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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(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(length(nil)) → mark(0)
Used ordering:
Polynomial interpretation [25]:

POL(0) = 0   
POL(U11(x1, x2)) = 2·x1 + 2·x2   
POL(U12(x1, x2)) = 2·x1 + 2·x2   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + 2·x2   
POL(length(x1)) = 2·x1   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 0   




↳ 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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(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:

CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(U11(tt, L)) → U121(tt, L)
MARK(tt) → ACTIVE(tt)
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
U121(X1, active(X2)) → U121(X1, X2)
MARK(U11(X1, X2)) → U111(mark(X1), X2)
ACTIVE(U12(tt, L)) → S(length(L))
U111(X1, mark(X2)) → U111(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(length(X)) → MARK(X)
MARK(U12(X1, X2)) → U121(mark(X1), X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(zeros) → CONS(0, zeros)
LENGTH(mark(X)) → LENGTH(X)
U121(active(X1), X2) → U121(X1, X2)
LENGTH(active(X)) → LENGTH(X)
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
U111(X1, active(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
S(active(X)) → S(X)
S(mark(X)) → S(X)
U121(X1, mark(X2)) → U121(X1, X2)
MARK(s(X)) → S(mark(X))
MARK(U12(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(length(cons(N, L))) → U111(tt, L)
MARK(length(X)) → LENGTH(mark(X))
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(0) → ACTIVE(0)
U121(mark(X1), X2) → U121(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
MARK(nil) → ACTIVE(nil)
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → LENGTH(L)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
QDP
          ↳ DependencyGraphProof

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

CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(U11(tt, L)) → U121(tt, L)
MARK(tt) → ACTIVE(tt)
MARK(U11(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
U121(X1, active(X2)) → U121(X1, X2)
MARK(U11(X1, X2)) → U111(mark(X1), X2)
ACTIVE(U12(tt, L)) → S(length(L))
U111(X1, mark(X2)) → U111(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(length(X)) → MARK(X)
MARK(U12(X1, X2)) → U121(mark(X1), X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(zeros) → CONS(0, zeros)
LENGTH(mark(X)) → LENGTH(X)
U121(active(X1), X2) → U121(X1, X2)
LENGTH(active(X)) → LENGTH(X)
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
U111(X1, active(X2)) → U111(X1, X2)
U111(active(X1), X2) → U111(X1, X2)
MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
S(active(X)) → S(X)
S(mark(X)) → S(X)
U121(X1, mark(X2)) → U121(X1, X2)
MARK(s(X)) → S(mark(X))
MARK(U12(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(length(cons(N, L))) → U111(tt, L)
MARK(length(X)) → LENGTH(mark(X))
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(0) → ACTIVE(0)
U121(mark(X1), X2) → U121(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
U111(mark(X1), X2) → U111(X1, X2)
MARK(nil) → ACTIVE(nil)
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → LENGTH(L)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 6 SCCs with 13 less nodes.

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

S(mark(X)) → S(X)
S(active(X)) → S(X)

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ 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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
QDP
                ↳ UsableRulesProof
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ UsableRulesProof
QDP
                    ↳ QDPSizeChangeProof
              ↳ QDP
              ↳ QDP
              ↳ QDP

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

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

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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ 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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ 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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ 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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ 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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ 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))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(length(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(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)) = x1 + x2   
POL(U12(x1, x2)) = x1 + x2   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + x2   
POL(length(x1)) = 1 + x1   
POL(mark(x1)) = x1   
POL(nil) = 0   
POL(s(x1)) = x1   
POL(tt) = 1   
POL(zeros) = 0   



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
QDP
                    ↳ RuleRemovalProof

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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
MARK(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(zeros) → ACTIVE(zeros)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(zeros) → MARK(cons(0, zeros))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(U12(X1, X2)) → MARK(X1)
MARK(U11(X1, X2)) → MARK(X1)


Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 2·x1   
POL(MARK(x1)) = 2·x1   
POL(U11(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(U12(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 2·x1 + x2   
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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ 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(tt, L))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(zeros) → ACTIVE(zeros)
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


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

MARK(U11(X1, X2)) → ACTIVE(U11(mark(X1), X2))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(zeros) → ACTIVE(zeros)
MARK(s(X)) → MARK(X)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
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(U12(x1, x2)) = 1   
POL(active(x1)) = 0   
POL(cons(x1, x2)) = 0   
POL(length(x1)) = 1   
POL(mark(x1)) = 0   
POL(nil) = 0   
POL(s(x1)) = 0   
POL(tt) = 0   
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)
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)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
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(tt, L))
MARK(s(X)) → MARK(X)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(zeros) → ACTIVE(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(tt, L))
MARK(s(X)) → MARK(X)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
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(U12(x1, x2)) = 0   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = x1   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(s(x1)) = x1   
POL(tt) = 0   
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)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ 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(tt, L))
MARK(s(X)) → MARK(X)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ 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(tt, L))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = 0   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 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:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
QDP
                                        ↳ Narrowing

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(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
MARK(length(X)) → ACTIVE(length(mark(X)))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
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))



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
QDP
                                            ↳ Narrowing

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(tt, L))
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))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2))
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, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(U12(X1, X2)) → ACTIVE(U12(mark(X1), X2)) at position [0] we obtained the following new rules:

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
QDP
                                                ↳ Narrowing

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(U11(U11(x0, x1), y1)) → ACTIVE(U11(active(U11(mark(x0), x1)), y1))
MARK(s(X)) → MARK(X)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(X)) → ACTIVE(length(mark(X)))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
MARK(length(0)) → ACTIVE(length(active(0)))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(length(x0)) → ACTIVE(length(x0))



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
QDP
                                                    ↳ Narrowing

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

MARK(length(tt)) → ACTIVE(length(active(tt)))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, 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(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
QDP
                                                        ↳ DependencyGraphProof

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, 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(length(tt)) → ACTIVE(length(tt))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
QDP
                                                            ↳ Narrowing

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, 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(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
QDP
                                                                ↳ DependencyGraphProof

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

MARK(length(0)) → ACTIVE(length(0))
MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
QDP
                                                                    ↳ Narrowing

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(length(nil)) → ACTIVE(length(active(nil)))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
QDP
                                                                        ↳ DependencyGraphProof

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(length(nil)) → ACTIVE(length(nil))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ DependencyGraphProof
QDP
                                                                            ↳ QDPOrderProof

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U11(cons(x0, x1), y1)) → ACTIVE(U11(active(cons(mark(x0), x1)), y1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = x1   
POL(U12(x1, x2)) = 0   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 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:

U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ DependencyGraphProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
QDP
                                                                                ↳ QDPOrderProof

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(0, y1)) → ACTIVE(U12(active(0), y1))
The remaining pairs can at least be oriented weakly.

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = x1   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 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:

U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ DependencyGraphProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
QDP
                                                                                    ↳ QDPOrderProof

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(zeros, y1)) → ACTIVE(U12(active(zeros), y1))
The remaining pairs can at least be oriented weakly.

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 0   
POL(U12(x1, x2)) = x1   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 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:

U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ DependencyGraphProof
                                                                          ↳ QDP
                                                                            ↳ QDPOrderProof
                                                                              ↳ QDP
                                                                                ↳ QDPOrderProof
                                                                                  ↳ QDP
                                                                                    ↳ QDPOrderProof
QDP
                                                                                        ↳ QDPOrderProof

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
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(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(U11(nil, y1)) → ACTIVE(U11(active(nil), y1))
MARK(U11(0, y1)) → ACTIVE(U11(active(0), y1))
MARK(U11(zeros, y1)) → ACTIVE(U11(active(zeros), y1))
The remaining pairs can at least be oriented weakly.

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(length(x0)) → ACTIVE(length(x0))
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(ACTIVE(x1)) = 0   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = x1   
POL(U12(x1, x2)) = 0   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = x1 + x2   
POL(length(x1)) = 0   
POL(mark(x1)) = x1   
POL(nil) = 1   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(zeros) = 1   

The following usable rules [17] were oriented:

U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(nil, y1)) → ACTIVE(U12(active(nil), y1))
The remaining pairs can at least be oriented weakly.

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
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(U12(x1, x2)) = x1   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 0   
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:

U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
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(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(cons(x0, x1), y1)) → ACTIVE(U12(active(cons(mark(x0), x1)), y1))
The remaining pairs can at least be oriented weakly.

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
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)) = 0   
POL(U12(x1, x2)) = x1   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 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:

U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U11(length(x0), y1)) → ACTIVE(U11(active(length(mark(x0))), y1))
MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(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)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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))
MARK(U11(U12(x0, x1), y1)) → ACTIVE(U11(active(U12(mark(x0), x1)), y1))
The remaining pairs can at least be oriented weakly.

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 1 + x1   
POL(U12(x1, x2)) = 1   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = x1   
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:

U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(U11(x0, x1), y1)) → ACTIVE(U12(active(U11(mark(x0), x1)), y1))
MARK(U12(length(x0), y1)) → ACTIVE(U12(active(length(mark(x0))), y1))
MARK(U12(U12(x0, x1), y1)) → ACTIVE(U12(active(U12(mark(x0), x1)), y1))
The remaining pairs can at least be oriented weakly.

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0   
POL(ACTIVE(x1)) = 1   
POL(MARK(x1)) = x1   
POL(U11(x1, x2)) = 1   
POL(U12(x1, x2)) = 1 + x1   
POL(active(x1)) = x1   
POL(cons(x1, x2)) = 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:

U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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)
length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(s(x0), y1)) → ACTIVE(U12(active(s(mark(x0))), y1))
The remaining pairs can at least be oriented weakly.

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

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

M( tt ) =
/0\
\0/

M( zeros ) =
/0\
\0/

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

M( s(x1) ) =
/1\
\0/
+
/00\
\01/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\10/
·x1+
/00\
\00/
·x2

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

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

M( 0 ) =
/0\
\0/

M( nil ) =
/0\
\0/

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

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U11(s(x0), y1)) → ACTIVE(U11(active(s(mark(x0))), y1))
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

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

M( tt ) =
/0\
\0/

M( zeros ) =
/0\
\0/

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

M( s(x1) ) =
/0\
\1/
+
/10\
\00/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

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

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

M( 0 ) =
/0\
\0/

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:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(U11(y0, active(x1))) → ACTIVE(U11(mark(y0), x1))
MARK(U11(y0, mark(x1))) → ACTIVE(U11(mark(y0), x1))
The remaining pairs can at least be oriented weakly.

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( active(x1) ) =
/0\
\1/
+
/10\
\01/
·x1

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

M( tt ) =
/0\
\1/

M( zeros ) =
/0\
\0/

M( mark(x1) ) =
/0\
\1/
+
/10\
\01/
·x1

M( s(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/10\
\00/
·x2

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

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

M( 0 ) =
/0\
\0/

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:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
mark(length(X)) → active(length(mark(X)))
active(zeros) → mark(cons(0, zeros))
mark(zeros) → active(zeros)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
active(U12(tt, L)) → mark(s(length(L)))
active(U11(tt, L)) → mark(U12(tt, L))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(0) → active(0)
mark(tt) → active(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)
mark(nil) → active(nil)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(U12(y0, mark(x1))) → ACTIVE(U12(mark(y0), x1))
MARK(U12(y0, active(x1))) → ACTIVE(U12(mark(y0), x1))
The remaining pairs can at least be oriented weakly.

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( active(x1) ) =
/1\
\0/
+
/10\
\01/
·x1

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/01\
\11/
·x2

M( tt ) =
/0\
\0/

M( zeros ) =
/0\
\0/

M( mark(x1) ) =
/1\
\0/
+
/10\
\01/
·x1

M( s(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( U12(x1, x2) ) =
/0\
\1/
+
/00\
\00/
·x1+
/01\
\11/
·x2

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

M( length(x1) ) =
/0\
\1/
+
/01\
\01/
·x1

M( 0 ) =
/1\
\0/

M( nil ) =
/0\
\0/

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

M( ACTIVE(x1) ) = 1+
[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:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
mark(length(X)) → active(length(mark(X)))
active(zeros) → mark(cons(0, zeros))
mark(zeros) → active(zeros)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
active(U12(tt, L)) → mark(s(length(L)))
active(U11(tt, L)) → mark(U12(tt, L))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(0) → active(0)
mark(tt) → active(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)
mark(nil) → active(nil)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(length(length(x0))) → ACTIVE(length(active(length(mark(x0)))))
The remaining pairs can at least be oriented weakly.

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

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

M( tt ) =
/1\
\1/

M( zeros ) =
/0\
\0/

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

M( s(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/01\
\01/
·x1+
/11\
\11/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/10\
\10/
·x1+
/11\
\11/
·x2

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

M( 0 ) =
/0\
\0/

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:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
mark(length(X)) → active(length(mark(X)))
active(zeros) → mark(cons(0, zeros))
mark(zeros) → active(zeros)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
active(U12(tt, L)) → mark(s(length(L)))
active(U11(tt, L)) → mark(U12(tt, L))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(0) → active(0)
mark(tt) → active(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)
mark(nil) → active(nil)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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


The following pairs can be oriented strictly and are deleted.


MARK(length(U12(x0, x1))) → ACTIVE(length(active(U12(mark(x0), x1))))
MARK(length(U11(x0, x1))) → ACTIVE(length(active(U11(mark(x0), x1))))
The remaining pairs can at least be oriented weakly.

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
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(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
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( zeros ) =
/0\
\0/

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

M( s(x1) ) =
/0\
\0/
+
/10\
\00/
·x1

M( U12(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/11\
\11/
·x2

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

M( length(x1) ) =
/1\
\0/
+
/11\
\01/
·x1

M( 0 ) =
/0\
\0/

M( nil ) =
/0\
\0/

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

M( ACTIVE(x1) ) = 1+
[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:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
mark(length(X)) → active(length(mark(X)))
active(zeros) → mark(cons(0, zeros))
mark(zeros) → active(zeros)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
active(U12(tt, L)) → mark(s(length(L)))
active(U11(tt, L)) → mark(U12(tt, L))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(0) → active(0)
mark(tt) → active(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)
mark(nil) → active(nil)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
QDP
                                                                                                                                    ↳ QDPOrderProof

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(length(s(x0))) → ACTIVE(length(active(s(mark(x0)))))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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)))))
The remaining pairs can at least be oriented weakly.

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
MARK(s(X)) → MARK(X)
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( active(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

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

M( tt ) =
/1\
\1/

M( zeros ) =
/1\
\0/

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

M( s(x1) ) =
/0\
\1/
+
/10\
\00/
·x1

M( U12(x1, x2) ) =
/0\
\0/
+
/10\
\10/
·x1+
/11\
\11/
·x2

M( U11(x1, x2) ) =
/0\
\0/
+
/01\
\01/
·x1+
/11\
\11/
·x2

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

M( 0 ) =
/0\
\0/

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:

length(active(X)) → length(X)
length(mark(X)) → length(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
mark(length(X)) → active(length(mark(X)))
active(zeros) → mark(cons(0, zeros))
mark(zeros) → active(zeros)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
active(U12(tt, L)) → mark(s(length(L)))
active(U11(tt, L)) → mark(U12(tt, L))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(0) → active(0)
mark(tt) → active(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)
mark(nil) → active(nil)
U12(mark(X1), X2) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
U12(X1, mark(X2)) → U12(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
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
QDP
                                                                                                                                        ↳ SemLabProof
                                                                                                                                        ↳ SemLabProof2

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(cons(x0, x1))) → ACTIVE(length(active(cons(mark(x0), x1))))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
MARK(s(X)) → MARK(X)
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
MARK(length(x0)) → ACTIVE(length(x0))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.mark: 0
U12: 0
U11: 0
0: 0
ACTIVE: 0
active: 0
cons: 0
MARK: 0
tt: 0
zeros: 1
s: 0
length: 0
nil: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

MARK.0(length.0(cons.1-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.1(x0), x1))))
MARK.0(length.1(x0)) → ACTIVE.0(length.1(x0))
ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
MARK.0(length.0(x0)) → ACTIVE.0(length.0(x0))
ACTIVE.0(length.0(cons.1-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(length.0(cons.0-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.0(x0), x1))))
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
MARK.0(U11.0-0(tt., y1)) → ACTIVE.0(U11.0-0(active.0(tt.), y1))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))
MARK.0(length.0(cons.0-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.0(x0), x1))))
MARK.0(U11.1-0(x0, x1)) → ACTIVE.0(U11.1-0(x0, x1))
ACTIVE.0(U11.0-0(tt., L)) → MARK.0(U12.0-0(tt., L))
ACTIVE.0(length.0(cons.0-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(U12.0-0(x0, x1)) → ACTIVE.0(U12.0-0(x0, x1))
ACTIVE.0(U12.0-0(tt., L)) → MARK.0(s.0(length.0(L)))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
MARK.0(length.0(cons.1-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.1(x0), x1))))
MARK.0(s.1(X)) → MARK.1(X)
MARK.0(U12.1-0(x0, x1)) → ACTIVE.0(U12.1-0(x0, x1))
MARK.0(U12.0-0(tt., y1)) → ACTIVE.0(U12.0-0(active.0(tt.), y1))
MARK.0(U11.0-0(x0, x1)) → ACTIVE.0(U11.0-0(x0, x1))
ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(U12.1-1(x0, x1)) → ACTIVE.0(U12.1-1(x0, x1))
MARK.0(U11.1-1(x0, x1)) → ACTIVE.0(U11.1-1(x0, x1))
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))

The TRS R consists of the following rules:

active.0(length.0(cons.0-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.1-0(X1, active.1(X2)) → U12.1-1(X1, X2)
U12.0-0(mark.1(X1), X2) → U12.1-0(X1, X2)
U11.0-0(X1, active.1(X2)) → U11.0-1(X1, X2)
cons.1-0(X1, active.0(X2)) → cons.1-0(X1, X2)
mark.0(cons.1-1(X1, X2)) → active.0(cons.0-1(mark.1(X1), X2))
mark.1(zeros.) → active.1(zeros.)
cons.0-0(X1, active.1(X2)) → cons.0-1(X1, X2)
cons.0-1(active.0(X1), X2) → cons.0-1(X1, X2)
active.0(length.0(cons.1-0(N, L))) → mark.0(U11.0-0(tt., L))
U11.0-0(mark.1(X1), X2) → U11.1-0(X1, X2)
U11.0-0(X1, active.0(X2)) → U11.0-0(X1, X2)
U11.0-1(mark.0(X1), X2) → U11.0-1(X1, X2)
cons.0-0(X1, mark.1(X2)) → cons.0-1(X1, X2)
U11.0-1(mark.1(X1), X2) → U11.1-1(X1, X2)
U11.0-1(active.0(X1), X2) → U11.0-1(X1, X2)
cons.1-0(X1, active.1(X2)) → cons.1-1(X1, X2)
cons.0-0(active.0(X1), X2) → cons.0-0(X1, X2)
mark.0(cons.0-1(X1, X2)) → active.0(cons.0-1(mark.0(X1), X2))
mark.0(U12.1-1(X1, X2)) → active.0(U12.0-1(mark.1(X1), X2))
mark.0(s.0(X)) → active.0(s.0(mark.0(X)))
active.0(U11.0-1(tt., L)) → mark.0(U12.0-1(tt., L))
U11.0-0(active.1(X1), X2) → U11.1-0(X1, X2)
active.0(U11.0-0(tt., L)) → mark.0(U12.0-0(tt., L))
cons.0-1(mark.0(X1), X2) → cons.0-1(X1, X2)
mark.0(0.) → active.0(0.)
cons.0-0(X1, mark.0(X2)) → cons.0-0(X1, X2)
U12.1-0(X1, mark.0(X2)) → U12.1-0(X1, X2)
mark.0(length.0(X)) → active.0(length.0(mark.0(X)))
length.0(mark.1(X)) → length.1(X)
cons.0-0(active.1(X1), X2) → cons.1-0(X1, X2)
U11.0-1(active.1(X1), X2) → U11.1-1(X1, X2)
mark.0(U12.0-1(X1, X2)) → active.0(U12.0-1(mark.0(X1), X2))
length.0(active.0(X)) → length.0(X)
U12.0-1(mark.0(X1), X2) → U12.0-1(X1, X2)
length.0(active.1(X)) → length.1(X)
active.0(U12.0-1(tt., L)) → mark.0(s.0(length.1(L)))
U11.1-0(X1, active.0(X2)) → U11.1-0(X1, X2)
cons.1-0(X1, mark.0(X2)) → cons.1-0(X1, X2)
U12.0-0(X1, active.0(X2)) → U12.0-0(X1, X2)
cons.0-0(X1, active.0(X2)) → cons.0-0(X1, X2)
mark.0(s.1(X)) → active.0(s.0(mark.1(X)))
mark.0(U11.0-1(X1, X2)) → active.0(U11.0-1(mark.0(X1), X2))
mark.0(cons.0-0(X1, X2)) → active.0(cons.0-0(mark.0(X1), X2))
cons.1-0(X1, mark.1(X2)) → cons.1-1(X1, X2)
mark.0(U12.1-0(X1, X2)) → active.0(U12.0-0(mark.1(X1), X2))
U11.0-0(active.0(X1), X2) → U11.0-0(X1, X2)
active.0(length.0(cons.1-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.0(X1), X2) → U12.0-0(X1, X2)
U12.1-0(X1, active.0(X2)) → U12.1-0(X1, X2)
mark.0(tt.) → active.0(tt.)
cons.0-1(mark.1(X1), X2) → cons.1-1(X1, X2)
U12.0-0(X1, mark.1(X2)) → U12.0-1(X1, X2)
mark.0(nil.) → active.0(nil.)
s.0(active.0(X)) → s.0(X)
length.0(mark.0(X)) → length.0(X)
U11.0-0(X1, mark.1(X2)) → U11.0-1(X1, X2)
cons.0-1(active.1(X1), X2) → cons.1-1(X1, X2)
mark.0(U11.0-0(X1, X2)) → active.0(U11.0-0(mark.0(X1), X2))
active.0(length.0(cons.0-0(N, L))) → mark.0(U11.0-0(tt., L))
s.0(mark.1(X)) → s.1(X)
U11.1-0(X1, mark.0(X2)) → U11.1-0(X1, X2)
active.1(zeros.) → mark.0(cons.0-1(0., zeros.))
mark.0(length.1(X)) → active.0(length.0(mark.1(X)))
mark.0(U12.0-0(X1, X2)) → active.0(U12.0-0(mark.0(X1), X2))
U12.0-1(mark.1(X1), X2) → U12.1-1(X1, X2)
U12.0-0(X1, active.1(X2)) → U12.0-1(X1, X2)
U12.0-1(active.1(X1), X2) → U12.1-1(X1, X2)
s.0(active.1(X)) → s.1(X)
U12.0-0(active.1(X1), X2) → U12.1-0(X1, X2)
U11.1-0(X1, mark.1(X2)) → U11.1-1(X1, X2)
U12.1-0(X1, mark.1(X2)) → U12.1-1(X1, X2)
cons.0-0(mark.1(X1), X2) → cons.1-0(X1, X2)
U11.0-0(X1, mark.0(X2)) → U11.0-0(X1, X2)
mark.0(cons.1-0(X1, X2)) → active.0(cons.0-0(mark.1(X1), X2))
active.0(U12.0-0(tt., L)) → mark.0(s.0(length.0(L)))
U12.0-0(active.0(X1), X2) → U12.0-0(X1, X2)
U12.0-0(X1, mark.0(X2)) → U12.0-0(X1, X2)
s.0(mark.0(X)) → s.0(X)
U11.0-0(mark.0(X1), X2) → U11.0-0(X1, X2)
mark.0(U11.1-1(X1, X2)) → active.0(U11.0-1(mark.1(X1), X2))
U12.0-1(active.0(X1), X2) → U12.0-1(X1, X2)
U11.1-0(X1, active.1(X2)) → U11.1-1(X1, X2)
cons.0-0(mark.0(X1), X2) → cons.0-0(X1, X2)
mark.0(U11.1-0(X1, X2)) → active.0(U11.0-0(mark.1(X1), X2))

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

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ SemLabProof
QDP
                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                        ↳ SemLabProof2

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

MARK.0(length.0(cons.1-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.1(x0), x1))))
MARK.0(length.1(x0)) → ACTIVE.0(length.1(x0))
ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
MARK.0(length.0(x0)) → ACTIVE.0(length.0(x0))
ACTIVE.0(length.0(cons.1-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(length.0(cons.0-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.0(x0), x1))))
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
MARK.0(U11.0-0(tt., y1)) → ACTIVE.0(U11.0-0(active.0(tt.), y1))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))
MARK.0(length.0(cons.0-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.0(x0), x1))))
MARK.0(U11.1-0(x0, x1)) → ACTIVE.0(U11.1-0(x0, x1))
ACTIVE.0(U11.0-0(tt., L)) → MARK.0(U12.0-0(tt., L))
ACTIVE.0(length.0(cons.0-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(U12.0-0(x0, x1)) → ACTIVE.0(U12.0-0(x0, x1))
ACTIVE.0(U12.0-0(tt., L)) → MARK.0(s.0(length.0(L)))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
MARK.0(length.0(cons.1-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.1(x0), x1))))
MARK.0(s.1(X)) → MARK.1(X)
MARK.0(U12.1-0(x0, x1)) → ACTIVE.0(U12.1-0(x0, x1))
MARK.0(U12.0-0(tt., y1)) → ACTIVE.0(U12.0-0(active.0(tt.), y1))
MARK.0(U11.0-0(x0, x1)) → ACTIVE.0(U11.0-0(x0, x1))
ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(U12.1-1(x0, x1)) → ACTIVE.0(U12.1-1(x0, x1))
MARK.0(U11.1-1(x0, x1)) → ACTIVE.0(U11.1-1(x0, x1))
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))

The TRS R consists of the following rules:

active.0(length.0(cons.0-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.1-0(X1, active.1(X2)) → U12.1-1(X1, X2)
U12.0-0(mark.1(X1), X2) → U12.1-0(X1, X2)
U11.0-0(X1, active.1(X2)) → U11.0-1(X1, X2)
cons.1-0(X1, active.0(X2)) → cons.1-0(X1, X2)
mark.0(cons.1-1(X1, X2)) → active.0(cons.0-1(mark.1(X1), X2))
mark.1(zeros.) → active.1(zeros.)
cons.0-0(X1, active.1(X2)) → cons.0-1(X1, X2)
cons.0-1(active.0(X1), X2) → cons.0-1(X1, X2)
active.0(length.0(cons.1-0(N, L))) → mark.0(U11.0-0(tt., L))
U11.0-0(mark.1(X1), X2) → U11.1-0(X1, X2)
U11.0-0(X1, active.0(X2)) → U11.0-0(X1, X2)
U11.0-1(mark.0(X1), X2) → U11.0-1(X1, X2)
cons.0-0(X1, mark.1(X2)) → cons.0-1(X1, X2)
U11.0-1(mark.1(X1), X2) → U11.1-1(X1, X2)
U11.0-1(active.0(X1), X2) → U11.0-1(X1, X2)
cons.1-0(X1, active.1(X2)) → cons.1-1(X1, X2)
cons.0-0(active.0(X1), X2) → cons.0-0(X1, X2)
mark.0(cons.0-1(X1, X2)) → active.0(cons.0-1(mark.0(X1), X2))
mark.0(U12.1-1(X1, X2)) → active.0(U12.0-1(mark.1(X1), X2))
mark.0(s.0(X)) → active.0(s.0(mark.0(X)))
active.0(U11.0-1(tt., L)) → mark.0(U12.0-1(tt., L))
U11.0-0(active.1(X1), X2) → U11.1-0(X1, X2)
active.0(U11.0-0(tt., L)) → mark.0(U12.0-0(tt., L))
cons.0-1(mark.0(X1), X2) → cons.0-1(X1, X2)
mark.0(0.) → active.0(0.)
cons.0-0(X1, mark.0(X2)) → cons.0-0(X1, X2)
U12.1-0(X1, mark.0(X2)) → U12.1-0(X1, X2)
mark.0(length.0(X)) → active.0(length.0(mark.0(X)))
length.0(mark.1(X)) → length.1(X)
cons.0-0(active.1(X1), X2) → cons.1-0(X1, X2)
U11.0-1(active.1(X1), X2) → U11.1-1(X1, X2)
mark.0(U12.0-1(X1, X2)) → active.0(U12.0-1(mark.0(X1), X2))
length.0(active.0(X)) → length.0(X)
U12.0-1(mark.0(X1), X2) → U12.0-1(X1, X2)
length.0(active.1(X)) → length.1(X)
active.0(U12.0-1(tt., L)) → mark.0(s.0(length.1(L)))
U11.1-0(X1, active.0(X2)) → U11.1-0(X1, X2)
cons.1-0(X1, mark.0(X2)) → cons.1-0(X1, X2)
U12.0-0(X1, active.0(X2)) → U12.0-0(X1, X2)
cons.0-0(X1, active.0(X2)) → cons.0-0(X1, X2)
mark.0(s.1(X)) → active.0(s.0(mark.1(X)))
mark.0(U11.0-1(X1, X2)) → active.0(U11.0-1(mark.0(X1), X2))
mark.0(cons.0-0(X1, X2)) → active.0(cons.0-0(mark.0(X1), X2))
cons.1-0(X1, mark.1(X2)) → cons.1-1(X1, X2)
mark.0(U12.1-0(X1, X2)) → active.0(U12.0-0(mark.1(X1), X2))
U11.0-0(active.0(X1), X2) → U11.0-0(X1, X2)
active.0(length.0(cons.1-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.0(X1), X2) → U12.0-0(X1, X2)
U12.1-0(X1, active.0(X2)) → U12.1-0(X1, X2)
mark.0(tt.) → active.0(tt.)
cons.0-1(mark.1(X1), X2) → cons.1-1(X1, X2)
U12.0-0(X1, mark.1(X2)) → U12.0-1(X1, X2)
mark.0(nil.) → active.0(nil.)
s.0(active.0(X)) → s.0(X)
length.0(mark.0(X)) → length.0(X)
U11.0-0(X1, mark.1(X2)) → U11.0-1(X1, X2)
cons.0-1(active.1(X1), X2) → cons.1-1(X1, X2)
mark.0(U11.0-0(X1, X2)) → active.0(U11.0-0(mark.0(X1), X2))
active.0(length.0(cons.0-0(N, L))) → mark.0(U11.0-0(tt., L))
s.0(mark.1(X)) → s.1(X)
U11.1-0(X1, mark.0(X2)) → U11.1-0(X1, X2)
active.1(zeros.) → mark.0(cons.0-1(0., zeros.))
mark.0(length.1(X)) → active.0(length.0(mark.1(X)))
mark.0(U12.0-0(X1, X2)) → active.0(U12.0-0(mark.0(X1), X2))
U12.0-1(mark.1(X1), X2) → U12.1-1(X1, X2)
U12.0-0(X1, active.1(X2)) → U12.0-1(X1, X2)
U12.0-1(active.1(X1), X2) → U12.1-1(X1, X2)
s.0(active.1(X)) → s.1(X)
U12.0-0(active.1(X1), X2) → U12.1-0(X1, X2)
U11.1-0(X1, mark.1(X2)) → U11.1-1(X1, X2)
U12.1-0(X1, mark.1(X2)) → U12.1-1(X1, X2)
cons.0-0(mark.1(X1), X2) → cons.1-0(X1, X2)
U11.0-0(X1, mark.0(X2)) → U11.0-0(X1, X2)
mark.0(cons.1-0(X1, X2)) → active.0(cons.0-0(mark.1(X1), X2))
active.0(U12.0-0(tt., L)) → mark.0(s.0(length.0(L)))
U12.0-0(active.0(X1), X2) → U12.0-0(X1, X2)
U12.0-0(X1, mark.0(X2)) → U12.0-0(X1, X2)
s.0(mark.0(X)) → s.0(X)
U11.0-0(mark.0(X1), X2) → U11.0-0(X1, X2)
mark.0(U11.1-1(X1, X2)) → active.0(U11.0-1(mark.1(X1), X2))
U12.0-1(active.0(X1), X2) → U12.0-1(X1, X2)
U11.1-0(X1, active.1(X2)) → U11.1-1(X1, X2)
cons.0-0(mark.0(X1), X2) → cons.0-0(X1, X2)
mark.0(U11.1-0(X1, X2)) → active.0(U11.0-0(mark.1(X1), X2))

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 4 less nodes.

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ SemLabProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ DependencyGraphProof
QDP
                                                                                                                                                ↳ RuleRemovalProof
                                                                                                                                        ↳ SemLabProof2

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

MARK.0(length.0(cons.1-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.1(x0), x1))))
ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
ACTIVE.0(length.0(cons.1-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(length.0(x0)) → ACTIVE.0(length.0(x0))
MARK.0(length.0(cons.0-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.0(x0), x1))))
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(U11.0-0(tt., y1)) → ACTIVE.0(U11.0-0(active.0(tt.), y1))
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))
MARK.0(length.0(cons.0-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.0(x0), x1))))
MARK.0(U11.1-0(x0, x1)) → ACTIVE.0(U11.1-0(x0, x1))
ACTIVE.0(U11.0-0(tt., L)) → MARK.0(U12.0-0(tt., L))
ACTIVE.0(length.0(cons.0-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(U12.0-0(x0, x1)) → ACTIVE.0(U12.0-0(x0, x1))
ACTIVE.0(U12.0-0(tt., L)) → MARK.0(s.0(length.0(L)))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
MARK.0(length.0(cons.1-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.1(x0), x1))))
MARK.0(U11.0-0(x0, x1)) → ACTIVE.0(U11.0-0(x0, x1))
MARK.0(U12.0-0(tt., y1)) → ACTIVE.0(U12.0-0(active.0(tt.), y1))
MARK.0(U12.1-0(x0, x1)) → ACTIVE.0(U12.1-0(x0, x1))
ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))

The TRS R consists of the following rules:

active.0(length.0(cons.0-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.1-0(X1, active.1(X2)) → U12.1-1(X1, X2)
U12.0-0(mark.1(X1), X2) → U12.1-0(X1, X2)
U11.0-0(X1, active.1(X2)) → U11.0-1(X1, X2)
cons.1-0(X1, active.0(X2)) → cons.1-0(X1, X2)
mark.0(cons.1-1(X1, X2)) → active.0(cons.0-1(mark.1(X1), X2))
mark.1(zeros.) → active.1(zeros.)
cons.0-0(X1, active.1(X2)) → cons.0-1(X1, X2)
cons.0-1(active.0(X1), X2) → cons.0-1(X1, X2)
active.0(length.0(cons.1-0(N, L))) → mark.0(U11.0-0(tt., L))
U11.0-0(mark.1(X1), X2) → U11.1-0(X1, X2)
U11.0-0(X1, active.0(X2)) → U11.0-0(X1, X2)
U11.0-1(mark.0(X1), X2) → U11.0-1(X1, X2)
cons.0-0(X1, mark.1(X2)) → cons.0-1(X1, X2)
U11.0-1(mark.1(X1), X2) → U11.1-1(X1, X2)
U11.0-1(active.0(X1), X2) → U11.0-1(X1, X2)
cons.1-0(X1, active.1(X2)) → cons.1-1(X1, X2)
cons.0-0(active.0(X1), X2) → cons.0-0(X1, X2)
mark.0(cons.0-1(X1, X2)) → active.0(cons.0-1(mark.0(X1), X2))
mark.0(U12.1-1(X1, X2)) → active.0(U12.0-1(mark.1(X1), X2))
mark.0(s.0(X)) → active.0(s.0(mark.0(X)))
active.0(U11.0-1(tt., L)) → mark.0(U12.0-1(tt., L))
U11.0-0(active.1(X1), X2) → U11.1-0(X1, X2)
active.0(U11.0-0(tt., L)) → mark.0(U12.0-0(tt., L))
cons.0-1(mark.0(X1), X2) → cons.0-1(X1, X2)
mark.0(0.) → active.0(0.)
cons.0-0(X1, mark.0(X2)) → cons.0-0(X1, X2)
U12.1-0(X1, mark.0(X2)) → U12.1-0(X1, X2)
mark.0(length.0(X)) → active.0(length.0(mark.0(X)))
length.0(mark.1(X)) → length.1(X)
cons.0-0(active.1(X1), X2) → cons.1-0(X1, X2)
U11.0-1(active.1(X1), X2) → U11.1-1(X1, X2)
mark.0(U12.0-1(X1, X2)) → active.0(U12.0-1(mark.0(X1), X2))
length.0(active.0(X)) → length.0(X)
U12.0-1(mark.0(X1), X2) → U12.0-1(X1, X2)
length.0(active.1(X)) → length.1(X)
active.0(U12.0-1(tt., L)) → mark.0(s.0(length.1(L)))
U11.1-0(X1, active.0(X2)) → U11.1-0(X1, X2)
cons.1-0(X1, mark.0(X2)) → cons.1-0(X1, X2)
U12.0-0(X1, active.0(X2)) → U12.0-0(X1, X2)
cons.0-0(X1, active.0(X2)) → cons.0-0(X1, X2)
mark.0(s.1(X)) → active.0(s.0(mark.1(X)))
mark.0(U11.0-1(X1, X2)) → active.0(U11.0-1(mark.0(X1), X2))
mark.0(cons.0-0(X1, X2)) → active.0(cons.0-0(mark.0(X1), X2))
cons.1-0(X1, mark.1(X2)) → cons.1-1(X1, X2)
mark.0(U12.1-0(X1, X2)) → active.0(U12.0-0(mark.1(X1), X2))
U11.0-0(active.0(X1), X2) → U11.0-0(X1, X2)
active.0(length.0(cons.1-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.0(X1), X2) → U12.0-0(X1, X2)
U12.1-0(X1, active.0(X2)) → U12.1-0(X1, X2)
mark.0(tt.) → active.0(tt.)
cons.0-1(mark.1(X1), X2) → cons.1-1(X1, X2)
U12.0-0(X1, mark.1(X2)) → U12.0-1(X1, X2)
mark.0(nil.) → active.0(nil.)
s.0(active.0(X)) → s.0(X)
length.0(mark.0(X)) → length.0(X)
U11.0-0(X1, mark.1(X2)) → U11.0-1(X1, X2)
cons.0-1(active.1(X1), X2) → cons.1-1(X1, X2)
mark.0(U11.0-0(X1, X2)) → active.0(U11.0-0(mark.0(X1), X2))
active.0(length.0(cons.0-0(N, L))) → mark.0(U11.0-0(tt., L))
s.0(mark.1(X)) → s.1(X)
U11.1-0(X1, mark.0(X2)) → U11.1-0(X1, X2)
active.1(zeros.) → mark.0(cons.0-1(0., zeros.))
mark.0(length.1(X)) → active.0(length.0(mark.1(X)))
mark.0(U12.0-0(X1, X2)) → active.0(U12.0-0(mark.0(X1), X2))
U12.0-1(mark.1(X1), X2) → U12.1-1(X1, X2)
U12.0-0(X1, active.1(X2)) → U12.0-1(X1, X2)
U12.0-1(active.1(X1), X2) → U12.1-1(X1, X2)
s.0(active.1(X)) → s.1(X)
U12.0-0(active.1(X1), X2) → U12.1-0(X1, X2)
U11.1-0(X1, mark.1(X2)) → U11.1-1(X1, X2)
U12.1-0(X1, mark.1(X2)) → U12.1-1(X1, X2)
cons.0-0(mark.1(X1), X2) → cons.1-0(X1, X2)
U11.0-0(X1, mark.0(X2)) → U11.0-0(X1, X2)
mark.0(cons.1-0(X1, X2)) → active.0(cons.0-0(mark.1(X1), X2))
active.0(U12.0-0(tt., L)) → mark.0(s.0(length.0(L)))
U12.0-0(active.0(X1), X2) → U12.0-0(X1, X2)
U12.0-0(X1, mark.0(X2)) → U12.0-0(X1, X2)
s.0(mark.0(X)) → s.0(X)
U11.0-0(mark.0(X1), X2) → U11.0-0(X1, X2)
mark.0(U11.1-1(X1, X2)) → active.0(U11.0-1(mark.1(X1), X2))
U12.0-1(active.0(X1), X2) → U12.0-1(X1, X2)
U11.1-0(X1, active.1(X2)) → U11.1-1(X1, X2)
cons.0-0(mark.0(X1), X2) → cons.0-0(X1, X2)
mark.0(U11.1-0(X1, X2)) → active.0(U11.0-0(mark.1(X1), X2))

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:

ACTIVE.0(U11.0-0(tt., L)) → MARK.0(U12.0-0(tt., L))

Strictly oriented rules of the TRS R:

U11.0-0(X1, active.1(X2)) → U11.0-1(X1, X2)
cons.0-0(X1, active.1(X2)) → cons.0-1(X1, X2)
cons.0-0(X1, mark.1(X2)) → cons.0-1(X1, X2)
cons.1-0(X1, active.1(X2)) → cons.1-1(X1, X2)
active.0(U11.0-0(tt., L)) → mark.0(U12.0-0(tt., L))
cons.1-0(X1, mark.1(X2)) → cons.1-1(X1, X2)
U11.0-0(X1, mark.1(X2)) → U11.0-1(X1, X2)
U11.1-0(X1, mark.1(X2)) → U11.1-1(X1, X2)
U11.1-0(X1, active.1(X2)) → U11.1-1(X1, X2)

Used ordering: POLO with Polynomial interpretation [25]:

POL(0.) = 0   
POL(ACTIVE.0(x1)) = x1   
POL(MARK.0(x1)) = x1   
POL(U11.0-0(x1, x2)) = 1 + x1 + x2   
POL(U11.0-1(x1, x2)) = x1 + x2   
POL(U11.1-0(x1, x2)) = 1 + x1 + x2   
POL(U11.1-1(x1, x2)) = x1 + x2   
POL(U12.0-0(x1, x2)) = x1 + x2   
POL(U12.0-1(x1, x2)) = x1 + x2   
POL(U12.1-0(x1, x2)) = x1 + x2   
POL(U12.1-1(x1, x2)) = x1 + x2   
POL(active.0(x1)) = x1   
POL(active.1(x1)) = x1   
POL(cons.0-0(x1, x2)) = 1 + x1 + x2   
POL(cons.0-1(x1, x2)) = x1 + x2   
POL(cons.1-0(x1, x2)) = 1 + x1 + x2   
POL(cons.1-1(x1, x2)) = x1 + x2   
POL(length.0(x1)) = x1   
POL(length.1(x1)) = x1   
POL(mark.0(x1)) = x1   
POL(mark.1(x1)) = x1   
POL(nil.) = 0   
POL(s.0(x1)) = x1   
POL(s.1(x1)) = x1   
POL(tt.) = 0   
POL(zeros.) = 0   



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ SemLabProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ RuleRemovalProof
QDP
                                                                                                                                                    ↳ RuleRemovalProof
                                                                                                                                        ↳ SemLabProof2

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

ACTIVE.0(length.0(cons.0-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(length.0(cons.1-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.1(x0), x1))))
MARK.0(U12.0-0(x0, x1)) → ACTIVE.0(U12.0-0(x0, x1))
ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
ACTIVE.0(U12.0-0(tt., L)) → MARK.0(s.0(length.0(L)))
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
ACTIVE.0(length.0(cons.1-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(length.0(x0)) → ACTIVE.0(length.0(x0))
MARK.0(length.0(cons.1-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.1(x0), x1))))
MARK.0(U12.1-0(x0, x1)) → ACTIVE.0(U12.1-0(x0, x1))
MARK.0(U12.0-0(tt., y1)) → ACTIVE.0(U12.0-0(active.0(tt.), y1))
MARK.0(U11.0-0(x0, x1)) → ACTIVE.0(U11.0-0(x0, x1))
MARK.0(length.0(cons.0-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.0(x0), x1))))
ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(U11.0-0(tt., y1)) → ACTIVE.0(U11.0-0(active.0(tt.), y1))
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))
MARK.0(length.0(cons.0-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.0(x0), x1))))
MARK.0(U11.1-0(x0, x1)) → ACTIVE.0(U11.1-0(x0, x1))

The TRS R consists of the following rules:

active.0(length.0(cons.0-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.1-0(X1, active.1(X2)) → U12.1-1(X1, X2)
U12.0-0(mark.1(X1), X2) → U12.1-0(X1, X2)
cons.1-0(X1, active.0(X2)) → cons.1-0(X1, X2)
mark.0(cons.1-1(X1, X2)) → active.0(cons.0-1(mark.1(X1), X2))
mark.1(zeros.) → active.1(zeros.)
cons.0-1(active.0(X1), X2) → cons.0-1(X1, X2)
active.0(length.0(cons.1-0(N, L))) → mark.0(U11.0-0(tt., L))
U11.0-0(mark.1(X1), X2) → U11.1-0(X1, X2)
U11.0-0(X1, active.0(X2)) → U11.0-0(X1, X2)
U11.0-1(mark.0(X1), X2) → U11.0-1(X1, X2)
U11.0-1(mark.1(X1), X2) → U11.1-1(X1, X2)
U11.0-1(active.0(X1), X2) → U11.0-1(X1, X2)
cons.0-0(active.0(X1), X2) → cons.0-0(X1, X2)
mark.0(cons.0-1(X1, X2)) → active.0(cons.0-1(mark.0(X1), X2))
mark.0(U12.1-1(X1, X2)) → active.0(U12.0-1(mark.1(X1), X2))
mark.0(s.0(X)) → active.0(s.0(mark.0(X)))
active.0(U11.0-1(tt., L)) → mark.0(U12.0-1(tt., L))
U11.0-0(active.1(X1), X2) → U11.1-0(X1, X2)
cons.0-1(mark.0(X1), X2) → cons.0-1(X1, X2)
mark.0(0.) → active.0(0.)
cons.0-0(X1, mark.0(X2)) → cons.0-0(X1, X2)
U12.1-0(X1, mark.0(X2)) → U12.1-0(X1, X2)
mark.0(length.0(X)) → active.0(length.0(mark.0(X)))
length.0(mark.1(X)) → length.1(X)
cons.0-0(active.1(X1), X2) → cons.1-0(X1, X2)
U11.0-1(active.1(X1), X2) → U11.1-1(X1, X2)
mark.0(U12.0-1(X1, X2)) → active.0(U12.0-1(mark.0(X1), X2))
length.0(active.0(X)) → length.0(X)
U12.0-1(mark.0(X1), X2) → U12.0-1(X1, X2)
length.0(active.1(X)) → length.1(X)
active.0(U12.0-1(tt., L)) → mark.0(s.0(length.1(L)))
U11.1-0(X1, active.0(X2)) → U11.1-0(X1, X2)
cons.1-0(X1, mark.0(X2)) → cons.1-0(X1, X2)
U12.0-0(X1, active.0(X2)) → U12.0-0(X1, X2)
cons.0-0(X1, active.0(X2)) → cons.0-0(X1, X2)
mark.0(s.1(X)) → active.0(s.0(mark.1(X)))
mark.0(U11.0-1(X1, X2)) → active.0(U11.0-1(mark.0(X1), X2))
mark.0(cons.0-0(X1, X2)) → active.0(cons.0-0(mark.0(X1), X2))
mark.0(U12.1-0(X1, X2)) → active.0(U12.0-0(mark.1(X1), X2))
U11.0-0(active.0(X1), X2) → U11.0-0(X1, X2)
active.0(length.0(cons.1-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.0(X1), X2) → U12.0-0(X1, X2)
U12.1-0(X1, active.0(X2)) → U12.1-0(X1, X2)
mark.0(tt.) → active.0(tt.)
cons.0-1(mark.1(X1), X2) → cons.1-1(X1, X2)
U12.0-0(X1, mark.1(X2)) → U12.0-1(X1, X2)
mark.0(nil.) → active.0(nil.)
s.0(active.0(X)) → s.0(X)
length.0(mark.0(X)) → length.0(X)
cons.0-1(active.1(X1), X2) → cons.1-1(X1, X2)
mark.0(U11.0-0(X1, X2)) → active.0(U11.0-0(mark.0(X1), X2))
active.0(length.0(cons.0-0(N, L))) → mark.0(U11.0-0(tt., L))
s.0(mark.1(X)) → s.1(X)
U11.1-0(X1, mark.0(X2)) → U11.1-0(X1, X2)
active.1(zeros.) → mark.0(cons.0-1(0., zeros.))
mark.0(length.1(X)) → active.0(length.0(mark.1(X)))
mark.0(U12.0-0(X1, X2)) → active.0(U12.0-0(mark.0(X1), X2))
U12.0-1(mark.1(X1), X2) → U12.1-1(X1, X2)
U12.0-0(X1, active.1(X2)) → U12.0-1(X1, X2)
U12.0-1(active.1(X1), X2) → U12.1-1(X1, X2)
s.0(active.1(X)) → s.1(X)
U12.0-0(active.1(X1), X2) → U12.1-0(X1, X2)
U12.1-0(X1, mark.1(X2)) → U12.1-1(X1, X2)
cons.0-0(mark.1(X1), X2) → cons.1-0(X1, X2)
U11.0-0(X1, mark.0(X2)) → U11.0-0(X1, X2)
mark.0(cons.1-0(X1, X2)) → active.0(cons.0-0(mark.1(X1), X2))
active.0(U12.0-0(tt., L)) → mark.0(s.0(length.0(L)))
U12.0-0(active.0(X1), X2) → U12.0-0(X1, X2)
U12.0-0(X1, mark.0(X2)) → U12.0-0(X1, X2)
s.0(mark.0(X)) → s.0(X)
U11.0-0(mark.0(X1), X2) → U11.0-0(X1, X2)
mark.0(U11.1-1(X1, X2)) → active.0(U11.0-1(mark.1(X1), X2))
U12.0-1(active.0(X1), X2) → U12.0-1(X1, X2)
cons.0-0(mark.0(X1), X2) → cons.0-0(X1, X2)
mark.0(U11.1-0(X1, X2)) → active.0(U11.0-0(mark.1(X1), X2))

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:

ACTIVE.0(U12.0-0(tt., L)) → MARK.0(s.0(length.0(L)))

Strictly oriented rules of the TRS R:

U12.1-0(X1, active.1(X2)) → U12.1-1(X1, X2)
U12.0-0(X1, mark.1(X2)) → U12.0-1(X1, X2)
U12.0-0(X1, active.1(X2)) → U12.0-1(X1, X2)
U12.1-0(X1, mark.1(X2)) → U12.1-1(X1, X2)
active.0(U12.0-0(tt., L)) → mark.0(s.0(length.0(L)))

Used ordering: POLO with Polynomial interpretation [25]:

POL(0.) = 0   
POL(ACTIVE.0(x1)) = x1   
POL(MARK.0(x1)) = x1   
POL(U11.0-0(x1, x2)) = 1 + x1 + x2   
POL(U11.0-1(x1, x2)) = x1 + x2   
POL(U11.1-0(x1, x2)) = 1 + x1 + x2   
POL(U11.1-1(x1, x2)) = x1 + x2   
POL(U12.0-0(x1, x2)) = 1 + x1 + x2   
POL(U12.0-1(x1, x2)) = x1 + x2   
POL(U12.1-0(x1, x2)) = 1 + x1 + x2   
POL(U12.1-1(x1, x2)) = x1 + x2   
POL(active.0(x1)) = x1   
POL(active.1(x1)) = x1   
POL(cons.0-0(x1, x2)) = 1 + x1 + x2   
POL(cons.0-1(x1, x2)) = x1 + x2   
POL(cons.1-0(x1, x2)) = 1 + x1 + x2   
POL(cons.1-1(x1, x2)) = x1 + x2   
POL(length.0(x1)) = x1   
POL(length.1(x1)) = x1   
POL(mark.0(x1)) = x1   
POL(mark.1(x1)) = x1   
POL(nil.) = 0   
POL(s.0(x1)) = x1   
POL(s.1(x1)) = x1   
POL(tt.) = 0   
POL(zeros.) = 0   



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ SemLabProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ RuleRemovalProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ RuleRemovalProof
QDP
                                                                                                                                                        ↳ RuleRemovalProof
                                                                                                                                        ↳ SemLabProof2

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

ACTIVE.0(length.0(cons.0-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(length.0(cons.1-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.1(x0), x1))))
MARK.0(U12.0-0(x0, x1)) → ACTIVE.0(U12.0-0(x0, x1))
ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
ACTIVE.0(length.0(cons.1-0(N, L))) → MARK.0(U11.0-0(tt., L))
MARK.0(length.0(x0)) → ACTIVE.0(length.0(x0))
MARK.0(length.0(cons.1-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.1(x0), x1))))
MARK.0(U12.1-0(x0, x1)) → ACTIVE.0(U12.1-0(x0, x1))
MARK.0(U12.0-0(tt., y1)) → ACTIVE.0(U12.0-0(active.0(tt.), y1))
MARK.0(U11.0-0(x0, x1)) → ACTIVE.0(U11.0-0(x0, x1))
MARK.0(length.0(cons.0-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.0(x0), x1))))
ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
MARK.0(U11.0-0(tt., y1)) → ACTIVE.0(U11.0-0(active.0(tt.), y1))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))
MARK.0(length.0(cons.0-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.0(x0), x1))))
MARK.0(U11.1-0(x0, x1)) → ACTIVE.0(U11.1-0(x0, x1))

The TRS R consists of the following rules:

active.0(length.0(cons.0-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.1(X1), X2) → U12.1-0(X1, X2)
cons.1-0(X1, active.0(X2)) → cons.1-0(X1, X2)
mark.0(cons.1-1(X1, X2)) → active.0(cons.0-1(mark.1(X1), X2))
mark.1(zeros.) → active.1(zeros.)
cons.0-1(active.0(X1), X2) → cons.0-1(X1, X2)
active.0(length.0(cons.1-0(N, L))) → mark.0(U11.0-0(tt., L))
U11.0-0(mark.1(X1), X2) → U11.1-0(X1, X2)
U11.0-0(X1, active.0(X2)) → U11.0-0(X1, X2)
U11.0-1(mark.0(X1), X2) → U11.0-1(X1, X2)
U11.0-1(mark.1(X1), X2) → U11.1-1(X1, X2)
U11.0-1(active.0(X1), X2) → U11.0-1(X1, X2)
cons.0-0(active.0(X1), X2) → cons.0-0(X1, X2)
mark.0(cons.0-1(X1, X2)) → active.0(cons.0-1(mark.0(X1), X2))
mark.0(U12.1-1(X1, X2)) → active.0(U12.0-1(mark.1(X1), X2))
mark.0(s.0(X)) → active.0(s.0(mark.0(X)))
active.0(U11.0-1(tt., L)) → mark.0(U12.0-1(tt., L))
U11.0-0(active.1(X1), X2) → U11.1-0(X1, X2)
cons.0-1(mark.0(X1), X2) → cons.0-1(X1, X2)
mark.0(0.) → active.0(0.)
cons.0-0(X1, mark.0(X2)) → cons.0-0(X1, X2)
U12.1-0(X1, mark.0(X2)) → U12.1-0(X1, X2)
mark.0(length.0(X)) → active.0(length.0(mark.0(X)))
length.0(mark.1(X)) → length.1(X)
cons.0-0(active.1(X1), X2) → cons.1-0(X1, X2)
U11.0-1(active.1(X1), X2) → U11.1-1(X1, X2)
mark.0(U12.0-1(X1, X2)) → active.0(U12.0-1(mark.0(X1), X2))
length.0(active.0(X)) → length.0(X)
U12.0-1(mark.0(X1), X2) → U12.0-1(X1, X2)
length.0(active.1(X)) → length.1(X)
active.0(U12.0-1(tt., L)) → mark.0(s.0(length.1(L)))
U11.1-0(X1, active.0(X2)) → U11.1-0(X1, X2)
cons.1-0(X1, mark.0(X2)) → cons.1-0(X1, X2)
U12.0-0(X1, active.0(X2)) → U12.0-0(X1, X2)
cons.0-0(X1, active.0(X2)) → cons.0-0(X1, X2)
mark.0(s.1(X)) → active.0(s.0(mark.1(X)))
mark.0(U11.0-1(X1, X2)) → active.0(U11.0-1(mark.0(X1), X2))
mark.0(cons.0-0(X1, X2)) → active.0(cons.0-0(mark.0(X1), X2))
mark.0(U12.1-0(X1, X2)) → active.0(U12.0-0(mark.1(X1), X2))
U11.0-0(active.0(X1), X2) → U11.0-0(X1, X2)
active.0(length.0(cons.1-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.0(X1), X2) → U12.0-0(X1, X2)
U12.1-0(X1, active.0(X2)) → U12.1-0(X1, X2)
mark.0(tt.) → active.0(tt.)
cons.0-1(mark.1(X1), X2) → cons.1-1(X1, X2)
mark.0(nil.) → active.0(nil.)
s.0(active.0(X)) → s.0(X)
length.0(mark.0(X)) → length.0(X)
cons.0-1(active.1(X1), X2) → cons.1-1(X1, X2)
mark.0(U11.0-0(X1, X2)) → active.0(U11.0-0(mark.0(X1), X2))
active.0(length.0(cons.0-0(N, L))) → mark.0(U11.0-0(tt., L))
s.0(mark.1(X)) → s.1(X)
U11.1-0(X1, mark.0(X2)) → U11.1-0(X1, X2)
active.1(zeros.) → mark.0(cons.0-1(0., zeros.))
mark.0(length.1(X)) → active.0(length.0(mark.1(X)))
mark.0(U12.0-0(X1, X2)) → active.0(U12.0-0(mark.0(X1), X2))
U12.0-1(mark.1(X1), X2) → U12.1-1(X1, X2)
U12.0-1(active.1(X1), X2) → U12.1-1(X1, X2)
s.0(active.1(X)) → s.1(X)
U12.0-0(active.1(X1), X2) → U12.1-0(X1, X2)
cons.0-0(mark.1(X1), X2) → cons.1-0(X1, X2)
U11.0-0(X1, mark.0(X2)) → U11.0-0(X1, X2)
mark.0(cons.1-0(X1, X2)) → active.0(cons.0-0(mark.1(X1), X2))
U12.0-0(active.0(X1), X2) → U12.0-0(X1, X2)
U12.0-0(X1, mark.0(X2)) → U12.0-0(X1, X2)
s.0(mark.0(X)) → s.0(X)
U11.0-0(mark.0(X1), X2) → U11.0-0(X1, X2)
mark.0(U11.1-1(X1, X2)) → active.0(U11.0-1(mark.1(X1), X2))
U12.0-1(active.0(X1), X2) → U12.0-1(X1, X2)
cons.0-0(mark.0(X1), X2) → cons.0-0(X1, X2)
mark.0(U11.1-0(X1, X2)) → active.0(U11.0-0(mark.1(X1), X2))

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:

ACTIVE.0(length.0(cons.0-0(N, L))) → MARK.0(U11.0-0(tt., L))
ACTIVE.0(length.0(cons.1-0(N, L))) → MARK.0(U11.0-0(tt., L))

Strictly oriented rules of the TRS R:

active.0(length.0(cons.1-0(N, L))) → mark.0(U11.0-0(tt., L))
active.0(length.0(cons.0-0(N, L))) → mark.0(U11.0-0(tt., L))

Used ordering: POLO with Polynomial interpretation [25]:

POL(0.) = 0   
POL(ACTIVE.0(x1)) = x1   
POL(MARK.0(x1)) = x1   
POL(U11.0-0(x1, x2)) = x1 + x2   
POL(U11.0-1(x1, x2)) = x1 + x2   
POL(U11.1-0(x1, x2)) = x1 + x2   
POL(U11.1-1(x1, x2)) = x1 + x2   
POL(U12.0-0(x1, x2)) = x1 + x2   
POL(U12.0-1(x1, x2)) = x1 + x2   
POL(U12.1-0(x1, x2)) = x1 + x2   
POL(U12.1-1(x1, x2)) = x1 + x2   
POL(active.0(x1)) = x1   
POL(active.1(x1)) = x1   
POL(cons.0-0(x1, x2)) = 1 + x1 + x2   
POL(cons.0-1(x1, x2)) = x1 + x2   
POL(cons.1-0(x1, x2)) = 1 + x1 + x2   
POL(cons.1-1(x1, x2)) = x1 + x2   
POL(length.0(x1)) = x1   
POL(length.1(x1)) = x1   
POL(mark.0(x1)) = x1   
POL(mark.1(x1)) = x1   
POL(nil.) = 0   
POL(s.0(x1)) = x1   
POL(s.1(x1)) = x1   
POL(tt.) = 0   
POL(zeros.) = 0   



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ SemLabProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ RuleRemovalProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ RuleRemovalProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ RuleRemovalProof
QDP
                                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                        ↳ SemLabProof2

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

MARK.0(length.0(cons.1-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.1(x0), x1))))
MARK.0(U12.0-0(x0, x1)) → ACTIVE.0(U12.0-0(x0, x1))
ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
MARK.0(length.0(x0)) → ACTIVE.0(length.0(x0))
MARK.0(length.0(cons.1-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.1(x0), x1))))
MARK.0(U12.1-0(x0, x1)) → ACTIVE.0(U12.1-0(x0, x1))
MARK.0(U12.0-0(tt., y1)) → ACTIVE.0(U12.0-0(active.0(tt.), y1))
MARK.0(U11.0-0(x0, x1)) → ACTIVE.0(U11.0-0(x0, x1))
MARK.0(length.0(cons.0-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.0(x0), x1))))
ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(U11.0-0(tt., y1)) → ACTIVE.0(U11.0-0(active.0(tt.), y1))
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))
MARK.0(length.0(cons.0-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.0(x0), x1))))
MARK.0(U11.1-0(x0, x1)) → ACTIVE.0(U11.1-0(x0, x1))

The TRS R consists of the following rules:

active.0(length.0(cons.0-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.1(X1), X2) → U12.1-0(X1, X2)
cons.1-0(X1, active.0(X2)) → cons.1-0(X1, X2)
mark.0(cons.1-1(X1, X2)) → active.0(cons.0-1(mark.1(X1), X2))
mark.1(zeros.) → active.1(zeros.)
cons.0-1(active.0(X1), X2) → cons.0-1(X1, X2)
U11.0-0(mark.1(X1), X2) → U11.1-0(X1, X2)
U11.0-0(X1, active.0(X2)) → U11.0-0(X1, X2)
U11.0-1(mark.0(X1), X2) → U11.0-1(X1, X2)
U11.0-1(mark.1(X1), X2) → U11.1-1(X1, X2)
U11.0-1(active.0(X1), X2) → U11.0-1(X1, X2)
cons.0-0(active.0(X1), X2) → cons.0-0(X1, X2)
mark.0(cons.0-1(X1, X2)) → active.0(cons.0-1(mark.0(X1), X2))
mark.0(U12.1-1(X1, X2)) → active.0(U12.0-1(mark.1(X1), X2))
mark.0(s.0(X)) → active.0(s.0(mark.0(X)))
active.0(U11.0-1(tt., L)) → mark.0(U12.0-1(tt., L))
U11.0-0(active.1(X1), X2) → U11.1-0(X1, X2)
cons.0-1(mark.0(X1), X2) → cons.0-1(X1, X2)
mark.0(0.) → active.0(0.)
cons.0-0(X1, mark.0(X2)) → cons.0-0(X1, X2)
U12.1-0(X1, mark.0(X2)) → U12.1-0(X1, X2)
mark.0(length.0(X)) → active.0(length.0(mark.0(X)))
length.0(mark.1(X)) → length.1(X)
cons.0-0(active.1(X1), X2) → cons.1-0(X1, X2)
U11.0-1(active.1(X1), X2) → U11.1-1(X1, X2)
mark.0(U12.0-1(X1, X2)) → active.0(U12.0-1(mark.0(X1), X2))
length.0(active.0(X)) → length.0(X)
U12.0-1(mark.0(X1), X2) → U12.0-1(X1, X2)
length.0(active.1(X)) → length.1(X)
active.0(U12.0-1(tt., L)) → mark.0(s.0(length.1(L)))
U11.1-0(X1, active.0(X2)) → U11.1-0(X1, X2)
cons.1-0(X1, mark.0(X2)) → cons.1-0(X1, X2)
U12.0-0(X1, active.0(X2)) → U12.0-0(X1, X2)
cons.0-0(X1, active.0(X2)) → cons.0-0(X1, X2)
mark.0(s.1(X)) → active.0(s.0(mark.1(X)))
mark.0(U11.0-1(X1, X2)) → active.0(U11.0-1(mark.0(X1), X2))
mark.0(cons.0-0(X1, X2)) → active.0(cons.0-0(mark.0(X1), X2))
mark.0(U12.1-0(X1, X2)) → active.0(U12.0-0(mark.1(X1), X2))
U11.0-0(active.0(X1), X2) → U11.0-0(X1, X2)
active.0(length.0(cons.1-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.0(X1), X2) → U12.0-0(X1, X2)
U12.1-0(X1, active.0(X2)) → U12.1-0(X1, X2)
mark.0(tt.) → active.0(tt.)
cons.0-1(mark.1(X1), X2) → cons.1-1(X1, X2)
mark.0(nil.) → active.0(nil.)
s.0(active.0(X)) → s.0(X)
length.0(mark.0(X)) → length.0(X)
cons.0-1(active.1(X1), X2) → cons.1-1(X1, X2)
mark.0(U11.0-0(X1, X2)) → active.0(U11.0-0(mark.0(X1), X2))
s.0(mark.1(X)) → s.1(X)
U11.1-0(X1, mark.0(X2)) → U11.1-0(X1, X2)
active.1(zeros.) → mark.0(cons.0-1(0., zeros.))
mark.0(length.1(X)) → active.0(length.0(mark.1(X)))
mark.0(U12.0-0(X1, X2)) → active.0(U12.0-0(mark.0(X1), X2))
U12.0-1(mark.1(X1), X2) → U12.1-1(X1, X2)
U12.0-1(active.1(X1), X2) → U12.1-1(X1, X2)
s.0(active.1(X)) → s.1(X)
U12.0-0(active.1(X1), X2) → U12.1-0(X1, X2)
cons.0-0(mark.1(X1), X2) → cons.1-0(X1, X2)
U11.0-0(X1, mark.0(X2)) → U11.0-0(X1, X2)
mark.0(cons.1-0(X1, X2)) → active.0(cons.0-0(mark.1(X1), X2))
U12.0-0(active.0(X1), X2) → U12.0-0(X1, X2)
U12.0-0(X1, mark.0(X2)) → U12.0-0(X1, X2)
s.0(mark.0(X)) → s.0(X)
U11.0-0(mark.0(X1), X2) → U11.0-0(X1, X2)
mark.0(U11.1-1(X1, X2)) → active.0(U11.0-1(mark.1(X1), X2))
U12.0-1(active.0(X1), X2) → U12.0-1(X1, X2)
cons.0-0(mark.0(X1), X2) → cons.0-0(X1, X2)
mark.0(U11.1-0(X1, X2)) → active.0(U11.0-0(mark.1(X1), X2))

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.0(U12.0-0(x0, x1)) → ACTIVE.0(U12.0-0(x0, x1))
MARK.0(U12.0-0(tt., y1)) → ACTIVE.0(U12.0-0(active.0(tt.), y1))
MARK.0(U11.1-0(x0, x1)) → ACTIVE.0(U11.1-0(x0, x1))
The remaining pairs can at least be oriented weakly.

MARK.0(length.0(cons.1-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.1(x0), x1))))
ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
MARK.0(length.0(x0)) → ACTIVE.0(length.0(x0))
MARK.0(length.0(cons.1-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.1(x0), x1))))
MARK.0(U12.1-0(x0, x1)) → ACTIVE.0(U12.1-0(x0, x1))
MARK.0(U11.0-0(x0, x1)) → ACTIVE.0(U11.0-0(x0, x1))
MARK.0(length.0(cons.0-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.0(x0), x1))))
ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(U11.0-0(tt., y1)) → ACTIVE.0(U11.0-0(active.0(tt.), y1))
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))
MARK.0(length.0(cons.0-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.0(x0), x1))))
Used ordering: Polynomial interpretation [25]:

POL(0.) = 0   
POL(ACTIVE.0(x1)) = 1   
POL(MARK.0(x1)) = 1 + x1   
POL(U11.0-0(x1, x2)) = 0   
POL(U11.0-1(x1, x2)) = 0   
POL(U11.1-0(x1, x2)) = 1 + x1 + x2   
POL(U11.1-1(x1, x2)) = 1 + x1   
POL(U12.0-0(x1, x2)) = 1 + x1 + x2   
POL(U12.0-1(x1, x2)) = 0   
POL(U12.1-0(x1, x2)) = x1 + x2   
POL(U12.1-1(x1, x2)) = 1 + x1 + x2   
POL(active.0(x1)) = 0   
POL(active.1(x1)) = x1   
POL(cons.0-0(x1, x2)) = x1 + x2   
POL(cons.0-1(x1, x2)) = x1   
POL(cons.1-0(x1, x2)) = 1 + x1   
POL(cons.1-1(x1, x2)) = 1 + x1 + x2   
POL(length.0(x1)) = 0   
POL(length.1(x1)) = 0   
POL(mark.0(x1)) = x1   
POL(mark.1(x1)) = x1   
POL(nil.) = 0   
POL(s.0(x1)) = x1   
POL(s.1(x1)) = 1 + x1   
POL(tt.) = 0   
POL(zeros.) = 1   

The following usable rules [17] were oriented: none



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ SemLabProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ RuleRemovalProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ RuleRemovalProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ RuleRemovalProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ QDPOrderProof
QDP
                                                                                                                                                                ↳ QDPOrderProof
                                                                                                                                        ↳ SemLabProof2

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

MARK.0(length.0(cons.1-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.1(x0), x1))))
ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
MARK.0(length.0(x0)) → ACTIVE.0(length.0(x0))
MARK.0(length.0(cons.1-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.1(x0), x1))))
MARK.0(U12.1-0(x0, x1)) → ACTIVE.0(U12.1-0(x0, x1))
MARK.0(U11.0-0(x0, x1)) → ACTIVE.0(U11.0-0(x0, x1))
MARK.0(length.0(cons.0-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.0(x0), x1))))
ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
MARK.0(U11.0-0(tt., y1)) → ACTIVE.0(U11.0-0(active.0(tt.), y1))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))
MARK.0(length.0(cons.0-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.0(x0), x1))))

The TRS R consists of the following rules:

active.0(length.0(cons.0-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.1(X1), X2) → U12.1-0(X1, X2)
cons.1-0(X1, active.0(X2)) → cons.1-0(X1, X2)
mark.0(cons.1-1(X1, X2)) → active.0(cons.0-1(mark.1(X1), X2))
mark.1(zeros.) → active.1(zeros.)
cons.0-1(active.0(X1), X2) → cons.0-1(X1, X2)
U11.0-0(mark.1(X1), X2) → U11.1-0(X1, X2)
U11.0-0(X1, active.0(X2)) → U11.0-0(X1, X2)
U11.0-1(mark.0(X1), X2) → U11.0-1(X1, X2)
U11.0-1(mark.1(X1), X2) → U11.1-1(X1, X2)
U11.0-1(active.0(X1), X2) → U11.0-1(X1, X2)
cons.0-0(active.0(X1), X2) → cons.0-0(X1, X2)
mark.0(cons.0-1(X1, X2)) → active.0(cons.0-1(mark.0(X1), X2))
mark.0(U12.1-1(X1, X2)) → active.0(U12.0-1(mark.1(X1), X2))
mark.0(s.0(X)) → active.0(s.0(mark.0(X)))
active.0(U11.0-1(tt., L)) → mark.0(U12.0-1(tt., L))
U11.0-0(active.1(X1), X2) → U11.1-0(X1, X2)
cons.0-1(mark.0(X1), X2) → cons.0-1(X1, X2)
mark.0(0.) → active.0(0.)
cons.0-0(X1, mark.0(X2)) → cons.0-0(X1, X2)
U12.1-0(X1, mark.0(X2)) → U12.1-0(X1, X2)
mark.0(length.0(X)) → active.0(length.0(mark.0(X)))
length.0(mark.1(X)) → length.1(X)
cons.0-0(active.1(X1), X2) → cons.1-0(X1, X2)
U11.0-1(active.1(X1), X2) → U11.1-1(X1, X2)
mark.0(U12.0-1(X1, X2)) → active.0(U12.0-1(mark.0(X1), X2))
length.0(active.0(X)) → length.0(X)
U12.0-1(mark.0(X1), X2) → U12.0-1(X1, X2)
length.0(active.1(X)) → length.1(X)
active.0(U12.0-1(tt., L)) → mark.0(s.0(length.1(L)))
U11.1-0(X1, active.0(X2)) → U11.1-0(X1, X2)
cons.1-0(X1, mark.0(X2)) → cons.1-0(X1, X2)
U12.0-0(X1, active.0(X2)) → U12.0-0(X1, X2)
cons.0-0(X1, active.0(X2)) → cons.0-0(X1, X2)
mark.0(s.1(X)) → active.0(s.0(mark.1(X)))
mark.0(U11.0-1(X1, X2)) → active.0(U11.0-1(mark.0(X1), X2))
mark.0(cons.0-0(X1, X2)) → active.0(cons.0-0(mark.0(X1), X2))
mark.0(U12.1-0(X1, X2)) → active.0(U12.0-0(mark.1(X1), X2))
U11.0-0(active.0(X1), X2) → U11.0-0(X1, X2)
active.0(length.0(cons.1-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.0(X1), X2) → U12.0-0(X1, X2)
U12.1-0(X1, active.0(X2)) → U12.1-0(X1, X2)
mark.0(tt.) → active.0(tt.)
cons.0-1(mark.1(X1), X2) → cons.1-1(X1, X2)
mark.0(nil.) → active.0(nil.)
s.0(active.0(X)) → s.0(X)
length.0(mark.0(X)) → length.0(X)
cons.0-1(active.1(X1), X2) → cons.1-1(X1, X2)
mark.0(U11.0-0(X1, X2)) → active.0(U11.0-0(mark.0(X1), X2))
s.0(mark.1(X)) → s.1(X)
U11.1-0(X1, mark.0(X2)) → U11.1-0(X1, X2)
active.1(zeros.) → mark.0(cons.0-1(0., zeros.))
mark.0(length.1(X)) → active.0(length.0(mark.1(X)))
mark.0(U12.0-0(X1, X2)) → active.0(U12.0-0(mark.0(X1), X2))
U12.0-1(mark.1(X1), X2) → U12.1-1(X1, X2)
U12.0-1(active.1(X1), X2) → U12.1-1(X1, X2)
s.0(active.1(X)) → s.1(X)
U12.0-0(active.1(X1), X2) → U12.1-0(X1, X2)
cons.0-0(mark.1(X1), X2) → cons.1-0(X1, X2)
U11.0-0(X1, mark.0(X2)) → U11.0-0(X1, X2)
mark.0(cons.1-0(X1, X2)) → active.0(cons.0-0(mark.1(X1), X2))
U12.0-0(active.0(X1), X2) → U12.0-0(X1, X2)
U12.0-0(X1, mark.0(X2)) → U12.0-0(X1, X2)
s.0(mark.0(X)) → s.0(X)
U11.0-0(mark.0(X1), X2) → U11.0-0(X1, X2)
mark.0(U11.1-1(X1, X2)) → active.0(U11.0-1(mark.1(X1), X2))
U12.0-1(active.0(X1), X2) → U12.0-1(X1, X2)
cons.0-0(mark.0(X1), X2) → cons.0-0(X1, X2)
mark.0(U11.1-0(X1, X2)) → active.0(U11.0-0(mark.1(X1), X2))

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.0(length.0(cons.1-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.1(x0), x1))))
MARK.0(length.0(x0)) → ACTIVE.0(length.0(x0))
MARK.0(length.0(cons.1-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.1(x0), x1))))
MARK.0(U12.1-0(x0, x1)) → ACTIVE.0(U12.1-0(x0, x1))
MARK.0(U11.0-0(x0, x1)) → ACTIVE.0(U11.0-0(x0, x1))
MARK.0(length.0(cons.0-0(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-0(mark.0(x0), x1))))
MARK.0(U11.0-0(tt., y1)) → ACTIVE.0(U11.0-0(active.0(tt.), y1))
MARK.0(length.0(cons.0-1(x0, x1))) → ACTIVE.0(length.0(active.0(cons.0-1(mark.0(x0), x1))))
The remaining pairs can at least be oriented weakly.

ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))
Used ordering: Polynomial interpretation [25]:

POL(0.) = 0   
POL(ACTIVE.0(x1)) = 0   
POL(MARK.0(x1)) = x1   
POL(U11.0-0(x1, x2)) = 1 + x1   
POL(U11.0-1(x1, x2)) = 0   
POL(U11.1-0(x1, x2)) = 1 + x1 + x2   
POL(U11.1-1(x1, x2)) = 1 + x1 + x2   
POL(U12.0-0(x1, x2)) = x2   
POL(U12.0-1(x1, x2)) = 0   
POL(U12.1-0(x1, x2)) = 1 + x1   
POL(U12.1-1(x1, x2)) = 1 + x1 + x2   
POL(active.0(x1)) = 0   
POL(active.1(x1)) = 1 + x1   
POL(cons.0-0(x1, x2)) = x2   
POL(cons.0-1(x1, x2)) = 1   
POL(cons.1-0(x1, x2)) = 0   
POL(cons.1-1(x1, x2)) = x1   
POL(length.0(x1)) = 1 + x1   
POL(length.1(x1)) = 0   
POL(mark.0(x1)) = x1   
POL(mark.1(x1)) = 1 + x1   
POL(nil.) = 0   
POL(s.0(x1)) = x1   
POL(s.1(x1)) = 0   
POL(tt.) = 0   
POL(zeros.) = 1   

The following usable rules [17] were oriented: none



↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ SemLabProof
                                                                                                                                          ↳ QDP
                                                                                                                                            ↳ DependencyGraphProof
                                                                                                                                              ↳ QDP
                                                                                                                                                ↳ RuleRemovalProof
                                                                                                                                                  ↳ QDP
                                                                                                                                                    ↳ RuleRemovalProof
                                                                                                                                                      ↳ QDP
                                                                                                                                                        ↳ RuleRemovalProof
                                                                                                                                                          ↳ QDP
                                                                                                                                                            ↳ QDPOrderProof
                                                                                                                                                              ↳ QDP
                                                                                                                                                                ↳ QDPOrderProof
QDP
                                                                                                                                        ↳ SemLabProof2

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

ACTIVE.0(U11.0-1(tt., L)) → MARK.0(U12.0-1(tt., L))
ACTIVE.0(length.0(cons.1-1(N, L))) → MARK.0(U11.0-1(tt., L))
MARK.0(U12.0-1(x0, x1)) → ACTIVE.0(U12.0-1(x0, x1))
MARK.0(s.0(X)) → MARK.0(X)
MARK.0(U11.0-1(tt., y1)) → ACTIVE.0(U11.0-1(active.0(tt.), y1))
MARK.0(length.1(zeros.)) → ACTIVE.0(length.0(active.1(zeros.)))
MARK.0(U11.0-1(x0, x1)) → ACTIVE.0(U11.0-1(x0, x1))
MARK.0(U12.0-1(tt., y1)) → ACTIVE.0(U12.0-1(active.0(tt.), y1))
ACTIVE.0(length.0(cons.0-1(N, L))) → MARK.0(U11.0-1(tt., L))
ACTIVE.0(U12.0-1(tt., L)) → MARK.0(s.0(length.1(L)))

The TRS R consists of the following rules:

active.0(length.0(cons.0-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.1(X1), X2) → U12.1-0(X1, X2)
cons.1-0(X1, active.0(X2)) → cons.1-0(X1, X2)
mark.0(cons.1-1(X1, X2)) → active.0(cons.0-1(mark.1(X1), X2))
mark.1(zeros.) → active.1(zeros.)
cons.0-1(active.0(X1), X2) → cons.0-1(X1, X2)
U11.0-0(mark.1(X1), X2) → U11.1-0(X1, X2)
U11.0-0(X1, active.0(X2)) → U11.0-0(X1, X2)
U11.0-1(mark.0(X1), X2) → U11.0-1(X1, X2)
U11.0-1(mark.1(X1), X2) → U11.1-1(X1, X2)
U11.0-1(active.0(X1), X2) → U11.0-1(X1, X2)
cons.0-0(active.0(X1), X2) → cons.0-0(X1, X2)
mark.0(cons.0-1(X1, X2)) → active.0(cons.0-1(mark.0(X1), X2))
mark.0(U12.1-1(X1, X2)) → active.0(U12.0-1(mark.1(X1), X2))
mark.0(s.0(X)) → active.0(s.0(mark.0(X)))
active.0(U11.0-1(tt., L)) → mark.0(U12.0-1(tt., L))
U11.0-0(active.1(X1), X2) → U11.1-0(X1, X2)
cons.0-1(mark.0(X1), X2) → cons.0-1(X1, X2)
mark.0(0.) → active.0(0.)
cons.0-0(X1, mark.0(X2)) → cons.0-0(X1, X2)
U12.1-0(X1, mark.0(X2)) → U12.1-0(X1, X2)
mark.0(length.0(X)) → active.0(length.0(mark.0(X)))
length.0(mark.1(X)) → length.1(X)
cons.0-0(active.1(X1), X2) → cons.1-0(X1, X2)
U11.0-1(active.1(X1), X2) → U11.1-1(X1, X2)
mark.0(U12.0-1(X1, X2)) → active.0(U12.0-1(mark.0(X1), X2))
length.0(active.0(X)) → length.0(X)
U12.0-1(mark.0(X1), X2) → U12.0-1(X1, X2)
length.0(active.1(X)) → length.1(X)
active.0(U12.0-1(tt., L)) → mark.0(s.0(length.1(L)))
U11.1-0(X1, active.0(X2)) → U11.1-0(X1, X2)
cons.1-0(X1, mark.0(X2)) → cons.1-0(X1, X2)
U12.0-0(X1, active.0(X2)) → U12.0-0(X1, X2)
cons.0-0(X1, active.0(X2)) → cons.0-0(X1, X2)
mark.0(s.1(X)) → active.0(s.0(mark.1(X)))
mark.0(U11.0-1(X1, X2)) → active.0(U11.0-1(mark.0(X1), X2))
mark.0(cons.0-0(X1, X2)) → active.0(cons.0-0(mark.0(X1), X2))
mark.0(U12.1-0(X1, X2)) → active.0(U12.0-0(mark.1(X1), X2))
U11.0-0(active.0(X1), X2) → U11.0-0(X1, X2)
active.0(length.0(cons.1-1(N, L))) → mark.0(U11.0-1(tt., L))
U12.0-0(mark.0(X1), X2) → U12.0-0(X1, X2)
U12.1-0(X1, active.0(X2)) → U12.1-0(X1, X2)
mark.0(tt.) → active.0(tt.)
cons.0-1(mark.1(X1), X2) → cons.1-1(X1, X2)
mark.0(nil.) → active.0(nil.)
s.0(active.0(X)) → s.0(X)
length.0(mark.0(X)) → length.0(X)
cons.0-1(active.1(X1), X2) → cons.1-1(X1, X2)
mark.0(U11.0-0(X1, X2)) → active.0(U11.0-0(mark.0(X1), X2))
s.0(mark.1(X)) → s.1(X)
U11.1-0(X1, mark.0(X2)) → U11.1-0(X1, X2)
active.1(zeros.) → mark.0(cons.0-1(0., zeros.))
mark.0(length.1(X)) → active.0(length.0(mark.1(X)))
mark.0(U12.0-0(X1, X2)) → active.0(U12.0-0(mark.0(X1), X2))
U12.0-1(mark.1(X1), X2) → U12.1-1(X1, X2)
U12.0-1(active.1(X1), X2) → U12.1-1(X1, X2)
s.0(active.1(X)) → s.1(X)
U12.0-0(active.1(X1), X2) → U12.1-0(X1, X2)
cons.0-0(mark.1(X1), X2) → cons.1-0(X1, X2)
U11.0-0(X1, mark.0(X2)) → U11.0-0(X1, X2)
mark.0(cons.1-0(X1, X2)) → active.0(cons.0-0(mark.1(X1), X2))
U12.0-0(active.0(X1), X2) → U12.0-0(X1, X2)
U12.0-0(X1, mark.0(X2)) → U12.0-0(X1, X2)
s.0(mark.0(X)) → s.0(X)
U11.0-0(mark.0(X1), X2) → U11.0-0(X1, X2)
mark.0(U11.1-1(X1, X2)) → active.0(U11.0-1(mark.1(X1), X2))
U12.0-1(active.0(X1), X2) → U12.0-1(X1, X2)
cons.0-0(mark.0(X1), X2) → cons.0-0(X1, X2)
mark.0(U11.1-0(X1, X2)) → active.0(U11.0-0(mark.1(X1), X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
As can be seen after transforming the QDP problem by semantic labelling [33] and then some rule deleting processors, only certain labelled rules and pairs can be used. Hence, we only have to consider all unlabelled pairs and rules (without the decreasing rules for quasi-models).

↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
              ↳ QDP
                ↳ RuleRemovalProof
                  ↳ QDP
                    ↳ RuleRemovalProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ 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
                                                                                                                                  ↳ QDP
                                                                                                                                    ↳ QDPOrderProof
                                                                                                                                      ↳ QDP
                                                                                                                                        ↳ SemLabProof
                                                                                                                                        ↳ SemLabProof2
QDP

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

MARK(U12(tt, y1)) → ACTIVE(U12(active(tt), y1))
MARK(U12(x0, x1)) → ACTIVE(U12(x0, x1))
MARK(length(zeros)) → ACTIVE(length(active(zeros)))
ACTIVE(length(cons(N, L))) → MARK(U11(tt, L))
MARK(s(X)) → MARK(X)
MARK(U11(tt, y1)) → ACTIVE(U11(active(tt), y1))
MARK(U11(x0, x1)) → ACTIVE(U11(x0, x1))
ACTIVE(U11(tt, L)) → MARK(U12(tt, L))
ACTIVE(U12(tt, L)) → MARK(s(length(L)))

The TRS R consists of the following rules:

active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(cons(N, L))) → mark(U11(tt, L))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(U11(X1, X2)) → active(U11(mark(X1), X2))
mark(tt) → active(tt)
mark(U12(X1, X2)) → active(U12(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(length(X)) → active(length(mark(X)))
mark(nil) → active(nil)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
U11(mark(X1), X2) → U11(X1, X2)
U11(X1, mark(X2)) → U11(X1, X2)
U11(active(X1), X2) → U11(X1, X2)
U11(X1, active(X2)) → U11(X1, X2)
U12(mark(X1), X2) → U12(X1, X2)
U12(X1, mark(X2)) → U12(X1, X2)
U12(active(X1), X2) → U12(X1, X2)
U12(X1, active(X2)) → U12(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
length(mark(X)) → length(X)
length(active(X)) → length(X)

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