Termination w.r.t. Q proof of /tmp/tmpz_mCVO/LengthOfFiniteLists_nosorts_noand

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:

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ 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)

↳ 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)

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ 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)

↳ 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)

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

U12¹(X1, mark(X2)) → U12¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U12¹(mark(X1), X2) → U12¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
U12¹(X1, active(X2)) → U12¹(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
U12¹(active(X1), X2) → U12¹(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ 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)

↳ 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)

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ 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)

↳ 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)

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

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

↳ QTRS

  ↳ RRRPoloQTRSProof

    ↳ QTRS

      ↳ 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)

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	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( zeros ) =

/	0	\
\	0	/

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U12(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	1	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

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

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

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	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	0	/

M( zeros ) =

/	0	\
\	0	/

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( U12(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	0	/

x₁

/	0	0	\
\	0	0	/

x₂

M( length(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

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	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( tt ) =

/	0	\
\	1	/

M( zeros ) =

/	0	\
\	0	/

M( mark(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U12(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( length(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

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	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	1	1	/

x₂

M( tt ) =

/	0	\
\	0	/

M( zeros ) =

/	0	\
\	0	/

M( mark(x₁) ) =

/	1	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( U12(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( length(x₁) ) =

/	0	\
\	1	/

/	0	1	\
\	0	1	/

x₁

M( 0 ) =

/	1	\
\	0	/

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

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	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( tt ) =

/	1	\
\	1	/

M( zeros ) =

/	0	\
\	0	/

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U12(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

/	1	1	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( length(x₁) ) =

/	1	\
\	1	/

/	1	1	\
\	1	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

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	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( tt ) =

/	0	\
\	0	/

M( zeros ) =

/	0	\
\	0	/

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( U12(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( length(x₁) ) =

/	1	\
\	0	/

/	1	1	\
\	0	1	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

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	/

/	1	0	\
\	0	1	/

x₁

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( tt ) =

/	1	\
\	1	/

M( zeros ) =

/	1	\
\	0	/

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( U12(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	1	0	/

x₁

/	1	1	\
\	1	1	/

x₂

M( U11(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

/	1	1	\
\	1	1	/

x₂

M( length(x₁) ) =

/	1	\
\	1	/

/	1	1	\
\	1	0	/

x₁

M( 0 ) =

/	0	\
\	0	/

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( MARK(x₁) ) =

[

]

x₁

M( ACTIVE(x₁) ) =

[

]

x₁

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))

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))

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))

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.