Termination proof of /tmp/tmplVJsTQ/LengthOfFiniteLists

Termination of the following Term Rewriting System could be proven:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set
nil: empty set

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

Context-sensitive rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
s: {1}
length: {1}
and: {1}
isNat: empty set
isNatList: empty set
isNatIList: empty set
nil: empty set

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

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

  ↳ Incomplete Giesl Middeldorp-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, and, U11¹, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList, ISNATLIST, ISNAT, ISNATILIST, U} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

U11¹(tt, L) → LENGTH(L)
ISNAT(length(V1)) → ISNATLIST(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNATILIST(V) → ISNATLIST(V)
ISNATILIST(cons(V1, V2)) → AND(isNat(V1), isNatIList(V2))
ISNATILIST(cons(V1, V2)) → ISNAT(V1)
ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))
ISNATLIST(cons(V1, V2)) → ISNAT(V1)
LENGTH(cons(N, L)) → U11¹(and(isNatList(L), isNat(N)), L)
LENGTH(cons(N, L)) → AND(isNatList(L), isNat(N))
LENGTH(cons(N, L)) → ISNATLIST(L)

The collapsing dependency pairs are DP_c:

U11¹(tt, L) → L
AND(tt, X) → X

The hidden terms of R are:

zeros
isNatIList(V2)
isNatList(V2)

Every hiding context is built from:none

Hence, the new unhiding pairs DP_u are :

U11¹(tt, L) → U(L)
AND(tt, X) → U(X)
U(zeros) → ZEROS
U(isNatIList(V2)) → ISNATILIST(V2)
U(isNatList(V2)) → ISNATLIST(V2)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs with 4 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length} are replacing on all positions.
For all symbols f in {cons, U11, and, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList, ISNATLIST, U, ISNATILIST, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))
AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATILIST(V) → ISNATLIST(V)
ISNATLIST(cons(V1, V2)) → ISNAT(V1)
ISNAT(length(V1)) → ISNATLIST(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNATILIST(cons(V1, V2)) → AND(isNat(V1), isNatIList(V2))
ISNATILIST(cons(V1, V2)) → ISNAT(V1)
U(isNatList(V2)) → ISNATLIST(V2)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

Using the order
Polynomial interpretation [25]:

POL(0) = 2
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = 0
POL(ISNATILIST(x₁)) = 1
POL(ISNATLIST(x₁)) = 0
POL(U(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + 2·x₁
POL(and(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 0
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 1
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 2
POL(nil) = 0
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 2

the following usable rules

isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
zeros → cons(0, zeros)

could all be oriented weakly.
Furthermore, the pairs

ISNATILIST(V) → ISNATLIST(V)
ISNATILIST(cons(V1, V2)) → ISNAT(V1)

could be oriented strictly and thus removed.
The pairs

ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))
AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATLIST(cons(V1, V2)) → ISNAT(V1)
ISNAT(length(V1)) → ISNATLIST(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNATILIST(cons(V1, V2)) → AND(isNat(V1), isNatIList(V2))
U(isNatList(V2)) → ISNATLIST(V2)

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))
AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATLIST(cons(V1, V2)) → ISNAT(V1)
ISNAT(length(V1)) → ISNATLIST(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNATILIST(cons(V1, V2)) → AND(isNat(V1), isNatIList(V2))
U(isNatList(V2)) → ISNATLIST(V2)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

Using the order
Polynomial interpretation [25]:

POL(0) = 0
POL(AND(x₁, x₂)) = x₂
POL(ISNAT(x₁)) = 2·x₁
POL(ISNATILIST(x₁)) = 2·x₁
POL(ISNATLIST(x₁)) = x₁
POL(U(x₁)) = x₁
POL(U11(x₁, x₂)) = 1 + 2·x₂
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(isNat(x₁)) = 2·x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

the following usable rules

isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
zeros → cons(0, zeros)

could all be oriented weakly.
Furthermore, the pairs

ISNAT(length(V1)) → ISNATLIST(V1)

could be oriented strictly and thus removed.
The pairs

ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))
AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATLIST(cons(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNATILIST(cons(V1, V2)) → AND(isNat(V1), isNatIList(V2))
U(isNatList(V2)) → ISNATLIST(V2)

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))
AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATLIST(cons(V1, V2)) → ISNAT(V1)
ISNAT(s(V1)) → ISNAT(V1)
ISNATILIST(cons(V1, V2)) → AND(isNat(V1), isNatIList(V2))
U(isNatList(V2)) → ISNATLIST(V2)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs with 1 less node.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ AND

                        ↳ QCSDP

                          ↳ QCSDPSubtermProof

                        ↳ QCSDP

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length} are replacing on all positions.
For all symbols f in {cons, U11, and} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList, ISNAT} are not replacing on any position.

The TRS P consists of the following rules:

ISNAT(s(V1)) → ISNAT(V1)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

ISNAT(s(V1)) → ISNAT(V1)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
ISNAT(x1) = x1

Subterm Order

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ AND

                        ↳ QCSDP

                          ↳ QCSDPSubtermProof

                            ↳ QCSDP

                              ↳ PIsEmptyProof

                        ↳ QCSDP

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

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

The TRS P consists of the following rules:
none

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ AND

                        ↳ QCSDP

                        ↳ QCSDP

                          ↳ QCSDPNarrowingProcessor

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length} are replacing on all positions.
For all symbols f in {cons, U11, and, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList, U, ISNATILIST, ISNATLIST} are not replacing on any position.

The TRS P consists of the following rules:

AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATILIST(cons(V1, V2)) → AND(isNat(V1), isNatIList(V2))
U(isNatList(V2)) → ISNATLIST(V2)
ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

Using the Context-Sensitive Narrowing Processor
the pair ISNATILIST(cons(V1, V2)) → AND(isNat(V1), isNatIList(V2))
was transformed to the following new pairs:

ISNATILIST(cons(0, y1)) → AND(tt, isNatIList(y1))
ISNATILIST(cons(length(x0), y1)) → AND(isNatList(x0), isNatIList(y1))
ISNATILIST(cons(s(x0), y1)) → AND(isNat(x0), isNatIList(y1))

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ AND

                        ↳ QCSDP

                        ↳ QCSDP

                          ↳ QCSDPNarrowingProcessor

                            ↳ QCSDP

                              ↳ QCSDPReductionPairProof

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length} are replacing on all positions.
For all symbols f in {cons, U11, and, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList, ISNATILIST, ISNATLIST, U} are not replacing on any position.

The TRS P consists of the following rules:

ISNATILIST(cons(s(x0), y1)) → AND(isNat(x0), isNatIList(y1))
ISNATILIST(cons(0, y1)) → AND(tt, isNatIList(y1))
U(isNatList(V2)) → ISNATLIST(V2)
AND(tt, X) → U(X)
ISNATILIST(cons(length(x0), y1)) → AND(isNatList(x0), isNatIList(y1))
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

Using the order
Polynomial interpretation [25]:

POL(0) = 0
POL(AND(x₁, x₂)) = 2·x₂
POL(ISNATILIST(x₁)) = 2·x₁
POL(ISNATLIST(x₁)) = 0
POL(U(x₁)) = x₁
POL(U11(x₁, x₂)) = 0
POL(and(x₁, x₂)) = 2·x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 2
POL(nil) = 2
POL(s(x₁)) = 0
POL(tt) = 0
POL(zeros) = 0

the following usable rules

isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
zeros → cons(0, zeros)

could all be oriented weakly.
Furthermore, the pairs

ISNATILIST(cons(length(x0), y1)) → AND(isNatList(x0), isNatIList(y1))

could be oriented strictly and thus removed.
The pairs

ISNATILIST(cons(s(x0), y1)) → AND(isNat(x0), isNatIList(y1))
ISNATILIST(cons(0, y1)) → AND(tt, isNatIList(y1))
U(isNatList(V2)) → ISNATLIST(V2)
AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ AND

                        ↳ QCSDP

                        ↳ QCSDP

                          ↳ QCSDPNarrowingProcessor

                            ↳ QCSDP

                              ↳ QCSDPReductionPairProof

                                ↳ QCSDP

                                  ↳ QCSDPReductionPairProof

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length} are replacing on all positions.
For all symbols f in {cons, U11, and, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList, ISNATILIST, ISNATLIST, U} are not replacing on any position.

The TRS P consists of the following rules:

ISNATILIST(cons(s(x0), y1)) → AND(isNat(x0), isNatIList(y1))
ISNATILIST(cons(0, y1)) → AND(tt, isNatIList(y1))
U(isNatList(V2)) → ISNATLIST(V2)
AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

Using the order
Polynomial interpretation [25]:

POL(0) = 0
POL(AND(x₁, x₂)) = 2·x₂
POL(ISNATILIST(x₁)) = 2·x₁
POL(ISNATLIST(x₁)) = 0
POL(U(x₁)) = 2·x₁
POL(U11(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(and(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = 0
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = 0
POL(length(x₁)) = 2 + 2·x₁
POL(nil) = 1
POL(s(x₁)) = 1
POL(tt) = 0
POL(zeros) = 0

the following usable rules

isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
zeros → cons(0, zeros)

could all be oriented weakly.
Furthermore, the pairs

ISNATILIST(cons(s(x0), y1)) → AND(isNat(x0), isNatIList(y1))

could be oriented strictly and thus removed.
The pairs

ISNATILIST(cons(0, y1)) → AND(tt, isNatIList(y1))
U(isNatList(V2)) → ISNATLIST(V2)
AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ AND

                        ↳ QCSDP

                        ↳ QCSDP

                          ↳ QCSDPNarrowingProcessor

                            ↳ QCSDP

                              ↳ QCSDPReductionPairProof

                                ↳ QCSDP

                                  ↳ QCSDPReductionPairProof

                                    ↳ QCSDP

                                      ↳ QCSDPNarrowingProcessor

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length} are replacing on all positions.
For all symbols f in {cons, U11, and, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList, ISNATILIST, ISNATLIST, U} are not replacing on any position.

The TRS P consists of the following rules:

ISNATILIST(cons(0, y1)) → AND(tt, isNatIList(y1))
U(isNatList(V2)) → ISNATLIST(V2)
AND(tt, X) → U(X)
U(isNatIList(V2)) → ISNATILIST(V2)
ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

Using the Context-Sensitive Narrowing Processor
the pair ISNATLIST(cons(V1, V2)) → AND(isNat(V1), isNatList(V2))
was transformed to the following new pairs:

ISNATLIST(cons(0, y1)) → AND(tt, isNatList(y1))
ISNATLIST(cons(length(x0), y1)) → AND(isNatList(x0), isNatList(y1))
ISNATLIST(cons(s(x0), y1)) → AND(isNat(x0), isNatList(y1))

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ AND

                        ↳ QCSDP

                        ↳ QCSDP

                          ↳ QCSDPNarrowingProcessor

                            ↳ QCSDP

                              ↳ QCSDPReductionPairProof

                                ↳ QCSDP

                                  ↳ QCSDPReductionPairProof

                                    ↳ QCSDP

                                      ↳ QCSDPNarrowingProcessor

                                        ↳ QCSDP

                                          ↳ QCSDPReductionPairProof

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length} are replacing on all positions.
For all symbols f in {cons, U11, and, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList, ISNATLIST, U, ISNATILIST} are not replacing on any position.

The TRS P consists of the following rules:

ISNATLIST(cons(s(x0), y1)) → AND(isNat(x0), isNatList(y1))
U(isNatList(V2)) → ISNATLIST(V2)
ISNATILIST(cons(0, y1)) → AND(tt, isNatIList(y1))
AND(tt, X) → U(X)
ISNATLIST(cons(0, y1)) → AND(tt, isNatList(y1))
ISNATLIST(cons(length(x0), y1)) → AND(isNatList(x0), isNatList(y1))
U(isNatIList(V2)) → ISNATILIST(V2)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

Using the order
Polynomial interpretation [25]:

POL(0) = 0
POL(AND(x₁, x₂)) = 2·x₂
POL(ISNATILIST(x₁)) = 2·x₁
POL(ISNATLIST(x₁)) = x₁
POL(U(x₁)) = 2·x₁
POL(U11(x₁, x₂)) = 2 + x₂
POL(and(x₁, x₂)) = 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatIList(x₁)) = 2·x₁
POL(isNatList(x₁)) = x₁
POL(length(x₁)) = 2 + x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0

the following usable rules

isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
zeros → cons(0, zeros)

could all be oriented weakly.
Furthermore, the pairs

ISNATLIST(cons(length(x0), y1)) → AND(isNatList(x0), isNatList(y1))

could be oriented strictly and thus removed.
The pairs

ISNATLIST(cons(s(x0), y1)) → AND(isNat(x0), isNatList(y1))
U(isNatList(V2)) → ISNATLIST(V2)
ISNATILIST(cons(0, y1)) → AND(tt, isNatIList(y1))
AND(tt, X) → U(X)
ISNATLIST(cons(0, y1)) → AND(tt, isNatList(y1))
U(isNatIList(V2)) → ISNATILIST(V2)

could only be oriented weakly and must be analyzed further.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ QCSDPReductionPairProof

              ↳ QCSDP

                ↳ QCSDPReductionPairProof

                  ↳ QCSDP

                    ↳ QCSDependencyGraphProof

                      ↳ AND

                        ↳ QCSDP

                        ↳ QCSDP

                          ↳ QCSDPNarrowingProcessor

                            ↳ QCSDP

                              ↳ QCSDPReductionPairProof

                                ↳ QCSDP

                                  ↳ QCSDPReductionPairProof

                                    ↳ QCSDP

                                      ↳ QCSDPNarrowingProcessor

                                        ↳ QCSDP

                                          ↳ QCSDPReductionPairProof

                                            ↳ QCSDP

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {s, length} are replacing on all positions.
For all symbols f in {cons, U11, and, AND} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList, ISNATLIST, U, ISNATILIST} are not replacing on any position.

The TRS P consists of the following rules:

ISNATLIST(cons(s(x0), y1)) → AND(isNat(x0), isNatList(y1))
U(isNatList(V2)) → ISNATLIST(V2)
ISNATILIST(cons(0, y1)) → AND(tt, isNatIList(y1))
AND(tt, X) → U(X)
ISNATLIST(cons(0, y1)) → AND(tt, isNatList(y1))
U(isNatIList(V2)) → ISNATILIST(V2)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-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, and, U11¹} we have µ(f) = {1}.
The symbols in {isNat, isNatList, isNatIList} are not replacing on any position.

The TRS P consists of the following rules:

LENGTH(cons(N, L)) → U11¹(and(isNatList(L), isNat(N)), L)
U11¹(tt, L) → LENGTH(L)

The TRS R consists of the following rules:

zeros → cons(0, zeros)
U11(tt, L) → s(length(L))
and(tt, X) → X
isNat(0) → tt
isNat(length(V1)) → isNatList(V1)
isNat(s(V1)) → isNat(V1)
isNatIList(V) → isNatList(V)
isNatIList(zeros) → tt
isNatIList(cons(V1, V2)) → and(isNat(V1), isNatIList(V2))
isNatList(nil) → tt
isNatList(cons(V1, V2)) → and(isNat(V1), isNatList(V2))
length(nil) → 0
length(cons(N, L)) → U11(and(isNatList(L), isNat(N)), L)

Q is empty.

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

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
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) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → isNatListActive(V1)
isNatActive(s(V1)) → isNatActive(V1)
isNatIListActive(V) → isNatListActive(V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatIList(V2))
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
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) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → isNatListActive(V1)
isNatActive(s(V1)) → isNatActive(V1)
isNatIListActive(V) → isNatListActive(V)
isNatIListActive(zeros) → tt
isNatIListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatIList(V2))
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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:

isNatIListActive(V) → isNatListActive(V)
isNatIListActive(zeros) → tt

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(U11Active(x₁, x₂)) = x₁ + 2·x₂
POL(and(x₁, x₂)) = x₁ + x₂
POL(andActive(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatActive(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + 2·x₁
POL(isNatIListActive(x₁)) = 1 + 2·x₁
POL(isNatList(x₁)) = 2·x₁
POL(isNatListActive(x₁)) = 2·x₁
POL(length(x₁)) = 2·x₁
POL(lengthActive(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
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) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → isNatListActive(V1)
isNatActive(s(V1)) → isNatActive(V1)
isNatIListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatIList(V2))
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
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) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → isNatListActive(V1)
isNatActive(s(V1)) → isNatActive(V1)
isNatIListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatIList(V2))
isNatListActive(nil) → tt
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(nil) → 0
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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:

isNatListActive(nil) → tt
lengthActive(nil) → 0

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(U11Active(x₁, x₂)) = x₁ + 2·x₂
POL(and(x₁, x₂)) = x₁ + x₂
POL(andActive(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatActive(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatIListActive(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(isNatListActive(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(lengthActive(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 1
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
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) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → isNatListActive(V1)
isNatActive(s(V1)) → isNatActive(V1)
isNatIListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatIList(V2))
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
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) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(length(V1)) → isNatListActive(V1)
isNatActive(s(V1)) → isNatActive(V1)
isNatIListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatIList(V2))
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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:

isNatActive(length(V1)) → isNatListActive(V1)

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(and(x₁, x₂)) = x₁ + x₂
POL(andActive(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatActive(x₁)) = x₁
POL(isNatIList(x₁)) = x₁
POL(isNatIListActive(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(isNatListActive(x₁)) = x₁
POL(length(x₁)) = 1 + 2·x₁
POL(lengthActive(x₁)) = 1 + 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
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) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatIListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatIList(V2))
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(isNat(x1)) → isNatActive(x1)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
isNatListActive(x1) → isNatList(x1)
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) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatIListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatIList(V2))
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(isNat(x1)) → isNatActive(x1)
isNatIListActive(x1) → isNatIList(x1)
mark(isNatList(x1)) → isNatListActive(x1)
mark(0) → 0
mark(tt) → tt
mark(nil) → nil
isNatIListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatIList(V2))

Used ordering:
Polynomial interpretation [25]:

POL(0) = 1
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U11Active(x₁, x₂)) = x₁ + x₂
POL(and(x₁, x₂)) = x₁ + x₂
POL(andActive(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatActive(x₁)) = x₁
POL(isNatIList(x₁)) = 1 + 2·x₁
POL(isNatIListActive(x₁)) = 2 + 2·x₁
POL(isNatList(x₁)) = x₁
POL(isNatListActive(x₁)) = x₁
POL(length(x₁)) = x₁
POL(lengthActive(x₁)) = x₁
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0
POL(zerosActive) = 1

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
isNatActive(x1) → isNat(x1)
mark(isNatIList(x1)) → isNatIListActive(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(isNatIList(x1)) → isNatIListActive(x1)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + 2·x₂
POL(U11Active(x₁, x₂)) = x₁ + 2·x₂
POL(and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(andActive(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatActive(x₁)) = x₁
POL(isNatIList(x₁)) = 2 + x₁
POL(isNatIListActive(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(isNatListActive(x₁)) = x₁
POL(length(x₁)) = 2·x₁
POL(lengthActive(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
isNatActive(x1) → isNat(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
isNatActive(x1) → isNat(x1)
isNatListActive(x1) → isNatList(x1)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → s(lengthActive(mark(L)))
andActive(tt, X) → mark(X)
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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:

isNatActive(x1) → isNat(x1)
isNatListActive(x1) → isNatList(x1)
andActive(tt, X) → mark(X)

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(and(x₁, x₂)) = x₁ + 2·x₂
POL(andActive(x₁, x₂)) = x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatActive(x₁)) = 1 + 2·x₁
POL(isNatList(x₁)) = x₁
POL(isNatListActive(x₁)) = 1 + x₁
POL(length(x₁)) = 2 + 2·x₁
POL(lengthActive(x₁)) = 2 + 2·x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → s(lengthActive(mark(L)))
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → s(lengthActive(mark(L)))
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))
lengthActive(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), 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(cons(N, L)) → U11Active(andActive(isNatListActive(L), isNat(N)), L)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 0
POL(U11(x₁, x₂)) = x₁ + x₂
POL(U11Active(x₁, x₂)) = x₁ + x₂
POL(and(x₁, x₂)) = x₁ + x₂
POL(andActive(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(isNat(x₁)) = x₁
POL(isNatActive(x₁)) = 1 + x₁
POL(isNatList(x₁)) = x₁
POL(isNatListActive(x₁)) = x₁
POL(length(x₁)) = x₁
POL(lengthActive(x₁)) = x₁
POL(mark(x₁)) = 1 + x₁
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0
POL(zerosActive) = 1

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → s(lengthActive(mark(L)))
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
U11Active(tt, L) → s(lengthActive(mark(L)))
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))

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(tt, L) → s(lengthActive(mark(L)))

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(and(x₁, x₂)) = 2·x₁ + 2·x₂
POL(andActive(x₁, x₂)) = 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(isNatActive(x₁)) = 1 + 2·x₁
POL(isNatList(x₁)) = x₁
POL(isNatListActive(x₁)) = 2 + 2·x₁
POL(length(x₁)) = 1 + x₁
POL(lengthActive(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(s(x₁)) = x₁
POL(tt) = 1
POL(zeros) = 0
POL(zerosActive) = 0

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))

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(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))

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)
mark(s(x1)) → s(mark(x1))
zerosActive → cons(0, zeros)
isNatActive(0) → tt
isNatActive(s(V1)) → isNatActive(V1)

Used ordering:
Polynomial interpretation [25]:

POL(0) = 1
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(U11Active(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(and(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(andActive(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(cons(x₁, x₂)) = x₁ + 2·x₂
POL(isNatActive(x₁)) = x₁
POL(isNatList(x₁)) = 2·x₁
POL(isNatListActive(x₁)) = 2 + 2·x₁
POL(length(x₁)) = x₁
POL(lengthActive(x₁)) = x₁
POL(mark(x₁)) = 2 + 2·x₁
POL(s(x₁)) = 1 + x₁
POL(tt) = 0
POL(zeros) = 0
POL(zerosActive) = 2

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

                                        ↳ QTRS

                                          ↳ RRRPoloQTRSProof

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

mark(zeros) → zerosActive
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))

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)
mark(and(x1, x2)) → andActive(mark(x1), x2)
andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)
mark(cons(x1, x2)) → cons(mark(x1), x2)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))

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(zeros) → zerosActive
mark(U11(x1, x2)) → U11Active(mark(x1), x2)
mark(and(x1, x2)) → andActive(mark(x1), x2)
mark(cons(x1, x2)) → cons(mark(x1), x2)
isNatListActive(cons(V1, V2)) → andActive(isNatActive(V1), isNatList(V2))

Used ordering:
Polynomial interpretation [25]:

POL(U11(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(U11Active(x₁, x₂)) = 1 + x₁ + x₂
POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(andActive(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(cons(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(isNatActive(x₁)) = x₁
POL(isNatList(x₁)) = x₁
POL(isNatListActive(x₁)) = 2·x₁
POL(length(x₁)) = x₁
POL(lengthActive(x₁)) = x₁
POL(mark(x₁)) = 2 + 2·x₁
POL(zeros) = 2
POL(zerosActive) = 1

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

                                        ↳ QTRS

                                          ↳ RRRPoloQTRSProof

                                            ↳ QTRS

                                              ↳ RRRPoloQTRSProof

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

andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)

Q is empty.

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

andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))
lengthActive(x1) → length(x1)

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(and(x₁, x₂)) = 2 + x₁ + x₂
POL(andActive(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(length(x₁)) = 1 + 2·x₁
POL(lengthActive(x₁)) = 2 + 2·x₁
POL(mark(x₁)) = 2·x₁

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

                                        ↳ QTRS

                                          ↳ RRRPoloQTRSProof

                                            ↳ QTRS

                                              ↳ RRRPoloQTRSProof

                                                ↳ QTRS

                                                  ↳ RRRPoloQTRSProof

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

andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))

Q is empty.

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

andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))

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

andActive(x1, x2) → and(x1, x2)
mark(length(x1)) → lengthActive(mark(x1))

Used ordering:
Polynomial interpretation [25]:

POL(and(x₁, x₂)) = 1 + x₁ + x₂
POL(andActive(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(length(x₁)) = 2 + 2·x₁
POL(lengthActive(x₁)) = 1 + x₁
POL(mark(x₁)) = 2 + x₁

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ RRRPoloQTRSProof

        ↳ QTRS

          ↳ RRRPoloQTRSProof

            ↳ QTRS

              ↳ RRRPoloQTRSProof

                ↳ QTRS

                  ↳ RRRPoloQTRSProof

                    ↳ QTRS

                      ↳ RRRPoloQTRSProof

                        ↳ QTRS

                          ↳ RRRPoloQTRSProof

                            ↳ QTRS

                              ↳ RRRPoloQTRSProof

                                ↳ QTRS

                                  ↳ RRRPoloQTRSProof

                                    ↳ QTRS

                                      ↳ RRRPoloQTRSProof

                                        ↳ QTRS

                                          ↳ RRRPoloQTRSProof

                                            ↳ QTRS

                                              ↳ RRRPoloQTRSProof

                                                ↳ QTRS

                                                  ↳ RRRPoloQTRSProof

                                                    ↳ QTRS

                                                      ↳ RisEmptyProof

Q restricted rewrite system:
R is empty.
Q is empty.

The TRS R is empty. Hence, termination is trivially proven.