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:



↳ 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)

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
                      ↳ 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)

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:



↳ 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)

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
                      ↳ 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)

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:



↳ 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)

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
                      ↳ 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)

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:



↳ 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)

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
                      ↳ 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)

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:



↳ 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)

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
                      ↳ 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)

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:



↳ 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)

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
                      ↳ 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)

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:



↳ 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)

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
                      ↳ 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)

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:



↳ 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/
+
/00\
\11/
·x1

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\11/
·x1+
/00\
\11/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( active(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

M( cons(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/00\
\01/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/1\
\0/

M( isNatIList(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( s(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( length(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[0,1]
·x1

M( ACTIVE(x1) ) = 0+
[0,1]
·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/
+
/11\
\00/
·x1

M( and(x1, x2) ) =
/0\
\0/
+
/11\
\00/
·x1+
/11\
\00/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/1\
\0/

M( active(x1) ) =
/0\
\0/
+
/11\
\00/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/11\
\00/
·x1+
/10\
\00/
·x2

M( tt ) =
/1\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( zeros ) =
/0\
\1/

M( isNatIList(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( s(x1) ) =
/0\
\0/
+
/11\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\10/
·x1

M( length(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[1,0]
·x1

M( ACTIVE(x1) ) = 0+
[1,0]
·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
                                                  ↳ 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/
+
/10\
\01/
·x1

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( active(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/10\
\00/
·x1+
/00\
\00/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/0\
\0/
+
/00\
\10/
·x1

M( s(x1) ) =
/1\
\0/
+
/11\
\01/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

M( length(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[0,1]
·x1

M( ACTIVE(x1) ) = 0+
[0,1]
·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:

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/
+
/11\
\01/
·x1

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/01\
\00/
·x1+
/00\
\00/
·x2

M( 0 ) =
/0\
\0/

M( active(x1) ) =
/0\
\0/
+
/11\
\11/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/11\
\00/
·x1

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\1/
+
/10\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( length(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[1,0]
·x1

M( ACTIVE(x1) ) = 0+
[0,0]
·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)
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/
+
/10\
\01/
·x1

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\11/
·x2

M( 0 ) =
/0\
\0/

M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/1\
\1/
+
/00\
\10/
·x1

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/1\
\1/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( length(x1) ) =
/0\
\0/
+
/11\
\10/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[1,0]
·x1

M( ACTIVE(x1) ) = 0+
[0,1]
·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)
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/
+
/10\
\01/
·x1

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\11/
·x2

M( 0 ) =
/0\
\0/

M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\11/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( isNat(x1) ) =
/1\
\0/
+
/00\
\00/
·x1

M( length(x1) ) =
/0\
\0/
+
/11\
\01/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[1,0]
·x1

M( ACTIVE(x1) ) = 0+
[0,1]
·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)
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/
+
/10\
\11/
·x1

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( U11(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( 0 ) =
/0\
\0/

M( active(x1) ) =
/0\
\1/
+
/11\
\10/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/11\
\00/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( length(x1) ) =
/1\
\0/
+
/10\
\00/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[1,0]
·x1

M( ACTIVE(x1) ) = 0+
[1,0]
·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)
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/
+
/10\
\01/
·x1

M( and(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\01/
·x2

M( U11(x1, x2) ) =
/1\
\1/
+
/00\
\00/
·x1+
/11\
\11/
·x2

M( 0 ) =
/0\
\0/

M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/01\
\01/
·x1+
/01\
\11/
·x2

M( tt ) =
/0\
\0/

M( isNatList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( zeros ) =
/0\
\0/

M( isNatIList(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/1\
\0/
+
/00\
\01/
·x1

M( isNat(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( length(x1) ) =
/1\
\1/
+
/01\
\11/
·x1

M( nil ) =
/0\
\0/

Tuple symbols:
M( MARK(x1) ) = 0+
[0,1]
·x1

M( ACTIVE(x1) ) = 0+
[1,0]
·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)
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.