43(x1)) = x_1   
POL(ACTIVE(x1)) = (1/4)x_1   
POL(cons(x1, x2)) = x_1 + (2)x_2   
POL(MARK(x1)) = (1/4)x_1   
POL(U22(x1)) = (7/4)x_1   
POL(U53(x1)) = (2)x_1   
POL(U61(x1, x2)) = (2)x_1 + (4)x_2   
POL(U32(x1)) = 0   
POL(length(x1)) = (2)x_1   
The value of delta used in the strict ordering is 1/8.
The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(length(X)) → MARK(X)
MARK(U61(X1, X2)) → MARK(X1)
The remaining pairs can at least be oriented weakly.

MARK(U12(X)) → MARK(X)
MARK(U41(X1, X2, X3)) → ACTIVE(U41(mark(X1), X2, X3))
MARK(U52(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(isNatIListKind(X)) → ACTIVE(isNatIListKind(X))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L))
ACTIVE(U42(tt, V2)) → MARK(U43(isNatIList(V2)))
MARK(U31(X1, X2)) → ACTIVE(U31(mark(X1), X2))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(U52(tt, V2)) → MARK(U53(isNatList(V2)))
ACTIVE(isNatList(cons(V1, V2))) → MARK(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
MARK(U61(X1, X2)) → ACTIVE(U61(mark(X1), X2))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2))
ACTIVE(isNat(length(V1))) → MARK(U11(isNatIListKind(V1), V1))
ACTIVE(U31(tt, V)) → MARK(U32(isNatList(V)))
MARK(and(X1, X2)) → MARK(X1)
MARK(U51(X1, X2, X3)) → ACTIVE(U51(mark(X1), X2, X3))
MARK(U51(X1, X2, X3)) → MARK(X1)
ACTIVE(isNatKind(s(V1))) → MARK(isNatKind(V1))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(U53(X)) → MARK(X)
ACTIVE(isNat(s(V1))) → MARK(U21(isNatKind(V1), V1))
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U52(X1, X2)) → ACTIVE(U52(mark(X1), X2))
ACTIVE(isNatKind(length(V1))) → MARK(isNatIListKind(V1))
ACTIVE(U21(tt, V1)) → MARK(U22(isNat(V1)))
ACTIVE(U11(tt, V1)) → MARK(U12(isNatList(V1)))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U61(tt, L)) → MARK(s(length(L)))
MARK(isNatKind(X)) → ACTIVE(isNatKind(X))
ACTIVE(U41(tt, V1, V2)) → MARK(U42(isNat(V1), V2))
MARK(U22(X)) → MARK(X)
ACTIVE(isNatIListKind(cons(V1, V2))) → MARK(and(isNatKind(V1), isNatIListKind(V2)))
MARK(U43(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(U21(X1, X2)) → MARK(X1)
ACTIVE(and(tt, X)) → MARK(X)
ACTIVE(U51(tt, V1, V2)) → MARK(U52(isNat(V1), V2))
MARK(U42(X1, X2)) → ACTIVE(U42(mark(X1), X2))
MARK(U21(X1, X2)) → ACTIVE(U21(mark(X1), X2))
Used ordering: Polynomial interpretation [25,35]:

POL(U52(x1, x2)) = x_1   
POL(U41(x1, x2, x3)) = 0   
POL(mark(x1)) = x_1   
POL(and(x1, x2)) = (4)x_1 + x_2   
POL(U21(x1, x2)) = (2)x_1   
POL(isNatIListKind(x1)) = 0   
POL(active(x1)) = x_1   
POL(tt) = 0   
POL(isNatList(x1)) = 0   
POL(zeros) = 0   
POL(isNatIList(x1)) = 0   
POL(U12(x1)) = (2)x_1   
POL(s(x1)) = x_1   
POL(isNat(x1)) = 0   
POL(U51(x1, x2, x3)) = (4)x_1   
POL(nil) = 0   
POL(U31(x1, x2)) = 0   
POL(U42(x1, x2)) = 0   
POL(U11(x1, x2)) = (4)x_1   
POL(isNatKind(x1)) = 0   
POL(0) = 0   
POL(U43(x1)) = (4)x_1   
POL(ACTIVE(x1)) = (1/2)x_1   
POL(cons(x1, x2)) = (2)x_1 + (3/2)x_2   
POL(MARK(x1)) = (1/2)x_1   
POL(U22(x1)) = (2)x_1   
POL(U53(x1)) = (2)x_1   
POL(U61(x1, x2)) = 5/4 + (2)x_1 + (4)x_2   
POL(U32(x1)) = (2)x_1   
POL(length(x1)) = 5/4 + (4)x_1   
The value of delta used in the strict ordering is 5/8.
The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP

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

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

The TRS R consists of the following rules:

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

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