Termination proof of /tmp/tmpF0p6F6/LengthOfFiniteLists

Termination of the following Term Rewriting System could not be shown:

Context-sensitive 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

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Context-sensitive 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

The CSR is orthogonal. By [10] we can switch to innermost.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Context-sensitive 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

Innermost Strategy.

Using Improved CS-DPs we result in the following initial Q-CSDP problem.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length, LENGTH} are replacing on all positions.
For all symbols f in {cons, U11, U12, U12¹, U11¹} we have µ(f) = {1}.
The symbols in {U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

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

The collapsing dependency pairs are DP_c:

U12¹(tt, L) → L

The hidden terms of R are:

zeros

Every hiding context is built from:none

Hence, the new unhiding pairs DP_u are :

U12¹(tt, L) → U(L)
U(zeros) → ZEROS

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 set Q consists of the following terms:

zeros
U11(tt, x0)
U12(tt, x0)
length(nil)
length(cons(x0, x1))

The approximation of the Context-Sensitive Dependency Graph contains 1 SCC with 2 less nodes.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ ConvertedToQDPProblemProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

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

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 set Q consists of the following terms:

zeros
U11(tt, x0)
U12(tt, x0)
length(nil)
length(cons(x0, x1))

Converted QDP Problem, but could not keep Q or minimality.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ ConvertedToQDPProblemProof

                ↳ QDP

                  ↳ NonTerminationProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

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

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

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)

Q is empty.
We have to consider all (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:

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

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)

s = U11¹(tt, zeros) evaluates to t =U11¹(tt, zeros)

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

U11¹(tt, zeros) → U11¹(tt, cons(0, zeros))
with rule zeros → cons(0, zeros) at position [1] and matcher [ ]

U11¹(tt, cons(0, zeros)) → U12¹(tt, cons(0, zeros))
with rule U11¹(tt, L') → U12¹(tt, L') at position [] and matcher [L' / cons(0, zeros)]

U12¹(tt, cons(0, zeros)) → LENGTH(cons(0, zeros))
with rule U12¹(tt, L') → LENGTH(L') at position [] and matcher [L' / cons(0, zeros)]

LENGTH(cons(0, zeros)) → U11¹(tt, zeros)
with rule LENGTH(cons(N, L)) → U11¹(tt, L)

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 transformation [11] to transform the context-sensitive TRS to a usual TRS.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

  ↳ Trivial-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

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

  ↳ Trivial-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

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

  ↳ Trivial-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

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

  ↳ Trivial-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

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

  ↳ Trivial-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

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

  ↳ Trivial-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 + 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

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

  ↳ Trivial-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(0) → 0
mark(tt) → tt

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 + x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 1

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

  ↳ Trivial-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(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(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)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11Active(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U12Active(x₁, x₂)) = 1 + 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

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ Overlay + Local Confluence

  ↳ Trivial-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

  ↳ CSRInnermostProof

  ↳ 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

  ↳ Trivial-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:

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

  ↳ CSRInnermostProof

  ↳ 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

  ↳ Trivial-Transformation

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

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

  ↳ CSRInnermostProof

  ↳ 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

  ↳ Trivial-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

  ↳ CSRInnermostProof

  ↳ 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

  ↳ Trivial-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

  ↳ CSRInnermostProof

  ↳ 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

  ↳ Trivial-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

  ↳ CSRInnermostProof

  ↳ 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

  ↳ Trivial-Transformation

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

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

  ↳ CSRInnermostProof

  ↳ 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

  ↳ Trivial-Transformation

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

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

  ↳ CSRInnermostProof

  ↳ 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

                                                            ↳ Narrowing

  ↳ Trivial-Transformation

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

LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)
U12ACTIVE(tt, L) → LENGTHACTIVE(mark(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 narrowing [15] the rule U12ACTIVE(tt, L) → LENGTHACTIVE(mark(L)) at position [0] we obtained the following new rules:

U12ACTIVE(tt, s(x0)) → LENGTHACTIVE(s(mark(x0)))
U12ACTIVE(tt, zeros) → LENGTHACTIVE(zerosActive)

↳ CSR

  ↳ CSRInnermostProof

  ↳ 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

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

U12ACTIVE(tt, s(x0)) → LENGTHACTIVE(s(mark(x0)))
LENGTHACTIVE(cons(N, L)) → U11ACTIVE(tt, L)
U11ACTIVE(tt, L) → U12ACTIVE(tt, L)
U12ACTIVE(tt, zeros) → LENGTHACTIVE(zerosActive)

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.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ CSR

  ↳ CSRInnermostProof

  ↳ 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

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

  ↳ Trivial-Transformation

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

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

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

↳ CSR

  ↳ CSRInnermostProof

  ↳ 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

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

  ↳ Trivial-Transformation

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

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

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

  ↳ CSRInnermostProof

  ↳ 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

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

  ↳ Trivial-Transformation

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

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

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 narrowing [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

  ↳ CSRInnermostProof

  ↳ 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

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

  ↳ Trivial-Transformation

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

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

zerosActive

↳ CSR

  ↳ CSRInnermostProof

  ↳ 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

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

  ↳ Trivial-Transformation

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

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

  ↳ CSRInnermostProof

  ↳ 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

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ UsableRulesProof

                                                                      ↳ QDP

                                                                        ↳ QReductionProof

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ UsableRulesProof

                                                                                  ↳ QDP

                                                                                    ↳ QReductionProof

                                                                                      ↳ QDP

  ↳ Trivial-Transformation

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We applied the Trivial transformation to transform the context-sensitive TRS to a usual TRS.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

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)

Q is empty.

The following Q TRS is given: 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)

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

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ Overlay + Local Confluence

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(cons(N, L)) → U11(tt, L)

Q is empty.

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

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

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(cons(N, L)) → U11(tt, L)

The set Q consists of the following terms:

zeros
U11(tt, x0)
U12(tt, x0)
length(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:

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

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(cons(N, L)) → U11(tt, L)

The set Q consists of the following terms:

zeros
U11(tt, x0)
U12(tt, x0)
length(cons(x0, x1))

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

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

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

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

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(cons(N, L)) → U11(tt, L)

The set Q consists of the following terms:

zeros
U11(tt, x0)
U12(tt, x0)
length(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ QDP

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

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

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(cons(N, L)) → U11(tt, L)

The set Q consists of the following terms:

zeros
U11(tt, x0)
U12(tt, x0)
length(cons(x0, x1))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                      ↳ QDP

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

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

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

zeros
U11(tt, x0)
U12(tt, x0)
length(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.

zeros
U11(tt, x0)
U12(tt, x0)
length(cons(x0, x1))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ QDP

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

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

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

U11¹(tt, L) → U12¹(tt, L)
The graph contains the following edges 1 >= 1, 2 >= 2
U12¹(tt, L) → LENGTH(L)
The graph contains the following edges 2 >= 1
LENGTH(cons(N, L)) → U11¹(tt, L)
The graph contains the following edges 1 > 2

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

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

ZEROS → ZEROS

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(cons(N, L)) → U11(tt, L)

The set Q consists of the following terms:

zeros
U11(tt, x0)
U12(tt, x0)
length(cons(x0, x1))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

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

ZEROS → ZEROS

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

zeros
U11(tt, x0)
U12(tt, x0)
length(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.

zeros
U11(tt, x0)
U12(tt, x0)
length(cons(x0, x1))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

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

ZEROS → ZEROS

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