Termination w.r.t. Q proof of /tmp/tmpaU1aCy/LengthOfFiniteLists_nokinds

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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.

↳ QTRS

  ↳ RRRPoloQTRSProof

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(nil)) → mark(tt)
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

active(isNatList(nil)) → mark(tt)
active(length(nil)) → mark(0)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + 2·x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

active(isNatIList(V)) → mark(isNatList(V))
active(isNatIList(zeros)) → mark(tt)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 2·x₁ + 2·x₂
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(length(V1))) → mark(isNatList(V1))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

active(isNat(length(V1))) → mark(isNatList(V1))

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

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

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(tt) → ACTIVE(tt)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNat(N))
U11¹(X1, mark(X2)) → U11¹(X1, X2)
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
U11¹(X1, active(X2)) → U11¹(X1, X2)
S(active(X)) → S(X)
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILIST(V2)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
AND(X1, active(X2)) → AND(X1, X2)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNat(V1), isNatIList(V2))
ACTIVE(U11(tt, L)) → S(length(L))
MARK(zeros) → ACTIVE(zeros)
ISNATILIST(mark(X)) → ISNATILIST(X)
MARK(length(X)) → LENGTH(mark(X))
AND(active(X1), X2) → AND(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
ISNATLIST(active(X)) → ISNATLIST(X)
ACTIVE(and(tt, X)) → MARK(X)
U11¹(mark(X1), X2) → U11¹(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ISNATLIST(mark(X)) → ISNATLIST(X)
MARK(U11(X1, X2)) → U11¹(mark(X1), X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNat(0)) → MARK(tt)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
U11¹(active(X1), X2) → U11¹(X1, X2)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
S(mark(X)) → S(X)
MARK(s(X)) → S(mark(X))
MARK(and(X1, X2)) → AND(mark(X1), X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)
ACTIVE(isNatList(cons(V1, V2))) → AND(isNat(V1), isNatList(V2))
ACTIVE(U11(tt, L)) → LENGTH(L)
ACTIVE(isNatList(cons(V1, V2))) → ISNATLIST(V2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
AND(mark(X1), X2) → AND(X1, X2)
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(length(cons(N, L))) → U11¹(and(isNatList(L), isNat(N)), L)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(0) → ACTIVE(0)
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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

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

MARK(tt) → ACTIVE(tt)
MARK(U11(X1, X2)) → MARK(X1)
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
AND(X1, mark(X2)) → AND(X1, X2)
ACTIVE(length(cons(N, L))) → AND(isNatList(L), isNat(N))
U11¹(X1, mark(X2)) → U11¹(X1, X2)
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
ACTIVE(zeros) → CONS(0, zeros)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(isNatIList(cons(V1, V2))) → ISNAT(V1)
MARK(isNatList(X)) → ACTIVE(isNatList(X))
U11¹(X1, active(X2)) → U11¹(X1, X2)
S(active(X)) → S(X)
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
ACTIVE(isNatIList(cons(V1, V2))) → ISNATILIST(V2)
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(isNatList(cons(V1, V2))) → ISNAT(V1)
AND(X1, active(X2)) → AND(X1, X2)
ACTIVE(isNatIList(cons(V1, V2))) → AND(isNat(V1), isNatIList(V2))
ACTIVE(U11(tt, L)) → S(length(L))
MARK(zeros) → ACTIVE(zeros)
ISNATILIST(mark(X)) → ISNATILIST(X)
MARK(length(X)) → LENGTH(mark(X))
AND(active(X1), X2) → AND(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
ISNATLIST(active(X)) → ISNATLIST(X)
ACTIVE(and(tt, X)) → MARK(X)
U11¹(mark(X1), X2) → U11¹(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ISNATLIST(mark(X)) → ISNATLIST(X)
MARK(U11(X1, X2)) → U11¹(mark(X1), X2)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNat(0)) → MARK(tt)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
LENGTH(mark(X)) → LENGTH(X)
LENGTH(active(X)) → LENGTH(X)
U11¹(active(X1), X2) → U11¹(X1, X2)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
S(mark(X)) → S(X)
MARK(s(X)) → S(mark(X))
MARK(and(X1, X2)) → AND(mark(X1), X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(length(cons(N, L))) → ISNATLIST(L)
ISNAT(active(X)) → ISNAT(X)
ISNAT(mark(X)) → ISNAT(X)
ACTIVE(isNatList(cons(V1, V2))) → AND(isNat(V1), isNatList(V2))
ACTIVE(U11(tt, L)) → LENGTH(L)
ACTIVE(isNatList(cons(V1, V2))) → ISNATLIST(V2)
ACTIVE(isNat(s(V1))) → ISNAT(V1)
AND(mark(X1), X2) → AND(X1, X2)
ACTIVE(length(cons(N, L))) → ISNAT(N)
ACTIVE(length(cons(N, L))) → U11¹(and(isNatList(L), isNat(N)), L)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(0) → ACTIVE(0)
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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 9 SCCs with 23 less nodes.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ 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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                      ↳ 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)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

ISNATILIST(mark(X)) → ISNATILIST(X)
The graph contains the following edges 1 > 1
ISNATILIST(active(X)) → ISNATILIST(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ 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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ 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)

From the DPs we obtained the following set of size-change graphs:

ISNATLIST(mark(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1
ISNATLIST(active(X)) → ISNATLIST(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ 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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                      ↳ 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)

From the DPs we obtained the following set of size-change graphs:

ISNAT(active(X)) → ISNAT(X)
The graph contains the following edges 1 > 1
ISNAT(mark(X)) → ISNAT(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ 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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                      ↳ 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)

From the DPs we obtained the following set of size-change graphs:

AND(mark(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(active(X1), X2) → AND(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
AND(X1, mark(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
AND(X1, active(X2)) → AND(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ 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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                      ↳ 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)

From the DPs we obtained the following set of size-change graphs:

LENGTH(mark(X)) → LENGTH(X)
The graph contains the following edges 1 > 1
LENGTH(active(X)) → LENGTH(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ 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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

From the DPs we obtained the following set of size-change graphs:

S(mark(X)) → S(X)
The graph contains the following edges 1 > 1
S(active(X)) → S(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ 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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

U11¹(X1, mark(X2)) → U11¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U11¹(X1, active(X2)) → U11¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U11¹(active(X1), X2) → U11¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U11¹(mark(X1), X2) → U11¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ 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, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                      ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

CONS(X1, active(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(active(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(X1, mark(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(U11(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
MARK(length(X)) → MARK(X)
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(zeros) → ACTIVE(zeros)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MARK(U11(X1, X2)) → MARK(X1)
MARK(length(X)) → MARK(X)

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(zeros) → MARK(cons(0, zeros))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(zeros) → ACTIVE(zeros)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(isNatIList(X)) → active(isNatIList(X))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(and(tt, X)) → mark(X)
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatList(X)) → active(isNatList(X))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(V1))) → mark(isNat(V1))
active(U11(tt, L)) → mark(s(length(L)))
mark(isNat(X)) → active(isNat(X))
mark(tt) → active(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNat(0)) → mark(tt)
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)
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)
mark(nil) → active(nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(zeros) → ACTIVE(zeros)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(cons(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 1 + x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(isNatIList(X)) → active(isNatIList(X))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(and(tt, X)) → mark(X)
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatList(X)) → active(isNatList(X))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(V1))) → mark(isNat(V1))
active(U11(tt, L)) → mark(s(length(L)))
mark(isNat(X)) → active(isNat(X))
mark(tt) → active(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNat(0)) → mark(tt)
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)
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)
mark(nil) → active(nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

/	0	0	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	1	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

Matrix type:
We used a basic matrix type which is not further parametrizeable.

As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(isNatIList(X)) → active(isNatIList(X))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(and(tt, X)) → mark(X)
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatList(X)) → active(isNatList(X))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(V1))) → mark(isNat(V1))
active(U11(tt, L)) → mark(s(length(L)))
mark(isNat(X)) → active(isNat(X))
mark(tt) → active(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNat(0)) → mark(tt)
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)
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)
mark(nil) → active(nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(isNatIList(X)) → ACTIVE(isNatIList(X))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(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)
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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(and(tt, X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	1	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( tt ) =

/	1	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	1	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(isNatIList(X)) → active(isNatIList(X))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(and(tt, X)) → mark(X)
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatList(X)) → active(isNatList(X))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(V1))) → mark(isNat(V1))
active(U11(tt, L)) → mark(s(length(L)))
mark(isNat(X)) → active(isNat(X))
mark(tt) → active(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
active(isNat(0)) → mark(tt)
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)
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)
mark(nil) → active(nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(isNatList(X)) → ACTIVE(isNatList(X))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(isNatList(X)) → ACTIVE(isNatList(X))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₁
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 0
POL(U11(x₁, x₂)) = 0
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 1 + x₁
POL(length(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
length(active(X)) → length(X)
length(mark(X)) → length(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)
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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(and(X1, X2)) → MARK(X1)
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(length(X)) → ACTIVE(length(mark(X)))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
MARK(length(X)) → ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(isNat(s(V1))) → MARK(isNat(V1))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	1	1	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(isNat(X)) → ACTIVE(isNat(X))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = x₁
POL(isNat(x₁)) = 1 + x₁
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 1

The following usable rules [17] were oriented:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                        ↳ QDPOrderProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(X)) → ACTIVE(length(mark(X)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2)) at position [0] we obtained the following new rules:

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                        ↳ QDPOrderProof

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(length(X)) → ACTIVE(length(mark(X))) at position [0] we obtained the following new rules:

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(x0)) → ACTIVE(length(x0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(length(tt)) → ACTIVE(length(active(tt))) at position [0] we obtained the following new rules:

MARK(length(tt)) → ACTIVE(length(tt))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(tt)) → ACTIVE(length(tt))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(length(0)) → ACTIVE(length(active(0))) at position [0] we obtained the following new rules:

MARK(length(0)) → ACTIVE(length(0))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                        ↳ QDPOrderProof

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

MARK(length(0)) → ACTIVE(length(0))
MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(length(nil)) → ACTIVE(length(active(nil))) at position [0] we obtained the following new rules:

MARK(length(nil)) → ACTIVE(length(nil))

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(nil)) → ACTIVE(length(nil))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0

The following usable rules [17] were oriented:

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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 1
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(0, y1)) → ACTIVE(U11(active(0), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 1
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 1
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 0
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 1
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))

The remaining pairs can at least be oriented weakly.

ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	1	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	1	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(isNatIList(X)) → active(isNatIList(X))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(and(tt, X)) → mark(X)
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatList(X)) → active(isNatList(X))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(V1))) → mark(isNat(V1))
active(U11(tt, L)) → mark(s(length(L)))
mark(isNat(X)) → active(isNat(X))
mark(tt) → active(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
active(isNat(0)) → 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)
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)
mark(nil) → active(nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(isNat(x0))) → ACTIVE(length(active(isNat(x0))))

The remaining pairs can at least be oriented weakly.

ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	0	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(isNatIList(X)) → active(isNatIList(X))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(and(tt, X)) → mark(X)
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatList(X)) → active(isNatList(X))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(V1))) → mark(isNat(V1))
active(U11(tt, L)) → mark(s(length(L)))
mark(isNat(X)) → active(isNat(X))
mark(tt) → active(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
active(isNat(0)) → 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)
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)
mark(nil) → active(nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))

The remaining pairs can at least be oriented weakly.

ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	1	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	1	/

/	1	1	\
\	1	0	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(isNatIList(X)) → active(isNatIList(X))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(and(tt, X)) → mark(X)
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatList(X)) → active(isNatList(X))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(V1))) → mark(isNat(V1))
active(U11(tt, L)) → mark(s(length(L)))
mark(isNat(X)) → active(isNat(X))
mark(tt) → active(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
active(isNat(0)) → 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)
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)
mark(nil) → active(nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))

The remaining pairs can at least be oriented weakly.

MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( and(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( U11(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( active(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

/	0	1	\
\	1	1	/

x₂

M( tt ) =

/	0	\
\	0	/

M( isNatList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( zeros ) =

/	0	\
\	0	/

M( isNatIList(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( isNat(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( length(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	1	1	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
mark(0) → active(0)
mark(isNatIList(X)) → active(isNatIList(X))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
active(and(tt, X)) → mark(X)
active(zeros) → mark(cons(0, zeros))
mark(length(X)) → active(length(mark(X)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNatList(X)) → active(isNatList(X))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(N)), L))
mark(s(X)) → active(s(mark(X)))
active(isNat(s(V1))) → mark(isNat(V1))
active(U11(tt, L)) → mark(s(length(L)))
mark(isNat(X)) → active(isNat(X))
mark(tt) → active(tt)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
active(isNat(0)) → 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)
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)
mark(nil) → active(nil)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ QDPOrderProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ QDPOrderProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ QDPOrderProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ QDPOrderProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ QDPOrderProof

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ QDPOrderProof

                                                                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                        ↳ QDPOrderProof

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

MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(U11(isNatList(x0), y1)) → ACTIVE(U11(active(isNatList(x0)), y1))

The remaining pairs can at least be oriented weakly.

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = 0
POL(MARK(x₁)) = x₁
POL(U11(x₁, x₂)) = x₁
POL(active(x₁)) = x₁
POL(and(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = x₁
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = x₁
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ QDPOrderProof

                                                                                                              ↳ QDP

                                                                                                                ↳ QDPOrderProof

                                                                                                                ↳ QDPOrderProof

                                                                                                                  ↳ QDP

                                        ↳ QDPOrderProof

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

MARK(length(isNatIList(x0))) → ACTIVE(length(active(isNatIList(x0))))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U11(isNat(x0), y1)) → ACTIVE(U11(active(isNat(x0)), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
MARK(length(and(x0, x1))) → ACTIVE(length(active(and(mark(x0), x1))))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(isNat(x0))) → ACTIVE(length(active(isNat(x0))))
MARK(U11(isNatIList(x0), y1)) → ACTIVE(U11(active(isNatIList(x0)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
MARK(length(isNatList(x0))) → ACTIVE(length(active(isNatList(x0))))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(and(x0, x1), y1)) → ACTIVE(U11(active(and(mark(x0), x1)), y1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(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(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(U11(x₁, x₂)) = 1
POL(active(x₁)) = 0
POL(and(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 1
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 1
POL(length(x₁)) = 1
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

isNatList(active(X)) → isNatList(X)
isNatList(mark(X)) → isNatList(X)
isNat(active(X)) → isNat(X)
isNat(mark(X)) → isNat(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(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)
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)

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ DependencyGraphProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                        ↳ QDPOrderProof

                                          ↳ QDP

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(and(isNatList(L), isNat(N)), L))
MARK(isNat(X)) → ACTIVE(isNat(X))
MARK(s(X)) → MARK(X)
MARK(isNatIList(X)) → ACTIVE(isNatIList(X))
ACTIVE(U11(tt, L)) → MARK(s(length(L)))
ACTIVE(isNatIList(cons(V1, V2))) → MARK(and(isNat(V1), isNatIList(V2)))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(and(X1, X2)) → ACTIVE(and(mark(X1), X2))
ACTIVE(isNatList(cons(V1, V2))) → MARK(and(isNat(V1), isNatList(V2)))
MARK(and(X1, X2)) → MARK(X1)
ACTIVE(isNat(s(V1))) → MARK(isNat(V1))
ACTIVE(and(tt, X)) → MARK(X)
MARK(isNatList(X)) → ACTIVE(isNatList(X))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(s(length(L)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(s(V1))) → mark(isNat(V1))
active(isNatIList(cons(V1, V2))) → mark(and(isNat(V1), isNatIList(V2)))
active(isNatList(cons(V1, V2))) → mark(and(isNat(V1), isNatList(V2)))
active(length(cons(N, L))) → mark(U11(and(isNatList(L), isNat(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(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(and(X1, X2)) → active(and(mark(X1), X2))
mark(isNat(X)) → active(isNat(X))
mark(isNatList(X)) → active(isNatList(X))
mark(isNatIList(X)) → active(isNatIList(X))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(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)
isNat(mark(X)) → isNat(X)
isNat(active(X)) → isNat(X)
isNatList(mark(X)) → isNatList(X)
isNatList(active(X)) → isNatList(X)
isNatIList(mark(X)) → isNatIList(X)
isNatIList(active(X)) → isNatIList(X)

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