Termination proof of ../tpdb/TRS/CSR_Maude/length-lazy-list/LengthOfFiniteLists

MAYBE Termination proof of ../tpdb/TRS/CSR_Maude/length-lazy-list/LengthOfFiniteLists_nosorts-noand.trs
Termination of the following Term Rewriting System could not be shown:

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

zeros → cons(0, zeros)
U11(tt, L) → U12(tt, L)
U12(tt, L) → s(length(L))
length(nil) → 0
length(cons(N, L)) → U11(tt, L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
nil: empty set

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

zeros → cons(0, zeros)
U11(tt, L) → U12(tt, L)
U12(tt, L) → s(length(L))
length(nil) → 0
length(cons(N, L)) → U11(tt, L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U12: {1}
s: {1}
length: {1}
nil: empty set

We applied the Zantema [34] to transform the context-sensitive TRS to an usual TRS.

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(nil) → 0
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

Q is empty.

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

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(nil) → 0
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

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

length(nil) → 0

Used ordering:
Polynomial interpretation [25]:

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

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

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:

U12¹(tt, L) → LENGTH(a(L))
U11¹(tt, L) → U12¹(tt, a(L))
LENGTH(cons(N, L)) → A(L)
U12¹(tt, L) → A(L)
A(zerosInact) → ZEROS
U11¹(tt, L) → A(L)
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

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

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U12¹(tt, L) → LENGTH(a(L))
U11¹(tt, L) → U12¹(tt, a(L))
LENGTH(cons(N, L)) → A(L)
U12¹(tt, L) → A(L)
A(zerosInact) → ZEROS
U11¹(tt, L) → A(L)
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

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.

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ UsableRulesProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U11¹(tt, L) → U12¹(tt, a(L))
U12¹(tt, L) → LENGTH(a(L))
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

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.

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ UsableRulesProof

                    ↳ QDP

                      ↳ Narrowing

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U12¹(tt, L) → LENGTH(a(L))
U11¹(tt, L) → U12¹(tt, a(L))
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

a(x) → x
a(zerosInact) → zeros
zeros → cons(0, zerosInact)
zeros → zerosInact

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U12¹(tt, L) → LENGTH(a(L)) at position [0] we obtained the following new rules:

U12¹(tt, x0) → LENGTH(x0)
U12¹(tt, zerosInact) → LENGTH(zeros)

↳ CSR

  ↳ Zantema-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ UsableRulesProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ NonTerminationProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U12¹(tt, x0) → LENGTH(x0)
U11¹(tt, L) → U12¹(tt, a(L))
U12¹(tt, zerosInact) → LENGTH(zeros)
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

a(x) → x
a(zerosInact) → zeros
zeros → cons(0, zerosInact)
zeros → zerosInact

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U12¹(tt, x0) → LENGTH(x0)
U11¹(tt, L) → U12¹(tt, a(L))
U12¹(tt, zerosInact) → LENGTH(zeros)
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

a(x) → x
a(zerosInact) → zeros
zeros → cons(0, zerosInact)
zeros → zerosInact

s = LENGTH(a(a(zerosInact))) evaluates to t =LENGTH(a(a(zerosInact)))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

LENGTH(a(a(zerosInact))) → LENGTH(a(zerosInact))
with rule a(x) → x at position [0] and matcher [x / a(zerosInact)]

LENGTH(a(zerosInact)) → LENGTH(zeros)
with rule a(zerosInact) → zeros at position [0] and matcher [ ]

LENGTH(zeros) → LENGTH(cons(0, zerosInact))
with rule zeros → cons(0, zerosInact) at position [0] and matcher [ ]

LENGTH(cons(0, zerosInact)) → U11¹(tt, a(zerosInact))
with rule LENGTH(cons(N, L')) → U11¹(tt, a(L')) at position [] and matcher [L' / zerosInact, N / 0]

U11¹(tt, a(zerosInact)) → U12¹(tt, a(a(zerosInact)))
with rule U11¹(tt, L) → U12¹(tt, a(L)) at position [] and matcher [L / a(zerosInact)]

U12¹(tt, a(a(zerosInact))) → LENGTH(a(a(zerosInact)))
with rule U12¹(tt, x0) → LENGTH(x0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

We applied the Incomplete Giesl Middeldorp [11] to transform the context-sensitive TRS to an usual TRS.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U11Active(tt, L)

Q is empty.

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

mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U11Active(tt, L)

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

lengthActive(nil) → 0

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U11Active(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U12(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U12Active(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(length(x₁)) = x₁
POL(lengthActive(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 2
POL(s(x₁)) = 2·x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

Q is empty.

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

mark(zeros) → zerosActive
zerosActive → zeros
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
mark(nil) → nil
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

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

zerosActive → zeros
mark(nil) → nil

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U11Active(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U12(x₁, x₂)) = x₁ + 2·x₂
POL(U12Active(x₁, x₂)) = x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(length(x₁)) = x₁
POL(lengthActive(x₁)) = x₁
POL(mark(x₁)) = 2·x₁
POL(nil) = 2
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 1
POL(zerosActive) = 2

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

Q is empty.

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

mark(zeros) → zerosActive
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
U11Active(x1, x2) → U11(x1, x2)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
U12Active(x1, x2) → U12(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

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

mark(U11(x1, x2)) → U11Active(mark(x1), x2)
mark(U12(x1, x2)) → U12Active(mark(x1), x2)
mark(length(x1)) → lengthActive(mark(x1))

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U11Active(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U12(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U12Active(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(length(x₁)) = 1 + x₁
POL(lengthActive(x₁)) = 1 + x₁
POL(mark(x₁)) = 2·x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
U11Active(x1, x2) → U11(x1, x2)
U12Active(x1, x2) → U12(x1, x2)
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

Q is empty.

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

mark(zeros) → zerosActive
U11Active(x1, x2) → U11(x1, x2)
U12Active(x1, x2) → U12(x1, x2)
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

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

lengthActive(x1) → length(x1)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U11Active(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U12(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U12Active(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(length(x₁)) = 1 + x₁
POL(lengthActive(x₁)) = 2 + 2·x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 2
POL(zerosActive) = 2

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
U11Active(x1, x2) → U11(x1, x2)
U12Active(x1, x2) → U12(x1, x2)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

Q is empty.

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

mark(zeros) → zerosActive
U11Active(x1, x2) → U11(x1, x2)
U12Active(x1, x2) → U12(x1, x2)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

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

U12Active(x1, x2) → U12(x1, x2)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(U11Active(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U12(x₁, x₂)) = 2·x₁ + x₂
POL(U12Active(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(lengthActive(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
U11Active(x1, x2) → U11(x1, x2)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

Q is empty.

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

mark(zeros) → zerosActive
U11Active(x1, x2) → U11(x1, x2)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

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

U11Active(x1, x2) → U11(x1, x2)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U11Active(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U12Active(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(lengthActive(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

Q is empty.

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

mark(zeros) → zerosActive
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(tt) → tt
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

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

mark(tt) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11Active(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U12Active(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(lengthActive(x₁)) = 2 + x₁
POL(mark(x₁)) = 2·x₁
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

Q is empty.

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

mark(zeros) → zerosActive
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

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

mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(0) → 0

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11Active(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U12Active(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(lengthActive(x₁)) = x₁
POL(mark(x₁)) = 1 + 2·x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 1

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

Q is empty.

The TRS is overlay and locally confluent. By [19] we can switch to innermost.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

mark(zeros) → zerosActive
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive
U11Active(tt, x0)
U12Active(tt, x0)
lengthActive(cons(x0, x1))

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

U12ACTIVE(tt, L) → MARK(L)
MARK(s(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U12ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
MARK(zeros) → ZEROSACTIVE
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

mark(zeros) → zerosActive
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive
U11Active(tt, x0)
U12Active(tt, x0)
lengthActive(cons(x0, x1))

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

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U12ACTIVE(tt, L) → MARK(L)
MARK(s(x1)) → MARK(x1)
LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U12ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
MARK(zeros) → ZEROSACTIVE
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

mark(zeros) → zerosActive
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive
U11Active(tt, x0)
U12Active(tt, x0)
lengthActive(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 2 less nodes.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                  ↳ QDP

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

MARK(s(x1)) → MARK(x1)

The TRS R consists of the following rules:

mark(zeros) → zerosActive
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive
U11Active(tt, x0)
U12Active(tt, x0)
lengthActive(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                  ↳ QDP

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

MARK(s(x1)) → MARK(x1)

R is empty.
The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive
U11Active(tt, x0)
U12Active(tt, x0)
lengthActive(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

mark(zeros)
mark(s(x0))
zerosActive
U11Active(tt, x0)
U12Active(tt, x0)
lengthActive(cons(x0, x1))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ QDPSizeChangeProof

                                                  ↳ QDP

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

MARK(s(x1)) → MARK(x1)

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:

MARK(s(x1)) → MARK(x1)
The graph contains the following edges 1 > 1

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U12ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

mark(zeros) → zerosActive
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → U12Active(tt, L)
U12Active(tt, L) → s(lengthActive(mark(L)))
lengthActive(cons(N, L)) → U11Active(tt, L)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive
U11Active(tt, x0)
U12Active(tt, x0)
lengthActive(cons(x0, x1))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U12ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

mark(zeros) → zerosActive
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive
U11Active(tt, x0)
U12Active(tt, x0)
lengthActive(cons(x0, x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

U11Active(tt, x0)
U12Active(tt, x0)
lengthActive(cons(x0, x1))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U12ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

mark(zeros) → zerosActive
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive

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 rules of the TRS R:

mark(s(x1)) → s(mark(x1))

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(LENGTHACTIVE(x₁)) = x₁
POL(U11ACTIVE(x₁, x₂)) = x₁ + 2·x₂
POL(U12ACTIVE(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(mark(x₁)) = 2·x₁
POL(s(x₁)) = 2 + 2·x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

                                                              ↳ QDP

                                                                ↳ Narrowing

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U12ACTIVE(tt, L) → LENGTHACTIVE(mark(L))
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

mark(zeros) → zerosActive
zerosActive → cons(0, zeros)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U12ACTIVE(tt, L) → LENGTHACTIVE(mark(L)) at position [0] we obtained the following new rules:

U12ACTIVE(tt, zeros) → LENGTHACTIVE(zerosActive)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U12ACTIVE(tt, zeros) → LENGTHACTIVE(zerosActive)
LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

mark(zeros) → zerosActive
zerosActive → cons(0, zeros)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U12ACTIVE(tt, zeros) → LENGTHACTIVE(zerosActive)
LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

zerosActive → cons(0, zeros)

The set Q consists of the following terms:

mark(zeros)
mark(s(x0))
zerosActive

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

mark(zeros)
mark(s(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U12ACTIVE(tt, zeros) → LENGTHACTIVE(zerosActive)
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

zerosActive → cons(0, zeros)

The set Q consists of the following terms:

zerosActive

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule U12ACTIVE(tt, zeros) → LENGTHACTIVE(zerosActive) at position [0] we obtained the following new rules:

U12ACTIVE(tt, zeros) → LENGTHACTIVE(cons(0, zeros))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U12ACTIVE(tt, zeros) → LENGTHACTIVE(cons(0, zeros))
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

The TRS R consists of the following rules:

zerosActive → cons(0, zeros)

The set Q consists of the following terms:

zerosActive

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U12ACTIVE(tt, zeros) → LENGTHACTIVE(cons(0, zeros))
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

R is empty.
The set Q consists of the following terms:

zerosActive

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

zerosActive

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U12ACTIVE(tt, zeros) → LENGTHACTIVE(cons(0, zeros))
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L) we obtained the following new rules:

LENGTHACTIVE(cons(0, zeros)) → U11ACTIVE(tt, zeros)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U12ACTIVE(tt, zeros) → LENGTHACTIVE(cons(0, zeros))
LENGTHACTIVE(cons(0, zeros)) → U11ACTIVE(tt, zeros)
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule U11ACTIVE(tt, L) → U12ACTIVE(tt, L) we obtained the following new rules:

U11ACTIVE(tt, zeros) → U12ACTIVE(tt, zeros)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

                                        ↳ QTRS

                                          ↳ DependencyPairsProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ AND

                                                  ↳ QDP

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ QReductionProof

                                                          ↳ QDP

                                                            ↳ RuleRemovalProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

                                                                                        ↳ Instantiation

                                                                                          ↳ QDP

                                                                                            ↳ Instantiation

                                                                                              ↳ QDP

                                                                                                ↳ NonTerminationProof

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

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

U12ACTIVE(tt, zeros) → LENGTHACTIVE(cons(0, zeros))
LENGTHACTIVE(cons(0, zeros)) → U11ACTIVE(tt, zeros)
U11ACTIVE(tt, zeros) → U12ACTIVE(tt, zeros)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

The TRS P consists of the following rules:

U12ACTIVE(tt, zeros) → LENGTHACTIVE(cons(0, zeros))
LENGTHACTIVE(cons(0, zeros)) → U11ACTIVE(tt, zeros)
U11ACTIVE(tt, zeros) → U12ACTIVE(tt, zeros)

The TRS R consists of the following rules:none

s = LENGTHACTIVE(cons(0, zeros)) evaluates to t =LENGTHACTIVE(cons(0, zeros))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

LENGTHACTIVE(cons(0, zeros)) → U11ACTIVE(tt, zeros)
with rule LENGTHACTIVE(cons(0, zeros)) → U11ACTIVE(tt, zeros) at position [] and matcher [ ]

U11ACTIVE(tt, zeros) → U12ACTIVE(tt, zeros)
with rule U11ACTIVE(tt, zeros) → U12ACTIVE(tt, zeros) at position [] and matcher [ ]

U12ACTIVE(tt, zeros) → LENGTHACTIVE(cons(0, zeros))
with rule U12ACTIVE(tt, zeros) → LENGTHACTIVE(cons(0, zeros))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

We applied the Improved Ferreira Ribeiro [5,11] to transform the context-sensitive TRS to an usual TRS.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

  ↳ Complete Giesl Middeldorp-Transformation

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

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(nil) → 0
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

Q is empty.

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

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(nil) → 0
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

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

length(nil) → 0

Used ordering:
Polynomial interpretation [25]:

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

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

  ↳ Complete Giesl Middeldorp-Transformation

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

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

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:

U12¹(tt, L) → LENGTH(a(L))
U11¹(tt, L) → U12¹(tt, a(L))
LENGTH(cons(N, L)) → A(L)
U12¹(tt, L) → A(L)
A(zerosInact) → ZEROS
U11¹(tt, L) → A(L)
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

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

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

  ↳ Complete Giesl Middeldorp-Transformation

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

U12¹(tt, L) → LENGTH(a(L))
U11¹(tt, L) → U12¹(tt, a(L))
LENGTH(cons(N, L)) → A(L)
U12¹(tt, L) → A(L)
A(zerosInact) → ZEROS
U11¹(tt, L) → A(L)
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ UsableRulesProof

  ↳ Complete Giesl Middeldorp-Transformation

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

U11¹(tt, L) → U12¹(tt, a(L))
U12¹(tt, L) → LENGTH(a(L))
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

zeros → cons(0, zerosInact)
U11(tt, L) → U12(tt, a(L))
U12(tt, L) → s(length(a(L)))
length(cons(N, L)) → U11(tt, a(L))
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ UsableRulesProof

                    ↳ QDP

                      ↳ Narrowing

  ↳ Complete Giesl Middeldorp-Transformation

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

U12¹(tt, L) → LENGTH(a(L))
U11¹(tt, L) → U12¹(tt, a(L))
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

a(x) → x
a(zerosInact) → zeros
zeros → cons(0, zerosInact)
zeros → zerosInact

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U12¹(tt, L) → LENGTH(a(L)) at position [0] we obtained the following new rules:

U12¹(tt, x0) → LENGTH(x0)
U12¹(tt, zerosInact) → LENGTH(zeros)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ UsableRulesProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ NonTerminationProof

  ↳ Complete Giesl Middeldorp-Transformation

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

U12¹(tt, x0) → LENGTH(x0)
U11¹(tt, L) → U12¹(tt, a(L))
U12¹(tt, zerosInact) → LENGTH(zeros)
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

a(x) → x
a(zerosInact) → zeros
zeros → cons(0, zerosInact)
zeros → zerosInact

U12¹(tt, x0) → LENGTH(x0)
U11¹(tt, L) → U12¹(tt, a(L))
U12¹(tt, zerosInact) → LENGTH(zeros)
LENGTH(cons(N, L)) → U11¹(tt, a(L))

The TRS R consists of the following rules:

a(x) → x
a(zerosInact) → zeros
zeros → cons(0, zerosInact)
zeros → zerosInact

s = LENGTH(a(a(zerosInact))) evaluates to t =LENGTH(a(a(zerosInact)))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

LENGTH(a(a(zerosInact))) → LENGTH(a(zerosInact))
with rule a(x) → x at position [0] and matcher [x / a(zerosInact)]

LENGTH(a(zerosInact)) → LENGTH(zeros)
with rule a(zerosInact) → zeros at position [0] and matcher [ ]

LENGTH(zeros) → LENGTH(cons(0, zerosInact))
with rule zeros → cons(0, zerosInact) at position [0] and matcher [ ]

LENGTH(cons(0, zerosInact)) → U11¹(tt, a(zerosInact))
with rule LENGTH(cons(N, L')) → U11¹(tt, a(L')) at position [] and matcher [L' / zerosInact, N / 0]

U11¹(tt, a(zerosInact)) → U12¹(tt, a(a(zerosInact)))
with rule U11¹(tt, L) → U12¹(tt, a(L)) at position [] and matcher [L / a(zerosInact)]

U12¹(tt, a(a(zerosInact))) → LENGTH(a(a(zerosInact)))
with rule U12¹(tt, x0) → LENGTH(x0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

We applied the Complete Giesl Middeldorp [11] to transform the context-sensitive TRS to an usual TRS.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ 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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

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

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ 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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

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

TOP(mark(x)) → PROPER(x)
ACTIVE(zeros) → CONS(0, zeros)
PROPER(U12(x1, x2)) → PROPER(x1)
LENGTH(ok(x1)) → LENGTH(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(U12(x1, x2)) → U12¹(proper(x1), proper(x2))
PROPER(U12(x1, x2)) → PROPER(x2)
U12¹(mark(x1), x2) → U12¹(x1, x2)
S(mark(x1)) → S(x1)
LENGTH(mark(x1)) → LENGTH(x1)
TOP(mark(x)) → TOP(proper(x))
PROPER(s(x1)) → S(proper(x1))
PROPER(U11(x1, x2)) → U11¹(proper(x1), proper(x2))
S(ok(x1)) → S(x1)
ACTIVE(s(x1)) → ACTIVE(x1)
CONS(mark(x1), x2) → CONS(x1, x2)
ACTIVE(cons(x1, x2)) → CONS(active(x1), x2)
ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(U12(x1, x2)) → U12¹(active(x1), x2)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
PROPER(cons(x1, x2)) → PROPER(x1)
ACTIVE(length(x1)) → LENGTH(active(x1))
U12¹(ok(x1), ok(x2)) → U12¹(x1, x2)
PROPER(length(x1)) → LENGTH(proper(x1))
U11¹(ok(x1), ok(x2)) → U11¹(x1, x2)
ACTIVE(U12(tt, L)) → S(length(L))
TOP(ok(x)) → ACTIVE(x)
U11¹(mark(x1), x2) → U11¹(x1, x2)
ACTIVE(s(x1)) → S(active(x1))
PROPER(U11(x1, x2)) → PROPER(x1)
PROPER(cons(x1, x2)) → CONS(proper(x1), proper(x2))
ACTIVE(U11(x1, x2)) → ACTIVE(x1)
PROPER(s(x1)) → PROPER(x1)
TOP(ok(x)) → TOP(active(x))
PROPER(cons(x1, x2)) → PROPER(x2)
ACTIVE(U12(x1, x2)) → ACTIVE(x1)
ACTIVE(U11(x1, x2)) → U11¹(active(x1), x2)
ACTIVE(U12(tt, L)) → LENGTH(L)
CONS(ok(x1), ok(x2)) → CONS(x1, x2)
ACTIVE(length(cons(N, L))) → U11¹(tt, L)
PROPER(U11(x1, x2)) → PROPER(x2)
ACTIVE(U11(tt, L)) → U12¹(tt, 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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

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

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

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

TOP(mark(x)) → PROPER(x)
ACTIVE(zeros) → CONS(0, zeros)
PROPER(U12(x1, x2)) → PROPER(x1)
LENGTH(ok(x1)) → LENGTH(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(U12(x1, x2)) → U12¹(proper(x1), proper(x2))
PROPER(U12(x1, x2)) → PROPER(x2)
U12¹(mark(x1), x2) → U12¹(x1, x2)
S(mark(x1)) → S(x1)
LENGTH(mark(x1)) → LENGTH(x1)
TOP(mark(x)) → TOP(proper(x))
PROPER(s(x1)) → S(proper(x1))
PROPER(U11(x1, x2)) → U11¹(proper(x1), proper(x2))
S(ok(x1)) → S(x1)
ACTIVE(s(x1)) → ACTIVE(x1)
CONS(mark(x1), x2) → CONS(x1, x2)
ACTIVE(cons(x1, x2)) → CONS(active(x1), x2)
ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(U12(x1, x2)) → U12¹(active(x1), x2)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
PROPER(cons(x1, x2)) → PROPER(x1)
ACTIVE(length(x1)) → LENGTH(active(x1))
U12¹(ok(x1), ok(x2)) → U12¹(x1, x2)
PROPER(length(x1)) → LENGTH(proper(x1))
U11¹(ok(x1), ok(x2)) → U11¹(x1, x2)
ACTIVE(U12(tt, L)) → S(length(L))
TOP(ok(x)) → ACTIVE(x)
U11¹(mark(x1), x2) → U11¹(x1, x2)
ACTIVE(s(x1)) → S(active(x1))
PROPER(U11(x1, x2)) → PROPER(x1)
PROPER(cons(x1, x2)) → CONS(proper(x1), proper(x2))
ACTIVE(U11(x1, x2)) → ACTIVE(x1)
PROPER(s(x1)) → PROPER(x1)
TOP(ok(x)) → TOP(active(x))
PROPER(cons(x1, x2)) → PROPER(x2)
ACTIVE(U12(x1, x2)) → ACTIVE(x1)
ACTIVE(U11(x1, x2)) → U11¹(active(x1), x2)
ACTIVE(U12(tt, L)) → LENGTH(L)
CONS(ok(x1), ok(x2)) → CONS(x1, x2)
ACTIVE(length(cons(N, L))) → U11¹(tt, L)
PROPER(U11(x1, x2)) → PROPER(x2)
ACTIVE(U11(tt, L)) → U12¹(tt, 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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 8 SCCs with 17 less nodes.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

LENGTH(mark(x1)) → LENGTH(x1)
LENGTH(ok(x1)) → LENGTH(x1)

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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

LENGTH(mark(x1)) → LENGTH(x1)
LENGTH(ok(x1)) → LENGTH(x1)

R is empty.
The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

LENGTH(ok(x1)) → LENGTH(x1)
LENGTH(mark(x1)) → LENGTH(x1)

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

LENGTH(mark(x1)) → LENGTH(x1)
The graph contains the following edges 1 > 1
LENGTH(ok(x1)) → LENGTH(x1)
The graph contains the following edges 1 > 1

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

S(ok(x1)) → S(x1)
S(mark(x1)) → S(x1)

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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

S(ok(x1)) → S(x1)
S(mark(x1)) → S(x1)

R is empty.
The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

S(ok(x1)) → S(x1)
S(mark(x1)) → S(x1)

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

S(ok(x1)) → S(x1)
The graph contains the following edges 1 > 1
S(mark(x1)) → S(x1)
The graph contains the following edges 1 > 1

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

U12¹(ok(x1), ok(x2)) → U12¹(x1, x2)
U12¹(mark(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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

U12¹(ok(x1), ok(x2)) → U12¹(x1, x2)
U12¹(mark(x1), x2) → U12¹(x1, x2)

R is empty.
The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

U12¹(ok(x1), ok(x2)) → U12¹(x1, x2)
U12¹(mark(x1), x2) → U12¹(x1, x2)

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

U12¹(ok(x1), ok(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

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

U11¹(mark(x1), x2) → U11¹(x1, x2)
U11¹(ok(x1), ok(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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

U11¹(mark(x1), x2) → U11¹(x1, x2)
U11¹(ok(x1), ok(x2)) → U11¹(x1, x2)

R is empty.
The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

U11¹(mark(x1), x2) → U11¹(x1, x2)
U11¹(ok(x1), ok(x2)) → U11¹(x1, x2)

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

U11¹(mark(x1), x2) → U11¹(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2
U11¹(ok(x1), ok(x2)) → U11¹(x1, x2)
The graph contains the following edges 1 > 1, 2 > 2

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

CONS(mark(x1), x2) → CONS(x1, x2)
CONS(ok(x1), ok(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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

CONS(mark(x1), x2) → CONS(x1, x2)
CONS(ok(x1), ok(x2)) → CONS(x1, x2)

R is empty.
The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

CONS(mark(x1), x2) → CONS(x1, x2)
CONS(ok(x1), ok(x2)) → CONS(x1, x2)

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

CONS(mark(x1), x2) → CONS(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(ok(x1), ok(x2)) → CONS(x1, x2)
The graph contains the following edges 1 > 1, 2 > 2

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

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

PROPER(U12(x1, x2)) → PROPER(x1)
PROPER(U11(x1, x2)) → PROPER(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(s(x1)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x2)
PROPER(U12(x1, x2)) → PROPER(x2)
PROPER(U11(x1, x2)) → PROPER(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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

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

PROPER(U12(x1, x2)) → PROPER(x1)
PROPER(U11(x1, x2)) → PROPER(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(s(x1)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x2)
PROPER(U12(x1, x2)) → PROPER(x2)
PROPER(U11(x1, x2)) → PROPER(x2)

R is empty.
The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

active(zeros)
proper(zeros)
active(cons(x0, x1))
proper(cons(x0, x1))
proper(0)
active(U11(x0, x1))
proper(U11(x0, x1))
proper(tt)
active(U12(x0, x1))
proper(U12(x0, x1))
active(s(x0))
proper(s(x0))
active(length(x0))
proper(length(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

PROPER(U11(x1, x2)) → PROPER(x1)
PROPER(U12(x1, x2)) → PROPER(x1)
PROPER(s(x1)) → PROPER(x1)
PROPER(length(x1)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x1)
PROPER(cons(x1, x2)) → PROPER(x2)
PROPER(U12(x1, x2)) → PROPER(x2)
PROPER(U11(x1, x2)) → PROPER(x2)

R is empty.
The set Q consists of the following terms:

cons(mark(x0), x1)
cons(ok(x0), ok(x1))
U11(mark(x0), x1)
U11(ok(x0), ok(x1))
U12(mark(x0), x1)
U12(ok(x0), ok(x1))
s(mark(x0))
s(ok(x0))
length(mark(x0))
length(ok(x0))

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:

PROPER(U12(x1, x2)) → PROPER(x1)
The graph contains the following edges 1 > 1
PROPER(U11(x1, x2)) → PROPER(x1)
The graph contains the following edges 1 > 1
PROPER(length(x1)) → PROPER(x1)
The graph contains the following edges 1 > 1
PROPER(s(x1)) → PROPER(x1)
The graph contains the following edges 1 > 1
PROPER(cons(x1, x2)) → PROPER(x1)
The graph contains the following edges 1 > 1
PROPER(cons(x1, x2)) → PROPER(x2)
The graph contains the following edges 1 > 1
PROPER(U12(x1, x2)) → PROPER(x2)
The graph contains the following edges 1 > 1
PROPER(U11(x1, x2)) → PROPER(x2)
The graph contains the following edges 1 > 1

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

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

ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
ACTIVE(U11(x1, x2)) → ACTIVE(x1)
ACTIVE(U12(x1, x2)) → ACTIVE(x1)
ACTIVE(s(x1)) → ACTIVE(x1)

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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

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

ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
ACTIVE(U11(x1, x2)) → ACTIVE(x1)
ACTIVE(U12(x1, x2)) → ACTIVE(x1)
ACTIVE(s(x1)) → ACTIVE(x1)

R is empty.
The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

active(zeros)
proper(zeros)
active(cons(x0, x1))
proper(cons(x0, x1))
proper(0)
active(U11(x0, x1))
proper(U11(x0, x1))
proper(tt)
active(U12(x0, x1))
proper(U12(x0, x1))
active(s(x0))
proper(s(x0))
active(length(x0))
proper(length(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

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

ACTIVE(length(x1)) → ACTIVE(x1)
ACTIVE(U11(x1, x2)) → ACTIVE(x1)
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
ACTIVE(U12(x1, x2)) → ACTIVE(x1)
ACTIVE(s(x1)) → ACTIVE(x1)

R is empty.
The set Q consists of the following terms:

cons(mark(x0), x1)
cons(ok(x0), ok(x1))
U11(mark(x0), x1)
U11(ok(x0), ok(x1))
U12(mark(x0), x1)
U12(ok(x0), ok(x1))
s(mark(x0))
s(ok(x0))
length(mark(x0))
length(ok(x0))

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:

ACTIVE(length(x1)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
ACTIVE(cons(x1, x2)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
ACTIVE(U11(x1, x2)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
ACTIVE(U12(x1, x2)) → ACTIVE(x1)
The graph contains the following edges 1 > 1
ACTIVE(s(x1)) → ACTIVE(x1)
The graph contains the following edges 1 > 1

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

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

TOP(ok(x)) → TOP(active(x))
TOP(mark(x)) → TOP(proper(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))
proper(zeros) → ok(zeros)
active(cons(x1, x2)) → cons(active(x1), x2)
cons(mark(x1), x2) → mark(cons(x1, x2))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(0) → ok(0)
active(U11(x1, x2)) → U11(active(x1), x2)
U11(mark(x1), x2) → mark(U11(x1, x2))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
proper(tt) → ok(tt)
active(U12(x1, x2)) → U12(active(x1), x2)
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(s(x1)) → s(active(x1))
s(mark(x1)) → mark(s(x1))
proper(s(x1)) → s(proper(x1))
s(ok(x1)) → ok(s(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
proper(length(x1)) → length(proper(x1))
length(ok(x1)) → ok(length(x1))
proper(nil) → ok(nil)
top(mark(x)) → top(proper(x))
top(ok(x)) → top(active(x))

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

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

TOP(ok(x)) → TOP(active(x))
TOP(mark(x)) → TOP(proper(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))
active(cons(x1, x2)) → cons(active(x1), x2)
active(U11(x1, x2)) → U11(active(x1), x2)
active(U12(x1, x2)) → U12(active(x1), x2)
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
U12(mark(x1), x2) → mark(U12(x1, x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U11(mark(x1), x2) → mark(U11(x1, x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(0) → ok(0)
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(nil) → ok(nil)

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)
top(mark(x0))
top(ok(x0))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

top(mark(x0))
top(ok(x0))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ Narrowing

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

TOP(ok(x)) → TOP(active(x))
TOP(mark(x)) → TOP(proper(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))
active(cons(x1, x2)) → cons(active(x1), x2)
active(U11(x1, x2)) → U11(active(x1), x2)
active(U12(x1, x2)) → U12(active(x1), x2)
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
U12(mark(x1), x2) → mark(U12(x1, x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U11(mark(x1), x2) → mark(U11(x1, x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(0) → ok(0)
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(nil) → ok(nil)

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)

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

TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(tt, x0)))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(tt, x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(ok(length(x0))) → TOP(length(active(x0)))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

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

TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(x)) → TOP(proper(x))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(tt, x0)))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(tt, x1)))
TOP(ok(length(x0))) → TOP(length(active(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))
active(cons(x1, x2)) → cons(active(x1), x2)
active(U11(x1, x2)) → U11(active(x1), x2)
active(U12(x1, x2)) → U12(active(x1), x2)
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
U12(mark(x1), x2) → mark(U12(x1, x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U11(mark(x1), x2) → mark(U11(x1, x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(0) → ok(0)
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(nil) → ok(nil)

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(mark(x)) → TOP(proper(x)) at position [0] we obtained the following new rules:

TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(tt)) → TOP(ok(tt))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

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

TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(tt, x1)))
TOP(mark(zeros)) → TOP(ok(zeros))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(mark(tt)) → TOP(ok(tt))
TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(tt, x0)))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(mark(0)) → TOP(ok(0))

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))
active(cons(x1, x2)) → cons(active(x1), x2)
active(U11(x1, x2)) → U11(active(x1), x2)
active(U12(x1, x2)) → U12(active(x1), x2)
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
U12(mark(x1), x2) → mark(U12(x1, x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U11(mark(x1), x2) → mark(U11(x1, x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(0) → ok(0)
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(nil) → ok(nil)

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)

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.

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

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

TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(tt, x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(tt, x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(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))
active(cons(x1, x2)) → cons(active(x1), x2)
active(U11(x1, x2)) → U11(active(x1), x2)
active(U12(x1, x2)) → U12(active(x1), x2)
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
U12(mark(x1), x2) → mark(U12(x1, x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U11(mark(x1), x2) → mark(U11(x1, x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(0) → ok(0)
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(nil) → ok(nil)

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)

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.

TOP(ok(zeros)) → TOP(mark(cons(0, zeros)))

The remaining pairs can at least be oriented weakly.

TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(tt, x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(tt, x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁, x₂)) = 0
POL(active(x₁)) = 1
POL(cons(x₁, x₂)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 1

The following usable rules [17] were oriented:

s(mark(x1)) → mark(s(x1))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
length(mark(x1)) → mark(length(x1))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
s(ok(x1)) → ok(s(x1))
U11(mark(x1), x2) → mark(U11(x1, x2))
length(ok(x1)) → ok(length(x1))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U12(mark(x1), x2) → mark(U12(x1, x2))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

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

TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(tt, x1)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(tt, x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(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))
active(cons(x1, x2)) → cons(active(x1), x2)
active(U11(x1, x2)) → U11(active(x1), x2)
active(U12(x1, x2)) → U12(active(x1), x2)
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
U12(mark(x1), x2) → mark(U12(x1, x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U11(mark(x1), x2) → mark(U11(x1, x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(0) → ok(0)
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(nil) → ok(nil)

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)

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.

TOP(ok(length(cons(x0, x1)))) → TOP(mark(U11(tt, x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))
TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(tt, x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

s(mark(x1)) → mark(s(x1))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
length(mark(x1)) → mark(length(x1))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
s(ok(x1)) → ok(s(x1))
U11(mark(x1), x2) → mark(U11(x1, x2))
length(ok(x1)) → ok(length(x1))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U12(mark(x1), x2) → mark(U12(x1, x2))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

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

TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U11(tt, x0))) → TOP(mark(U12(tt, x0)))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(length(x0))) → TOP(length(active(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))
active(cons(x1, x2)) → cons(active(x1), x2)
active(U11(x1, x2)) → U11(active(x1), x2)
active(U12(x1, x2)) → U12(active(x1), x2)
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
U12(mark(x1), x2) → mark(U12(x1, x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U11(mark(x1), x2) → mark(U11(x1, x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(0) → ok(0)
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(nil) → ok(nil)

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)

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.

TOP(ok(U11(tt, x0))) → TOP(mark(U12(tt, x0)))

The remaining pairs can at least be oriented weakly.

TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

active(U12(x1, x2)) → U12(active(x1), x2)
proper(zeros) → ok(zeros)
proper(length(x1)) → length(proper(x1))
s(mark(x1)) → mark(s(x1))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
cons(mark(x1), x2) → mark(cons(x1, x2))
active(zeros) → mark(cons(0, zeros))
length(mark(x1)) → mark(length(x1))
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
active(U11(x1, x2)) → U11(active(x1), x2)
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(tt) → ok(tt)
active(U12(tt, L)) → mark(s(length(L)))
s(ok(x1)) → ok(s(x1))
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
active(length(cons(N, L))) → mark(U11(tt, L))
U11(mark(x1), x2) → mark(U11(x1, x2))
length(ok(x1)) → ok(length(x1))
active(cons(x1, x2)) → cons(active(x1), x2)
proper(0) → ok(0)
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
active(U11(tt, L)) → mark(U12(tt, L))
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
U12(mark(x1), x2) → mark(U12(x1, x2))
proper(nil) → ok(nil)

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

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

TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(length(x0))) → TOP(length(active(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))
active(cons(x1, x2)) → cons(active(x1), x2)
active(U11(x1, x2)) → U11(active(x1), x2)
active(U12(x1, x2)) → U12(active(x1), x2)
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
U12(mark(x1), x2) → mark(U12(x1, x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U11(mark(x1), x2) → mark(U11(x1, x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(0) → ok(0)
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(nil) → ok(nil)

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)

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.

TOP(ok(U12(tt, x0))) → TOP(mark(s(length(x0))))

The remaining pairs can at least be oriented weakly.

TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(length(x0))) → TOP(length(active(x0)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(U12(x₁, x₂)) = 1
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(length(x₁)) = 0
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 0

The following usable rules [17] were oriented:

s(mark(x1)) → mark(s(x1))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
length(mark(x1)) → mark(length(x1))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
s(ok(x1)) → ok(s(x1))
U11(mark(x1), x2) → mark(U11(x1, x2))
length(ok(x1)) → ok(length(x1))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U12(mark(x1), x2) → mark(U12(x1, x2))

↳ CSR

  ↳ Zantema-Transformation

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Improved Ferreira Ribeiro-Transformation

  ↳ Complete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

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

TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(U12(x0, x1))) → TOP(U12(proper(x0), proper(x1)))
TOP(mark(U11(x0, x1))) → TOP(U11(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(U11(x0, x1))) → TOP(U11(active(x0), x1))
TOP(ok(U12(x0, x1))) → TOP(U12(active(x0), x1))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(length(x0))) → TOP(length(proper(x0)))
TOP(ok(length(x0))) → TOP(length(active(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))
active(cons(x1, x2)) → cons(active(x1), x2)
active(U11(x1, x2)) → U11(active(x1), x2)
active(U12(x1, x2)) → U12(active(x1), x2)
active(s(x1)) → s(active(x1))
active(length(x1)) → length(active(x1))
length(mark(x1)) → mark(length(x1))
length(ok(x1)) → ok(length(x1))
s(mark(x1)) → mark(s(x1))
s(ok(x1)) → ok(s(x1))
U12(mark(x1), x2) → mark(U12(x1, x2))
U12(ok(x1), ok(x2)) → ok(U12(x1, x2))
U11(mark(x1), x2) → mark(U11(x1, x2))
U11(ok(x1), ok(x2)) → ok(U11(x1, x2))
cons(mark(x1), x2) → mark(cons(x1, x2))
cons(ok(x1), ok(x2)) → ok(cons(x1, x2))
proper(zeros) → ok(zeros)
proper(cons(x1, x2)) → cons(proper(x1), proper(x2))
proper(0) → ok(0)
proper(U11(x1, x2)) → U11(proper(x1), proper(x2))
proper(tt) → ok(tt)
proper(U12(x1, x2)) → U12(proper(x1), proper(x2))
proper(s(x1)) → s(proper(x1))
proper(length(x1)) → length(proper(x1))
proper(nil) → ok(nil)

The set Q consists of the following terms:

active(zeros)
proper(zeros)
active(cons(x0, x1))
cons(mark(x0), x1)
proper(cons(x0, x1))
cons(ok(x0), ok(x1))
proper(0)
active(U11(x0, x1))
U11(mark(x0), x1)
proper(U11(x0, x1))
U11(ok(x0), ok(x1))
proper(tt)
active(U12(x0, x1))
U12(mark(x0), x1)
proper(U12(x0, x1))
U12(ok(x0), ok(x1))
active(s(x0))
s(mark(x0))
proper(s(x0))
s(ok(x0))
active(length(x0))
length(mark(x0))
proper(length(x0))
length(ok(x0))
proper(nil)

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